Spectral Efficiency and Energy Efficiency of OFDM Systems: Impact of Power Amplifiers and Countermeasures
aa r X i v : . [ c s . I T ] M a y Spectral Efficiency and Energy Efficiency ofOFDM Systems: Impact of Power Amplifiers andCountermeasures
Jingon Joung, Chin Keong Ho, and Sumei Sun
Abstract
In wireless communication systems, the nonlinear effect and inefficiency of power amplifier (PA) have posedpractical challenges for system designs to achieve high spectral efficiency (SE) and energy efficiency (EE). In thispaper, we analyze the impact of PA on the SE-EE tradeoff of orthogonal frequency division multiplex (OFDM)systems. An ideal PA that is always linear and incurs no additional power consumption can be shown to yield adecreasing convex function in the SE-EE tradeoff. In contrast, we show that a practical PA has an SE-EE tradeoffthat has a turning point and decreases sharply after its maximum EE point. In other words, the Pareto-optimaltradeoff boundary of the SE-EE curve is very narrow. A wide range of SE-EE tradeoff, however, is desired forfuture wireless communications that have dynamic demand depending on the traffic loads, channel conditions,and system applications, e.g., high-SE-with-low-EE for rate-limited systems and high-EE-with-low-SE for energy-limited systems. For the SE-EE tradeoff improvement, we propose a PA switching (PAS) technique. In a PAStransmitter, one or more PAs are switched on intermittently to maximize the EE and deliver an overall required SE.As a consequence, a high EE over a wide range SE can be achieved, which is verified by numerical evaluations:with
SE reduction for low SE demand, the PAS between a low power PA and a high power PA can improveEE by , while a single high power PA transmitter improves EE by only . Index Terms
Energy efficiency, spectral efficiency, power amplifier, power amplifier switching, OFDM.
Parts of this work have been presented at the
IEEE Global Telecommunications Conference (GLOBECOM) , Anaheim, CA, USA, Dec.2012 [1], and the
Asia Pacific Signal and Information Processing Association (APSIPA) , Annual Summit and Conference, Hollywood,CA, USA, Dec. 2012 [2].The authors are with the Institute for Infocomm Research (I R), A ⋆ STAR, Singapore 138632 (e-mail: { jgjoung, hock, sunsm } @i2r.a-star.edu.sg) . I NTRODUCTION
Wireless access communication networks consume significant amount of energy to overcome fading andinterference, compared to fixed line communication networks [3], [4]. In wireless networks, energy is mostlyconsumed at the base station (BS) [3], of which a substantial fraction of – of overall power is consumedat power amplifiers (PAs) [5]. A measure of the PA efficiency is given by the drain efficiency η that is the ratioof PA output power P out to PA power consumption P PA , i.e., η = P out /P PA . Fig. 1(a) plots PA maximum outputpower P maxout versus P PA , based on our survey of commercially available PAs for which we give a summary ofthe key parameters in Table II in Appendix A. From Fig. 1(a), we see that η at P maxout is typically between and , which confirms that the overhead incurred at PA is substantial. To ensure high energy efficiency (EE),the PA characteristics have to be carefully considered in system designs.On the other hand, high spectral efficiency (SE) is needed to support the growing demands of high-rateapplications. Orthogonal frequency division multiplex (OFDM) and orthogonal frequency division multiple access(OFDMA) are two popular spectral efficient systems. However, OFDM and OFDMA modulated signals exhibithigh peak-to-average power ratio (PAPR), thus suffering from severe nonlinearity effects [6], [7] as illustrated inFig. 