Bandwidth Analysis of Multiport Radio-Frequency Systems
11 Bandwidth Analysis of Multiport Radio-FrequencySystems—Part I
Ding Nie,
Member, IEEE, and Bertrand M. Hochwald,
Fellow, IEEE
Abstract —When multiple radio-frequency sources are con-nected to multiple loads through a passive multiport matchingnetwork, perfect power transfer to the loads across all frequenciesis generally impossible. In this two-part paper, we provideanalyses of bandwidth over which power transfer is possible.Our principal tools include broadband multiport matching upperbounds, presented herein, on the integral over all frequency ofthe logarithm of a suitably defined power loss ratio. In general,the larger the integral, the larger the bandwidth over whichpower transfer can be accomplished. We apply these bounds inseveral ways: We show how the number of sources and loads,and the coupling between loads, affect achievable bandwidth. Weanalyze the bandwidth of networks constrained to have certainarchitectures. We characterize systems whose bandwidths scaleas the ratio between the numbers of loads and sources.The first part of the paper presents the bounds and uses themto analyze loads whose frequency responses can be represented byanalytical circuit models. The second part analyzes the bandwidthof realistic loads whose frequency responses are available numer-ically. We provide applications to wireless transmitters where theloads are antennas being driven by amplifiers. The derivationsof the bounds are also included.
Index Terms —Bandwidth, Bode-Fano bounds, broadbandmatching bounds, non-reciprocal networks, passive matchingnetworks, radio-frequency coupling
I. I
NTRODUCTION
Analyses of the bandwidth over which power can betransferred in radio-frequency (RF) systems are often limitedto a single source and load because of the complexity inmanipulating multiport matching networks and multiple loads.Factors that contribute to the complexity include definingappropriate measures of bandwidth when there are manysources and loads, and the difficulty of analyzing couplingbetween loads. We propose methods of analysis that utilizebroadband performance bounds applicable to a wide classof passive networks and an arbitrary number of sources anddissipative loads.The ability to transfer power from sources to loads relies,in part, on the ability to match the impedance of the sourcesto the frequency-dependent impedance Z L ( jω ) of the loadsover a broad frequency range. Bandwidth upper bounds areof great help in determining the best achievable bandwidthperformance for a given load. Classical Bode-Fano results [1],[2] on the integral of the logarithm of the reflection coefficient D. Nie is with Apple Inc., Cupertino, CA 95014, USA (e-mail:[email protected]).B. M. Hochwald is with the Department of Electrical Engineering,University of Notre Dame, Notre Dame, IN 46556, USA (e-mail: [email protected]).This work was supported, in part, by NSF grants CCF-1403458 and ECCS-1509188. can be used for such bounds when there is a single sourceand load. When there are multiple loads, analyses are oftenlimited to special cases. For example, bandwidth bounds forloads with some specific structures are discussed in [3]–[6],and examples of analyses of multiport systems include [7], [8].Often, multiple reflection coefficients are defined and analyzedseparately using scalar Bode-Fano theory. However, as shownin [9], a physically-meaningful single reflection coefficient canbe defined and analyzed when there are N arbitrarily coupledloads driven by N sources ( N > ).In this two-part paper, we present a bandwidth analysisof matched multiport RF systems that builds on bandwidthbounds in [9]. The first part presents the bounds and appliesthem to systems that can be expressed in closed form. Thesecond part provides proofs of the bounds and applies themto systems whose scattering parameters are expressed numer-ically. The bounds in [9] apply to loads that are modeled asperfect reflectors as ω → ∞ . We extend these results, andpresent bounds that apply to loads that are reflectors at anyfrequency, including ω = 0 . We allow the matching networkand loads to be non-reciprocal. The network can also be lossy.We permit the number of sources and loads to be unequal.By applying bandwidth bounds, we demonstrate how thenumber of sources, loads, and the coupling between loadsaffect the achievable bandwidth of a matched multiport system.We prove that bandwidth bounds generally scale as N/M ,where M is the number of sources and N is the number ofloads. This result also holds in the presence of coupling, aslong as it is not “too strong”. This suggests that unlimitedbandwidth is theoretically achievable by simply adding moreloads for a fixed number of sources. As is shown, both theloads and the network architecture play an important role inachieving linear-in- N performance of the overall system, fora given M .We also propose a bandwidth analysis for situations where aportion of the network is constrained to have a certain structurewhile other portions are unconstrained. This situation occurs inbeamforming applications since a beamforming antenna arraycan be thought of as N loads driven with prescribed amplitudeand phase relationships by a single source.The basic premise of broadband matching is that whena source and load are connected to each other, even if thereflection coefficient is made small at a design frequency ω = ω d , it is generally not small for all ω [10]. When thereare multiple sources and loads, there is no single reflectioncoefficient since power sent from source i may, throughcoupling, return to source j = i . In [9], a definition of amultiport reflection coefficient that takes this phenomenon a r X i v : . [ c s . I T ] M a r into account is used to derive bounds on the ability to matchmultiple sources and loads over all ω with a lossless network.We expand this definition to include lossy networks.Of particular interest is the application to loads that areclosely-spaced antennas, such as may be found in multiple-input multiple-output (MIMO) communication systems. The“densification” of portable wireless communication devices,including cellular telephones, with multiple transmitter andreceiver chains in close proximity, makes coupling difficultto avoid. Furthermore, there are situations where there aremore antennas than RF chains [11]. Our analysis methodsquantify the bandwidth attainable in the transmitters of theseMIMO systems, where the RF amplifiers are treated as sourcesand the coupled antennas are the loads. Part II, in particular,shows how realistic antennas are modeled to obtain accuratebandwidth results.We consider an RF system where M sources drive N loads through an arbitrary passive ( M + N ) -port matchingnetwork. The M input ports on the multiport network connectto the sources, and N output ports connect to the loads.No relationship between M and N is assumed. Our goalis to transfer maximum power from sources with knowncharacteristic impedances to loads with known frequency-dependent impedances. The loads are dissipative and poten-tially non-reciprocal; the network can be lossy and also non-reciprocal. The quality of match between the sources and theloads at a frequency ω is then determined by the power losteither because it is returned to the sources or because it isdissipated in the network. We derive and utilize bounds onthis quality metric when an arbitrary passive ( M + N ) -portnetwork is used. Our bounds can readily be calculated fromthe frequency-dependent S-matrix of the loads.We do not consider noise in our analysis, and hence ourmethods apply more to transmitters driving loads than toreceivers, where the noise figures of the input amplifiers playan important role. We are interested in the “input bandwidth”of the loads and do not evaluate the ability of the loads to usetheir input power efficiently.When the loads are coupled, there is only a nominalassociation of source i with load i for i = 1 , . . . , N sincesource i potentially also stimulates load j = i if the twoloads are coupled. In minimizing the amount of lost power,a matching network between the sources and loads thereforecould connect any source with any of the loads, and converse-ly; examples include decoupling networks [12]–[15], whichensure all power from the sources is delivered to the loads ata design frequency. An effective network prevents reflection bydecoupling the loads from each other over as wide a frequencyrange as possible. Generally, for a given design frequency ω d , there are frequencies ω < ω d and ω > ω d , wherethe fractional power delivered to the loads falls below someprescribed threshold. The larger ω − ω is, the larger thebandwidth.Techniques involving active non-Foster circuits [16], [17]and adaptive matching [18]–[21] are not discussed. Non-Fostercircuits such as those used to realize negative capacitanceand inductance values can theoretically cancel the reactiveFoster behavior in the loads, and achieve an impedance match
0Z Passive(M+N)-portmatchingnetwork S... L(cid:0)(cid:1) N-portloads S... (cid:2)(cid:3)(cid:4)1(cid:5)(cid:6)(cid:7)b(cid:8) 2(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)u(cid:19)(cid:20)(cid:16)(cid:15) (cid:21)(cid:18)u(cid:22)(cid:17)(cid:16)(cid:21) (cid:23)(cid:23) (cid:23)(cid:24)(cid:24)(cid:23) (cid:24)(cid:24)(cid:25) (cid:26)(cid:27) (cid:27)(cid:25)(cid:26) (cid:27) (cid:27)(cid:28) (cid:29)(cid:30) (cid:31) ! "
Fig. 1. An RF system where M uncorrelated sources drive N loads having S-matrix S L through a passive ( M + N ) -port matching network S (the complex-frequency argument s is omitted). We use S LM to denote the S-matrix asseen at the input of the network, and S G = S to denote the S-matrix asseen at the output. between the sources and the loads over a wider frequencyband than passive networks. Adaptive matching uses tunablecapacitors and inductors to match antennas whose impedanceis affected by the environment or carrier frequency.The basic model assumptions are given in Section II. Thedefinitions and notations we use are similar to those in [9]. Theprincipal bounds we use are presented in Section III. Detailson the mechanics of how to apply these bounds are providedwithin the subsections of Section III. Analytical applicationsof the results appear in Section IV, where scaling laws arederived and network architectures are analyzed. Section VIconcludes.The results presented herein assume the loads can be accu-rately modeled by rational functions of frequency. Since notall loads are necessarily rational, Part II examines how rationalfunctions can be used as approximations for arbitrary loads.We present examples of how to use the bounds in practice.Proofs of all the results are also given.II. P ROBLEM D EFINITION AND N OTATIONS
A. System description
Figure 1 shows an RF system where N dissipative loadsare driven by M decoupled sources with real impedance Z ,the characteristic impedance of the system. Let S L ( s ) be the N × N S-matrix of the dissipative loads as a function of thecomplex frequency s = σ + jω , where σ and ω are real. S L ( s ) is obtained by extending the S-matrix of the loads S L ( jω ) asa function of the angular frequency ω to the whole complexplane (WCP). Mathematically, S L ( s ) can be thought of as thetransfer function between the N × incident and reflectedwaves ~a e st and ~b e st , where t represents time. Therefore,we have ~b ( s ) = S L ( s ) ~a ( s ) . If the loads are reciprocal then S L ( s ) is symmetric. If the loads are coupled, at least one off-diagonal entry of S L ( s ) is non-zero. The impedance matrix Z L ( jω ) of the loads can be obtained by Z L ( jω ) = Z ( I + S L ( jω ))( I − S L ( jω )) − .The M sources and N loads are matched by inserting apassive ( M + N ) -port network between them, as indicated inFigure 1. The M input ports of the network are connectedto the sources, and the N output ports are connected to theloads. The network is not necessarily lossless or reciprocal, so we allow standard passive capacitive and inductive elements aswell as non-reciprocal ferromagnetic components in its design.Let S ( s ) be the ( M + N ) × ( M + N ) S-matrix of themultiport network as a function of s , partitioned as in Figure 1 S ( s ) = M NM S ( s ) S ( s ) N S ( s ) S ( s ) ! . (1)Let S LM ( s ) denote the M × M S-matrix of the cascade ofthe network and the loads, and S G ( s ) be the N × N S-matrixseen from the output ports of the network. Since the inputto the network is terminated by sources with characteristicimpedance Z , it follows that S G ( s ) = S ( s ) , and S LM ( s ) = S ( s ) + S ( s ) S L ( s )( I − S G ( s ) S L ( s )) − S ( s ) . (2)As in Figure 1, let M × vector ~a ( s ) be the incidentwave to the network. Then the reflected wave is ~b ( s ) = S LM ( s ) ~a ( s ) . Since the network is potentially lossy, theincident power may be reflected to the sources (return loss),or dissipated in the network (insertion loss). We now define ameasure of performance that includes both of these effects. B. Power loss ratio
Our measure of performance of a matching network be-tween the sources and loads is given by the fraction of sourcepower that is not transferred to the loads, as a function offrequency. At frequency s = jω , the total instantaneous powerfrom the M sources is k ~a ( jω ) k , and the total instantaneouspower delivered to the N loads is k ~a ( jω ) k −k ~b ( jω ) k ≥ .The power lost due to dissipation and reflection is therefore k ~a ( jω ) k − ( k ~a ( jω ) k − k ~b ( jω ) k ) . Definition 1:
The power loss ratio at frequency jω is theratio between the expected power loss and the expected totalincident power at jω : r ( ω ) = E [ k ~a ( jω ) k − ( k ~a ( jω ) k − k ~b ( jω ) k )] E k ~a ( jω ) k , (3)where the expectation is over the random input signals.By convention, when we use r ( ω ) we mean the positivesquare root of (3), and by construction, ≤ r ( ω ) ≤ wherevalues close to zero indicate that little source power is beinglost and therefore most of it is being delivered to the loads.Values close to one indicate most of the power is lost todissipation or reflection. We note that r ( ω ) = 0 means that theloads are perfectly matched and decoupled from one another.When the matching network S ( s ) is lossless, the power lossratio is equivalent to the power reflection ratio defined in [9].By considering power dissipation in the network in additionto power reflection by the network, we handle situations thatappear favorable because the reflected energy is low, but areactually unfavorable because the network (rather than the load)is dissipating the incident power. This issue is raised in [22]. C. Experimental measurement of r ( ω ) Because (3) includes an expectation, we devote a few wordsto the measurement of r ( ω ) in practice. The incident signalsfrom the sources are assumed to have independent randomphases and instantaneous powers at each frequency, and theexpectation is taken over all such uncorrelated incident signals.Nevertheless, the expectation is unnecessary when M = 1 since the amplitude and phase of a single source does not affectthe power loss ratio. In this case, − r ( ω ) is equivalent to thestandard transducer power gain [8], [23], and the expectationsin the numerator and denominator of (3) may be dropped.When M > the expectations play the important roleof averaging over all amplitude and phase combinations ofthe input signals. Its importance can be demonstrated byexamining M = 2 , where it is well known that even-mode(in-phase) and odd-mode (out-of-phase) signals can elicit verydifferent reflective responses from a two-port system. Since thesources are uncorrelated, no preference is given to even or oddmode signals, and the expectation eliminates mode dependenceby averaging over both of them. Hence, r ( ω ) can be thought ofas an “average loss” experienced when the loads are stimulatedby uncorrelated sources.The value of r ( ω ) can be experimentally measured byforming the sample average of the numerator k ~a ( jω ) k − ( k ~a ( jω ) k − k ~b ( jω ) k ) for a variety of inputs ~a ( jω ) , andtaking the ratio of this average to the sample average of thedenominator k ~a ( jω ) k . This ratio of averages converges to r ( ω ) as more measurements are taken with all possible sourcephase and amplitude combinations. D. Definition of bandwidth
The network S ( s ) should be constructed to make r ( ω ) assmall as possible over a prescribed bandwidth, or make thebandwidth as wide as possible for a prescribed threshold.Usually, bandwidth is measured in the vicinity of a designfrequency, which we denote as ω d . A decoupling network [15]enforces r ( ω ) = 0 at ω = ω d . We can define the bandwidthof the combined network and loads using (3). Definition 2:
