11 Analog Signal Processing
Christophe Caloz,
Fellow, IEEE,
Shulabh Gupta,
Member, IEEE,
Qingfeng Zhang,
Member, IEEE, and Babak Nikfal,
Student Member, IEEE
Abstract —Analog signal processing (ASP) is presented as asystematic approach to address future challenges in high speedand high frequency microwave applications. The general conceptof ASP is explained with the help of examples emphasizingbasic ASP effects, such as time spreading and compression,chirping and frequency discrimination. Phasers, which repre-sent the core of ASP systems, are explained to be elementsexhibiting a frequency-dependent group delay response, andhence a nonlinear phase response versus frequency, and variousphaser technologies are discussed and compared. Real-timeFourier transformation (RTFT) is derived as one of the mostfundamental ASP operations. Upon this basis, the specificationsof a phaser – resolution, absolute bandwidth and magnitudebalance – are established, and techniques are proposed to enhancephasers for higher ASP performance. Novel closed-form synthesistechniques, applicable to all-pass transmission-type cascaded C-section phasers, all-pass reflection-type coupled resonator phasersand band-pass cross-coupled resonator phasers are described.Several applications using these phasers are presented, includinga tunable pulse delay system, a spectrum sniffer and a real-time spectrum analyzer (RTSA). Finally, future challenges andopportunities are discussed.
Index Terms —Analog signal processing (ASP), phaser,frequency-dependent group delay, dispersion engineering, dis-persive delay structure, chirping, real-time Fourier transform,C-section and D-section, cross-coupled resonators, filter synthesis.
I. I
NTRODUCTION AND M OTIVATION
Today’s exploding demand for faster, more reliable andubiquitous radio systems in communication, instrumentation,radar and sensors poses unprecedented challenges in mi-crowave and millimeter-wave engineering.Recently, the predominant trend has been to place anincreasing emphasis on digital signal processing (DSP). How-ever, while offering device compactness and processing flex-ibility, DSP suffers of fundamental drawbacks, such as high-cost analog-digital conversion, high power consumption andpoor performance at high frequencies.To overcome these drawbacks, and hence address the afore-mentioned challenges, one might possibly get inspiration fromultrafast optics [1]. In this area, ultra-short and thus huge-bandwidth electromagnetic pulses are efficiently processed inreal time using analog and dispersive materials and compo-nents [2], [3], [4].We speculate here that the same approach could be po-tentially applied to high-frequency and high-bandwidth mi-crowave signals, leading to systematic microwave analogsignal processing (ASP) as an alternative to DSP-based pro-cessing, with particular promise for millimeter and terahertzfrequency applications.The paper first explains more specifically what is meantby microwave ASP, covering the basic chirping and frequency discrimination effects, emphasizing the concept of group delayengineering, and presenting the fundamental concept of real-time Fourier transformation. Next, it addresses the topic ofthe “phaser”, which is the core of an ASP system; it reviewsphaser technologies, explains the characteristics of the mostpromising microwave phasers, and proposes correspondingenhancement techniques for higher ASP performance. Basedon ASP requirements, found to be phaser resolution, absolutebandwidth and magnitude balance, it then describes novelsynthesis techniques for the design of all-pass transmissionand reflection phasers and band-pass cross-coupled phasers.Finally, it presents some applications using these phasers,including a tunable pulse delay system for pulse positionmodulation, a real-time spectrum sniffer for cognitive radioand a real-time spectrogram analyzer for the characterizationand processing of nonstationary signals.II. W
HAT IS M ICROWAVE A NALOG S IGNAL P ROCESSING ?Microwave ASP might be defined as the manipulation ofsignals in their pristine analog form and in real time to realizespecific operations enabling microwave or millimeter-waveand terahertz applications .The essence of ASP might be best approached by consider-ing the two basic effects described in Fig. 1, chirping with timespreading and frequency discrimination in the time domain.Both effects involve a linear element with transfer function H ( ω ) = e jφ ( ω ) , which is assumed to be of unity magnitudeand whose phase, φ ( ω ) , is a nonlinear function of frequency, orwhose group delay, τ ( ω ) = − ∂φ ( ω ) /∂ω , assuming the time-harmonic phasor dependence e jωt , is a function of frequency.Note that there is no incompatibility between the linearity ofthe element and the nonlinearity of its phase: the former refersto the independence of the element’s response to the magnitudeof the input signal , whereas the latter refers to the nonlinearityof the element’s phase versus frequency , which is an inherentproperty of the element , independent of the input signal. Suchan element, with frequency-dependent group delay , is called temporally dispersive , or simply dispersive when the contextis unambiguous. It is to be noted that temporal dispersion contrasts with spatial dispersion , occurring for instance in thephenomena of radiation through an aperture or propagationacross a periodic structure, where different spatial frequencies,corresponding to scattering in different directions, are gener-ated. The spectrum of H ( ω ) is assumed to cover the entirespectrum of the input signal.In the chirping case, depicted in Fig. 1(a), a pulse (typicallybe a gaussian pulse, whose spectrum is also a gaussianfunction [5]) modulated at an angular frequency ω is passedthrough the element H ( ω ) , which is assumed here to ex-hibit a positive linear group delay slope over a frequency a r X i v : . [ phy s i c s . op ti c s ] A ug H ( ω ) = e jφ ( ω ) H ( ω ) = e jφ ( ω ) tttt ωωτ ( ω ) τ ( ω ) ω ω ω ω ∆ τ ∆ τ ω ω ω ω ω ω ω , ω T in = T T in = T T out > T τ τ τ τ ∆ ωω τ τ = τ T out = T T out = T (a)(b) Fig. 1: Basic effects in ASP. (a) Chirping with time spreading.(b) Frequency discrimination in the time domain.band centered at the frequency ω , corresponding to a groupdelay τ . Due to the dispersive nature of this element, thedifferent spectral components of the pulse experience differentdelays and therefore emerge at different times. Here, thelower-frequency components are less delayed and thereforeemerge earlier than the higher-frequency components, whilethe center-frequency component appears at the time τ = τ .This results in an output pulse whose instantaneous frequencyis progressively increasing (would be decreasing in the caseof a phaser with a negative group delay slope), a phenomenoncalled “chirping”, and which has experienced time spread-ing ( T out > T ), accompanied with reduced amplitude dueto energy conservation. When undesired, this time spreadingeffect can be compensated by either using a stepped groupdelay phaser, as will be shown in the next paragraph, orby compression , using phasers of opposite chirps, as will beshown in the tunable pulse delay line application of Sec. VII.It should be noted that in many applications, time spreading isactually useful. For instance, it belongs to the essence of real-time Fourier transformation, to be presented in Sec. IV. Asanother example, time spreading may be exploited to increasethe sampling rate of a signal at fixed clock [6],[7].In the latter case, depicted in Fig. 1(b), the input pulse ismodulated by a two-tone signal, with frequencies ω and ω ,and passed through a dispersive element H ( ω ) exhibiting apositive stepped group delay, with two steps, centered at ω and ω , respectively. Based on this dispersive characteristic,the part of the pulse modulated at the lower frequency, ω , isdelayed less than the part modulated at the higher frequency, ω , and hence emerges earlier in time. As a result, the twopulses are resolved (or separated) in the time domain, and theirrespective modulation frequencies may be deduced from theirrespective group delays from the dispersive law, H ( ω ) . Note that with the flat-step law considered here the pulses are not time-spread ( T out = T ), assuming that the pulse bandwidth fitsin the flat bands of the steps, since all the spectral componentswithin each band are delayed by the same amount of time.Reference [8] provides a fundamental explanation of groupdelay dispersion in terms of wave interference mechanisms.III. C ORE OF AN
ASP S
YSTEM : THE P HASER