1(b), in which PA output power P out is shown over the PA input power P in . Two commonly used models,namely the Rapp model [8] and the soft limiter model [9], are shown in Fig. 1(b). They are used to describe thenonlinear amplitude (i.e., signal power) distortion, especially at a high power region, while the phase is assumedto be undistorted (the details will be given in Section III, and for more nonlinearity models, refer to the referencesin [2]). In practice, to circumvent the resulting performance degradation, input backoff (IBO) is implemented byreducing the power of the input signal at the PA, so that the amplification stays within the linearity region asmuch as possible. While IBO allows high SE to be achieved, it can reduce the EE, because the PA efficiency istypically designed to peak near the saturation point and it usually drops rapidly as the input power decreases [10].Hence, a tradeoff between SE and EE is inevitable while optimizing with respect to the PA. It is thus importantto jointly characterize the role that a PA plays in both SE and EE of wireless communication systems. Recently,circuit power consumption has been taken into consideration for energy efficient system designs [11]–[13], butwithout consideration of the nonlinearity of the PA.In this paper, the tradeoff of SE and EE for OFDM systems is analyzed by taking into account the impact ofpractical PAs that is both inefficient and nonlinear . To provide tractable results, we assume that the nonlinearityof the PA is modeled by a soft limiter. To capture the PA inefficiency, we propose a nonlinear transmit power PSfrag replacements P PA dBm P m a x o u t d B m PAs from data sheets20% drain efficiency line30% drain efficiency line100% drain efficiency line P out P in ( P maxout , P maxin )1 dB saturation compression point soft limiter modelRapp modelperfectly linear model (a) PSfrag replacements P PA dBm P maxout dBm PAs from data sheets20% drain efficiency line30% drain efficiency line100% drain efficiency line P o u t P in ( P maxout , P maxin ) saturation compression point soft limiter modelRapp modelperfectly linear model (b)Fig. 1. Two fundamental characteristics of a practical PA. (a) Efficiency: maximum output power P maxout (at the linear region) versusconsumed power P PA . (b) Nonlinearity: PA output power P out versus PA input power P in . model depending on the PA types. We further provide theoretical results to achieve maximum SE and maximumEE from our analysis, and verify the theoretical results through simulations using real-life device parameters.Consequently, it is shown that the practical SE-EE tradeoff increases before a turning point and decreases rapidlyafter the turning point. In other words, the PA can support a narrow SE-EE tradeoff with only a limited rangeof SE. In cellular communications, however, a wide range of SE-EE tradeoff is desired because the BSs needhigh data rates intermittently, yet need to save energy whenever possible to save operation costs. To achieve awide Pareto-optimal SE-EE tradeoff region, we propose a PA switching (PAS) technique, in which one or morePAs are switched on at any time to maximize the EE while satisfying the required SE, resulting in a high EEover a wide SE range. For example, with SE reduction for low SE demand, the PAS between a low powerPA (
25 W maximum power) and a high power PA (
100 W maximum power) can improve EE by , while aingle high power PA transmitter improves EE by only . Specifically, our key contributions are summarizedas follows: • Practical SE : We obtain a closed-form expression of SE with consideration of PA nonlinearity, and showthat its approximation is a concave function with a unique maximum with respect to the input power of thePA. • Practical EE : We establish a PA-dependent nonlinear power consumption model from various recent studieson empirical power measurement and parameters for cellular and wireless local area networks. We show thatthe EE is a piecewise quasi-concave function with a unique maximum point if the PA is perfectly linear. • PAS : We observe that the practical SE-EE tradeoff decreases rapidly after a turning point, i.e., the limitedSE-EE tradeoff for dynamic traffic conditions. To circumvent this, we propose a PAS technique. Numericalresults show that the SE-EE tradeoff improvement is significant even though practical losses are considered,such as switch insertion loss and switching time overhead.II. P
ROLOGUE