The bandwidth is the frequency range forwhich r ( ω ) is no greater than a threshold τ > in the vicinityof a design frequency ω d : ω BW ( τ, ω d ) = max ω ≤ ω d ≤ ω r ( ω ) ≤ τ, ∀ ω ∈ [ ω ,ω ] ω − ω . (4)Let the elements of ~a ( jω ) , representing the incident signalfrom M decoupled sources at frequency jω , have equal ex-pected square-magnitude over all frequency, and have uniform-ly distributed random phases in [0 , π ) that are independentof the amplitudes and each other. Then (3) yields r ( ω ) = 1 − E tr { ~a H ( jω )( I − S HL ( jω ) S L ( jω )) ~a ( jω ) } E tr { ~a H ( jω ) ~a ( jω ) } . Because the phases are independent and uniformly distributed, E [ ~a ( jω ) ~a H ( jω )] is a multiple of the identity matrix. Weapply ~a ( jω ) = ( I − S G ( jω ) S L ( jω )) − S ( jω ) ~a ( jω ) toobtain r ( ω ) = 1 − tr { S H ( I − S G S L ) − H ( I − S HL S L )( I − S G S L ) − S } M , (5)where tr ( · ) denotes trace, H denotes Hermitian transpose; thefrequency argument jω is omitted on the right-hand side of(5).When M > N , r ( ω ) ≥ − N/M since the matrixinside the trace on the right-hand side of (5) is a rank- N positive semidefinite matrix whose eigenvalues are less thanor equal to one. Hence, a certain fraction of the source poweris always reflected or absorbed, and letting τ < p − N/M in (4) always obtains zero bandwidth. We therefore assume p − N/M < τ < for M > N ; for M ≤ N , we have < τ < .Note that “ganging” amplifiers through couplers to attainhigh output power with a single load appears to be an examplewhere M > N . But such ganged sources are correlatedsince each carries the same signal, possibly differing onlyin a constant relative phase or amplitude. Since we assumeuncorrelated sources, ganged amplifiers (and other correlatedsources) should be considered as a single source to apply ourresults.Although the incident signals from the sources are assumedto be uncorrelated, the output of the matching network willhave correlated components when
M < N since any networkdriving all N loads will necessarily derive its signals from the M sources. As an example, a beamformer can be modeledas M = 1 source driving N > loads in a fixed phaserelationship. We consider beamforming in Section V-A. E. Properties of S-matrices
We briefly summarize some properties of S-matrices sincethey play an important role. Passive real networks have S-matrices that are real-rational, Hurwitzian, and bounded; thisapplies, for example, to S L ( s ) and S ( s ) . The definitionsof these terms can be found in [23]–[25]. We also employdefinitions of poles and zeros of rational matrices that arewidely used in multivariable control theory [26]. For anarbitrary rational matrix A ( s ) : • Poles: are the roots of the pole polynomial of A ( s ) , wherethis polynomial is the monic least common multiple ofthe denominators of all minors of all dimensions of A ( s ) . • Zeros: are the roots of the zero polynomial of A ( s ) ,where this polynomial is the monic greatest commondivisor of the numerators of all minors of dimension L ,and L is the normal rank of A ( s ) . It is assumed that theseminors have the pole polynomial as their denominators.The normal rank of a matrix is its maximum rank amongall s ∈ C ; the use of “normal” refers to the rank almosteverywhere in the complex plane [24, A.70]. We use LHPto denote the left-half complex plane (Re { s } < ) and RHPto denote the right-half (Re { s } > ). Throughout, we use p L,i and z L,i , i = 1 , , . . . to represent the poles and zeros overthe WCP (whole complex plane) of S L ( s ) . Since S L ( s ) isHurwitzian, it has no RHP poles.We assume I − S TL ( − s ) S L ( s ) is full normal rank. This issatisfied if the loads are strictly dissipative, so that there is nocombination of load signals that is completely reflected for all s . On the other hand, we also assume that there exists an s with Re { s } ≥ such that S TL ( − s ) S L ( s ) = I. (6)In general, s is arbitrary and can be infinite.When s is purely imaginary s = jω , (6) has theinterpretation of modeling the loads as perfect reflectors atfrequency ω since S TL ( − jω ) = S HL ( jω ) and therefore thesingular values of S L ( jω ) are all unity. Because S L ( s ) is abounded matrix, (6) is equivalent to | det S L ( jω ) | = 1 .When s has positive real part, the physical interpretationof (6) is elusive but we still use the terminology “perfectreflector” at s . Examples of load structures with such an s are given in Section IV-B2.For any s , (6) must also hold if we replace s with − s . Therefore, without loss of generality, we only considerRe { s } ≥ . If there are multiple distinct values of s forwhich (6) holds then the bounds presented herein apply to eachvalue separately. We therefore consider only a single distinct s . III. B ROADBAND M ATCHING B OUNDS
We present the principal bounds used in this paper. They arederived in detail in Part II, but knowledge of the derivationsis not needed to apply the bounds. A description of how touse these bounds appears in Section III-B. Conditions forachieving equality are presented in Section III-C. The boundsdepend on zeros and poles of S L ( s ) , and techniques to obtainthese are discussed in Section III-D and Part II. A. Principal boundsBound 1:
For s = jω , Z ∞ ( ω − ω ) − + ( ω + ω ) − r ( ω ) dω ≤ − π M h X i ( p L,i − jω ) − + X i ( z L,i + jω ) − i . (7) Bound 2:
For Re { s } > , Z ∞ Re [( s − jω ) − + ( s + jω ) − ]2 log 1 r ( ω ) dω ≤ − π M log (cid:12)(cid:12)(cid:12)(cid:12) det S L ( s ) · Q i ( s + z L,i ) Q i ( s − z L,i ) (cid:12)(cid:12)(cid:12)(cid:12) . (8) Bound 3:
For s = ∞ , Z ∞ log 1 r ( ω ) dω ≤ − π M X i p L,i + X i z L,i ! . (9)Bound 1 is useful for loads that are modeled as electricallyopen or short at some frequency jω . For example, someantennas are capacitive relative to ground when they are“electrically small” compared to the signal wavelength. Thus,they are effectively an open circuit at s = 0 ; equivalently S L (0) = I .Bound 2 applies to loads that are modeled as a mixture ofresistive and reactive components [27]. An example of this isgiven in Section IV-B2. TABLE IF
ORMS OF f ( jω ) AND B IN (10) FOR DIFFERENT LOCATIONS OF s , WHERE p L,i , z
L,i
ARE THE POLES AND ZEROS OF S L ( s ) . s = jω f ( ω ) = [( ω − ω ) − + ( ω + ω ) − ] B = − π M (cid:2)P i ( p L,i − jω ) − + P i ( z L,i + jω ) − (cid:3) Re { s } > f ( ω ) = Re [( s − jω ) − + ( s + jω ) − ] B = − π M log (cid:12)(cid:12)(cid:12) det S L ( s ) · Q i ( s + z L,i ) Q i ( s − z L,i ) (cid:12)(cid:12)(cid:12) s = ∞ f ( ω ) = 1 B = − π M (cid:0)P i p L,i + P i z L,i (cid:1)
Bound 3 applies to loads that are modeled as open or shortcircuits at infinite frequency. The classical model of a parallelresistive and capacitive load used to demonstrate the Bode-Fano bound falls into this category. A bound similar to (9) ispresented in [9]; however, the bound in [9] requires M = N and the matching network to be lossless.In all three cases, the bounds have the form Z ∞ f ( ω ) log 1 r ( ω ) dω ≤ B, (10)where f ( ω ) ≥ . The form of f ( ω ) depends on the locationof s , and the computation of B depends also on the polesand zeros of S L ( s ) . Because ≤ r ( ω ) ≤ , we have log(1 /r ( ω )) ≥ . Hence, f ( ω ) log(1 /r ( ω )) ≥ for any ω .Clearly, B must be positive as well. We use (10) to genericallyindicate any of (7)–(9). The number of sources appears as M in the denominators of all three bounds. The forms of f ( ω ) and B are summarized in Table I. B. How to use bounds
Suppose that we wish to assess the achievable bandwidth [ ω , ω ] of a set of loads, where ω < ω . Our measure ofachievability is that for some threshold τ > , the overallsystem should obey r ( ω ) ≤ τ for ω ∈ [ ω , ω ] . Hence thecombined multiport network and loads reflects (or absorbs)no more than τ within the passband. We assume that S L ( s ) is available to us (we have more to say about this in SectionIII-D) and we would like to know ω BW ( τ, ω d ) defined in (4).Suppose the loads obey S TL (0) S L (0) = I so that Bound 1(first row of Table I) with ω = 0 applies to any passivenetwork used for these loads. Let the right-hand side of (7) bedenoted B > , which depends only on S L ( s ) . Then B ≥ Z ∞ ω − log 1 r ( ω ) dω ≥ log 1 τ Z ω ω ω − dω = log 1 τ (cid:18) ω − ω (cid:19) . The first inequality applies to any network, while the secondinequality applies to a network with the desired passbandcharacteristics. Hence, ω − ω ≤ B log(1 /τ ) . (11)Clearly, this inequality imposes a constraint on ( ω , ω ) pairs. Bounds 2 and 3 can be applied in a similar fashion. If S HL ( ∞ ) S L ( ∞ ) = I , (9) gives ω − ω ≤ B log(1 /τ ) , (12)where B is the right-hand side of (9). This gives us a directbound on the bandwidth ω BW ( τ, ω d ) achievable for all ω d .Equations (11) and (12) are complementary in that both canbe in force simultaneously.We now discuss conditions under which these bounds canbe achieved. C. Conditions for equality
We distinguish between conditions for equality in (7)–(9)and conditions for equality in (11), (12). The former applyto any passive real network and the latter apply to networkswith particular passband characteristics. Bounds (7)–(9) havea common set of conditions for achieving equality:1) S ( s ) S T ( − s ) + S G ( s ) S TG ( − s ) = I for all s
2) The M × M matrix S H ( I − S G S L ) − H ( I − S HL S L )( I − S G S L ) − S (13)has equal singular values for all s = jω I − S L ( s ) S G ( s ) is non-singular4) S TL ( − s ) − S G ( s ) has no zeros in the RHPwhere s is defined in (6). These four conditions correspond tofour possible impediments to achieving the bounds. Meetingall the conditions is sufficient to attain equality in the bounds,but Conditions 1, 2 and 4 are also necessary. The N × N matrix S G ( s ) (see Figure 1) plays a prominent role in theseconditions and can be readily measured or modeled by con-necting pairs of output ports of the matching network to anetwork analyzer while terminating its remaining ports withcharacteristic impedances.These conditions have physical interpretations. Condition 1is satisfied for lossless networks since we are guaranteed that S ( s ) S T ( − s ) + S ( s ) S T ( − s ) = I for all s because S ( s ) is a para-unitary matrix, and S G ( s ) = S ( s ) .Condition 2 requires the singular values of (13) to be equalfor all s = jω . When this is satisfied, the total dissipatedpower at the loads depends only on the total incident power k ~a ( jω ) k , and not the “direction” of ~a ( jω ) .Condition 3 is a “non-degenerate” condition that we illus-trate with an example: Suppose the loads are capacitive toground, and hence are reflective with S L ( ∞ ) = − I . If theoutput impedance of the matching network is also capacitive,then S G ( ∞ ) = − I and Condition 3 is violated. Hence, inthis example, a matching network that wants to achieve highbandwidth should avoid capacitive output impedance. It turnsout that Condition 3 is superfluous in Bound 2 because S L ( s ) and S G ( s ) are bounded matrices; hence S L ( s ) S G ( s ) is alsobounded and I − S L ( s ) S G ( s ) is always non-singular forRe { s } > [24, 7.22]. Additional details on Condition 3 canbe found in [9].Condition 4 is a “minimum-phase” condition since the RHPzeros of S TL ( − s ) − S G ( s ) are the same as the RHP zeros of S GM ( s ) , which involves a “Darlington equivalent” network representation of the loads [28], [29] as described in Part II.However, knowledge of the Darlington equivalent network isnot needed to check this condition. We have more to sayabout network architectures that cannot meet this conditionin Section IV-C.Assuming Conditions 1–4 are met by the matching network,we obtain equality in (11), (12) if the network also achievesthe ideal response r ( ω ) = τ for ω ∈ [ ω , ω ] and r ( ω ) = 1 elsewhere. Any network such that r ( ω ) < for ω [ ω , ω ] sustains a non-negative “shaping loss” which is the differencebetween the integral over all ω of the left-hand sides of (7)–(9)versus the integral over [ ω , ω ] :shaping loss = Z ∞ f ( ω ) log 1 r ( ω ) dω − Z ω ω f ( ω ) log 1 r ( ω ) dω. (14)For a network whose shaping loss is positive, some perfor-mance is lost in the band of interest because either ω − ω could potentially be made larger, or τ could be made s-maller, by redesigning the network to have larger r ( ω ) for ω [ ω , ω ] . We provide an example of the shaping losscomputation in Part II, Section III-A. D. How to obtain S L ( s ) and its poles and zeros If analytical circuit models for the loads are known, S L ( s ) is uniquely determined by the standard Laplace transform rep-resentations of the model impedance or admittance matrix andusing the formula S L ( s ) = ( Z L ( s ) + Z I ) − ( Z L ( s ) − Z I ) .Examples of this are presented in Section IV-B.Absent an analytical model, numerical methods that modelthe S L ( s ) of the loads are needed to extract its poles andzeros. The modeling methods are discussed and illustrated byexample in Part II.IV. B ANDWIDTH A NALYSIS
The bounds yield a variety of conclusions when they areapplied to various system configurations. We first examinedecoupled loads in Section IV-A and then coupled loads inIV-B. We then identify network architectures that can achievethe bounds in Section IV-C.
A. Decoupled loads
Let S L ( s ) = S l ( s ) I where S l ( s ) is a scalar and I is an N × N identity matrix. Thus, there are N decoupled identical loads.Let S l ( s ) satisfy (6) for some s , and let B l be computed usingTable I applied to S l ( s ) for M = 1 . Then S L ( s ) satisfies (6)at the same s , and has poles and zeros at the same locationsas S l ( s ) , each with multiplicity N . It follows from Table Ithat the bound for N loads is N times the bound for a singleload. We have proven the following theorem. Theorem 1:
For N identical decoupled loads driven by M sources Z ∞ f ( ω ) log 1 r ( ω ) dω ≤ NM B l . (15)If N = M , (15) is simply B l . This is not surprising becausewe have assumed the sources and loads are decoupled and therefore we are essentially examining N identical isolatedsystems, each with bound B l . However, (15) scales linearlywith N for a fixed M . This formula suggests that N identicaldecoupled loads can achieve N times the bandwidth of oneload as long as the matching network is designed properly.Matching network architectures can be thought of in termsof their ability to achieve linear-in- N bandwidth behaviorwhen used with a set of loads with linear-in- N behavior.In Section IV-C we show that certain network architecturesprovably have linear-in- N behavior, while others do not. Asshown in the next section, this linear-in- N behavior extendsto coupled loads under some conditions. B. Circulantly-coupled loads
Loads with circulant S L ( s ) are especially easy to manipu-late mathematically because, while the eigenvalues of S L ( s ) depend on s , its eigenvectors do not. This makes computing p L,i and z L,i straightforward. Load structures that lead to sym-metric circulant S L ( s ) are identical, and the coupling betweenload i and neighbor j depends on min {| i − j | , N − | i − j |} .Two identical loads automatically have circulant symmetry,as long as they have reciprocal coupling. Three or more loadscan be placed in a circular or spherical arrangement to yieldcirculant S L ( s ) .
1) Loads exemplifying Bound 1:
Figure 2(a) illustrates N circulantly-coupled loads. The loads consist of an N -portinductive network with impedance matrix sZ l and an N -portseries LC network with impedance matrix s /ω +12 s Z lc resonat-ing at jω . These networks are connected in parallel to eachother, and terminated by isolated characteristic impedances Z .The parallel N -port networks are shown in Figure 2(b,c); theinductive network has each port grounded through L andevery pair of ports i and j is connected through L ℓ , where ℓ = min {| i − j | , N − | i − j |} is the “distance” between ports i and j . The series LC network has each port grounded through L ′ and C ′ , and every pair of ports i and j is connected through L ′ ℓ and C ′ ℓ , where p L ′ C ′ = p L ′ ℓ C ′ ℓ = 1 /ω . The N × N matrices Z l and Z lc are circulant symmetric.Since circulant matrices have eigenvectors that are columnsof a discrete Fourier transform (DFT) matrix, we obtain thefollowing eigenvalue decompositions: Z l = W Λ l W H , Z lc = W Λ lc W H , where W = 1 √ N · · · e − j πN · · · e − j π ( N − N ... ... . . . ... e − j π ( N − N · · · e − j π ( N − N (16)is the N × N unitary DFT matrix, and Λ l and Λ lc are realpositive diagonal matrices representing the eigenvalues of Z l and Z lc , respectively.The impedance matrix of the loads is Z L ( s ) = (cid:18) s Z − l + 2 ss /ω + 1 Z − lc + 1 Z I (cid:19) − .