The dispersive element H ( ω ) in Fig. 1, which manipulatesthe spectral components of the input signal in the timedomain, is the core of an ASP system . We suggest to callsuch a dispersive element, in which a nonlinear phase versusfrequency response is necessary for ASP, a phaser . Severalterms have been used by the authors and discussed withcolleagues from various academic and industrial backgrounds.After much reflection, “phaser” imposed itself as the mostappropriate term. Table I lists the four main terms that wereconsidered with their pros and cons, and shows the superiorityof “phaser” over the other three terms. A phaser “phases” as afilter filters: it modifies the phase of the input signal followingphase – or more fundamentally group delay – specifications,with some magnitude considerations, just in the same way asa filter modifies the magnitude of the input signal followingmagnitude specifications, with some phase considerations.It has come to the attention of the authors that a few peopleseem to dislike the terminology “phaser.” One of them assertedthat “this term should be reserved to Star Trek” [sic]. Well, thefact that “phaser” refers to a directed-energy weapon in thispopular science fiction series [9] is not considered a majorconcern by the authors, who judge that confusion betweenASP phasers and Star Trek phasers is fairly unlikely!An ideal phaser is a phaser that would exhibit an arbitrarygroup delay with flat and lossless magnitude over a givenfrequency band, as illustrated in Fig. 2. Obviously, such anideal response is practically unrealizable. Realizing a responseas close as possible to this ideal one for specific applicationsis the objective of phaser synthesis, which will be discussedin Sec. VI. ω ω g r oupd e l a y , τ m a gn it ud e , | S | frequency, ω passband stopbandstopband τ ( ω ) | S ( ω ) | Fig. 2: Ideal phaser design: arbitrary group delay with flat andlossless magnitude response over a given frequency band.The phase versus frequency function of a phaser, withina certain specified bandwidth may be expanded in a Taylor
TABLE I: Comparison of the main terms considered todesignate an element of the type H ( ω ) in Fig. 1, followingfrequency-dependent group delay specifications for ASP. Pros ConsPhaser • natural and unambiguous • counterpart in acoustics [10] • general (transfer function- phasing, material, structure,device, network, circuit,component, system) • short and elegantDispersiveDelayStructure(DDS) • emphasizes that τ isfrequency-dependent • sometimes used inphotonics, surfaceacoustic waves andmagnetostatic waves • long ⇒ acronym DDS • DDS also stands forDirect DigitalSynthesizer • not general(only structure)All-passNetwork • used for a longtime in microwaves • not all phasers areall-pass (e.g. [11]) • confusion withdispersion-less (linearphase) all-pass • not general(only network)ArbitraryPhaseNetwork • emphasis on phasevs magnitude • phase is designablebut not arbitrary • not general(only network) • confusion withphase shifters(single frequency) series around a frequency (often the center frequency) withinthe phaser’s bandwidth, ω : φ ( ω ) = φ + φ ( ω − ω ) + φ ω − ω ) + φ ω − ω ) + . . . (1a)with φ k = ∂ k φ ( ω ) ∂ω k (cid:12)(cid:12)(cid:12)(cid:12) ω = ω . (1b)The first three coefficients of the series explicitly read φ = φ ( ω ) (rad) , (2a) φ = ∂φ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = − τ ( ω ) (s) , (2b) φ = ∂ φ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ∂τ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = s τ ( ω ) (s /rad) , (2c)and represent the phase, group delay and group delay disper-sion parameters of the phaser at ω , respectively. The groupdelay function of the phaser is then obtained from (1a) as τ ( ω ) = − ∂φ ( ω ) ∂ω = − φ − φ ( ω − ω ) − φ ω − ω ) − . . . , (3)where one notes, consistently with (2c), that φ is the slopeof the group delay function at ω .The most common phaser group delay response is the lineargroup delay one. This response corresponds to a second-ordernonlinear phase function, where φ (cid:54) = 0 ( φ ≷ ), φ beingan arbitrary phase reference and φ > depending on theelectrical length of the phaser, and φ k = 0 for k > in (1)and (3). It is the response used in Fig. 1(a) with φ > , the response required for real-time Fourier transforming, tobe presented in Sec. IV, and this is the function that will beconsidered in most of the examples of this paper. However,other group delay functions may be required, such as forinstance the stepped group delay function, that is shown inFig. 1(b) for the case of two steps and that will be used in thespectrum sniffer to be presented in Sec. VII.The realization of phasers requires controllable dispersionfor efficient group delay engineering. It may follow twodistinct but possibly combined [12] approaches: the mediumapproach and the network approach . The medium approachconsists in exploiting the natural dispersion of highly dis-persive materials [1], [13], [14] or the artificial dispersionof metamaterial structures [15], [16], [17] for phasing. Itessentially synthesizes constitutive parameters, which are mostgenerally bi-anisotropic [18] and which correspond to thefundamental space harmonic in the case of periodic meta-materials [19]. This approach is particularly indicated whenspatial dispersion or/and radiation is involved in addition totemporal dispersion, as will be shown in the case of the real-time Fourier transformer to be presented in Sec. VII. Thenetwork approach consists in using microwave-type couplerand filter structures and techniques [20],[21], [22] for phasing.These structures are most often aperiodic. They offer a higherdegree of temporal dispersion control compared to material ormetamaterial phasers, as will be shown in Sec. VI, and theyare therefore preferred for purely wave-guiding phasers.Figure 3 shows some of the most common microwavephasers reported to date. These phasers can be broadly classi-fied as reflection-type phasers and transmission-type phasers ,some of the formers being shown in Figs. 3(a)-(c) and some ofthe latter being shown in Figs. 3(d)-(i). Reflection-type phasersrequire a circulator or a directional coupler at their inputto provide the required functional two-port network responsesuggested in Fig. 1 [23], while transmission-type phasers areinherently two-port network components and may be thereforestraightforwardly integrated into ASP system. Bragg grating phasers are realized by cascading Bragggrating sections [Fig. 3(a)] of progressively varying periods,so as to reflect different frequencies at different locationsand hence with different delays [1], [24], [25]. Bragg gratingphasers are excessively bulky at low microwave frequencies,but might be sometimes replaced by non-uniform artificialsubstrate planar structures based on the same principle [26].
Chirped microstrip line phasers [Fig. 3(b)] represent anotherplanar technology with position-frequency dependent sections,with the benefit of some extra design flexibility provided bysmooth discontinuities but the drawback of relatively largesize, and hence, high loss [27], [28], [29].
Coupled-resonatorstructure phasers [Fig. 3(c)], to be discussed next, are morecompact (in a given technology, e.g. planar or waveguide)and can exactly follow specified group delay responses [23],as will be shown in Sec. VI.
Magnetostatic wave (MSW) phasers [Fig. 3(d)] utilize the inherent dispersion of quasi-static modes in ferrimagnetic films, which may be controlledvia layer, strip and boundary structural parameters [30], [31],[32]. Due to the quasi-static nature of magnetostatic modes,the MSW wavelength is about four orders of magnitude smaller than that of electromagnetic modes, which leads tovery compact devices. MSW technology was very popularin the 1980ies but it is nowadays rarely used, mostly due tothe requirement of a biasing magnet.
Surface acoustic wave(SAW) phasers [Fig. 3(e)] owe their dispersive properties tothe distribution of electrodes on piezoelectric substrates [33],[34]. They are even more compact than MSW devices, therelevant acoustic wavelength being over six orders of magni-tude smaller than that of electromagnetic waves. SAW devicesare abundantly used in the microwave industry but they aremostly restricted to frequencies below the X band, due tomaterial limitations, and they are inapplicable to millimeter-wave and terahertz frequencies. Composite right/left-handed(CRLH) transmission lines [Fig. 3(f)] and other metamaterial-type (sub-wavelength unit-cell) artificial transmission linesmay have their dispersion controlled in terms of the Taylorcoefficients of their wavenumber, under the constraint offixed Bloch impedance for broad-band matching [15], [17],[35], [36]. They are less compact than MSW-based andSAW-based phasers, but they may be scaled to virtually anyfrequency, feature size-bandwidth independent characteristic,due to their lumped unit cell, and are particularly suited toradiative and spatially dispersive applications [17].