This paper attempts to quantify analytically and numerically the degradation of both SE and EE caused bythe practical nonlinearities and energy consumption of the PA. Specifically, we define SE, in b/s/Hz , as theamount of bits that are reliably decoded per channel use (i.e., per unit time and per unit bandwidth). We defineEE, in b/J , as the total amount of reliably decoded bits normalized by the energy. Thus, SE and EE are givenrespectively by [14], [15] SE = I ( f X ; e Y ) N (1a) EE = T Ω SE T P c = Ω SE P c . (1b)Here, I ( f X ; e Y ) is the mutual information in b/s/Hz given the length- N transmitted and received vectors f X and e Y , representing an achievable sum rate over N channel uses [16]; Ω is the total bandwidth used; T is the totaltime used; and P c is the total power consumption including the PA power consumption P PA .For illustration, consider an ideal system without system overhead power consumption, i.e., P c = P PA .Furthermore, consider the ideal PA which is always perfectly efficient (dotted line in Fig. 1(a)), i.e,. P PA = P out ,and always perfectly linear (dotted line in Fig. 1(b)). Using Gaussian signalling, which is optimal for the ideal PA,we get SE = log (1 + P out /σ ) , where σ is the noise power [16]. For the ideal system with ideal PA, therefore,asymptotically as P c increases, SE increases proportionally with log ( P c ) and hence EE decreases proportionallySfrag replacements e x e y I D FT F add C P → P / S → D A C A DC → S / P → r e m o v e C P D FT F H FF H x PA L PA ( · ) { h , · · · , h L − } w z y Fig. 2. An OFDM system with a nonlinear memoryless PA represented by function L PA ( · ) , assuming perfect synchronization. with log ( P c ) /P c . In other words, the SE-EE tradeoff region is a decreasing convex as observed in [14], [15]when the system and PA are ideal. In contrast, for a practical system, asymptotically as P c increases, the outputsaturates and so SE saturates to some upper limit, hence EE decreases proportionally with /P c ; moreover,significant overhead power exists, because P c > P PA > P out . To account for the degradation of both SE and EEin practice, it is essential to consider practical overhead in system power consumption P c and have a sufficientlyaccurate, yet tractable, model for the PA (i) on its energy consumption to specify the relationship of P PA and P out , and (ii) on its nonlinearity behavior. In the sequel, we shall address both issues when we determine the SEand EE in (1). III. S YSTEM M ODEL
Consider the OFDM system with a nonlinear memoryless PA shown in Fig. 2. Without loss of generality(w.l.o.g.), we consider the transmission of one OFDM symbol which consists of N complex-valued data symbols,denoted by the data vector e x = [ e x , · · · , e x N − ] T . The data symbol e x n is sent on the n th orthogonal subcarrier.These N subcarriers occupy a total frequency band of Ω Hz . The data symbols are assumed to be identical andindependently distributed (i.i.d.) subject to the power constraint E[ | e x | ] ≤ P in ; here and subsequently, we dropthe subcarrier or time index if there is no dependence on it or when there is no ambiguity. We transform e x to the time domain signal vector x = [ x , · · · , x N − ] T according to x = F e x , where F is an N -by- N unitaryinverse discrete Fourier transform (IDFT) matrix. Thus, E[ | x | ] ≤ P in . Then a cyclic prefix (CP) of length N CP isadded to x and passed to a parallel-to-serial (P/S) converter, followed by a digital-to-analogue converter (DAC).We assume the DAC, the analogue-to-digital converter (ADC) and subsequent processing (such as timing andfrequency synchronization) are ideal such that, w.l.o.g., we use x t to represent the output of the DAC at discreteime index t . We rewrite x t = a t e jθ t where a t , | x t | is the amplitude and θ t is the phase of x t where ≤ θ t < π .Next, the DAC output x t is amplified through a memoryless PA described by a nonlinear function L PA ( · ) togive the output w t = L PA ( x t ) , denoted collectively by the vector w = [ w , · · · , w N + N CP − ] T . Under the Rappand soft limiter models illustrated in Fig. 1(b), we can write w t = b t e jθ t where b t , | w t | while the phase remainsthe same as that of x t . Specifically, the Rapp model describes the amplitude distortion according to L PA ( a t ) = √ ga t (cid:18) √ ga t b sat (cid:19) p ! − p , where √ g ≥ is a parameter interpreted as the desired linear gain; b sat is the saturation amplitude when a t → ∞ ;and p controls the smoothness of the transition from the linear region to the saturation region. Thus, the gain isnonlinear for all input signals. For the soft limiter