21 34N 21 34NlsZ (cid:0) (cid:0)0/ 1 lcs Z(cid:1)s(cid:2) (cid:3) 0L (cid:4)L(cid:0)L 0C(cid:5) (cid:4)C(cid:5)(cid:0)C(cid:5) 0L(cid:5) (cid:4)L(cid:5)(cid:0)L(cid:5)((cid:6)(cid:7) ((cid:8)(cid:7)((cid:9)(cid:7)21 34N 0Z
Fig. 2. (a) N circulantly-coupled loads consisting of an inductive networkand an LC network in parallel, terminated in a set of characteristic impedances Z . The inductive and the LC portions of the network are shown in (b,c), andhave impedance matrices sZ l and s /ω +12 s Z lc , respectively. The loads arecoupled to ground through L and series L ′ and C ′ , and every load pair iscoupled through L ℓ and series L ′ ℓ and C ′ ℓ , where ℓ is the “distance” betweenthe pair. Then S L ( s ) can be obtained from S L ( s ) = ( Z L ( s ) + Z I ) − ( Z L ( s ) − Z I ) , which is S L ( s ) = W − Z [(2Λ − lc ω +Λ − l ) s +Λ − l ω ]2 s + Z (2Λ − lc ω +Λ − l ) s +2 ω s + Z Λ − l ω W H . (17)Note, we write ( · ) − as · ) when we take inverses of diag-onal matrices. Because S L ( s ) is a circulant matrix, only itseigenvalues depend on s and the poles and zeros of S L ( s ) aretherefore the poles and zeros of the individual eigenvalues.All N loads present short circuits to ground at s = 0 and jω , where the loads become perfect reflectors. Hence, S L (0) = S L ( jω ) = − I , and (6) is satisfied. We can thenobtain two distinct bounds by applying Bound 1 for s = 0 and jω . Using p L,i , z
L,i calculated from (17), Bound 1 yields Z ∞ ω − log 1 r ( ω ) dω ≤ − π M X i − λ l,i Z = π · tr Z l M Z , (18a) Z ∞ ( ω − ω ) − + ( ω + ω ) − r ( ω ) dω ≤ − π M X i − λ lc,i Z ω = 2 π · tr Z lc M Z ω . (18b)The i th diagonal element of Z l represents the inductancerelative to ground of port i of the inductive network inFigure 2(b), measured with the remaining ports open. Because Z l is circulant its diagonal elements are all equal. Hence (1 /N ) tr Z l = L eq ,N , where L eq ,N is the inductance of anyport relative to ground. A similar conclusion holds for theLC network in Figure 2(c). Let the equivalent series LCbranch of any port relative to ground have inductance L ′ eq ,N and capacitance C ′ eq ,N , where L ′ eq ,N C ′ eq ,N = 1 /ω . Then (1 /N ) tr Z lc = ω L ′ eq ,N + 1 /C ′ eq ,N . We can rewrite (18) as Z ∞ ω − log 1 r ( ω ) dω ≤ N πL eq ,N M Z , (19a) Z ∞ ( ω − ω ) − + ( ω + ω ) − r ( ω ) dω ≤ N πM Z L ′ eq ,N + 1 ω C ′ eq ,N ! , (19b)Let L eq ,N , L ′ eq ,N and C ′ eq ,N approach respective limits L eq , L ′ eq and C ′ eq as N → ∞ . We have proven the following theorem. Theorem 2:
The linear-in- N behavior shown in Theorem1 for decoupled loads extends to circulantly-coupled loadsprovided L eq > and L ′ eq > (or C ′ eq < ∞ ).The conditions L eq > and L ′ eq > (or C ′ eq < ∞ ) areequivalent to ensuring that the inductance (or capacitance) ofany port relative to ground does not go to zero (or infinity) as N → ∞ , and hence there are not “too many” parallel pathsto ground from any port. This is equivalent to imposing acondition that the coupling between loads not be “too strong”.For a given N and M , the bounds in (19) for Figure 2increase as inductive and capacitive coupling components areremoved, since L eq ,N increases and approaches L as all cross-inductive elements are removed; similarly L ′ eq ,N increases to L ′ and C ′ eq ,N decreases to C ′ because L eq ,N , L ′ eq ,N and C ′ eq ,N are obtained as L , L ′ and C ′ in parallel with the remainingparts of the networks.However, we cannot conclude that coupling always has anegative effect on bandwidth. In fact, Part II of this papershows that coupling between dipole antennas has a non-monotonic effect on the bandwidth bound. This dichotomyin behavior between the model in Figure 2 and the dipoles inPart II is not contradictory because there is no requirementthat dipoles should be modeled by fixed L , L ′ , and C ′ asthe coupling between them changes.
2) Loads exemplifying Bound 2:
Figure 3 illustrates N resistors Z terminated with two parallel N -port networks:one is resistive with N × N circulant impedance matrix Z r ,and the other is capacitive with N × N circulant impedancematrix s Z c . The impedance matrix of the loads Z L ( s ) is Z L ( s ) = Z I + ( Z − r + sZ − c ) − . Let Λ r and Λ c be the eigenvalue matrices of Z r and Z c . Then S L ( s ) is S L ( s ) = W Λ r Λ c Z Λ r s + 2 Z Λ c + Λ r Λ c W H (20)where W is given in (16).We note that I − S TL ( − s ) S L ( s ) = W − Z Λ r s + 4 Z ( Z I + Λ r )Λ c − Z Λ r s + (2 Z I + Λ r ) Λ c W H . Let the component values satisfy Λ c √ Z ( Z I +Λ r ) Z Λ r = σ I forsome σ > . Then (6) is satisfied for s = σ . We substitute Λ c = Z Λ r σ √ Z ( Z I +Λ r ) into (20), and apply (8) to yield Z ∞ σ σ + ω log 1 r ( ω ) dω
21 34N rZ 1 cs Z0Z 0R (cid:0)R(cid:1)R 0C (cid:0)C(cid:1)C 21 34N 21 34N ((cid:2)(cid:3)((cid:4)(cid:3)((cid:5)(cid:3)
Fig. 3. (a) N circulantly-coupled loads consisting of N resistors Z in serieswith a parallel resistive network Z r and a parallel capacitive network s Z c .The resistive and the capacitive portions of the network are shown in (b,c),and have respective impedance matrices Z r and s Z c . The loads are coupledto ground through R and C , and every load pair is coupled through R ℓ and C ℓ , where ℓ is the “distance” between the pair. ≤ − π M log det Λ r Z + Λ r + 2 p Z ( Z + Λ r )= π M N X i =1 log 2 Z + λ r,i + 2 p Z ( Z + λ r,i ) λ r,i , (21)where λ r,i are the diagonal elements of Λ r . The linear-in- N behavior of these loads depends on λ r, , . . . , λ r,N ; we do notpursue this analysis any further here. C. Ability of network architecture to achieve bounds
Matching network architectures that cannot achieve equalityin the bounds should be identified whenever possible andremoved from consideration in large-bandwidth applications.The following theorem shows how S G ( s ) defined in Figure 1can be compared with S L ( s ) to identify such networks. Wethereby make Condition 4 in Section III-C more physicallytangible. Theorem 3:
For matching networks satisfying Condition 3,if S G ( s ) has an eigenvector in common with S L ( s ) , and theassociated eigenvalue λ G ( s ) satisfies | λ G ( jω ) | = 1 for all ω ,then Condition 4 cannot be satisfied and therefore the boundscannot be achieved. Proof:
See the Appendix.Theorem 3 can be applied to the example of decoupled loadsin Section IV-A. We assume ≤ M < N and Condition 3 issatisfied. Since S L ( s ) = S l ( s ) I , any vector is an eigenvectorof S L ( s ) . Theorem 3 then implies that any S G ( s ) that has | λ G ( jω ) | = 1 cannot achieve equality in (15).For example, networks where S G ( s ) is a real symmetricmatrix cannot achieve equality. This follows because even ifthe network is lossless and Condition 1 is satisfied, implyingthat S ( s ) S T ( − s ) + S G ( s ) S TG ( − s ) = I for all s , thenbecause S ( s ) is an N × M matrix and S G ( s ) is an N × N matrix, S G ( s ) has at least N − M unit singular values. Since (cid:6)(cid:7) P(cid:8)e(cid:9)(cid:10)(cid:8)ibed(cid:11)(cid:12)+(cid:13))-po(cid:8)tmat(cid:10)hingnetwo(cid:8)k SLM(cid:14) (cid:12)-po(cid:8)tload(cid:9) S... (cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22) (cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)Two-po(cid:8)tmat(cid:10)hingnetwo(cid:8)k S (cid:29)(cid:24)(cid:25)(cid:30)(cid:27)(cid:28) (cid:31) !" , Fig. 4. An RF system where one source drives N loads having S-matrix S L through a constrained matching network consisting of a prescribed ( N +1) -port network S and an arbitrary network S (the complex-frequencyargument s is omitted). We use S L, eq to denote the S-parameter seen fromthe input of S . the moduli of the eigenvalues of a real symmetric matrix areequal to its singular values, at least one eigenvalue satisfies | λ G ( jω ) | = 1 . We conclude that reciprocal broadband splittersor couplers that yield real symmetric S G ( s ) cannot be usedto achieve equality in (15) when M < N and the loads aredecoupled. We see an example of this in the next section.V. O NE S OURCE AND M ULTIPLE L OADS
We probe the special case of one source more deeplywith some examples in this section. We analyze constrainednetworks in Section V-A. Decoupled and coupled loads areexamined in Sections V-B and V-C, leading to a class of non-reciprocal networks called “determinant” networks.
A. Constrained matching network
Let a portion of the network be constrained or prescribedto have a particular structure. For example, the constrainedportion might include ideal circulators or power splitters.The prescribed portion and loads then together constitute an“equivalent load” and the achievable bandwidth is determinedby the characteristics of this load.Figure 4 illustrates an example of such a structure for M =1 , where a prescribed ( N + 1) -port network is combined withthe loads and forms an equivalent one-port load. Let the S-matrix of the prescribed network be S ( s ) = N S , ( s ) S , ( s ) N S , ( s ) S , ( s ) ! . Then the S-parameter of the equivalent load is S L, eq ( s ) = S , ( s )+ S , ( s ) S L ( s )( I − S , ( s ) S L ( s )) − S , ( s ) . (22)The remaining unspecified portion of the network S ( s ) isconnected to the input port of S L, eq ( s ) , and we may ask whatbandwidth is attainable by S ( s ) .Generally, we cannot apply the bounds directly to S L, eq ( s ) .Unlike S L ( s ) where power is either dissipated or reflectedby the loads, power in S L, eq ( s ) can also be absorbed by theconstrained portion of the network when S ( s ) is lossy. Let η ( ω ) denote the ratio between the power dissipated by S L ( s ) and the power delivered to S L, eq ( s ) , defined as η ( ω ) = k ~a ( jω ) k − k ~b ( jω ) k k a ( jω ) k − k b ( jω ) k , where ( ~a ( jω ) , ~b ( jω )) and ( a ( jω ) , b ( jω )) are the (inci-dent, reflected) signals from S L ( s ) and S L, eq ( s ) , respectively.Then ≤ η ( ω ) ≤ and η ( ω ) = S H , ( I − S , S L ) − H ( I − S HL S L )( I − S , S L ) − S , − S HL, eq S L, eq , (23)which depends only on the prescribed network and the loads.We assume S L, eq ( s ) satisfies (6) for some s ; this doesnot require S L ( s ) to satisfy (6) for the same s . Then (10)becomes the following bound. Theorem 4:
For loads matched by a constrained network, Z ∞ f ( ω ) log s η ( ω ) r ( ω ) + η ( ω ) − dω ≤ B eq (24)where B eq is the right-hand side of (10) applied to S L, eq ( s ) . Proof:
Using Figure 4, we define r eq ( ω ) = 1 − k a ( jω ) k − k b ( jω ) k k a ( jω ) k as the ratio between power not delivered to S L, eq ( s ) and theincident power from the source. Then it follows that r ( ω ) = 1 − k ~a ( jω ) k − k ~b ( jω ) k k a ( jω ) k = 1 − k a ( jω ) k − k b ( jω ) k k a ( jω ) k · k ~a ( jω ) k − k ~b ( jω ) k k a ( jω ) k − k b ( jω ) k = 1 − (1 − r eq ( ω )) · η ( ω ) . Equivalently, we have r eq ( ω ) = q r ( ω )+ η ( ω ) − η ( ω ) .We apply (10) to the equivalent load S L, eq ( s ) . The result isan inequality on r eq ( ω ) : Z ∞ f ( ω ) log 1 r eq ( ω ) dω ≤ B eq , where B eq is the right-hand side of (10) applied to S L, eq ( s ) .Replacing r eq ( ω ) with q r ( ω )+ η ( ω ) − η ( ω ) gives us (24).We use beamforming as an example application. Let thedesired amplitude and phase relationship of the antennas bedenoted by the N × unit real-rational vector ~v ( s ) for s = jω .Then S ( s ) = (cid:20) ~v T ( s ) ~v ( s ) 0 (cid:21) (25)denotes the S-matrix of a reciprocal one-to- N power dividerthat constrains the incident signal ~a ( jω ) to be alignedwith ~v ( jω ) . We are not concerned with the so-called gain-bandwidth product of a phased array [30] which measures itsability to maintain far-field gain across a range of frequencies. Theorem 5:
Let the beamforming vector ~v ( s ) be a realconstant unit eigenvector of S L ( s ) . Then Z ∞ f ( ω ) log 1 r ( ω ) dω ≤ B eq , (26)where B eq is calculated using S L, eq ( s ) = ~v T ( s ) S L ( s ) ~v ( s ) . Proof:
We substitute (25) into (22) and (23), where S , = 0 , S , = ~v T ( s ) , S , = ~v ( s ) and S , = 0 .Then (22) yields S L, eq ( s ) = ~v T ( s ) S L ( s ) ~v ( s ) , and (23) yields η ( ω ) = ~v H ( I − S HL S L ) ~v − S HL, eq S L, eq = ~v H ~v − ~v H S HL S L ~v − ~v H S HL ~v ∗ ~v T S L ~v . Because ~v ( s ) is a real unit eigenvector of S L ( s ) , it is readilyverified that η ( ω ) = 1 . The result then follows by applyingTheorem 4.Section III-B in Part II of this paper provides an examplewhere B eq in (26) varies with the choice of ~v ( s ) .We note that η ( ω ) = 1 is obtained for any lossless S ( s ) ,in which case (26) also applies. When η ( ω ) = 1 and S L ( s ) satisfies (6) for some s , then S L, eq ( s ) also satisfies (6) forthe same s ; the converse is not true.Compared with (10), (26) is smaller for N > . To seethis more explicitly, let the loads be decoupled as in SectionIV-A, whence S L ( s ) = S l ( s ) I . Then any real constant unitvector ~v ( s ) satisfies η ( ω ) = 1 , and the equivalent load satisfies S L, eq ( s ) = S l ( s ) . Then Theorem 5 yields Z ∞ f ( ω ) log 1 r ( ω ) dω ≤ B l , (27)where B l is the right-hand side of (10) applied to S l ( s ) for M = 1 , Clearly, B l < N B l , which is the bound obtained in(15) for M = 1 . We conclude that linear-in- N behavior is notgenerally achieved for the beamforming structure (25). Thisconclusion is consistent with Theorem 3. B. Decoupled loads
Section IV-C says that full bandwidth cannot be achievedwhen reciprocal broadband splitters are used to drive
N > loads when M = 1 . We instead consider a matching networkconsisting of non-reciprocal couplers in Figure 5(a) where an ( N + 1) -port network comprising N − circulators as theconstrained part of the network is displayed. The resulting ( N + 1) × ( N + 1) S-matrix is S ( s ) = · · · · · · · · · ... . . . . . . ... · · · . The lower right N × N block of this matrix, which cor-responds to S G ( s ) , has N zero eigenvalues and thereforedoes not satisfy the conditions of Theorem 3. Let S L, eq ( s ) be the S-matrix of the equivalent one-port load seen at theinput of the circulators. Because the circulators are lossless,
0Z C(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:1) L(cid:7)(cid:5)(cid:8)1L(cid:7)(cid:5)(cid:8)2L(cid:7)(cid:5)(cid:8)N(cid:9)( )r (cid:9)(cid:10) B(cid:11)(cid:12)(cid:13)(cid:14) ...(cid:15) (cid:9)( )r (cid:9)(cid:10) B(cid:11)(cid:12)(cid:13)(cid:16)...(cid:15)A(cid:17)(cid:17) (cid:18)(cid:19)(cid:20)(cid:21)(cid:22) (cid:9)( )r (cid:9)(cid:10) ...(cid:15) (cid:23)(cid:23)(cid:23) (cid:9)( )r (cid:9)(cid:10) B(cid:11)(cid:12)(cid:13)(cid:24) ...(cid:15) M(cid:19)(cid:25)(cid:26)(cid:27)(cid:28)(cid:20)(cid:29)(cid:30)(cid:31)(cid:25) !"