CascadedC-section structure phasers [Fig. 3(g)], to be discussed next,offer the benefit of higher compactness and, as coupled-resonator phasers, can exactly follow specified group delayresponses [37], [38], [39], as will be shown in Sec. VI.
Cas-caded CRLH C-section structure phasers incorporate CRLHC-sections [Fig. 3(h)] so as to benefit from the enhanced cou-pling characteristics of CRLH coupled-line couplers [40] forhigher dispersion and broader bandwidth [41]. Finally, cross-coupled resonator structure phasers [Fig. 3(i)] exhibit the sameconfiguration types as cross-coupled filters [22]. As in filters,the addition of cross-coupling to sequential coupling providesextra degrees of freedom in phasers, and the correspondingphaser responses can follow exact specifications [11], as willbe shown in Sec. VI.Based on the above comparisons, coupled-resonator phasers[Fig. 3(c)], cascaded C-section phasers [Fig. 3(g)-(h)] andcross-coupled resonator phasers [Fig. 3(i)], which are allnetwork-type phasers, seem the most promising technologiesfor ASP systems restricted to guided-wave requirements, whileartificial (e.g. CRLH) transmission line phasers seem thebest option for radiated and spatially dispersive phasers. Thelatter have been largely covered in recent literature. We shalltherefore focus here on the former. Let us first explain theirbasic principle with the help of Fig. 4.Figure 4(a) represents a three-section cascaded C-sectionphaser as a simplified version of the phaser shown inFig. 3(g). A C-section is a coupled-line coupler with twoend ports interconnected so as to form a two-port network.It is an all-pass network ( | S ( ω ) | = 1 , ∀ ω ), whose trans-fer function reads S ( θ ) = ( a − j tan θ ) / ( a + j tan θ ) = e jφ with a = (cid:112) (1 + k ) / (1 − k ) , where θ = β(cid:96) is the electricallength of the structure, β and (cid:96) being its wavenumber andphysical length, respectively, and k is the coupler’s cou-pling coefficient [42]. The corresponding group delay is then τ ( ω ) = − dφ/dω = (cid:8) a/ [1 + ( a −
1) cos θ ] (cid:9) dθ/dω , and is (a) optical Bragggrating [24] (b) chirped microstripline [29] (c) coupled-resonatorstructure [23](d) magnetostatic wavedevice [31] (e) surface acousticwave device [42] (f) CRLH transmissionline [15](g) cascaded C-sectionstructure [37] (h) cascaded CRLHC-section structure [41] (i) cross-coupled resonatorstructure [11] R EFLECTION - TYPE PHASERS T RANSMISSION - TYPE PHASERS
Fig. 3: Some of the most common microwave phasers. realizedspecified τ SL ωω ω ω ω ω ω ω ω ω τ τ τ (a) (b)(c) (d) σjω s k s − s k s φ k Fig. 4: Principles of the selected phasers. (a) CascadedC-section phaser [Fig. 3(g)]. (b) Coupled-resonator phaser[Fig. 3(c)]. (c) Group delay response formation for (a) and(b). (d) Cross-coupled phaser [Fig. 3(i)].seen to reach maxima at θ = (2 m + 1) π ( m integer) or (cid:96) = (2 m + 1) λ/ , the first maximum being at (cid:96) = λ/ , asrepresented in the color curves of Fig. 4(c) for the different C-section lengths. A detailed wave-interference explanation forthe C-section’s response is provided in [41]. Cascading C-sections of different lengths forms then a transmission-type network, whose total group delay at each frequency is thesum of the delays incurred by each C-section, as representedby the black curve in Fig. 4(c). A phaser following a prescribedgroup delay function can then be realized by controlling thesizes of the different C-sections, their coupling coefficient andtheir number, as will be shown in Sec. VI. The coupled-resonator phaser represented in Fig. 4(b), cor-responding to the component shown in Fig. 3(c), is an all-pass network, as the previous phaser, but it is of reflection-type . Its operation is essentially similar to that of a Bragggrating [Fig. 3(a)] phaser, except that its response is achievedby aperture-resonator pairs rather than by a modulated periodiclattice, which provides higher design flexibility, allows perfectsynthesis accuracy and leads to much more compact devices.Intuitively, the lowest frequency components of the inputsignal are reflected at the largest apertures (diaphragms oririses in a waveguide) whereas the highest frequencies arereflected at the smallest apertures, as illustrated in Fig. 4(b),which leads to the desired total delay versus frequency curvein black in Fig. 4(c) with proper synthesis. Compared to thecascaded C-section phaser, this phaser has the drawback ofrequiring a circulator or a coupler, since it is a reflection-type as opposed to a transmission-type component. However,it provides higher dispersion (i.e. higher | φ | ) for narrow-bandapplications, as will be illustrated in the results of Sec. VI andshown from first principles elsewhere. It may be realized inplanar technology as well as in waveguide technology.Finally, the cross-resonator phaser represented in Fig. 4(d),corresponding to the component shown in Fig. 3(i), is thelatest one investigated by the authors [11]. Being of trans-mission type, it does not require a circulator or coupler,as the cascaded C-section phaser, while, exploiting cross-coupling, it is likely to provide maximal resolution afteroptimization. Its operation principle is less intuitive than thatof the other two phasers, but may be understood with the helpof Fig. 4(d). In a cross-coupled filter , the different wave pathsare typically designed so as to produce destructive interferenceat the frequencies of desired attenuation. The transmissionzeros, s k , of the transfer function, which may be generallywritten S = Π k ( s − s k ) /H ( s ) where H ( s ) is a Hurwitzpolynomial [22], are then placed on the imaginary axis of thecomplex s plane. As s spans the imaginary axis, correspondingto real frequencies ( s = jω ), it crosses the transmissionzeros, which builds the magnitude response at the cutoff andin the stopband. In contrast, in a cross-coupled phaser , thetransmission zeros are not intended to generate attenuation orstop-bands but to shape the phase response. Therefore, theyare placed off the imaginary axis and they are paired withtransmission poles, as illustrated at the bottom of Fig. 4(d),which cancels the attenuation effect while ensuring a phaseeffect. Note that this phaser is not an all-pass but a band-pass phaser, where the presence of the stopbands will relax thepassband constraints for more flexible phaser design, as willbe shown in Sec. VI.The cancelation by transmission poles of transmission zerosof the transfer function placed off the imaginary axis of thecomplex frequency plane is characteristic of all-pass functions.This effect may be verified as follows in the case of cascadedC-section phasers. Substituting the frequency mapping func-tion s = j tan θ into the C-section transfer function [38], [41], [42] S ( θ ) = a − j tan θa + j tan θ yields S ( s ) = − s − as + a . In the last relation, the transmission zero, s = a , and thetransmission pole, s = − a , are here both real and are placedanti-symmetrically with respect to the imaginary axis, whichleads, upon the substitution s = jω , to the all-pass response, | S | = 1 , ∀ ω .In a D-section , which represents the next higher-order all-pass function above the C-section, the two transmission zerosand poles are both complex and are placed symmetrically withrespect to the imaginary axis [39], [42], as represented inFig. 4(d), which also ensures an all-pass response.IV. R
EAL -T IME F OURIER T RANSFORMING