model, the amplitude distortion follows L PA ( a t ) = √ ga t , if a t < a max b max , if a t ≥ a max , where a max , p P maxin and b max , p P maxout . Thus, the output of soft limiter is clipped to a constant b max if theinput signal exceeds a threshold value a max , and experiences a linear scaling of its input with gain √ g otherwise.Finally, the PA output is transmitted through an L -tap multipath channel { h , h , · · · , h L − } . Assuming L ≤ N CP and perfect timing synchronization, the CP is removed and the received signal is given by y t = h t ⊗ w t + z t , r t e jφ , (2)for t = 0 , · · · , N − (for convenience, we shift the time indices to start from ). Here, ⊗ is the circularconvolution operator, z t ∼ CN (0 , σ z ) is an additive white Gaussian noise (AWGN), and r t and φ t represent theamplitude and phase of y t . The received signal vector y = [ y , · · · , y N − ] T is transformed via a DFT (i.e., aHermitian transpose of F ) to give the frequency domain signal vector e y = [ e y , · · · , e y N − ] T = F H y .In practice, the time-domain signal after IDFT typically produces a Gaussian-like signal with a high PAPR. Itis well known that the nonlinearity of a PA can thus result in significant degradation of the achievable rate ofthe signal [6]. To analytically model the high PAPR and the nonlinearities, we make the following assumptions: A : We assume that the data symbols are i.i.d. with complex normal distribution with zero mean and P in variance,denoted as e x ∼ CN (0 , P in ) . Hence the time-domain signals are also i.i.d. with distribution x ∼ CN (0 , P in ) .The time domain signals have very high PAPR and thus they are representatives of the scenario when ahigh-order modulation is used or when N is large. : For tractability of subsequent analysis, we employ the soft limiter model for the PA. A good approximationof the maximum power output P maxout is given by the one-dB input compression output, where the outputpower drops below the desired power output if the gain is linear as illustrated in Fig. 1(b). Thus, themaximum power input is P maxin = P maxout /g . We shall use data sheets of commercially available products(e.g., Table II in Appendix A) to extract suitable parameters for g and P maxin to obtain numerical results. Thesoft limiter model is analytically tractable, and it can capture the clipping effect in the high power regionas the Rapp model (the Rapp model approaches the soft limiter model as p increases). Nonlinearity in lowpower region of the soft limiter can be assumed to be mitigated by applying linearization techniques, suchas feedforward, feedback, and predistortion (refer to the references in [2]), which is the same as the Rappmodel.The assumption A is independent to the assumption of the soft limiter model in A , because the probabilitydensity functions (pdfs) of x and e x do not change regardless of the PA model. In this paper, we focus on point-to-point communications. The spectral regrowth arisen from the nonlinearity of the PA, which increases the adjacentchannel interferences to neighboring bands, is not considered explicitly.Typically, an IBO is performed to mitigate the degradation resulting from PA nonlinearities, by reducing theinput signal power P in such that it is much less than P maxin . To reflect this, we write P in = ξP maxin , where ξ ≥ isa power loading factor and is related to the IBO as IBO ,
10 log ( ξ − ) dB . By varying ξ , we can then performIBO to tradeoff between EE and SE.Based on assumptions A and A , we shall obtain tractable results which offer insights on how the PA affectsthe SE and EE in Sections IV and V, respectively. Then we study how this leads to the analysis of a new PAarchitecture in Section VI, which improves SE and EE tradeoff.IV. S PECTRAL E FFICIENCY