Fig. 5. (a) N − circulators are used to match N decoupled loads to M = 1 source, and achieve linear-in- N behavior. We use S L, eq ( s ) to denotethe S-parameter of the equivalent one-port load seen at the input of thecirculators. (b) N decoupled loads are driven by M = 1 source through N two-port networks and N − broadband circulators. Each of the two-portsmatches /N of the total bandwidth; network i passes band i and reflects theremaining portions. The circulators combine all N passbands and achieve atotal bandwidth N times of the bandwidth of each two-port network. η ( ω ) = 1 in this constrained network. It is readily calculatedthat S L, eq ( s ) = [ S l ( s )] N , which has poles and zeros at thesame locations as S l ( s ) , each with multiplicity N . Therefore,according to (10), the bandwidth achievable with this equiv-alent S L, eq ( s ) is N times that achievable by S l ( s ) . Hence,linear-in- N network behavior for N decoupled loads can beachieved by ideal circulators.This circulator-based network architecture presents “multi-ple opportunities” for the energy that is reflected from anyone load to be forwarded to the next for another attempt attransmission. We are ignoring the insertion losses associatedwith cascading circulators in such an arrangement.Figure 5(b) has a structure similar to Figure 5(a) but matches /N of the total bandwidth in an orderly fashion. The firstnetwork passes the lowest portion of the band and reflectsall the remaining portions, which are passed to the nextcirculator which matches the next portion, and so on. Thisstructure resembles a channelizer [31] in the sense that thereare multiple output ports where each port contains a portion ofthe total bandwidth. Although we have drawn these networksas having near-ideal flat frequency responses, this aspect is notcrucial. An explicit example of this type of network for a pairof dipoles is presented in Section III-A in Part II. C. Circulantly-coupled loads: Determinant networks
The previous section gave a circulator-based architecturefor achieving linear-in- N bandwidth for N uncoupled loads.We now show how to handle circulantly coupled loads suchas examined in Section IV-B, by demonstrating a networkarchitecture that converts the multiport system with S-matrix S L ( s ) into a single-port system with S-parameter S L, eq ( s ) =det S L ( s ) . Since det S L ( s ) has the same poles and zeros as S L ( s ) if S L ( s ) has no cancelling poles and zeros, S L, eq ( s ) hasthe same bound as S L ( s ) . We denote any network that converts S L ( s ) into det S L ( s ) a “determinant network”. Determinantnetworks are linear-in- N .Let W be defined as in (16) and have columns ~w , . . . ~w N .Define W = [ ~w ~w · · · ~w N ~ and the ( N + 1) × ( N + 1) matrix S ( s ) = (cid:20) ~w HN ~w W W H (cid:21) . (28)Notice that W is missing the column ~w . Then S ( s ) isconstant and lossless. The following theorem says that it isalso a determinant network. Theorem 6:
Let the network described by (28) have its N outputs connected to any circulantly-coupled set of loads withS-matrix S L ( s ) . Then its input has equivalent S-parameter S L, eq ( s ) = det S L ( s ) . Proof:
We let L = W H W and use the eigenval-ue decomposition S L ( s ) = W Λ( s ) W H where Λ( s ) = diag ( λ ( s ) , . . . , λ N ( s )) is a diagonal matrix of eigenvalues.Then (22) yields S L, eq ( s ) = ~w HN S L ( s )( I − W W H S L ( s )) − ~w = ~w HN W Λ( s ) W H ( I − W L Λ( s ) W H ) − ~w = (cid:2) · · · λ N ( s ) (cid:3) ( I − L Λ( s )) − × (cid:2) · · · (cid:3) T , where I − L Λ( s ) = · · · − λ ( s ) 1 0 · · · − λ ( s ) 1 0 ... . . . . . . ... · · · − λ N − ( s ) 1 . It follows that S L, eq ( s ) is λ N ( s ) times the ( N, entry of ( I − L Λ( s )) − . This gives S L, eq ( s ) = Q Nn =1 λ n ( s ) .An intuitive explanation of the operation of the networkis as follows. The network (28) orients the power from thesource along the first eigenvector ~w . Energy is then reflectedby the loads with amplitude λ ( s ) along ~w , at which pointthe determinant network reflects it entirely back to the loads,but with orientation ~w . This is reflected by the loads withamplitude λ ( s ) which is returned by the network to theloads reoriented along ~w , and so on. The last eigenvector ~w N reflected by the loads is returned to the source. The resultis an overall S-parameter with value Q Nn =1 λ n ( s ) , which isthe determinant of S L ( s ) . This description also explains whya determinant network is not unique.For example, for N = 2 , S ( s ) = √ − √ √ √ − − . (29)As can by seen by this S-matrix, the network works by firstexciting the even mode of the coupled loads. Any reflectedpower, which is also in an even mode, is then converted intoan odd-mode excitation.One possible realization of (29) is given in Figure 6, whichutilizes an ideal transformer and gyrator. A gyrator withresistance Z is transparent to forward propagating waves but
0Z M(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)N(cid:7)(cid:1)(cid:8)(cid:9)(cid:10)(cid:11) M(cid:12)(cid:1)(cid:12)(cid:0)(cid:13)(cid:2)(cid:9)(cid:12)c(cid:13)(cid:4)(cid:5)(cid:6)21 21 (cid:14)(cid:15)
Fig. 6. Realization of the network (29) that includes a transformer andnon-reciprocal gyrator connected to two coupled antennas. behaves as a π phase shift for reverse waves. The networkin Figure 6 converts the wave incident on the transformerprimary into an even-mode excitation for the loads on itssecondaries. Any power reflected by the loads, which is also inan even mode, is converted into an odd mode by the gyrator.The secondaries of the transformer acts like an open circuitto this odd-mode signal. The signal is therefore reflected bythe transformer back to the loads as an odd-mode excitation.Finally, any odd-mode excitation reflected by the antennas isthen converted by the gyrator into an even-mode signal andpassed by the transformer back to the source.The network in Figure 6 can achieve the bound in (10) fora single source and two identical loads that are reciprocallycoupled. This network would also work well for the decoupledloads in Section IV-A since decoupled loads are trivially recip-rocal. Note that the circulator structure proposed in Figure 5is a determinant network when the loads are decoupled but isotherwise not. Hence, the network in Figure 6 generalizes thenetworks in Figure 5 for N = 2 .VI. C ONCLUSIONS
We have presented bandwidth analyses for multiport RFsystems using bandwidth upper bounds that apply to anarbitrary number of sources and loads, and allows arbitrarycoupling between the loads. Conditions for achieving equalityin the bounds were discussed. We demonstrated that thebandwidth bounds scale generally as
N/M for M sourcesand N loads. We focused on loads whose scattering matricescan be expressed analytically with rational functions. The caseof one source and many loads was examined deeply.In Part II of this paper, we apply the upper bounds torealistic loads whose scattering parameters are expressed nu-merically and therefore need to be approximated by rationalfunctions. The accuracy needed of such approximations isanalyzed. Some of the effects of coupling are examinedin detail, and complete derivations of the bounds are alsoprovided. A PPENDIX P ROOF OF T HEOREM ~v ( s ) be the eigenvector of S L ( s ) and S G ( s ) associatedwith eigenvalues λ L ( s ) and λ G ( s ) . Because S L ( s ) , S G ( s ) are bounded matrices, λ L ( s ) , λ G ( s ) are bounded functions.Since | λ G ( jω ) | = 1 , λ G ( s ) is an all-pass function with allpoles in the LHP and zeros in the RHP. Moreover, because I − S TL ( − s ) S L ( s ) is full normal rank and S L ( s ) satisfies (6), λ L ( s ) is non-constant and satisfies λ L ( − s ) λ L ( s ) = 1 where s is defined in (6). Thus, | λ L ( s ) | < for Re { s } > , | λ G ( s ) | ≥ for Re { s } < and | λ G ( s ) | ≤ for Re { s } > ;the equalities holds if λ G ( s ) is a constant.We start by showing ~v H ( jω ) S G ( jω ) = λ G ( jω ) ~v H ( jω ) .Since λ G ( jω ) and ~v ( jω ) are an eigenvalue-eigenvector pairof S G ( jω ) , ~v H ( jω )( S G ( jω ) − λ G ( jω ) I ) ~v ( jω ) = 0 . Suppose ~u H ( jω ) = ~v H ( jω )( S G ( jω ) − λ G ( jω ) I ) = 0 , then ~u ( jω ) ⊥ ~v ( jω ) , and k ~v H ( jω ) S G ( jω ) k = k λ G ( jω ) ~v H ( jω ) + ~u H ( jω ) k > k λ G ( jω ) ~v H ( jω ) k = k ~v H ( jω ) k . But because S G ( s ) is bounded, all singular values of S G ( jω ) are no greater than one. This contradicts the inequalityabove. Hence, ~u H ( jω ) = 0 , and we have ~v H ( jω ) S G ( jω ) = λ G ( jω ) ~v H ( jω ) .Let ~v ′ ( s ) be the extension of ~v ∗ ( jω ) to the WCP. We thenshow that Condition 4 cannot be achieved when Condition3 is satisfied. We need to show ~v ′ T ( s )( S TL ( − s ) − S G ( s )) =( λ L ( − s ) − λ G ( s )) ~v ′ T ( s ) = 0 for some s in the RHP. It sufficesto show λ L ( − s ) − λ G ( s ) = 0 .Since | λ L ( − s ) | < and | λ G ( s ) | ≥ for Re { s } < , allzeros of λ L ( − s ) − λ G ( s ) = 0 must locate in Re { s } ≥ .Both λ L ( s ) and λ G ( s ) are rational functions, so we let theirdegrees be n and n , respectively. Since all the poles of λ L ( s ) and λ G ( s ) are in the LHP, the poles of λ L ( − s ) and λ G ( s ) do not coincide. Hence λ L ( − s ) − λ G ( s ) has degree n + n . Moreover, | λ L ( jω ) | = 1 has at most n solutions;this is because − λ L ( − s ) λ L ( s ) has degree at most n ,and any zeros of − λ L ( − s ) λ L ( s ) on the imaginary axishave multiplicity at least two. Since | λ G ( jω ) | = 1 for all ω , λ L ( − s ) − λ G ( s ) has at most n zeros on the imaginary axis.This leaves at least n zeros in the RHP, and we have proventhe theorem when n ≥ .When n = 0 , we prove by contradiction. Suppose all n zeros of λ L ( − s ) − λ G ( s ) are on the imaginary axis. Then thezeros of − λ L ( − s ) λ L ( s ) are at the same locations as the zerosof λ L ( − s ) − λ G ( s ) . Because we assume λ L ( − s ) λ L ( s ) = 1 for some s , s must be one of the zeros of λ L ( − s ) − λ G ( s ) .Thus, we have λ G ( s ) = λ L ( − s ) , and − λ G ( s ) λ L ( s ) =0 . This contradicts Condition 3. Hence, λ L ( − s ) − λ G ( s ) haszeros in the RHP when n = 0 . Thus, Condition 4 is not met,and since this condition is necessary for achieving the bounds,the bounds cannot be met with equality.R EFERENCES[1] H. W. Bode,
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Ding Nie
Ding Nie (M’16-) was born in Nanchang,Jiangxi, Peoples Republic of China. He received theB.S. degree in electronic engineering from Shang-hai Jiao Tong University, Shanghai, in 2010. Hereceived the M.S. degree and the Ph.D. degree inelectrical engineering from the University of NotreDame, Notre Dame, IN, in 2016. He then joinedApple Inc., Cupertino, CA, USA, in 2016. He wonthe 2016 Outstanding Young Author Award for thepaper he co-authored in the IEEE Transactions onCircuits and Systems. His research interests includecommunications, radio-frequency circuits and antennas.
Bertrand M. Hochwald
Bertrand M. Hochwald(F’08-) was born in New York, NY. He receivedhis undergraduate education from Swarthmore Col-lege, Swarthmore, PA. He received the M.S. de-gree in electrical engineering from Duke University,Durham, NC, and the M.A. degree in statistics andthe Ph.D. degree in electrical engineering from YaleUniversity, New Haven, CT.From 1986 to 1989, he worked for the Departmentof Defense, Fort Meade, MD. After completinggraduate school he was a Research Associate andVisiting Assistant Professor at the Coordinated Science Laboratory, Universityof Illinois, Urbana-Champaign. In September 1996, he joined the Mathemat-ics of Communications Research Department at Bell Laboratories, LucentTechnologies, Murray Hill, NJ where he was a Distinguished Member of theTechnical Staff.In 2005 he joined Beceem Communications, Santa Clara, CA, as their ChiefScientist, and in 2009 as Vice-President of Systems Engineering. He servedconcurrently as a Consulting Professor in Electrical Engineering at StanfordUniversity, Palo Alto, CA. In 2011, he joined the faculty at the University ofNotre Dame as Freimann Professor of Electrical Engineering.He received several achievement awards while employed at the Departmentof Defense and the Prize Teaching Fellowship at Yale University. He hasserved as an Editor for several IEEE journals and given plenary and invitedtalks on various aspects of signal processing and communications. He hasco-invented several well-known multiple-antenna techniques, including adifferential method, linear dispersion codes, and multi-user vector perturbationmethods. He has forty-five patents.He received the 2006 Stephen O. Rice Prize for the best paper publishedin the IEEE Transactions on Communications. He co-authored a paper thatwon the 2016 Best Paper Award by a young author in the IEEE Transactionson Circuits and Systems. He is listed as a Thomson Reuters Most InfluentialScientific Mind in 2014 and 2015. He is a Fellow of the IEEE.
Bandwidth Analysis of Multiport Radio-FrequencySystems—Part II
Ding Nie,
Member, IEEE, and Bertrand M. Hochwald,
Fellow, IEEE
Abstract —We analyze the bandwidth over which power canbe transferred from multiple radio-frequency sources to multipleloads through a passive multiport matching network. This is thesecond part of a two-part paper. In the first part we introducebroadband multiport matching upper bounds and apply them todetermine the bandwidth of loads and network structures whosescattering parameters can be expressed analytically with rationalfunctions. In this second part, we apply the bounds to loads,such as antennas, whose scattering parameters are obtained bymeasurement or simulation. We focus on the effects of couplingon bandwidth. Since the bounds require frequency responsesthat are rational functions, we provide guidelines on how toobtain rational approximations for arbitrary loads. Completederivations of the bounds are also provided.
Index Terms —Bandwidth, Bode-Fano bounds, broadbandmatching bounds, non-reciprocal networks, passive matchingnetworks, radio-frequency coupling
I. I
NTRODUCTION
We analyze the bandwidth over which power can be trans-ferred from M sources through a passive matching networkto N loads in radio-frequency (RF) systems. Our principaltools include broadband matching upper bounds, which arepresented in Part I of this paper [1], and briefly repeated hereinfor easy reference. These bounds extend the classical Bode-Fano results [2], [3], which apply to a single source and load.The bounds depend only on the scattering matrix (S-matrix)of the loads. The loads may be electromagnetically coupled toeach other.In this part, we apply the bounds to loads whose S-matrix isexpressed numerically, found through either measurements orsimulations. The bounds require rational models of the loadsthat can be obtained through fitting of the numerical data.We provide guidelines for assessing the accuracy of a rationalmodel.Of particular interest is the effect of coupling on thebandwidth of multiple antennas. One example demonstratesthat the bandwidth bound of a pair of dipoles fluctuates non-monotonically with the distance between them, and, with theright amount of coupling, can be significantly greater than fordecoupled dipoles. We focus on the “input bandwidth” of theloads, and are not concerned with how efficiently the loadsuse their input power. D. Nie is with Apple Inc., Cupertino, CA 95014, USA (e-mail:[email protected]).B. M. Hochwald is with the Department of Electrical Engineering,University of Notre Dame, Notre Dame, IN 46556, USA (e-mail: [email protected]).This work was supported, in part, by NSF grants CCF-1403458 and ECCS-1509188.