The Fourier transform [5] is one of the most fundamentaloperations in science and technology. In today’s microwavesystems, it is most often performed digitally using the fast-Fourier transform (FFT) algorithm [43], and it represents forinstance the basis of orthogonal frequency division multiplex-ing (OFDM) in wireless communications [44]. In the realm ofASP, Fourier transformation may be performed in real time andis then called real-time Fourier transformation (RTFT) [45].We shall describe here the principle of RTFT, although it is notdirectly used in many ASP applications, because it highlightssome fundamental characteristics of ASP systems, that willbe further detailed in terms of phasers in Sec. V and that willsubsequently guide synthesis specifications in Sec. VI.Figure 5(a) shows the block diagram of a microwave RTFTsystem. RTFT requires a linear group delay (or quadraticphase) phaser response, as represented in the figure. Theimpulse response of such a phaser, h ( t ) , is derived in theAppendix, where it is given by Eq. (15). As shown in Fig. 5(a),the base-band input signal, ψ in ( t ) , to be Fourier-analyzedin the time domain, is first up-converted to the microwavefrequency band of the phaser, which is centered at ω , to yield ψ m ( t ) = ψ in ( t ) e jω t . This modulated signal is then passedthrough the phaser, which is considered here linear and whoseoutput response, ψ h ( t ) , is hence the convolution of ψ m ( t ) with h ( t ) . On has thus, using (15): ψ h ( t ) = [ ψ in ( t ) e jω t ] ∗ h ( t )= (cid:90) + ∞−∞ (cid:2) ψ in ( τ ) e jω τ (cid:3) (cid:20) γe − j β ( t − τ ) φ e − j ( t − τ )22 φ (cid:21) dτ = γe − j βtφ e − j t φ (cid:90) + ∞−∞ ψ in ( τ ) e jω τ e j βτφ e j tτφ e − j τ φ dτ = γe − j βtφ e − j t φ (cid:90) + ∞−∞ ψ in ( τ ) e j ( φ t ) τφ e − j τ φ dτ. (4)In the last expression of (4), the second exponential of theintegrand reduces to unity if τ max / (2 | φ | ) (cid:28) π , or [46] T ψ in π | φ | (cid:28) , (5)
55 60 6500.20.40.60.81 4 5 6 71 2 3 4 5 600.51 ψ in ( t ) ψ in ( t ) e jω t ψ m ( t ) ψ h ( t ) LPF ψ out ( t ) h ( t ) ω τ ω mathematical FT ψ out ( t ) no r m a li ze d m a gn it ud e time (ns) time (ns)frequency (GHz) (a)(b) ∆ T T Fig. 5: Real-time Fourier transformer (RTFT). (a) Microwaveimplementation. (b) Result example using a measured cas-caded C-section phaser of the type of Fig. 3(g).where T ψ in represents the duration of the input signal (assumedto extend from t = 0 to t = T ψ in ), which limits the span ofthe integral (4). Note that condition (5) does not eliminate thefirst exponential of the integrand in (4), despite the presenceof the φ factor in its argument, because the numerator ofits argument is enhanced by the group delay term φ and ismultiplied by t , which is much greater than τ over most of theduration of ψ h ( t ) , due to the assumed large dispersion ( φ )in (5). In the interval of ψ h ( t ) ’s duration, we have t ∼ τ , but φ may still ensure the dominance of the first exponential ofthe integrand thanks to the term φ .Under condition (5), the absolute value of (4) reduces thento the output signal of Fig. 5(a), after envelope detection, ψ out [ ω ( t )] = | ψ h ( t ) | = (cid:114) πφ (cid:90) + ∞−∞ ψ in ( τ ) e jω ( t ) τ dτ (6a)with ω ( t ) = φ + tφ . (6b)The signal ψ out [ ω ( t )] in (6a) is essentially the Fourier trans-form of ψ in ( t ) in the time domain : its shape versus timeis identical to the Fourier transform of ψ in ( t ) , whose actualfrequencies are obtained by the mapping function (6b). Fig-ure 5(b) shows an example of an RTFT result involving ameasured cascaded C-section phaser.V. P HASER C HARACTERISTICS AND E NHANCEMENT
The performance of an ASP system directly depends onthe characteristics of the phaser that it utilizes. The threemost important phaser characteristics in this regard are the resolution , the absolute bandwidth and the magnitude balance .The resolution characteristic can be understood intuitivelywith the help of Fig. 1(b), which represents in essence a frequency discriminator [47], one of the most basic ASPsystems [48]. If the time difference, ∆ τ , between the groupdelays of the two input modulation frequencies, ω and ω ,is less than the duration T of the input pulse, the two pulsesat the output are not fully separated in time, i.e. resolved ,and can therefore not be unambiguously discriminated. Thenthe frequencies ω and ω cannot be clearly detected intime: the resolution of the ASP system is insufficient. Thisindicates that the resolution , or capability of a phaser to resolvethe frequency components of a signal, is proportional to thegroup delay difference (or swing), ∆ τ . At the same time, if T is reduced, then ∆ τ can also be reduced without loosingdiscrimination; the shorter T is, the smaller ∆ τ can be. So,the resolution of an ASP system is also inversely proportionalto duration of its input pulse, T . So, the resolution, that weshall denote (cid:37) , should follow (cid:37) ∝ ∆ τ /T . But T is inverselyproportional to the pulse bandwidth, B , so that (cid:37) ∝ ∆ τ B .Now, the spectrum of the pulse must clearly be fully containedin the bandwidth of the phaser, | ∆ ω | , or B ≤ | ∆ ω | , so thatwe also have B ∝ | ∆ ω | . Finally, the ASP resolution of a phaser may thus be defined as the unit-less quantity (cid:37) = | ∆ τ | · | ∆ ω | , (7)where ∆ τ represents the group delay swing provided by thephaser over its frequency bandwidth ∆ ω .It is instructive to consider the particular case of a lineargroup delay phaser at this point. This is case considered inthe system of Fig. 1(a), which may operate as a frequencymeter [49] where the unknown frequency to be measured, ω , is found from the time position of the maximum of theoutput chirped pulse using the phaser law, τ ( ω ) . In this case,from (2c), we have φ = ∂τ /∂ω | ω = ∆ τ / ∆ ω , and using thisrelation to eliminate ∆ τ in (7), yields (cid:37) lin = | φ | · | ∆ ω | . (8)This equation reveals that the resolution is proportional tothe group delay dispersion parameter, | φ | ( φ > (resp. φ < ) for a positive (resp. negative) group delay slope) andto the square of the phaser bandwidth. It is fully consistentwith the RTFT condition (5), both directly in terms of | φ | and indirectly in terms of ∆ ω ∝ /T = 1 /T ψ in , leadingto (cid:37) lin = | φ | / | T ψ in | : the accuracy of RTFT is proportionalto | φ | , or equivalently, for a given accuracy, increasing | φ | allows one to analyze signals with larger duration, T ψ in .If the ASP resolution of a given phaser (e.g. any phaserin Fig. 3) is insufficient, it may be enhanced by using thefeedback loop circuit shown in Fig. 6(a) [48]. In this system,the signal to be processed, x ( t ) , is passed N times througha feedback loop including the phaser plus and amplifier, re-generating the signal level, and a constant delay line, avoidingsignal self-overlapping. The group delay slope is progressivelyincreased as the signal loops in the system, as shown inFig. 6(b). After the second pass, all the delays experienced bythe different frequency components of x ( t ) have been doubled,and hence the slope has been doubled. After the N th pass, theslope has been multiplied by N , and thus, the bandwidth ∆ ω being unchanged, the resolution as defined by (7) or (8) hasbeen multiplied by the same factor: (cid:37) N = N (cid:37) = N | φ | . The processed signal, y N ( t ) , is then extracted from the loopusing an appropriate counting-switching scheme. A brute-forceresolution enhancement approach would consist in cascading N times the same phaser, with the amplifier at the output of thechain. However, the resulting system would have prohibitivedrawbacks: i) it would be excessively large; ii) it would sufferfrom high insertion loss due to multiple-reflection mismatchbetween the different phasers; iii) it would have a poor signal-to-noise due to the low level of the signal reaching theamplifier. x ( t ) phaser τ ( ω ) amplifier Gτ dividerdelay line y N ( t ) isolator (a) ωτ N ∆ ω ∆ τ ∆ τ N = N ∆ τ × N (b) Fig. 6: Feedback loop resolution enhancement principle [48].