In this section, we determine the SE in (1a) under assumptions A and A . To this end, we obtain the mutualinformation I ( f X ; e Y ) for flat fading channels in Section IV-A, and for multipath channels in Section IV-B.For simplicity, we ignore the throughput loss due to the addition of the CP. We fix the following PA-relatedparameters: the power loading factor ξ , the gain g in the linearity region and the maximum power output P maxout .Thus the maximum input power P maxin = g − P maxout is also fixed; for convenience, let γ , P maxout /σ z > be themaximum power output normalized by the noise variance σ z .We use upper case letters to represent random variables, such as X , W , and Y , and lower case letters toepresent their realizations, such as x , w , and y . The pdf of random variable X is denoted by f X ( · ) . Recall thatthe signals are written in terms of their amplitudes and phases as x = ae jθ , w = be jθ , and y = re jφ . A. Mutual Information in Flat Fading Channel
Consider the flat fading channel where the number of multipath is L = 1 . Let h = 1 , w.l.o.g., as the actualchannel attenuation and any fixed energy losses incurred can be reflected by adjusting the noise variance suchthat the signal-to-noise ratio (SNR) is maintained. Given input X = Ae jθ , the channel model at time index t is Y t = W t + Z t , where W t = L PA ( A t ) e θ t . (3)The SE, which is given by the achievable rate averaged over N transmissions, is I ( f X ; e Y ) /N ( a ) = I ( X ; Y ) /N ( b ) = N − X t =0 I ( X t ; Y t ) /N ( c ) = I ( X ; Y ) ( d ) = H ( Y ) − log πeσ z [b/s] . (4)Here, (a) follows from the facts that the frequency-domain signals (transmitted and received vectors f X and e Y )and time-domain signals (transmitted and received vectors X and Y ) are related by a unitary transform, whichdoes not change the mutual information; (b) follows from the independence of the signals in the time domain(because of the memoryless PA and the i.i.d. transmitted signals and noise); (c) follows from the fact that themutual information is identical over time, and so the time index can be dropped; and (d) follows from the factsthat I ( X ; Y ) = H ( Y ) − H ( Y | X ) , the conditional entropy H ( Y | X ) = H ( N ) , and H ( N ) is the differentialentropy of a complex Gaussian random variable with variance σ z derived by log πeσ z . The entropy of Y in (4)is given by [16] H ( Y ) = − Z y f Y ( y ) log f Y ( y ) dy. (5)Nonlinear distortion at the transmitter makes it difficult to derive f Y ( y ) in (5) directly. To tackle this problem,we define a binary random variable S that denotes whether clipping at the PA occurs, i.e., S = 0 if A ≤ a max and S = 1 otherwise, and rewrite the pdf of y as f Y ( y ) = P i =0 , f Y ( y, S = i ) . Since X = Ae jθ ∼ CN (0 , P in ) ,he random variable A follows the Rayleigh distribution. Thus, we get the probability of S as Pr( S = 0) = Pr ( A ≤ a max ) = 1 − exp (cid:0) − a P − (cid:1) = 1 − exp (cid:0) − ξ − (cid:1) Pr( S = 1) = 1 − Pr( S = 0) = exp (cid:0) − ξ − (cid:1) . (6)The numerical computation of the entropy (5) is straightforward with a closed-form expression of f Y ( y, S = 0) and f Y ( y, S = 1) , which are derived respectively as follows (see Appendix B): f Y ( y, S = 0) = N ( y ) h − Q (cid:16)p µ ( y ) , √ ρ max (cid:17)i (7a) f Y ( y, S = 1) = N ( y ) " Pr( S = 1) exp (cid:18) − b max y Re σ z (cid:19) I (cid:18) b max | y | σ z (cid:19) (7b)where N ( y ) denotes the pdf of CN (cid:0) , gP in + σ z (cid:1) ; Q ( · , · ) is the Marcum-Q-function [17] with parameters ρ max , gP in + σ z ) gP in √ b max and µ ( y ) , gP in ( gP in + σ z ) σ z | y | ; N ( y ) is the pdf of CN (cid:0) b max , σ z (cid:1) ; y Re is the real partof y ; and I ( · ) is the modified Bessel function of first kind [17]. B. Mutual Information in Multipath Channel
We now consider the general case of an L -tap multipath channel, where ≤ L ≤ N CP . The received signal inthe time domain is given by (2). If the amplification of PA is perfectly linear, then the mutual informationis given equivalently in the frequency domain as I ( X ; Y ) /N = P N − k =0 I ( e X k ; e Y k ) /N = P N − k =0 log (1 + | e H k | gP in /σ z ) /N , where e H k is the frequency domain channel, see e.g., [16]. In our case of interest, however,the nonlinear PA makes the exact analysis of the mutual information intractable, because the PA nonlinearitiesresult in a correlated interference in the frequency domain which is not formulated as a closed-form expression.Instead, we obtain a lower bound for the mutual information (see Appendix C): I ( f X ; e Y ) /N ≥ L − X t =0 I ( X t ; Y t , · · · , Y t + L − | X , · · · , X t − ) /N + N − X t = L I LB t /N (8)where I LB t is the mutual information