We also include complete derivations of the bounds. Thesederivations are not needed to apply the bounds but are of usein understanding how the conditions for equality in the boundapply.The notations and definitions are identical to those inPart I. We assume M decoupled sources with characteristicimpedance Z drive N loads that have a frequency-dependent N × N S-matrix S L ( s ) , where s = σ + jω is the extensionof frequency jω to the whole complex plane (WCP). An ( M + N ) -port passive matching network is inserted betweenthe sources and loads. Our measure of the quality of matchingis the power loss ratio r ( ω ) ∈ [0 , , defined as the ratiobetween the total power lost, including insertion and returnlosses, and the total incident power from the sources atfrequency ω . The insertion loss is the power dissipated in thematching network; the return loss is the power reflected to thesources. Since the loads are potentially coupled, the quantity r ( ω ) captures the possibility that incident power from onesource can reflect back to itself or to other sources. We thendefine the bandwidth ω BW ( τ, ω d ) as the frequency range forwhich r ( ω ) (the positive square root of r ( ω ) ) is no greaterthan a threshold τ > in the vicinity of a design frequency ω d . The mathematical definitions of r ( ω ) and ω BW ( τ, ω d ) canbe found in Part I.We generally employ the properties of S-matrices of passivereal networks to prove the bounds; these properties includereal-rational, Hurwitzian and bounded. We also use the defi-nitions of poles and zeros of rational matrices. We use LHPto denote the left-half complex plane (Re { s } < ) and RHPto denote the right-half (Re { s } > ). A. Summary of bounds
There are three bounds which are summarized as follows.If S TL ( − s ) S L ( s ) = I (1)is satisfied for some Re { s } ≥ , then the following inequalityholds for any passive matching network: Z ∞ f ( ω ) log 1 r ( ω ) dω ≤ B, (2)where f ( ω ) and B depend on the location of s and are givenin Table I. To compute the bound B , the poles and zeros of S L ( s ) , denoted as p L,i , z
L,i , are also needed. We refer to rows1, 2, and 3 in Table I as Bounds 1, 2, and 3.The bounds depend on the poles and zeros of the real-rational matrix S L ( s ) , which we explain how to obtain in TABLE IF
ORMS OF f ( jω ) AND B IN (2) FOR DIFFERENT LOCATIONS OF s , WHERE p L,i , z
L,i
ARE THE POLES AND ZEROS OF S L ( s ) . s = jω f ( ω ) = [( ω − ω ) − + ( ω + ω ) − ] B = − π M (cid:2)P i ( p L,i − jω ) − + P i ( z L,i + jω ) − (cid:3) Re { s } > f ( ω ) = Re [( s − jω ) − + ( s + jω ) − ] B = − π M log (cid:12)(cid:12)(cid:12) det S L ( s ) · Q i ( s + z L,i ) Q i ( s − z L,i ) (cid:12)(cid:12)(cid:12) s = ∞ f ( ω ) = 1 B = − π M (cid:0)P i p L,i + P i z L,i (cid:1)
Section II. We focus on the steps needed to fit realistic loadswith rational models. Bandwidth analyses of various loads arethen presented in Section III. The bounds are corollaries ofTheorems 2 and 3, which are presented in Section IV. Weconclude in Section V.II. O
BTAINING S L ( s ) AND ITS P OLES AND Z EROS
The matrix S L ( s ) = ( Z L ( s ) + Z I ) − ( Z L ( s ) − Z I ) isuniquely determined if the impedance matrix of the loads Z L ( s ) is known analytically. For many realistic loads, thisis generally impossible. Instead, either Z L ( jω ) or S L ( jω ) is often obtained as the result of a numerical simulation ormeasurement, where ω is contained to some frequency rangeof interest. The next section gives an example of how to usethe results of simulation to find S L ( s ) . A. Finding S L ( s ) from numerical simulations of loads The following steps can be used to find S L ( s ) :a) Measure or simulate the loads in the frequency range ofinterest [ ω , ω ] . Denote the measured response S ′ L ( jω ) .b) Find a passive rational S L ( s ) such that S L ( jω ) is “closeenough” to S ′ L ( jω ) for ω ∈ [ ω , ω ] .Step a) can be done with standard modeling software such asAnsys HFSS in the case of simulations, or a network analyzerin the case of measurements. Step b) can be accomplished byfitting rational functions to the individual entries of S ′ L ( jω ) using, for instance, the Matrix Fitting Toolbox [4]–[8] inMATLAB. We have more to say about this fitting in the nextsection.To compute the right-hand side of (2):c) Find an s where (1) is satisfied for the computed S L ( s ) .d) Calculate the poles and zeros p L,i , z
L,i of S L ( s ) .Step c) requires us to solve S TL ( − s ) S L ( s ) = I . For s = jω , we can instead solve | det S L ( jω ) | = 1 . For Stepd), p L,i , z
L,i can be obtained from the definitions of polesand zeros of matrices. In some cases the poles and zeros of S L ( s ) coincide with the poles and zeros of det S L ( s ) . Oneway of checking this is presented in [9], which we do notrepeat here. Then p L,i , z
L,i can be obtained by applying root-finding algorithms to / det S L ( s ) = 0 and det S L ( s ) = 0 ,respectively. The examples shown in Sections III-B–III-E usethis approach.Multiple rational models may exist within the error toler-ance of S ′ L ( jω ) for ω ∈ [ ω , ω ] . Each model could satisfy (1) for different s , and generate different bounds. When thesemodels yield consistent results, one can build confidence thatthe bounds and models are physically meaningful. When thesemodels contradict each other, further investigation is neededto determine the source of the inconsistency. An example ofmultiple models is shown in Section III-C. B. Analysis of distributed-element loads using rational models
Since S L ( s ) is a real-rational matrix, accurate modelingof lumped-circuit loads is straightforward and can be donewithout error. However, many loads such as antennas, trans-mission lines, and other distributed-element systems includetime-delays, stubs, and other structures that are often modeledusing non-rational functions of s . A real-rational S L ( s ) is thenneeded that “approximates” the loads with sufficient accuracyto obtain a meaningful bandwidth bound.The numerical and analytical approximation of distributed-element circuits by lumped circuits over a fixed bandwidth is atopic that has been studied in many contexts. An early exampleis [10]. A summary of some practical techniques can be foundin [11]. Recent advances in antenna modeling include [12].Other fields such as control engineering use rational fittingextensively [13].The mathematical problem of rational approximation andmodeling is addressed by Runge’s theorem in complex analy-sis [14], which states that any analytical function on a subset ofthe complex plane can be fit arbitrarily precisely with rationalfunctions. For example, the Pad´e approximation [15], [16] isoften used to model time delays using rational functions overa frequency band of interest.We are interested in obtaining an upper bound on Z ω ω f ( ω ) log 1 r ′ ( ω ) dω, (3)where r ′ ( ω ) is the loss experienced when the matchingnetwork is used with the actual loads S ′ L ( jω ) . The processoutlined in Section II-A creates a rational approximation S L ( jω ) for ω ∈ [ ω , ω ] . For an arbitrary matching network,(2) gives Z ω ω f ( ω ) log 1 r ( ω ) dω ≤ Z ∞ f ( ω ) log 1 r ( ω ) dω ≤ B, (4)where B is computed from S L ( s ) and r ( ω ) is the power lossratio experienced with S L ( s ) . The difference between the left-hand sides of (3) and (4) is Z ω ω f ( ω ) log r ( ω ) r ′ ( ω ) dω, and represents the error in the integral introduced by theapproximation of S ′ L ( jω ) . Then choosing S L ( s ) such that Z ω ω f ( ω ) log r ( ω ) r ′ ( ω ) dω ≤ δB (5)for some desired tolerance δB > yields the bound Z ω ω f ( ω ) log 1 r ′ ( ω ) dω ≤ B + δB. (6) The following theorem provides a first-order approximationof δB for S L ( s ) that is close to S ′ L ( jω ) in the frequency bandof interest. Theorem 1:
Let δS L ( jω ) = S L ( jω ) − S ′ L ( jω ) and define ρ ( ω ) = 2 σ δ, max ( ω )1 − σ ′ L, max ( ω ) · s σ ′ L, max ( ω ) − σ ′ L, min ( ω )(1 − σ ′ L, max ( ω )) , (7)where σ δ, max ( ω ) is the maximum singular value of δS L ( jω ) ,and ≤ σ ′ L, min ( ω ) < σ ′ L, max ( ω ) < are the minimum and themaximum singular values of S ′ L ( jω ) , respectively. Then δB → Z ω ω f ( ω )2 log (cid:18) − r ′ ( ω ) r ′ ( ω ) · ρ ( ω ) (cid:19) dω (8)as k δS L ( jω ) k F → for ω ∈ [ ω , ω ] . Proof:
See Appendix A.In (8), f ( ω ) is given in Table I, and ρ ( ω ) is determinedby S ′ L ( jω ) and δS L ( jω ) . The notation k · k F refers to theFrobenius norm (sum of squared magnitude of entries) of amatrix. In the limit of a perfect model when δS L ( jω ) = 0 ,then ρ ( ω ) = 0 and δB = 0 . In general, δB depends on r ′ ( ω ) ,which depends on the matching network. To eliminate thedependence of δB on r ′ ( ω ) , we choose the desired goal of r ′ ( ω ) ≈ τ for ω ∈ [ ω , ω ] , whence δB ≈ Z ω ω f ( ω )2 log (cid:18) − τ τ · ρ ( ω ) . (cid:19) dω. (9)Note that even if r ′ ( ω ) does not achieve τ for ω ∈ [ ω , ω ] ,(9) is still a useful quantity because it upper-bounds δB .The steps for evaluating goodness-of-fit are then:e) Compute δB in (9) numerically.f) Evaluate if the desired tolerance on δB/B is met.If the desired tolerance is not met, a smaller δS L ( jω ) isneeded. Finally,g) Compute the bound (6).Theorem 1 can also be used to examine cases wherethe rational fit may fail to generate an accurate bound. Forexample, if σ ′ L, max ( ω ) ≈ in the neighborhood of some ω ,meaning S ′ L ( jω ) is very reflective, (7) indicates that ρ ( ω ) becomes large. Any attempt to make r ′ ( ω ) small around this ω could then potentially make δB in (8) large. Thus, attemptingto match a load where it is naturally very reflective may leadto a loose bound. C. Hazard of over-fitting
In attempting to make δB/B small, care should be taken notto “over-fit” the loads. Over-fitting or over-modeling occurswhen the degrees of the polynomials in the numerator anddenominator of the rational functions in S L ( s ) are larger thanneeded to obtain the desired accuracy. As the polynomialdegrees are increased, the relative error δB/B is decreasedsince δB is decreased. But there is also the hazard that B is made unnecessarily large because of the excessive numberof poles and zeros in S L ( s ) . As a result, (6) becomes looseand Condition 4 for achieving B cannot be satisfied. There isa tension between making the rational model accurate whilestill having a small number of poles and zeros. One approach to determine if the system is over-fitted is toexamine the poles and zeros of the entries of S L ( s ) . Any polesand zeros that nearly cancel can have a small effect on δB buta large effect on B ; they should be removed and δB/B re-checked. Simpler models yield tighter bounds, and thus there isan incentive to find the minimal model that adequately capturesthe frequency response of the loads in the band of interest.A detailed discussion of modeling poles is found in [17]. Adiscussion of the sensitivity of poles and zeros to perturbationsof the data is found in [18].In the remainder, we use the notation r ( ω ) when we areconsidering the rational model, and we use r ′ ( ω ) when weare considering the actual loads. In Section III-D, we presentan example where time delay in a circuit is modeled usingrational functions. Section III-E computes (6) for four realisticcoupled antennas.III. B ANDWIDTH A NALYSIS OF A NTENNAS
We now calculate broadband matching bounds for fiveexamples, all involving amplifiers driving antennas. Aspects ofcoupling and rational approximation are emphasized. The firstthree examples assume that the rational models are accurateand therefore compute only (2), while the last two examplesincorporate the effects of rational fitting and therefore compute(6). We begin with two identical decoupled dipoles whoserational models are taken from [9].
A. Two decoupled dipoles
We design a broadband network of the type in Figure 5(b)in Part I for two decoupled dipoles ( N = 2 ) with one source( M = 1 ). From Theorem 1 in Part I, the bound for twodecoupled dipoles is twice as large as for a single dipole whenthere is only one source. The geometry of a dipole is shown inFigure 1(a), which is half-wavelength at 2.4 GHz. In [9], thesame dipole is simulated in the range 1–5 GHz, and is modeledusing an S-parameter model S l ( s ) normalized by characteristicimpedance Z = 50 Ω that satisfies S l ( ∞ ) = − . We applythe model here, and write the S-matrix of two such dipoles as S L ( s ) = diag ( S l ( s ) , S l ( s )) .Clearly, S L ( ∞ ) = − I , and (1) is satisfied at s = ∞ . Thepoles and zeros of S l ( s ) are listed in Table III in [9], andBound 3, using (2), for M = 1 gives Z ω ω log 1 r ( ω ) dω ≤ Z ∞ log 1 r ( ω ) dω ≤ . × , (10)where ω = 2 π × and ω = 10 π × are the lower andupper modeling frequencies.We are interested in matching the decoupled dipoles to onesource in the 2–4 GHz range. The structure in Figure 2 isemployed, where Matching Network 1 is tuned for the 2–3GHz band and Matching Network 2 is tuned for the 3–4 GHzband and one circulator is used. These two-port networks aredesigned using the Real Frequency Technique [19], followedby a realization using Darlington synthesis [20]. In Figure 1(b),the dashed and dash-dot curves show r ( ω ) of each networkwhen connected by itself to a dipole (typically through a balunthat is not shown). The r ( ω ) of the entire system, including ((cid:0)(cid:1) ((cid:2)(cid:1)62.5mm2.5mm −35−30−25−20−15−10−50 Frequency (Hz) r( ω ) ( d B ) IsolatedBand 1Band 2
Fig. 1. (a) Geometry of a dipole that is half-wavelength at 2.4 GHz. (b) Twodecoupled dipoles are matched by the structure shown in Figure 2. MatchingNetwork 1 handles 2–3 GHz and Network 2 handles 3–4 GHz; their resulting r ( ω ) ’s are given in the blue dashed and red dash-dot curves, respectively. Theoverall system r ( ω ) is shown by the solid curve. circulator, is the product of the r ( ω ) achieved by each, andis shown as the solid curve. We seek a power reflection ratioof r ( ω ) ≤ . ( − dB). From the figure, we see that thebandwidth achievable is ω BW (0 . , GHz ) = 2 . GHz. Theachieved integral is Z ω ω log 1 r ( ω ) dω = 4 . × . (11)There is only a . × gap between (10) and (11), whichis due entirely to shaping loss defined in Part I, Section III-C.We therefore achieve excellent performance for one sourceand two dipoles with this non-reciprocal network. Accordingto Theorem 3 in Part I, since M < N , using a reciprocalsplitter or coupler as part of the network would not performas well.An experimental measurement of the effectiveness of thematching network in Figure 2 would require standard S-parameter measurements by a network analyzer at the inputport of the circulator. Since there is only one source, noaveraging is needed over phase differences, as described inPart I, Section II-C. By the definition of r ( ω ) , and becausethe network is (theoretically) lossless, small values of r ( ω ) in 2–4 GHz imply that the power from the source is beingaccepted by the antennas.We also note that for two sources ( M = 2 ), we would beable to achieve only half the bandwidth for the same − dB threshold. Clearly, the matching network for two sourceswould differ considerably from the one presented in Figure2 and could be reciprocal. We do not design such a networkhere. B. Two coupled dipoles
To illustrate the effect of coupling between two paralleldipoles, we examine their bandwidth as a function of thedistance between them. We assume each dipole has the same2.4 GHz half-wavelength structure as Figure 1(a). The centerfeeding points of the parallel dipoles are at the same verticallevel, and the distance between them is d , as measured at any Fig. 2. Matching network of the type in Figure 5(b) in Part I is used toconnect a source (on the left) to two decoupled dipoles (connected on theright, typically through baluns that are not shown). Matching networks 1and 2 are designed using the Real Frequency Technique [19], followed byrealizations using Darlington synthesis [20]. Spacing (wavelength) B ound M=1 boundM=2 boundEven modeOdd mode M=1decoupledM=2decoupled
Fig. 3. Bound 3 for two parallel dipoles versus spacing d , for d between . λ and . λ ( λ =
125 mm is the wavelength at 2.4 GHz). Each dipole hasthe structure shown in Figure 1(a), the center feeding points of the dipoles areat the same vertical level, and the distance d is maintained along their lengths.The horizontal dotted lines indicate Bound 3 for two decoupled dipoles. Alsoshown are the even- and odd-mode beamforming bounds. place along their lengths. For each value of d , we apply themodeling recipe for the S-matrix S L ( jω ) detailed in Section IIin the range 1–5 GHz. Since we wish to compare the boundsfor coupled dipoles with (10), we model their S-matrices usingsix poles and six zeros, and enforce S L ( ∞ ) = − I . Then theresulting model S L ( s ) satisfies (1) at s = ∞ for every d , andBound 3 can be applied to compare with (10).We let d range from . λ to . λ , with step size . λ ,where λ = 125 mm is the wavelength at 2.4 GHz. The polesand zeros p L,i , z
L,i are computed from S L ( s ) , and Bound 3 iscomputed for each d . For M = 1 and M = 2 , the bounds areshown in Figure 3 by the black and blue curves. The blackcurve values are twice the blue curve. Horizontal dotted linesindicate the bounds for decoupled dipoles. Figure 3 shows oscillatory behavior and suggests that cou-pled dipoles can have large bandwidths at certain distancesfrom each other. The maxima appear near . λ and are 20%larger than their decoupled-dipole counterparts. As d increases,the bounds approach the decoupled-dipole limits. On the otherhand, when d approaches zero, the two dipoles merge into asingle dipole, and the bounds in Figure 3 approach the boundswhere M sources drive a single dipole.We already know from Part I, Section V-A, that usingthese dipoles in a beamformer configuration with M = 1 generally cannot achieve the bound for any d . But the beam-former configuration is still worth analyzing briefly. Thereare two beamformers that are of special interest: even-modeand odd-mode, corresponding to ~v ( s ) = [1 / √ , / √ T and [1 / √ , − / √ T in (25) in Part I. Since the dipoles arereciprocal and have a symmetric structure, the resulting S L ( s ) is symmetric circulant, and both even and odd ~v ( s ) are realconstant unit eigenvectors of S L ( s ) . Thus Theorem 5 in Part Iapplies.Figure 3 shows the results. Both even and odd-mode boundsare strictly smaller than the bound for M = 1 , as expected.Their sum, however, equals the M = 1 bound becausethe incident and reflected waves to and from the loads areorthogonal for the two modes. Therefore, the modes can betreated as decoupled loads, each with its own equivalent S-parameter and bound. C. Multiple models for the same loads
For a set of loads with a measured S ′ L ( jω ) , more than onemodel S L ( s ) may be created within a given error tolerance.To illustrate this, we take the parallel dipoles in Section III-Bwith S ′ L ( jω ) measured in 1–5 GHz as examples. For each d ,we model the S-matrix using the Matrix Fitting Toolbox withsix poles and six zeros and enforce S L (0) = I . The resulting S L ( s ) then satisfies (1) at s = 0 , and Bound 1 is appliedto the coupled dipoles. We contrast the results with SectionIII-B, where S L ( ∞ ) = − I is enforced and Bound 3 applies.For M = 1 , Figure 4 compares Bound 1 (left y -axis) withBound 3 (right y -axis) for the parallel dipoles when d variesfrom . λ to . λ . The horizontal dotted line indicates thelimiting values as the dipoles become decoupled. Both curvesshow similar trends when the dipoles are closely spaced, andsuggest a bandwidth peak at . λ . For d > . λ , the curvesseem out of phase with each other. Further exploration isneeded to see if there is any physical significance to this. D. Rational approximation of time delays
Any characterization of physically separated antennasshould account for the time of propagation of the signal be-tween the antennas. Time delays associated with this distancecan be modeled as a non-rational (exponential) function of s .A simple idealized analytical model of time delay is shownin Figure 5, which consist of two RC loads and a couplingbranch between them. The coupling is capacitive and includesa delay component. We examine our ability to match to thisload in the 100–400 MHz band. As a reference value, without Spacing (wavelength) B ound −10 B ound Bound 3Bound 1Decoupled bounds
Fig. 4. For one source and two parallel dipoles with spacing d , Bound 3(left y -axis) is compared with Bound 1 (right y -axis). Bound 3 is the sameas the black curve shown in Figure 3. The dotted horizontal line indicates thelimiting values for decoupled dipoles.