(a) Circuit block diagram. (b) Group delay slope ( | φ | ) mul-tiplication.In applications where the relevant frequency bandwidth isvery small, such as for instance in Doppler radar [50] or innarrow-band signal RTFT, the absolute frequency bandwidth is another essential figure of merit. Several phasers, such asthe transmission-type C-section phaser to be quantitativelycharacterized in Sec. VI, provide a group delay swing inthe order of 1 ns over a bandwidth in the order of 1 ns,corresponding to the relatively high resolution of (cid:37) = 1 ,according to (7). However, if the relevant frequency bandis much smaller (e.g. 10 MHz), the corresponding relevant resolution, corresponding to this frequency band, is muchsmaller (e.g. 1 ns · = 0 . ), and may be largely in-sufficient. In this case, another phaser technology must beused, such as for instance that of reflection-type coupled-resonator phasers, which offer an even higher group delayswing over a much smaller bandwidth, as will be quantitativelyexemplified in Sec. VI.Let us now consider the third and last phaser characteristic,magnitude balance. Magnitude balance is defined as a flattransmission magnitude, | S | , over the frequency band ofthe phaser, as represented in Fig. 2. In contrast to what isshown for the ideal case represented in Fig. 2, the magnitude cannot be of 0 dB, due to material and structural losses, but itmay be amplified, as in any processing system, if required.However, magnitude balance must be achieved as well aspossible to ensure high ASP performance. We shall next seehow magnitude imbalance comes about, how it affects ASP,and how to remedy it for higher ASP performance.The problematic of magnitude imbalance is described inFig. 7. Magnitude imbalance, i.e. a frequency varying trans-mission magnitude, | S | , in a phaser is caused by loss.The main loss contributors in a network phaser of effectivelength (cid:96) are material loss, e − α m ( ω ) (cid:96) , including conductor loss, e − α c ( ω ) (cid:96) , and dielectric loss, e − α d ( ω ) (cid:96) , although radiation lossmay also play a significant role in open structures at highfrequencies. It is well-known that these losses increase withincreasing frequencies in any microwave structure [51], asschematically represented by the green curves in Figs. 7(a)and (b), which represent the responses of a positive linearchirp (up-chirp) phaser and a negative linear chirp (down-chirp) phaser, respectively. However, an effect that is specificto phasers must also be considered, the phasing loss effect . Theamount of dissipation produced by any type of loss mechanismis necessarily proportional to the amount of time that thesignal spends in the system. In a phaser, this time greatlyvaries across the phaser bandwidth, and this variation is evendesired to be relatively large for high resolution, as seen in(7). Therefore, phasing loss, which may be modeled by theattenuation factor e − α τ ( ω ) (cid:96) , strongly depends on frequency,and increases (resp. decreases) with frequency in an up-chirp(resp. down-chirp) phaser, as represented by the blue curves inFig. 7(a) [resp. Figs 7(b)]. As a result, an unequalized up-chirpphaser will feature a negative-slope magnitude imbalance,as represented by the black curve in Fig. 7(a), while anunequalized down-chirp phaser may exhibit a negative-slopeor a positive-slop magnitude depending which is the dominanteffect between e − α τ ( ω ) (cid:96) and e − α m ( ω ) (cid:96) . Figures 7(c) and (d)show the group delay and magnitude responses, respectively,of a practical phaser including a both a positive slope (lowerfrequency range) and a negative slope (upper frequency range)of the group delay, each of which are independently utilizable,to illustrate this effect.Phaser magnitude imbalance unbalances ASP systems invarious fashions. For instance, in the frequency discriminatorof Fig. 1(b), the pulse associated with the lossier frequencymay fall below the noise floor. As an other example, Fig. 7illustrates the effect of magnitude imbalance in an RTFTsystem (Sec. IV) for a rectangular input base-band pulse. Inan ideal system, the output signal would be a perfect sincfunction of time, properly representing the Fourier transformof the input rectangular signal according to (6), with the centerfrequency, ω , appearing at t = ω φ − φ according to (6b).This is the situation is represented in Fig. 8(a). In the caseof the lossy transform, represented in Fig. 8(b), magnitudeimbalance has distorted the Fourier transform result.Magnitude imbalance can be remedied using various tech-niques. A possible technique would consist in using a resistivemagnitude equalization network, having negligible effect onthe phase response and sacrificing on the transmission levelof the less lossy parts of the spectrum [53]. A more subtle −40−30−20−1000 1 2 3 4246810 0 1 2 3 4 frequency (GHz)frequency (GHz) g r oupd e l a y ( n s ) S - p a r a m e t e r s ( d B ) ττ ωω | S || S | | S || S | up-chirp down-chirp τ ( ω ) α m = α c + α d α τ α = α m + α τ (a) (b)(c) (d) Fig. 7: Magnitude imbalance due to loss in an unequalizedphaser. (a) Trends of material and dispersion losses for anup-chirp phaser. (b) Same for a down-chirp phaser. (c) Groupdelay of the multilayer C-section phaser shown in the inset(solid: measurement, dashed: full-wave) [52]. (d) Correspond-ing S-parameter magnitudes. t tt (positive frequency chirping) unequal amplitudes ψ in ( t ) ψ ou t ( t ) ∝ F { ψ i n ( t ) } ψ ou t ( t ) ∝ F { ψ i n ( t ) } dB dB ψ h ( t ) ψ out ( t ) t = ω φ − φ t = ω φ − φ (a)(b) Fig. 8: Effect of magnitude imbalance in the case of real-time Fourier transformation [Fig. 5 and Eq. (6)]. (a) LosslessFourier transform (log scale) of the rect signal shown in theinset. (b) Corresponding lossy transform with negative slopemagnitude imbalance and exaggerated loss.technique is to account for the frequency-dependent lossesin the phaser synthesis, which is possible in cross-coupledresonator phasers, to be presented in Sec. VI. VI. P
HASER S YNTHESIS
As mentioned in Sec. III, the objective of phaser synthesis isto realize a response as close as possible to the ideal responserepresented in Fig. 2. Specifically, according to Sec. V, thephaser resolution, given by (7), must be sufficiently high andits absolute bandwidth must be appropriate for the targetedapplication, and its transmission magnitude must be carefullyequalized to combat frequency-dependent losses. The shape ofthe required phase response, represented by the Taylor series(1) is most often quadratic, corresponding to a linear groupdelay, as required for instance in RTFT (Sec. IV), but it mayalso be cubic, quartic or of higher order, or more complex,as in the stepped group delay response used in the spectrumsniffer to be presented in Sec. VII.The synthesis of medium-type phasers is beyond the scopeof this paper. We shall restrict here our attention to the case ofnetwork-type phasers. Little efforts have been dedicated to thesynthesis such phasers to date, since filter synthesis systemati-cally targets linear phase or zero-slope group delay responses,which exhibit no ASP capability. However, the important bodyof knowledge that has been built over the past decades inthe synthesis of linear phase filters and all-pass equalizersmay now be exploited and extended to the synthesis of actualphasers. The works reported in this area are too numerous to beexhaustively cited. They are described in many textbooks, suchas for instance [20], [21], [22]. Some of the contributions mostspecifically related to phasers include the following. In 1963,Steenaart introduced the C-sections and D-sections, under thenames of “all-pass networks of the first order and second or-der”, respectively [42], as the first distributed implementationsof the lumped-element all-pass lattice network [22]. Cristalproposed a transmission-type cascaded coupled-line equalizerin 1966 [54] and a reflection-type equalizer using a circulatorin 1969 [55]. Rhodes did