of flat fading channel (3) with the SNR given by the equivalent channel(C.4). As N → ∞ , the first term approaches zero, while the second term equals approaches I LB t which is in factindependent of t (we drop the index subsequently). Thus, the lower bound in (8) is given asymptotically by I LB for N ≫ L . Note that I LB can be computed from (4) directly. Numerical results (not included) show that thebound is typically tight if the power of the multipath decreases exponentially over the channel delay. . Analytical Results on SE Using (7a) and (7b) into (5), we find H ( Y ) and get I ( X ; Y ) from (4) in flat fading channels. Similarly, from(C.3), we can obtain the mutual information of the signals in multipath channels. Accordingly, we derive the SEin (1a) as a function of ξ as SE ( ξ ) = H ( Y ) − log πeσ z , (9)where note that the entropy H ( Y ) is a function of ξ as the conditional probabilities in (7a) and (7b) are functionsof P in = ξP maxin and b max = p gP in ξ − .If the PA is perfectly linear, i.e., b max → ∞ and thus a max → ∞ , it can be easily checked that f Y ( y, S =0) = N ( y ) from (B.1)–(B.3) and f Y ( y, S = 1) = 0 from (B.5) as P ( S = 0) = 1 and P ( S = 1) = 0 in (6).Thus, H ( Y ) = log πe (cid:0) gP in + σ z (cid:1) and we recover the well-known SE for ideal PA as SE ideal ( ξ ) = log (1 + γξ ) . (10)For tractable analysis, the SE in (9) is approximated under the assumption of low power input signal to PA,i.e., small ξ . If ξ ≪ , we can approximate the joint pdfs in (7a) and (7b) as follows: f Y ( y, S = 0) ≈ N ( y ) f Y ( y, S = 1) ≈ N ( y ) Pr( S = 1) . (11)The approximations comes from the observation that ξ ≪ implies that the received signal y is also aroundzero with high probability, i.e., f Y ( y ) is significant only for | y | ≪ . Thus, µ ( y ) ≪ , Q ( p µ ( y ) , √ ρ max ) ≈ in (7a), exp( · ) ≈ , and I ( · ) ≈ in (7b), which leads to (11). Thus, we approximate (5) as e H ( Y ) = − Z y ( N ( y ) + N ( y ) Pr( S = 1)) log ( N ( y ) + N ( y ) Pr( S = 1)) dy ≈ − Z y N ( y ) log N ( y ) dy − Z y N ( y ) Pr( S = 1) log N ( y ) Pr( S = 1) dy (12a) = log πe (1 + γξ ) − e − ξ log e − ξ + e − ξ log πeσ z (12b)where the approximation in (12a) follows from the further observation that the domains of N ( y ) and N ( y ) are approximately disjoint as the gap of their mean values is much larger than their variances, i.e., b max ≫{ gP in + σ z , σ z } . For example, see Fig. 3 where N ( y ) and N ( y ) are shown for φ = { , π } . We note thattypically this holds if ξ ≪ , when IBO is used. We thus call the resulting SE as SE IBO which is obtained bysubstituting (12b) to (9) as SE IBO ( ξ ) = e H ( Y ) − log πeσ z . (13) PSfrag replacements r pd f N ( y ) for φ = 0 and φ = π N ( y ) for φ = 0 and φ = π − b max b max Fig. 3. The pdfs of N ( y ) and N ( y ) for φ = 0 and φ = π , where ξ = 0 . . The details for the simulation environment are given inSection IV-D. The following theorems for the approximated SE, SE IBO ( ξ ) , allow us to obtain insights on the structured propertiesof the actual SE, SE ( ξ ) , at least for ξ ≪ . The proofs are given in Appendix D. Theorem 1:
The approximated SE, SE IBO ( ξ ) , is a concave function over max (cid:16) , − πσ z (cid:17) < ξ ≤ .Using Theorem 1 , we obtain the SE-aware optimal power loading factor ξ ⋆ SE . Theorem 2:
The SE-aware optimal power loading factor ξ ⋆ SE which maximizes SE IBO ( ξ ) is obtained by thesolution of the following equality: γ γξ = e − ξ − ξ − (cid:0) − ξ − + 1 − ln πeσ z (cid:1) . (14) Proposition 3:
A closed form approximation of ξ ⋆ SE is given by ξ ⋆ SE ≈ e ξ ⋆ SE , − W (cid:16) πeσ z ) (cid:17) , (15)here W ( · ) denotes the Lambert W function that satisfies q = W ( q ) e W ( q ) [18].Interestingly, the approximated e ξ ⋆ SE depends only on σ z ; intuitively, this is because we assume ξ ≪ . Thismakes e ξ ⋆ SE independent of other PA parameters. The typical values of IBO are between and
12 dB forlarge (e.g., macro) and small (e.g., femto) cell base stations, respectively, which include an additional margin forfading channels [19], [20]. The numerical results in the subsequent subsection show that the analytical resultswith ξ ≪ are accurate for ξ ≤ . , i.e., IBO ≥ . D. Numerical Results on SE