50 (cid:2)20 pF50 pF50 pF 50 (cid:2)0.5 ns D
Fig. 5. Two capacitively coupled loads. The coupling includes an ideal delaycomponent of 0.5 ns. the time delay (zero delay), the S-matrix of the loads is real-rational and satisfies S L ( ∞ ) = − I , has two poles at − . × and − . × , and has a zero at with multiplicitytwo. We apply Bound 3 and obtain B = 9 . × .In order to apply the broadband bounds to Figure 5, weneed to approximate the delay using a rational function. Weconsider two ways of doing this. The first way models the timedelay term e − × − s using the Pad´e approximation. We usethe fraction s − × s +1 . × s − . × s +1 . × s +2 × s +1 . × s +8 . × s +1 . × . This fraction is then inserted in place of the time delay, thusmaking the rational matrix S L ( s ) that satisfies S L ( ∞ ) = − I ,and has poles and zeros shown in Table II(a). (We omit theexpression of S L ( s ) because of its complexity.) Comparedwith the true S-matrix of the loads, the rational S-matrix hasan average error of − dB in its entries between 100 MHzand 400 MHz. We apply Bound 3 and obtain . × . Using(9) with τ = 0 . , we obtain δB ≈ . × , which is 8%of the bound. Hence (6) is Z × × log 1 r ′ ( ω ) dω ≤ . × . (12)This bound is larger than without the delay.A second way to model delay is to avoid the analyticalPad´e step, and directly fit numerical simulations using rational I(cid:0)(cid:1)(cid:2)(cid:3) O(cid:2)(cid:3)(cid:1)(cid:2)(cid:3)1234 567 8(cid:4)(cid:5)(cid:4) (cid:6)(cid:7) (cid:8)(cid:5)(cid:9) (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)0(cid:13)(cid:17)(cid:18)(cid:19) ((cid:20)(cid:21) ((cid:22)(cid:21)((cid:23)(cid:21)c(cid:24)(cid:24) 9.(cid:25)(cid:26) nH c(cid:27)(cid:28) (cid:25).(cid:29)(cid:30) pFc(cid:24)(cid:31) (cid:25). ! pF c(cid:27)" (cid:25).(cid:26)! pFc −8−6−4−20 Frequency (Hz) r’( ω ) ( d B ) Network (a)Network (b)Q=10Q=30Q=100
Fig. 7. (a) A two-port matching network for a single Skycross iMAT-1115antenna (duplicated four times) that does not account for antenna coupling.(b) A decoupling network at 2.5 GHz designed using Method 4 in [21], whereeach line represents a capacitor or inductor, and each port is grounded througha component not drawn in figure. The capacitance and inductance values arelisted in the table to the right; c ii i = 1 , . . . , are the components connectingport i to ground, and c ij i = j = 1 , . . . , are the components connectingport i and j . (c) The r ( ω ) for networks (a) and (b) are shown in 2–4 GHz.Also plotted is r ( ω ) for (b) when the reactive components have Q factors 10,30 and 100. Although the decoupling network has a better bandwidthperformance than the network in Figure 7(a), there still existsa significant gap between the achieved integrals and (14),indicating that much better bandwidth performance with theseantennas is still possible. The r ′ ( ω ) for both networks isplotted in Figure 7(c), where we see narrow bandwidth forboth networks. Neither network comes close to achieving thedesired threshold of τ = 0 . (power reflection ratio − dB)over 2–4 GHz, which should be possible according to (14).The design of a network to achieve (14) remains an openproblem.The reactive components in Figure 7(b) are ideal in thatthey have no resistive properties. To illustrate the bandwidthlost when the components have finite Q factors, we let thereactive components behave as ideal reactive components inparallel with resistances. All of the components in Figure 7(b)are assumed to have the same Q factors for all frequency,where Q is defined as the ratio between the susceptance andthe conductance of the components. Figure 7(c) shows the r ′ ( ω ) for different Q factors.Finally, we show that the grounded coplanar waveguide feedlines used for these antennas have minimal effect on the bound.We modify the S-matrix generated by Ansys HFSS by not de-embedding it from the transmission lines, thus including themin the frequency analysis. This S-matrix is then subjected to *Z Passive+M,N-/:ortmat;hi
The bounds are corollaries of Theorems 2 and 3, which arepresented in Section IV-B. The presentation of the theoremsrequires the Darlington representation for multiport loads,which is itself presented in Section IV-A. The bounds arederived in Section IV-C.
A. Darlington equivalent network
We consider the RF system shown in Figure 8, where M uncorrelated sources drive N loads through a passive ( M + N ) -port matching network. The network has an ( M + N ) × ( M + N ) S-matrix S ( s ) partitioned as S ( s ) = M NM S ( s ) S ( s ) N S ( s ) S ( s ) ! . In Figure 8, we transform the N dissipative real loads S L ( s ) into an equivalent lossless real N -port network S b ( s ) ter-minated by N isolated characteristic impedances Z . In [20],Darlington first verified that such an equivalent transformationis possible for N = 1 . The extension to N > is shown in[23]–[26]. The fact that I − S TL ( − s ) S L ( s ) has full normal rankensures that there are N resistors in the Darlington network[25, III.3.1]. We partition the N × N S-matrix S b ( s ) as S b ( s ) = (cid:20) S b ( s ) S b ( s ) S b ( s ) S b ( s ) (cid:21) , where S bij ( s ) are N × N submatrices. Then the necessary andsufficient condition for S b ( s ) being a Darlington equivalentnetwork for a real-rational S L ( s ) is that S b ( s ) is real-rational,Hurwitzian, bounded and para-unitary, and S b ( s ) = S L ( s ) .We do not need to know the exact form of the rest of S b ( s ) ,only its existence is needed.In Figure 8, let S c ( s ) be the ( M + N ) × ( M + N ) S-matrixof the concatenated network of S ( s ) and S b ( s ) , partitioned as S c ( s ) = M NM S c ( s ) S c ( s ) N S c ( s ) S c ( s ) ! . Let S GM ( s ) be the N × N S-matrix seen from the outputports of S b ( s ) , and S G ( s ) be the N × N S-matrix seen fromthe output ports of S ( s ) . It follows that S G ( s ) = S ( s ) , S GM ( s ) = S c ( s ) , and S GM ( s ) = S b ( s )+ S b ( s ) S G ( s )( I − S L ( s ) S G ( s )) − S b ( s ) . (15)We use the notation p × ,i and z × ,i , i = 1 , , . . . to representthe poles and zeros over the WCP of S × ( s ) , where S × isany of the S-matrices or submatrices shown in Figure 8. Inaddition, we use the subscript “ + ” to denote the zeros or polesthat are in the RHP, and “ − ” to denote those in the LHP.In Figure 8, let ~a ( s ) and ~b ( s ) be the N × incidentand reflected signal to and from the isolated impedances Z .Because S b ( s ) is lossless, the amount of power delivered tothe loads equals the power delivered to the resistive part of theDarlington network. Since ~b ( jω ) = 0 , the power delivered at jω is k ~a ( jω ) k , where ~a ( jω ) = S c ( jω ) ~a ( jω ) . So thetotal power lost to reflection and dissipation can be writtenas k ~a ( jω ) k − k ~a ( jω ) k , and the power loss ratio of thematching network and the loads becomes r ( ω ) = 1 − M tr { S Hc ( jω ) S c ( jω ) } . (16)In the rest of the presentation, we use a particular Darlingtonnetwork, which is given in the following lemma in [9]. Lemma 1:
There exists a Darlington network S b ( s ) suchthat z b ,i = − z L − ,i . (17)For such S b ( s ) , the RHP zeros of S GM ( s ) are identical to theRHP zeros of S TL ( − s ) − S G ( s ) . B. Integral log-determinant of S GM ( jω ) We now present the theorems on the integral of logarithm of det S GM ( jω ) . We assume that I − S TL ( − s ) S L ( s ) is full normalrank. We also assume (1) is satisfied for some Re { s } ≥ ,and choose positive integer m such that I − S TL ( − s ) S L ( s ) = A m ( s − s ) m + A m +1 ( s − s ) m +1 + . . . (18) if s is finite, or I − S TL ( − s ) S L ( s ) = A m s − m + A m +1 s − ( m +1) + . . . (19)if s = ∞ . In (18) and (19), A m = 0 is defined such thatthe entries of I − S TL ( − s ) S L ( s ) have a zero at s = s withmultiplicity at least m . When s = jω , it is shown in Lemma4 in Appendix C that m is even; when s = ∞ , m is alsoeven since the left-hand side of (19) is an even function. Thegeneral broadband matching theorems are as follows: Theorem 2:
Let S L ( s ) satisfy (18) for some Re { s } ≥ .Then for any passive network S ( s ) such that I − S L ( s ) S G ( s ) is non-singular, we have Z ∞ Re [( s − jω ) − + ( s + jω ) − ] log | det S GM ( jω ) | dω = π log (cid:12)(cid:12)(cid:12) det S L ( s ) · Q i ( s + z L,i ) Q i ( s + z GM + ,i ) Q i ( s − z L,i ) Q i ( s − z GM + ,i ) (cid:12)(cid:12)(cid:12) , (20)and Z ∞ [( s − jω ) − ( k +1) + ( s + jω ) − ( k +1) ] × log | det S GM ( jω ) | dω = ( − k πk h X i ( p L,i − s ) − k − X i ( − z L,i − s ) − k + (cid:16) X i ( z GM + ,i − s ) − k − X i ( − z GM + ,i − s ) − k (cid:17)i (21)for k = 1 , . . . , m − , where ± s , ± s ∗ are excluded in p L,i , z L,i in (21). If I − S L ( s ) S G ( s ) is singular, then Re { s } = 0 ,(20) holds, and (21) holds for k = 1 , . . . , m − ; for k = m − ,we let s = jω , and then have Z ∞ [( ω − ω ) − m + ( ω + ω ) − m ] log | det S GM ( jω ) | dω ≥ ( − m − πm − h X i ( p L,i − jω ) − ( m − − X i ( − z L,i − jω ) − ( m − + (cid:16) X i ( z GM + ,i − jω ) − ( m − − X i ( − z GM + ,i − jω ) − ( m − (cid:17)i . (22) Theorem 3:
Let S L ( s ) satisfy (19). Then Z ∞ ω k − log | det S GM ( jω ) | dω ≥ ( − k − π k h X i p kL,i + X i z kL,i + 2 X i z kGM + ,i i (23)for k = 1 , , . . . , m − . Equality in (23) holds if k = m − or I − S L ( s ) S G ( s ) is non-singular at s = ∞ .In Theorem 2, although (18) holds, s may still be a zeroof S L ( s ) when Re { s } > . This happens if and only if S L ( s ) has a pole at − s , which cancels the zero at s in(18). Hence, the poles and zeros of S L ( s ) at ± s , ± s ∗ areexcluded in the sums in (21). The proof of Theorem 2 appears in Appendix D; the proof uses several preliminary lemmaswhich are introduced in Appendices B and C.Theorem 3 is adapted from Theorem 1 in [9], which isproven with det S LM ( jω ) in place of det S GM ( jω ) in (23).Note S LM ( s ) is the M × M S-matrix seen from the inputports of S ( s ) (see Figure 8). Although it is assumed in [9]that M = N , and that the matching network is lossless andreciprocal, and the loads are also reciprocal, these assumptionscan be relaxed. In fact, Theorem 1 in [9] applies withoutchange to non-reciprocal networks and loads; the reciprocityof the networks and loads is an unnecessary restriction in themodel. Our version is needed to handle lossy networks, whichare not handled in [9]. The proof of (23) follows the samearguments used in the proof of Theorem 2 and is omitted. C. Proof of bounds
We relate det S GM ( jω ) to r ( ω ) in (16) using the arithmetic-geometric mean inequality: r ( ω ) = 1 M tr { I − S Hc ( jω ) S c ( jω ) }≥ det( I − S c ( jω ) S Hc ( jω )) /M ≥ det( S GM ( jω ) S HGM ( jω )) /M = | det S GM ( jω ) | /M . The first equality holds if and only if the eigenvalues of I − S Hc ( jω ) S c ( jω ) are all equal; this is equivalent to S H ( I − S G S L ) − H ( I − S HL S L )( I − S G S L ) − S havingequal singular values for all jω , which is Condition 2 forequality in [1]. The second equality holds if and only if S ( s ) satisfies S c ( s ) S Tc ( − s ) + S GM ( s ) S TGM ( − s ) = I . Since S b ( s ) is para-unitary, this is equivalent to S ( s ) S T ( − s ) + S G ( s ) S TG ( − s ) = I , which is Condition 1.Taking the logarithm on both sides of the inequality yields log r ( ω ) ≥ (1 /M ) log | det S GM ( jω ) | . (24)When s = jω , we apply (24) to (21) and (22) for k = 1 , andomit P i ( z GM + ,i − jω ) − − P i ( − z GM + ,i − jω ) − since itis non-negative. From Lemma 1, the RHP zeros of S GM ( s ) are identical to the RHP zeros of S TL ( − s ) − S G ( s ) ; thusCondition 4 is necessary and one of the sufficient conditions.This finishes the proof of Bound 1.When Re { s } > , we apply (24) to (20), and then omit Q i ( s + z GM + ,i ) Q i ( s − z GM + ,i ) since it has modulus no smaller than one. NoteRe [( s − jω ) − + ( s + jω ) − ] in (20) is positive for anyRe { s } > and ω . The result is Bound 2.We combine (24) and (23) for k = 1 , and then omit P i z GM + ,i since it is non-negative. The result is Bound 3.V. C ONCLUSIONS AND F UTURE W ORK
We have presented a bandwidth analysis for multiportmatching that applies to an arbitrary number of sourcesand coupled loads, using broadband matching bounds on theintegral of a power loss ratio. Part I presented the definitionsand bounds and applied them to settings where the loads couldbe described analytically using rational functions. Conditionswere given for when the bounds could be met with equality. Part II focused on realistic loads, including antennas, and therational fitting needed for accurate bandwidth calculations.Proofs of all results were also presented.Part I demonstrated that the bound scales generally as
N/M for M sources and N loads. This scaling is not affectedby coupling as long as it is not “too strong”. Hence, largebandwidth is possible even if the loads are coupled or closely-spaced. Some realistic examples in Part II showed how largebandwidths can be attained in practice. Although we touchedupon techniques to attain the bounds, the general practicaldesign problem remains open.For loads, such as antennas, whose frequency responsesare available numerically through simulation, rational fittingof the S-matrix is required to apply the bounds. We showedhow the rational matrix is then used to find a bandwidthbound for the system being fitted. Increasing the polynomialorders of the numerators and denominators of the rationalfunctions in the S-matrix generally improves the accuracyof the system approximation, but can cause the problem ofover-fitting, which results in a loose bound. Over-fitting is animportant issue that deserves further attention.Our examples included antennas that are matched near theirresonant frequency. It would be interesting to see how thebounds characterize the bandwidth for electrically small an-tennas, which are being used below their resonant frequenciesand are generally considered narrowband. The classical Chubound [28] and some recent advances [29]–[31] provide othermethods to describe the bandwidth for small antennas. Onefuture area of study could be to reconcile these methods withbandwidth results calculated using the modeling techniquesand bounds contained herein.The communication-theoretic implications of using broad-band multiport networks in wireless systems need study. Wehave been concerned primarily with the aspects of networkdesign that ensure efficient power delivery to loads such asan array of multiple coupled antennas. However, the choice ofnetwork can also affect the radiation efficiency and far-fieldpattern of an antenna array. Hence, a complete system designshould consider total power transfer from the transmitteramplifiers to a far-field receiver. We also did not examinethe implications of using the wideband matching networks forantennas that are intended for both transmission and reception.Although we have treated the case where uncorrelatedsources are driving coupled loads, the reverse situation wherecoupled sources are driving decoupled loads needs separateanalysis. Such a system could arise when the sources areclosely-spaced receiver antennas connected to isolated low-noise amplifiers. The noise figures of the amplifiers wouldplay a role in the criterion for determining bandwidth.Maximizing the data rate attainable with a prescribed setof N antennas is a potentially interesting problem. Conven-tional narrowband Shannon theory says that MIMO, with M independent data streams (sources) in a rich scatteringenvironment, achieves rates linear in N when M = N [32].But, as we have seen, the bandwidth attainable for M = N is only /N of that attainable for M = 1 . Thus, MIMOwith N streams achieves /N the bandwidth of MIMO witha single stream. Since the transmission data rate is directly proportional to bandwidth, one could therefore conjecturethat, with N antennas, one stream is as good as N . Thisbandwidth/multiplexing trade-off needs further study.VI. A CKNOWLEDGEMENTS
We thank the reviewers and editor for the detailed andinsightful comments on both parts of our manuscript. Inparticular, your technical suggestions on how to improve thereadability of the paper helped us to treat the issues of theexperimental determination of r ( ω ) , and the rational fitting ofrealistic loads. A PPENDIX
Theorem 1 is proven in Appendix A. We prove Theorem2 in Appendix D using several preliminary lemmas that areintroduced in Appendices B and C.