extensive work on the synthesisof all-pass equalizers and linear-phase filters. For instance, in1970, he proposed a method to synthesize linear-phase filtersbased on a recurrence formula generating Hurwitz polynomialswith arbitrary phase versus frequency response [56]. Shortlylater, he applied this method to synthesize a cross-coupledwaveguide structure [57]. However, this was before the adventof the coupling matrix technique [22], and the method led tohigh-order prototypes, due to the limitation of the synthesizedtransfer function. The phase polynomial generation procedureused in [56] was later refined by Henk [58], and seems to bethe only one of the kind available in the literature. One mayfinally cite the recent work of Atia’s group on the synthesis ofreflection-type equalizers using coupled-resonators [59]. Anexcellent overview of all the developments on linear-phasefilters and equalizers is available in [22].We shall next discuss closed-form synthesis techniques thatwere recently developed for the three phasers selected inSec. III. These techniques are all based on the algorithm givenin [56], [58], and derived explicitly in [38], for the least-mean-square approximate construction of a Hurwitz polynomialexhibiting a desired phase function. Assume a specified phasefunction φ (Ω) , where Ω will next represent the frequencyvariable in the lowpass domain. The aforementioned algorithm requires this phase function to be discretized in N points,which leads to the frequency and phaser sets { Ω , Ω , . . . Ω N } and { φ , φ , . . . φ N } , respectively. Based on these two sets, thealgorithm, which reads H ( s ) = 1 H ( s ) = s + α ... H N ( s ) = α N − H N − ( s ) + (cid:0) s + Ω N − (cid:1) H N − ( s ) (9a)with α i = Ω i +1 − Ω i α i − − Ω i +1 − Ω i − α i − − Ω i +1 − Ω i − . . . α − Ω i +1 / tan( φ i +1 / ,α = Ω / tan( φ / , (9b)iteratively builds the N th degree Hurwitz polynomial H N ( s ) exhibiting the specified phase φ (Ω) , i.e. ∠ [ H N ( j Ω)] = φ (Ω) .The order, N , of the polynomial will correspond to the order,and hence to the size, of the phaser structure, as will be shownnext. Therefore, as the number of discretization points is pro-portional to the bandwidth over which the approximation (9)must hold, the size of the phaser structure will tend to beproportional to its bandwidth.Let us first consider the synthesis of the all-pass reflection-type coupled-resonator phaser, shown in Fig. 3(c) and ex-plained in Fig. 4(b). Its synthesis procedure was introducedin [23] and is described by the flow chart of Fig. 9. Start-ing from a specified group delay function, τ ( ω ) , of thecomponent’s reflection function, S ( ω ) , one first computes,analytically if possible or otherwise numerically, the corre-sponding phase function, φ ( ω ) , by integration. Next, usingthe band-pass ( ω ) to low-pass ( Ω ) mapping function cor-responding to the phaser technology used, one obtains thephase function in the low-pass domain, φ (Ω) . Since theall-pass transfer function will read S = H ∗ N /H N , thephase of the sought Hurwitz polynomial is minus half thatof φ (Ω) , i.e. φ H (Ω) = − φ (Ω) / . The function φ H (Ω) is then discretized over a reasonable (as small as possible)number N of points, and the iterative algorithm of (9) isapplied to the resulting frequency and phase sets, yielding thepolynomial H N ( s = j Ω) , and hence S (Ω) , from which theinput impedance Z in (Ω) = (1 + S ) / (1 − S ) follows. Thisimpedance, expressed in terms of the ratio of two polynomialsin Ω , is then written in the form of a long division, and mappedto the long division expression of the input impedance of thelow-pass filter prototype, which provides the prototype’s g k parameters. From this point, conventional filter synthesis tech-niques [21] are applied to provide the final phaser structure.The all-pass transmission-type cascaded C-section phaser,of the type shown in Fig. 3(g) and explained in Fig. 4(a)may be synthesized using the technique presented in [38]. Thefirst part of this synthesis technique follows exactly the firstrow of the flow chart of Fig. 9 after replacing the subscript pair “11” by “21”, while its second part consists in determiningthe C-section lengths and coupling coefficients using formulasbased on the analytical transfer functions of the C-section (seesynthesis flow chart of Fig. 7 in [38]). It should be noted that asynthesis technique for an alternative to this phaser, involvingboth commensurate C-sections and
D-sections, was recentlypresented in [39]. In this technique, the Hurwitz polynomialof the transfer function includes real roots corresponding tothe C-sections and complex roots corresponding to the D-sections, and the C-section and D-section lengths and couplingcoefficients are also determined from corresponding analyticaltransfer function.Figures 10(a) and (b) show the synthesized group delayresponses for a cascaded C-section phaser and a coupled-resonator phaser, respectively. As announced in Sec. III, thethe latter allows operation over a much smaller bandwidth thanthe former, because of its reflection-type versus transmission-type nature. frequency (GHz)frequency (GHz) g r oupd e l a y ( n s ) g r oupd e l a y ( n s ) specifiedspecified synthesizedsynthesized ̺ = ∆ τ · ∆ f = 1 ns · GHz = 1 ̺ = ∆ τ · ∆ f = 10 ns · MHz = 0 . Fig. 10: Example of linear group delay all-pass phaser syn-thesis results. (a) Cascaded C-section phaser of the typein Fig. 3(g) but with alternating C-sections to avoid inter-coupling [38] (synthesized: analytical, using a C-section basedtransfer function). (b) Coupled-resonator phaser of the type inFig. 3(c) [23] (synthesized: full-wave).In the previous two all-pass type phasers, some magnitudeimbalance is generally unavoidable and a resistive equalizationnetwork may have to be used to suppress it (Sec. V). The band-pass transmission-type cross-coupled phaser, of the type shownin Fig. 3(i) and explained in Fig. 4(d), does not suffer fromthis drawback, because its phase and magnitude responsescan be synthesized independently. Moreover, it may offerhigher design flexibility, due to the absence (or minimality)of specification constraints in the stop-bands. Its synthesisprocedure is described by the flow chart of Fig. 11, where thetransmission and reflection functions take the usual polynomialratio forms S ( s ) = P ( s ) H ( s ) , (10a) S ( s ) = F ( s ) H ( s ) , (10b)where H ( s ) is a Hurwitz polynomial. In these relations,the phase and magnitude responses will be exclusively con-trolled by H ( s ) and P ( s ) , respectively, so that we will have | H ( s ) | = 1 and P ( s ) ∈ (cid:60) [11]. Once the low-pass phasefunction, φ (Ω) , has been determined from the prescribedband-pass group delay function, τ ( ω ) , the required phase τ ( ω ) φ ( ω ) φ (Ω) φ H (Ω) H N (Ω) S (Ω) Z in (Ω) R dω ω → Ω φ H = − φ φ H k (Ω k ) , k = 1 , . . . , N algorithm of Eq. (9) S = H ∗ N H N Z in = 1 + S − S ω ← Ω long division g g g g n − g n g n +1 Fig. 9: Coupled-resonator (all-pass reflection-type) phaser synthesis procedure [23]. The first row also applies to the the all-passcascaded C-section (all-pass transmission-type) phaser design [38], where the subscript pair “11” is then to be replaced by “21”.for H ( s = j Ω) is found as φ H (Ω) = − φ (Ω) , since H ( s ) is in the denominator of S in (10a). The next step is tobuild the corresponding phase polynomial, H ( s ) , using theiterative algorithm (9), exactly as in the synthesis of the all-pass phasers. One may then specify the magnitude of S (Ω) , P (Ω) , in the phaser’s pass-band, and this magnitude mayapproximate various functions, such as for instance that ofa Chebyshev polynomial, P (Ω) ≈ (cid:112) H N (Ω) H ∗ N (Ω) . Finally,the numerator of S (Ω) , F (Ω) in (10b), is determined byapplying the energy conservation condition, from the initialassumption of a lossless system. From this point, the con-ventional coupling matrix technique can be applied and thecorresponding phaser structure can be designed [22].Figure 12 presents a cross-coupled phaser synthesis ex-ample. This design uses the simply folded topology shownin Fig. 12(a) with the corresponding coupling matrix ofFig. 