A. Proof of Theorem 1
Let ε ( ω ) = r ( ω ) − r ′ ( ω ) . Then Z ω ω f ( ω ) log r ( ω ) r ′ ( ω ) dω = Z ω ω f ( ω )2 log (cid:18) ε ( ω ) r ′ ( ω ) (cid:19) dω = Z ω ω f ( ω )2 log (cid:18) − r ′ ( ω ) r ′ ( ω ) · ε ( ω )1 − r ′ ( ω ) (cid:19) dω. We show that ε ( ω ) / (1 − r ′ ( ω )) ≤ ρ ( ω ) .From (5) in Part I, r ( ω ) = 1 − tr { S H ( I − S G S L ) − H ( I − S HL S L )( I − S G S L ) − S } M , and a similar relation between r ′ ( ω ) and S ′ L ( jω ) holds. Forsimplicity of the presentation, we omit the argument jω forthe S-matrices in the remainder of the proof. We perturb S L ( jω ) = S ′ L ( jω ) + δS L ( jω ) ; a first-order expansion yields ε ( ω ) = r ( ω ) − r ′ ( ω ) ≈− M tr { S H ( I − S G S ′ L ) − H A ( I − S G S ′ L ) − S } , where A is the Hermitian matrix A = δS HL ( I − S ′ L S G ) − H ( S HG − S ′ L )+ ( S G − S ′ HL )( I − S ′ L S G ) − δS L . Because A is Hermitian, the matrix S H ( I − S G S ′ L ) − H A ( I − S G S ′ L ) − S is also Hermitian. Some manipulations of thetrace give us ε ( ω ) ≤ σ , max M tr { S H ( I − S G S ′ L ) − H ( I − S G S ′ L ) − S } , where σ , max is the maximum singular value of A . Similarly, − r ′ ( ω )= 1 M tr { S H ( I − S G S ′ L ) − H ( I − S ′ HL S ′ L )( I − S G S ′ L ) − S }≥ − σ ′ L, max ( ω ) M tr { S H ( I − S G S ′ L ) − H ( I − S G S ′ L ) − S } . Hence, ε ( ω )1 − r ′ ( ω ) ≤ σ , max − σ ′ L, max ( ω ) . We now seek an upper bound on σ , max , which depends onthe maximum singular values of both δS L and ( S G − S ′ HL )( I − S ′ L S G ) − . The maximum singular value of δS L is σ δ, max ( ω ) .To obtain the maximum singular value of ( S G − S ′ HL )( I − S ′ L S G ) − , we try to prove that I + ( σ ′ L, max ( ω ) − σ ′ L, min ( ω ))( I − S ′ L S G ) − H ( I − S ′ L S G ) − − ( I − S ′ L S G ) − H ( S HG − S ′ L )( S G − S ′ HL )( I − S ′ L S G ) − (25)is a positive semidefinite matrix, which means the maximumeigenvalue of the positive definite matrix I + ( σ ′ L, max ( ω ) − σ ′ L, min ( ω ))( I − S ′ L S G ) − H ( I − S ′ L S G ) − is larger than or equalto the maximum eigenvalue of the positive semidefinite matrix ( I − S ′ L S G ) − H ( S HG − S ′ L )( S G − S ′ HL )( I − S ′ L S G ) − . Then thesquare root of the maximum eigenvalue of I + ( σ ′ L, max ( ω ) − σ ′ L, min ( ω ))( I − S ′ L S G ) − H ( I − S ′ L S G ) − is larger than or equalto the maximum singular value of ( S G − S ′ HL )( I − S ′ L S G ) − .To prove that (25) is positive semidefinite, we simplify it I + ( σ ′ L, max ( ω ) − σ ′ L, min ( ω ))( I − S ′ L S G ) − H ( I − S ′ L S G ) − − ( I − S ′ L S G ) − H ( S HG − S ′ L )( S G − S ′ HL )( I − S ′ L S G ) − = ( I − S ′ L S G ) − H A ( I − S ′ L S G ) − , where A = ( σ ′ L, max ( ω ) − σ ′ L, min ( ω )) I + ( I − S ′ L S ′ HL ) − S HG ( I − S ′ HL S ′ L ) S G .A is a positive semidefinite matrix because the singu-lar values of S ′ L and S G are no larger than one, andtherefore the minimum eigenvalue of the positive definitematrix ( σ ′ L, max ( ω ) − σ ′ L, min ( ω )) I + ( I − S ′ L S ′ HL ) is largerthan or equal to the maximum eigenvalue of the positivesemidefinite matrix S HG ( I − S ′ HL S ′ L ) S G . Therefore, we haveproven that the square root of the maximum eigenvalue of I + ( σ ′ L, max ( ω ) − σ ′ L, min ( ω ))( I − S ′ L S G ) − H ( I − S ′ L S G ) − is larger than or equal to the maximum singular value of ( S G − S ′ HL )( I − S ′ L S G ) − . The maximum eigenvalue of I + ( σ ′ L, max ( ω ) − σ ′ L, min ( ω ))( I − S ′ L S G ) − H ( I − S ′ L S G ) − is smaller than or equal to σ ′ L, max ( ω ) − σ ′ L, min ( ω )(1 − σ ′ L, max ( ω )) . Therefore,the maximum singular value of ( S G − S ′ HL )( I − S ′ L S G ) − issmaller than or equal to r σ ′ L, max ( ω ) − σ ′ L, min ( ω )(1 − σ ′ L, max ( ω )) . We conclude σ , max ≤ σ δ, max ( ω ) s σ ′ L, max ( ω ) − σ ′ L, min ( ω )(1 − σ ′ L, max ( ω )) , which yields ε ( ω ) / (1 − r ′ ( ω )) ≤ ρ ( ω ) . This finishes theproof. B. Preliminary lemmas on real-rational functionsLemma 2:
Let f ( s ) be a real-rational function with f ( s ) = c = 0 . Then the series expansion of log f ( s ) around s = s can be written as log f ( s ) = log c + a ( s − s ) + a ( s − s ) + . . . + a ℓ ( s − s ) ℓ + . . . , (26)where a ℓ = 1 ℓ X i ( p i − s ) − ℓ − X i ( z i − s ) − ℓ ! (27)for ℓ = 1 , , . . . , where p i , z i are the poles and zeros of f ( s ) . Proof:
Since f ( s ) = 0 is finite, log f ( s ) = log c isfinite. So the expansion of log f ( s ) around s = s can beobtained using Taylor series expansion. The result is (27). Lemma 3:
Let f ( s ) be a real-rational function. For anyRe { s } ≥ , if f ( s ) is finite, non-zero, and − f ( − s ) f ( s ) has a zero at s = s with multiplicity m , then Z ∞ Re [( s − jω ) − + ( s + jω ) − ] log | f ( jω ) | dω = π log (cid:12)(cid:12)(cid:12)(cid:12) f ( s ) Q i ( s − p + ,i ) Q i ( s + z + ,i ) Q i ( s + p + ,i ) Q i ( s − z + ,i ) (cid:12)(cid:12)(cid:12)(cid:12) , (28)and Z ∞ [( s − jω ) − ( k +1) + ( s + jω ) − ( k +1) ] log | f ( jω ) | dω = ( − k πk h X i ( p i − s ) − k − X i ( z i − s ) − k − (cid:16) X i ( p + ,i − s ) − k − X i ( − p + ,i − s ) − k (cid:17) + (cid:16) X i ( z + ,i − s ) − k − X i ( − z + ,i − s ) − k (cid:17)i (29)for k = 1 , . . . , m − , where p i , z i are the poles and zerosof f ( s ) in the WCP, and p + ,i , z + ,i are the poles and zeros of f ( s ) in the RHP. Proof:
Since f ( s ) = 0 , we begin by applying Lemma2 to write the expansion of log f ( s ) as (26) and (27). Forconvenience, we define another real-rational function ˆ f ( s ) as ˆ f ( s ) = f ( s ) Q i ( s − p + ,i ) Q i ( s + p + ,i ) Q i ( s + z + ,i ) Q i ( s − z + ,i ) , (30)where p + ,i and z + ,i are the poles and zeros of f ( s ) in the RHP.Then ˆ f ( s ) has no poles or zeros in the RHP, and satisfies ˆ f ( − s ) ˆ f ( s ) = f ( − s ) f ( s ) and | ˆ f ( jω ) | = | f ( jω ) | . We applyLemma 2 to ˆ f ( s ) and get log ˆ f ( s ) = log ˆ c + ˆ a ( s − s ) + ˆ a ( s − s ) + . . . + ˆ a ℓ ( s − s ) ℓ + . . . , (31)where ˆ c = ˆ f ( s ) = f ( s ) Q i ( s − p + ,i ) Q i ( s + p + ,i ) Q i ( s + z + ,i ) Q i ( s − z + ,i ) , (32)and ˆ a ℓ = a ℓ − ℓ X i ( p + ,i − s ) − ℓ − X i ( − p + ,i − s ) − ℓ ! Fig. 9. The contours for the integrals (35), (37), (39) and (41) are shown in(a)–(d), respectively. The contours are in the clockwise direction. The sectionsof the contours are labeled C , C , . . . and their detailed descriptions are givenin the proof of Lemma 3. + 1 ℓ X i ( z + ,i − s ) − ℓ − X i ( − z + ,i − s ) − ℓ ! . (33)We can expand log f ( s ) and log ˆ f ( s ) at s = s ∗ in similarforms as (26) and (31). Since f ( s ) and ˆ f ( s ) are real-rational,the coefficients for the expansion of log f ( s ) and log ˆ f ( s ) at s = s ∗ are a ∗ ℓ and ˆ a ∗ ℓ , respectively.The next step is to separate our discussions into four cases: s = 0 , jω , σ , σ + jω . Since the cases are similar to oneanother, we elaborate more on the s = 0 case than the others. s = 0 : Since − f ( − s ) f ( s ) is an even real-rationalfunction and has a zero of multiplicity m at s = 0 , m mustbe an even integer. We take the logarithm of ˆ f ( − s ) ˆ f ( s ) = f ( − s ) f ( s ) = 1 + O ( s m ) and use the expansion (31) at s =0 . Because ˆ f ( s ) is real-rational, the coefficients in (33) for s = 0 are real. We therefore obtain | ˆ c | = 1 , ˆ a = ˆ a = . . . =ˆ a m − = 0 and Im { ˆ a } = Im { ˆ a } = . . . = Im { ˆ a m − } = 0 .For s = 0 , (28) is trivial since both sides are zero. Because ˆ a = ˆ a = . . . = ˆ a m − = 0 , (29) is also trivial for k =2 , , . . . , m − . We show (29) for k = 1 , , . . . , m − bytaking the contour integral of the function s − ( k +1) log ˆ f ( s ) (34)along the closed curve shown in Figure 9(a). This functionis analytic in the RHP; it is also analytic on the imaginaryaxis except the origin and possible zeros and poles of ˆ f ( s ) onthe imaginary axis, which we denote as jω ℓ . Therefore, thefollowing contour integral is zero: Z C + C + C + C s − ( k +1) log ˆ f ( s ) ds = 0 , (35) where C is the line segment between − jR and jR excluding [ − ε, ε ] and [ j ( ω ℓ − ε ) , j ( ω ℓ + ε )] ; C is the right semicirclewith radius R centered at the origin; C is the right semicirclewith radius ε centered at the origin; and C includes the rightsemicircles with radius ε centered at jω ℓ . We evaluate theintegral of (34) as follows: Z C s − ( k +1) log ˆ f ( s ) ds = ( − k +12 j Z R − R ω − ( k +1) log | f ( jω ) | dω − ( − k +12 c ) kε k + O ( R − k ) , where the integral from − R to R excludes the intervals [ − ε, ε ] and [ ω ℓ − ε, ω ℓ + ε ] . Furthermore, Z C s − ( k +1) log ˆ f ( s ) ds = Z − π/ π/ j ( Re jθ ) − k log ˆ f ( Re jθ ) dθ = O (cid:18) log RR (cid:19) . Z C s − ( k +1) log ˆ f ( s ) ds = Z π/ − π/ j ( εe jθ ) − k log ˆ f ( εe jθ ) dθ = jπ ˆ a k + ( − k +12 c ) kε k + O ( ε m − k ) . Z C s − ( k +1) log ˆ f ( s ) ds = X ℓ Z π/ − π/ ( jω ℓ + εe jθ ) − ( k +1) × log ˆ f ( jω ℓ + εe jθ ) jεe jθ dθ = O ( ε log ε ) . Combining these path integrals and letting R → ∞ , ε → ,we have Z ∞−∞ ω − ( k +1) log | f ( jω ) | dω = ( − k − π ˆ a k . Since f ( s ) is real-rational, | f ( − jω ) | = | f ∗ ( jω ) | = | f ( jω ) | ,we have ω − ( k +1) log | f ( jω ) | is an even function for k =1 , , . . . , m − . Using (33), we get (29) for k = 1 , , . . . , m − .This finishes the proof for s = 0 . s = jω : Since − f ( − s ) f ( s ) is a real-rationalfunction and has a zero of multiplicity m at s = jω , italso has a zero of multiplicity m at s = − jω . We take thelogarithm of ˆ f ( − s ) ˆ f ( s ) = f ( − s ) f ( s ) = 1 + O (( s − jω ) m ) and use the expansion (31) at s = jω . We therefore obtain | ˆ c | = 1 , Im { a } = Im { a } = . . . = Im { a k } = 0 for odd k and k < m , and Re { a } = Re { a } = . . . = Re { a k } = 0 foreven k and k < m .For s = jω , (28) is trivial since both sides are zero. Weshow (29) by taking the contour integral of the function [( jω − s ) − ( k +1) + ( jω + s ) − ( k +1) ] log ˆ f ( s ) (36)along the closed curve shown in Figure 9(b). This functionis analytic in the RHP; it is also analytic on the imaginaryaxis except ± jω and possible zeros and poles of ˆ f ( s ) on theimaginary axis, which we denote as jω ℓ . Therefore, Z C + C + C + C [( jω − s ) − ( k +1) + ( jω + s ) − ( k +1) ] × log ˆ f ( s ) ds = 0 , (37)where C is the line segment between − jR and jR excluding [ j ( ω − ε ) , j ( ω + ε )] , [ j ( − ω − ε ) , j ( − ω + ε )] and [ j ( ω ℓ − ε ) , j ( ω ℓ + ε )] ; C is the right semicircle with radius R centeredat the origin; C includes the right semicircles with radius ε centered at the jω and − jω ; and C includes the rightsemicircles with radius ε centered at jω ℓ .By evaluating the integral paths in (37), and letting R → ∞ , ε → , we obtain Z ∞−∞ [( ω − ω ) − ( k +1) + ( ω + ω ) − ( k +1) ] log | f ( jω ) | dω = ( − k j k +1 π ˆ a k . We have [( ω − ω ) − ( k +1) + ( ω + ω ) − ( k +1) ] log | f ( jω ) | isan even function, and using (33) we get (29) for k =1 , , . . . , m − . This finishes the proof for s = jω . s = σ : Since f ( s ) is a real-rational function,Im { ˆ a } = Im { ˆ a } = . . . = Im { ˆ a m − } = 0 . For s = σ ,we show (28) and (29) by taking the contour integral of thefunction [( σ − s ) − ( k +1) + ( σ + s ) − ( k +1) ] log ˆ f ( s ) (38)along the close curve shown in Figure 9(c). This functionis analytic in the RHP except σ ; it is also analytic on theimaginary axis except possible zeros and poles of ˆ f ( s ) onthe imaginary axis, which we denote as jω ℓ . Therefore, thefollowing contour integral