12(b). The group delay response is plotted in Fig. 12(c),while the magnitude response is shown in Fig. 12(d), wherethe quasi-Chebyshev behavior is clearly apparent in S .The following comparisons between the three consideredphasers are useful, in connection with the discussions of Sec. Von phaser characteristics. The cascaded C-section [Fig. 10(a)]has the highest resolution, (cid:37) = 1 , but the bandwidth that itrequires to achieve a group delay swing in the order of 1 nsis in the order of 1 GHz. This may be appropriate if theapplication features a bandwidth of the same order, but ifthe application requires a resolution in the order of 10 MHz,then the relevant resolution collapses to . , which is may beinsufficient. In this case, the coupled-resonator [Fig. 10(a)] isclearly more indicated, with its resolution (cid:37) = 0 . for MHzor 0.1 for the relevant MHz bandwidth. The cross-coupledresonator phaser [Fig. 12(c)], for the same absolute bandwidth( (cid:37) = 0 . over 50 GHz or 0.025 for 10 GHz), that is fourtimes smaller, for the same number of resonators (six). How-ever, the topological degrees of freedom could be exploitedhere to increase the resolution, while avoiding a circulator.In all the aforementioned synthesis techniques, iterativeprocedures may be applied to increase the synthesis bandwidthfrom the initial closed-form synthesis, as shown for instancein [23]. VII. ASP A PPLICATIONS
Although the area of microwave ASP is relatively recent,several ASP applications have already been reported in the
S SS L LL
Source or loadResonatorsDirect couplingCross coupling − . − . − . − . − . − . . . − . − . − . − . − . − . . . . . frequency (GHz) frequency (GHz) g r oupd e l a y ( n s ) m a gn it ud e ( d B ) synthesizedsynthesized full-wavefull-wave measuredmeasured S S (a) (b)(c) (d) ̺ = ∆ τ · ∆ f = 2 . ns · MHz = 0 . quasi-Chebyshevresponse Fig. 12: Example of linear group delay band-pass phasersynthesis results, corresponding to the cross-coupled resonatorstructure of Fig. 3(i). (a) Topology. (b) Coupling matrix.(c) Up-chirp group delay response (white range). (d) Corre-sponding scattering parameters.past few years. We shall describe here three of them in somedetails and provide references to some others.Let us first consider the tunable pulse system that isdepicted in Fig. 13 and that was proposed in [36] with aCRLH transmission line as a phaser. This system delayspulses by continuous and controllable amounts via a pair ofsynchronized local oscillators, one at ω c and the other oneat ω d = 2 ω c , while avoiding pulse spreading using mixerinversion. The input signal is first mixed with the harmonicwave ω c , which yields the modulated pulse v ( t ) , with thesame duration, T , as the input pulse. This pulse is then passedthrough the first phaser, whose center frequency coincides with ω c . This phaser does not require a linear group delay response;it may have any type of group delay response, such as forinstance the CRLH-type one shown in the figure. The pulse atthe output of the phaser, v ( t ) , has been delayed by an averageamount τ = φ [Eq. (2b)] (neglecting the delay induced by themixer), chirped so as the exhibit the instantaneous frequency ω ( t ) and spread out in time to a duration T (cid:48) due to dispersion. τ ( ω ) φ ( ω ) φ (Ω) φ H (Ω) H N (Ω) P (Ω) F (Ω) R dω ω → Ω φ H = − φ φ H k (Ω k ) , k = 1 , . . . , N algorithm of Eq. (9) P ≈ p H N H ∗ N H N (Ω) H ∗ N (Ω) = F (Ω) F ∗ (Ω) + P (Ω) ω ← Ω coupling matrix energy conservationChebyshevapproximation Fig. 11: Band-pass cross-coupled resonator phaser synthesis procedure.Next, v ( t ) is mixed with the harmonic wave ω d = 2 ω c , andpassed through a band-pass filter which eliminates the upperside band (USB) to keep only the lower side band (LSB) of themixer’s output. The instantaneous frequency of the resultingpulse, v ( t ) , is ω ( t ) = 2 ω c − ω ( t ) , which is centered againat ω c , but whose chirp has now been inverted [minus signin front of ω ( t ) ]: this inversion process is called mixer chirpinversion [36]. The pulse v ( t ) is now pass through the secondphaser, which must be identical to the first one. Since thechirp of v ( t ) has the opposite slope and an otherwise exactlyidentical shape as the response of the phaser, the phaser willexactly equalize (“un-chirp”) all the frequency components of v c ( t ) , an effect called pulse compression in radar technologyleading to an output signal v ( t ) that has exactly retrieved theduration of the input pulse, T , but that has been delayed byan amount depending on ω c , based on the phaser’s response, t ( ω c ) . Finally, the modulated pulse is envelop detected forbased-band processing. Note that this pulse delay system doesnot suffer from any mismatch associated with tuning, likevaractor controlled lines or components, since the controlparameters, ω c and ω d , are external to the transmission blocks.This ASP tunable pulse delay system may be applied toultra wideband processing, pulse position modulation [60] andcompressive receivers [36]. mixer USBLSB LSB phaser v ( t ) v ( t ) v ( t ) v ( t ) v ( t ) v ( t ) v ( t ) v ( t ) ω c ω c ω c ω c ττ tt tt v out ( t ) ττ τ TT T ′ T ′ dispersion compensation τ ( ω ) τ ( ω ) t ( ω c ) ω c ω c ω c ω c ω c ω d = 2 ω c ττ ω c ωω Fig. 13: Tunable pulse system, using mixer inversion fordispersion compensation [36].The next application example is the spectrum sniffer that is described in Fig. 14 and that was proposed in [61]. Thissystem may be considered as a generalized frequency discrim-inator [Fig. 1(b)]. Functionally, it “listens” to its radio envi-ronment through an omnidirectional antenna, and determines,in real time, the presence or absence of active channels inthis environment so as to allow an associated communicationsystem to opportunistically reconfigure itself to transmit in theavailable bands in a cognitive radio sense. The signal from theenvironment, v in ( t ) , is multiplied with an auxiliary pulse, g ( t ) ,after band-pass filtering and amplification, yielding v mix ( t ) ,which includes all the information on the environment’s spec-trum mixed up in time. The signal v mix ( t ) is then passedthrough a phaser with a stepped group delay response, of thetype already considered in Fig. 1(b), but including here foursteps for four channels. In each frequency band, correspondingto an expected channel, centered at ω k ( k = 1 , , , ), thegroup delay is flat, and hence the corresponding signal isneither distorted (i.e. could be demodulated) nor spread outin time, which will avoid channel overlap and subsequentinterpretation errors at the output. The output signal, v out ( t ) ,is consists in a sequence of pulses that have been resolved intime, and that can next be passed through a Schmitt triggerto generate the binary information on the presence or absenceof the channels. In the case of Fig. 14, channels 1, 3 and 4are active, while channel 2 is not, at the observation time,and could therefore be temporarily used to maximize the datathroughput. The authors are currently developing a systemreplacing the omnidirectional antenna by an electronicallysteered CRLH leaky-wave antenna (LWA) [62], that is si-multaneously temporally and spatially dispersive, as shall beseen in the next application example; this system will add theinformation of the angle-of-arrival of the different channels foreven higher processing efficiency. It is worth noting that ASP,in the area of radiative systems, can also be used for RFIDcoding, where each tag consists in a different phaser with itsspecific group delay response [63].The last application example is the real-time spectrum ana-lyzer (RTSA) that is described in Fig. 15 and that was proposedin [35]. An RTSA is a system that determines, in real-time,the spectrogram of signal, i.e. its joint time–frequency repre-sentation , X ( τ, ω ) = S { x ( t ) } = (cid:82) + ∞−∞ x ( t ) w ( t − τ ) e − jωt dt ,using a gating function w ( t ) [64]. The spectrogram is mostuseful for nonstationary signals , such as radar, security, in-strumentation, electromagnetic interference and compatibilityand biological signals, where the sole frequency or timeconsideration of the signal may not provide a sufficiently .. t t tt t frequency (GHz) g r oupd e l a y ( n s ) antenna phaser Schmitttriggerch1 ch2 ch3 ch4 ω ,ω ,ω ,ω ω ω ω ω ω ω ω ω input pulse mixer v out ( t ) v out ( t ) multi-channelinput signal v in ( t ) τ ( ω ) v mix ( t ) v mix ( t ) TT τ τ τ τ τ τ τ τ g ( t ) thresholdLNA Fig. 14: Real-time spectrum sniffer, using a cascaded C-sectionstepped group delay phaser with the response shown at thecenter (dashed: specified, solid: experimental) [61].informative representation of it. Here, the key component isa CRLH leaky-wave antenna (LWA) [15], [65], of the typeshown in Fig. 3(f). As shown in Fig. 15(a), such a LWA canbe used to map the temporal frequencies ( ω ) of a broadbandsignal, whose spectrum lies within its radiation region, ontospatial frequencies ( k ) or angles, θ , via the antenna scanninglaw, θ = sin − ( cβ/ω ) , shown in the figure [65]; it is thusa simultaneously a temporally dispersive (phasing) and aspatially dispersive device. This double dispersive property isexploited in the RTSA as shown in Fig. 15(b). The signal toanalyze is injected into the CRLH LWA, which radiates itsdifferent time spectral components into different directions,following the scanning law of Fig. 15(a). The correspondingwaves are picked up with circularly arranged antenna probes,envelope detected, and processed in real time so as to buildthe spectrogram, shown in Fig. 15(c) for the case of asignal composed by the succession of two gaussian pulseswith opposite chirps. In contrast to digital spectrum analyzes,this RTSA only requires very light processing, features shortacquisition times, can handle truly ultra wideband signals, andis easily scalable to the millimeter-wave frequency range.The double dispersive and full-space properties of CRLHLWAs have recently led to a radiation pattern diversitymultiple-input multiple-output (MIMO) communication sys-tem with drastically enhanced channel reliability and datathroughput [66]. In this system, the LWAs are controlled bya processor, which monitors the received signal level in real-time and subsequently steers the LWAs so as achieve at alltimes the highest possible signal-to-noise ratio, and hence thehighest possible channel capacity, in a real fluctuating wirelessmobile environment.VIII. C HALLENGES AND O PPORTUNITIES
As pointed out in Sec. I, ASP is not an establishedtechnology to systematically process microwave signals. Thispaper has shown that ASP technology features promisingattributes and seems to possess a particularly great potentialfor millimeter-wave and terahertz applications, where conven-tional DSP-based systems are inefficient or even unapplicable.However, several challenges will have to be faced and various
LH RH0 -90 RH spatial-spectraldecomposition probing andmonitoring post-processingradiationregionair line guided-waveregionguided-waveregion s i gn a lt o a n a l y ze invisiblespaceinvisiblespace visible space A / D c onv e r s i onb l o c k D SP b l o c k CR L H L W A antenna probesenvelopedemodulators fr e qu e n c y ( GH z ) inputsignalto analyze time (1 ns/div) (a) β θ (b) (c) f ( GH z ) f (GHz) θ x θ x ω ω BF ω EF ω x t Fig. 15: Leaky-wave antenna (LWA) based real-time spectrumanalyzer [35]. (a) Frequency-space mapping associated withtemporal-spatial dispersion for the CRLH LWA employed inthe RTSA. (b) System schematic. (c) Spectrogram example fora signal composed by the succession of two gaussian pulseswith opposite chirps.opportunities will have to be considered for this technology tosucceed at a large scale.Some of the most apparent challenges may be listed as fol-lows: a) realization of phaser characteristics for all the requiredapplications (e.g. frequency resolutions finer than kHz,or fractional bandwidth broader than ); b) closed-formsynthesis of complex phasers (e.g. cross-coupled Cascaded C-section phasers); c) fabrication of phasers in high millimeter-wave and terahertz frequencies, given that phasers are moresensitive to tolerances than filters, since they follow derivative (phase derivative) as direct transfer function specifications;c) development of optimal ASP modulation schemes forcommunications.Some opportunities to meet these challenges and discovernovel possibilities may include: i) exploitation of other groupdelay functions (e.g. cubic, quartic or quintic, with variousTaylor coefficients); ii) utilization of novel, in particular nano-scale and multi-scale materials, for increased resolution; iii) in-troduction of active elements to avoid restrictions related to theFoster reactance theorem [67] for higher dispersion diversity;iv) introduction of nonlinearities, possibly using distributedsemiconductor structures, for Kerr-type chirping, or self-phasemodulation, in substitution to or in addition to dispersion, asin solitary waves [68]. IX. C
ONCLUSION
Based on the speculation that analog signal processing(ASP) may soon offer an alternative or a complement to domi-nantly digital radio schemes, especially at millimeter-wave andterahertz frequencies, we have presented this emerging area ofmicrowaves in a general perspective. The preliminary researchresults are most promising. Dramatic progress has been madein the technology and synthesis of phasers, which are thekey components in an ASP system. Several applications havealready been reported and shown to offer distinct benefits overconventional DSP-based technology. It seems that microwaveASP has a great potential for future microwave, millimeter-wave and terahertz applications.A
CKNOWLEDGMENT
This work was supported by NSERC Grant CRDPJ 402801-10 in partnership with Research In Motion (RIM).A
PPENDIX I MPULSE R ESPONSE OF A L INEAR -C HIRP P HASER
A phaser is considered here to be a linear system. Thetransfer function of a linear system is defined as [5] H ( ω ) = ψ out ( ω ) ψ in ( ω ) , (11)where ψ in ( ω ) and ψ out ( ω ) represent the input and output sig-nals, respectively, in the spectral domain. The phaser transferfunction, assuming unity magnitude, is obtained from thephase function (1) as H ( ω ) = e jφ ( ω ) = e jφ e jφ ( ω − ω ) e j φ ( ω − ω ) e j φ ( ω − ω ) . . . (12)A linear-chirp phaser is a phaser whose group delay (withina certain specified bandwidth) is a linear function of frequencyor, equivalently according to (3), whose phase is a quadraticfunction of frequency. Its transfer function (12) includes thenonly the first three exponentials (i.e. φ k = 0 for k > ),where φ (cid:54) = 0 to avoid the limit case of a non-dispersive(linear-phase) phaser: H ( ω ) = e jφ e jφ ( ω − ω ) e j φ ( ω − ω ) . (13)The corresponding impulse response, defined by [5], h ( t ) = (cid:90) + ∞−∞ H ( ω ) e jωt dω, (14)is calculated as h ( t ) = (cid:90) + ∞−∞ e jφ e jφ ( ω − ω ) e j φ ( ω − ω ) e jωt dω = γe j βtφ e − j t φ , (15a)with β = β ( φ ) = φ − ω φ , (15b) γ = γ ( φ ) = (cid:114) πφ e j (cid:18) π + φ + φ ω − φ φ + φ ω − φ ω (cid:19) , (15c)where the tabulated result (cid:82) + ∞−∞ e − ( aω + bω ) dω = (cid:112) π/ae b a with a = − jφ / and b = − j ( t − ω φ ) has been used. R EFERENCES[1] B. E. A. Saleh and M. C. Teich,
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