is zero for k = 0 , , , . . . , m − : Z C + C + C + C [( σ − s ) − ( k +1) + ( σ + s ) − ( k +1) ] × log ˆ f ( s ) ds = 0 , (39)where C is the line segment between − jR and jR excluding [ j ( ω ℓ − ε ) , j ( ω ℓ + ε )] ; C is the right semicircle with radius R centered at the origin; C is the circle with radius ε centeredat the σ ; and C includes the right semicircles with radius ε centered at jω ℓ .To show (28), we evaluate the integral paths in (39) for k = 0 , and let R → ∞ , ε → . The result is Z ∞−∞ [( σ − jω ) − + ( σ + jω ) − ] log | f ( jω ) | dω = 2 π log | ˆ c | . We have [( σ − jω ) − + ( σ + jω ) − ] log | f ( jω ) | is an evenfunction, and using (32) we get (28).To show (29), we evaluate the integral paths in (39) for k = 1 , , . . . , m − , and let R → ∞ , ε → . The result is Z ∞−∞ [( σ − jω ) − ( k +1) + ( σ + jω ) − ( k +1) ] log | f ( jω ) | dω = ( − k π ˆ a k . We have [( σ − jω ) − ( k +1) + ( σ + jω ) − ( k +1) ] log | f ( jω ) | is an even function, and using (33) we get (29) for k =1 , , . . . , m − . This finishes the proof for s = σ . s = σ + jω : For s = σ + jω , we show (28) and(29) by taking the contour integral of the following functions [( s − s ) − ( k +1) + ( s + s ) − ( k +1) + ( s ∗ − s ) − ( k +1) + ( s ∗ + s ) − ( k +1) ] log ˆ f ( s ) (40a) [( s − s ) − ( k +1) + ( s + s ) − ( k +1) − ( s ∗ − s ) − ( k +1) − ( s ∗ + s ) − ( k +1) ] log ˆ f ( s ) (40b)along the close curve shown in Figure 9(d). These functionsare analytic in the RHP except s and s ∗ ; they are also analyticon the imaginary axis except possible zeros and poles of ˆ f ( s ) on the imaginary axis, which we denote as jω ℓ . Therefore, thefollowing contour integral is zero for k = 0 , , , . . . , m − : Z C + C + C + C [( s − s ) − ( k +1) + ( s + s ) − ( k +1) + ( s ∗ − s ) − ( k +1) + ( s ∗ + s ) − ( k +1) ] log ˆ f ( s ) ds = 0 , (41a) Z C + C + C + C [( s − s ) − ( k +1) + ( s + s ) − ( k +1) − ( s ∗ − s ) − ( k +1) − ( s ∗ + s ) − ( k +1) ] log ˆ f ( s ) ds = 0 , (41b)where C is the line segment between − jR and jR excluding [ j ( ω ℓ − ε ) , j ( ω ℓ + ε )] ; C is the right semicircle with radius R centered at the origin; C includes the circles with radius ε centered at the s and s ∗ ; and C includes the right semicircleswith radius ε centered at jω ℓ .To show (28), we evaluate the integral paths in (41a) for k = 0 , and let R → ∞ , ε → . The result is Z ∞−∞ Re [( s − jω ) − + ( s + jω ) − ] log | f ( jω ) | dω = 2 π log | ˆ c | . We have Re [( s − jω ) − + ( s + jω ) − ] log | f ( jω ) | is an evenfunction, and using (32) we get (28).To show (29), we first evaluate the integral paths in (41) for k = 1 , , . . . , m − , and let R → ∞ , ε → . The result is Z ∞−∞ [( s − jω ) − ( k +1) + ( s + jω ) − ( k +1) ] log | f ( jω ) | dω = ( − k π ˆ a k . We have [( s − jω ) − ( k +1) + ( s + jω ) − ( k +1) ] log | f ( jω ) | isan even function, and using (33) we get (29). This finishes theproof for s = σ + jω .
5) Remark on lemma 3:
Because the formulas (28), (29)subtract or divide poles from zeros in equal quantity, when weapply Lemma 3 to f ( s ) = det A ( s ) , (28), (29) still hold if thepoles and zeros of the determinant are replaced by the polesand zeros of the matrix A ( s ) . C. Preliminary lemmas on S-matricesLemma 4:
If Re { s } > and the Darlington network givenin Lemma 1 is used, then det S GM ( s ) = det S b ( s ) = 0 , (42)and X i ( p GM,i − s ) − ℓ − X i ( z GM,i − s ) − ℓ ! = X i ( p b ,i − s ) − ℓ − X i ( z b ,i − s ) − ℓ ! (43)for ℓ = 1 , . . . , m − .If Re { s } = 0 , then m is even, (42) holds, and (43) holdsfor ℓ = 1 , . . . , m − .If Re { s } = 0 and I − S L ( s ) S G ( s ) is non-singular then(43) holds for ℓ = m − .If Re { s } = 0 and I − S L ( s ) S G ( s ) is singular then ( − m × X i ( p GM,i − s ) − ( m − − X i ( z GM,i − s ) − ( m − ! ≤ ( − m × X i ( p b ,i − s ) − ( m − − X i ( z b ,i − s ) − ( m − ! . (44) Proof:
We separate our discussion into two possibilities:Re { s } > and Re { s } = 0 .
1) Re { s } > : Because the Darlington network in Lemma1 is assumed, S b ( s ) has no zeros at s = s , for otherwise S L ( s ) would have zeros at s = − s , and therefore have polesat s = s in order to satisfy (18). This contradicts S L ( s ) beingHurwitzian. Hence, det S b ( s ) = 0 , and S L ( s ) has no zerosat s = − s .Since S b ( s ) is lossless and therefore S Tb ( − s ) S b ( s ) = I , weget S b ( s ) = − S − TL ( − s ) S Tb ( − s ) S b ( s ) . We manipulate(15) to get S GM ( s ) = [ I − S b ( s ) S G ( s )( I − S L ( s ) S G ( s )) − × S − TL ( − s ) S Tb ( − s )] S b ( s ) . Taking determinant on both sides yields det S GM ( s ) = det S b ( s ) det[ I − S b ( s ) S G ( s ) × ( I − S L ( s ) S G ( s )) − S − TL ( − s ) S Tb ( − s )]= det S b ( s ) det[ I − S Tb ( − s ) S b ( s ) S G ( s ) × ( I − S L ( s ) S G ( s )) − S − TL ( − s )] . Since S L ( s ) and S G ( s ) are bounded, S L ( s ) S G ( s ) is alsobounded and I − S L ( s ) S G ( s ) is non-singular for Re { s } > [24, 7.22]. Hence, S Tb ( − s ) S b ( s ) = I − S TL ( − s ) S L ( s ) = O (( s − s ) m ) , and S L ( − s ) and I − S L ( s ) S G ( s ) are non-singular. We then have det S GM ( s ) = det S b ( s )[1 + O (( s − s ) m )] . Thus det S b ( s ) = det S GM ( s ) = 0 , and (42) holds forRe { s } > .To show (43), we apply Lemma 2 to det S b ( s ) and det S GM ( s ) : log det S GM ( s ) = a + a ( s − s ) + . . . + a m − ( s − s ) m − + . . . log det S b ( s ) = b + b ( s − s ) + . . . + b m − ( s − s ) m − + . . . , (45)where a ℓ and b ℓ have the form (27). Because det S GM ( s ) =det S b ( s ) + O (( s − s ) m ) , a ℓ = b ℓ for ℓ = 0 , , . . . , m − .Writing a ℓ and b ℓ in the form of (27) yields (43).
2) Re { s } = 0 : Let s = jω . Since S b ( s ) is lossless and(18) is satisfied, S HL ( jω ) S L ( jω ) = S Hb ( jω ) S b ( jω ) = I . Hence, det S b ( jω ) = 0 .We begin by showing that m is even. We substitute s = j ( ω ± ε ) into (18): I − S TL ( − j ( ω ± ε )) S L ( j ( ω ± ε ))= I − S HL ( j ( ω ± ε )) S L ( j ( ω ± ε )) = A m ( ± jε ) m + . . . . Since S L ( s ) is bounded, the A m ( ± jε ) m is positive semidefi-nite. With A m = 0 , it is possible only when m is even.If I − S L ( jω ) S G ( jω ) is non-singular, we follow a methodsimilar to the Re { s } > case to get det S GM ( s ) =det S b ( s ) + O (( s − jω ) m ) . Thus (42) and (43) hold for ℓ = 1 , , . . . , m − .If I − S L ( jω ) S G ( jω ) is singular, (44) can be proven ina manner similar to the proof of Lemma 5 in [9]. These stepsare omitted. This finishes the proof of Lemma 4. D. Proof of Theorem 2
We use the Darlington network in Lemma 1. Then Lem-ma 4 gives det S b ( s ) = det S GM ( s ) = 0 . Because S b ( s ) is para-unitary, (18) implies det( S Tb ( − s ) S b ( s )) =det( S TL ( − s ) S L ( s )) = 1 + O (( s − s ) m ) . Hence, det S b ( s ) satisfies the conditions of Lemma 3.From Lemma 4, det S GM ( s ) = det S b ( s ) + O (( s − s ) m ) for Re { s } > , and det S GM ( s ) = det S b ( s ) + O (( s − s ) m − ) for Re { s } = 0 . When Re { s } = 0 , we let s = jω and consider s = j ( ω ± ε ) for ε > : det[ S TGM ( − j ( ω ± ε )) S GM ( j ( ω ± ε ))]= det[ S HGM ( j ( ω ± ε )) S GM ( j ( ω ± ε ))]= 1 + b m − ( ± jε ) m − + O ( ε m ) ≤ . The inequality is because S GM ( s ) is bounded. Since m − isodd, b m − = 0 . Hence, det( S TGM ( − s ) S GM ( s )) = 1 + O (( s − s ) m ) , and det S GM ( s ) satisfies the conditions of Lemma 3.Unfortunately, Lemma 3 does not apply to det S L ( s ) sincethere are possible zeros of S L ( s ) at s = s when Re { s } > .From (18), if S L ( s ) has zeros at s , it also has zeros at s ∗ and poles at − s , − s ∗ ; the multiplicities of these poles orzeros are equal. We therefore construct a function \ det S L ( s ) by removing the poles at − s , − s ∗ and zeros at s , s ∗ from det S L ( s ) , such that | \ det S L ( jω ) | = | det S L ( jω ) | . Thisfunction satisfies \ det S L ( − s ) \ det S L ( s ) = 1 + O (( s − s ) m ) and \ det S L ( s ) = 0 , thus the conditions of Lemma 3.Since S b ( s ) is para-unitary, | det S L ( jω ) | = | det S b ( jω ) | .It follows that | det S L ( jω ) | = | \ det S L ( jω ) | = | det S b ( jω ) | . (46)We first apply (28): Z ∞ Re [( s − jω ) − + ( s + jω ) − ] log | det S GM ( jω ) | dω = π log (cid:12)(cid:12)(cid:12)(cid:12) det S GM ( s ) · Q i ( s + z GM + ,i ) Q i ( s − z GM + ,i ) (cid:12)(cid:12)(cid:12)(cid:12) (47a) Z ∞ Re [( s − jω ) − + ( s + jω ) − ] log | det S b ( jω ) | dω = π log (cid:12)(cid:12)(cid:12)(cid:12) det S b ( s ) · Q i ( s + z b ,i ) Q i ( s − z b ,i ) (cid:12)(cid:12)(cid:12)(cid:12) (47b) Z ∞ Re [( s − jω ) − + ( s + jω ) − ] log | \ det S L ( jω ) | dω = π log (cid:12)(cid:12)(cid:12)(cid:12) \ det S L ( s ) · Q i ( s + z L + ,i ) Q i ( s − z L + ,i ) (cid:12)(cid:12)(cid:12)(cid:12) . (47c)Note s and s ∗ are excluded in z L + ,i in (47c). From thedefinition of \ det S L ( s ) , we can rewrite (47c) as Z ∞ Re [( s − jω ) − + ( s + jω ) − ] log | \ det S L ( jω ) | dω = π log (cid:12)(cid:12)(cid:12)(cid:12) det S L ( s ) · Q i ( s + z L + ,i ) Q i ( s − z L + ,i ) (cid:12)(cid:12)(cid:12)(cid:12) , (48)with s and s ∗ included in z L + ,i . Because of (46), theintegral in (47b) is the same if we replace | det S b ( jω ) | with | \ det S L ( jω ) | . Hence, the right-hand sides of (47b) and (48) areequal. We now apply (42) in Lemma 4 to the right-hand sidesof (47a) and (47b), and (17) in Lemma 1 to the right-handside of (47b). The result is (20).We then apply (29): Z ∞ [( s − jω ) − ( k +1) + ( s + jω ) − ( k +1) ] × log | det S GM ( jω ) | dω = ( − k πk h X i ( p GM,i − s ) − k − X i ( z GM,i − s ) − k + (cid:16) X i ( z GM + ,i − s ) − k − X i ( − z GM + ,i − s ) − k (cid:17)i (49a) Z ∞ [( s − jω ) − ( k +1) + ( s + jω ) − ( k +1) ] × log | det S b ( jω ) | dω = ( − k πk h X i ( p b ,i − s ) − k − X i ( z b ,i − s ) − k + (cid:16) X i ( z b ,i − s ) − k − X i ( − z b ,i − s ) − k (cid:17)i (49b) Z ∞ [( s − jω ) − ( k +1) + ( s + jω ) − ( k +1) ] × log | \ det S L ( jω ) | dω = ( − k πk h X i ( p L,i − s ) − k − X i ( z L,i − s ) − k + (cid:16) X i ( z L + ,i − s ) − k − X i ( − z L + ,i − s ) − k (cid:17)i , (49c)where k = 1 , , . . . , m − . Note − s and − s ∗ are excludedin p L,i , and s and s ∗ are excluded in z L,i and z L + ,i in (49c).Because of (46), the integral in (49b) is the same if we replace | det S b ( jω ) | with | \ det S L ( jω ) | . Hence, the right-hand sidesof (49b) and (49c) are equal.When k = m − or I − S L ( s ) S G ( s ) is non-singular, weapply (43) in Lemma 4 to the right-hand sides of (49a) and(49b), and (17) in Lemma 1 to the right-hand side of (49b).The result is (21). When k = m − and I − S L ( s ) S G ( s ) is singular, weapply (44) instead of (43) to the right-hand sides of (49a) and(49b). The result is (22).R EFERENCES[1] D. Nie and B. M. Hochwald, “Bandwidth analysis of multiport radio-frequency systems—Part I,”
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Ding Nie
Ding Nie (M’16-) was born in Nanchang,Jiangxi, Peoples Republic of China. He received theB.S. degree in electronic engineering from Shang-hai Jiao Tong University, Shanghai, in 2010. Hereceived the M.S. degree and the Ph.D. degree inelectrical engineering from the University of NotreDame, Notre Dame, IN, in 2016. He then joinedApple Inc., Cupertino, CA, USA, in 2016. He wonthe 2016 Outstanding Young Author Award for thepaper he co-authored in the IEEE Transactions onCircuits and Systems. His research interests includecommunications, radio-frequency circuits and antennas.