To verify the analytical results on SE, we evaluate SE ( ξ ) with respect to the power loading factor ξ . Thebandwidth is set to
10 MHz . For simplicity, Rayleigh fading channel is assumed with zero mean and unit variance.A more realistic multipath channels as given in [21] may also be used for verifying the results obtained in SectionIV-B. The channel attenuation is modeled as follows [22]: G −
128 + 10 log ( d − α ) dB where G includes thetransceiver feeder loss and antenna gains; and d − α is the path loss where d is the distance in kilometers between atransmitter and a receiver and α is a path loss exponent. In simulations, we set G = 5 dB , α = 3 . , d = 200 m ,and σ z = −
174 dBm / Hz and use a PA SM2122-44L ( P maxout = 44 dBm = 25 W and g = 55 dB ) in Table II.Fig. 4 shows the numerical evaluation of SE. As expected, SE ideal ( ξ ) in (10) achieved by a perfectly linearPA is an increasing concave (log-shape) function, while the practical SE SE ( ξ ) in (9) is a concave function witha unique maximum when ξ ≤ . The approximated SE SE IBO in (12b) matches well with practical SE SE ( ξ ) if ξ is low. The optimal e ξ ⋆ SE in (15) found from SE IBO ( ξ ) yields almost the highest SE SE ( e ξ ⋆ SE ) as marked by ‘ ◦ ’( Theorem 2 ). This illustrates the tightness of the approximation made to obtain SE IBO ( ξ ) , at least for obtainingthe optimal ξ ⋆ SE . On the other hand, the discrepancy between the practical SE SE ( ξ ) and the approximated SEs, SE ideal ( ξ ) and SE IBO ( ξ ) , increases as ξ (i.e., the PA input or output power) increases.V. E NERGY E FFICIENCY
To derive the EE, we first model the power consumption at the transmitter. As shown in Fig. 5, powerconsumption and losses at the transmitter can occur in five modules: a direct current (DC) power supply (PS)module, a base band (BB) module, a radio frequency (RF) module, a PA module, and an active cooler and batterybackup (CB) module. Power consumption at BB, RF, PA, and CB modules are denoted by P BB , P RF , P PA , and For q < , W( q ) can take multiple values. We assume W ( · ) ≤ − which is known as the lower branch of W ( · ) , so that e ξ ⋆ SE ≤ .This gives a unique value for W ( · ) . PSfrag replacements
Power loading factor, ξ S E ( ξ ) , b / s / H z SE ideal ( ξ ) SE ( ξ ) SE IBO ( ξ ) SE ( e ξ ⋆ SE ) : ideal SE in (10) with an ideal PA : practical SE in (9) with a practical PA : approximated SE in (13) : optimal SE from (15) Fig. 4. Spectral efficiency evaluation with P maxout = 25 W and g = 55 dB . P CB , respectively, see details in [19], [23]–[25]. After introducing two known power consumption models, wewill introduce a new model by taking the PA types (efficiency) and power loading factor ξ into consideration,and subsequently derive the corresponding EE. A. Existing Power Consumption Models
One empirical linear model given in many recent studies, such as [11], [19], and [23], is P c ( ξ ′ ) = P fix + cξ ′ P maxout , (16)where ξ ′ is a frequency loading factor in OFDMA systems ( < ξ ′ ≤ ); P fix is a power consumption whichis independent of the PA output signal power, i.e., ξ ′ P maxout ; and c is a scaling coefficient for the power loadingdependency. If ξ ′ = 0 , i.e., at the idle mode, P c ( ξ ′ ) = P idle . In Table I, we summarize the parameters P fix , P maxout , P idle , and c for various types of networks. The power coefficient in [23] is modeled as c = η + P BB P maxout + P RF P maxout . Theparameters depend on the various practical factors, such as the transmitter configuration, the network structure,and the semiconductor technologies employed. For further information, refer to [20].Sfrag replacements P BB P RF P PA P CB P in P out loss loss loss loss DCPowerSupply(PS)Module Active Cooler and Battery Back up (CB) Module (if present)Base Band(BB) Module RadioFrequency(RF) Module(except PA) PowerAmplifier (PA)Module
Fig. 5. Power consumption block diagram including DC power supply (PS), base band (BB), radio frequency (RF), power amplifier (PA),and active cooler and battery back up (CB) modules.
TABLE IP
OWER M ODEL P ARAMETERS FROM [11] † , ( [19], [20]) ‡ , [23] § .BS type P maxout W P fix W P idle W c Macro 20 ‡ , § ‡ † , 405 § ‡ ‡ , 21.45 † , 17.8 § RRH ⋄ ‡ ‡ ‡ ‡ Micro 2 § , 6.3 ‡ ‡ , 71.5 † , 106 § ‡ ‡ , 7.84 † , 108.3 § Pico 0.13 ‡ ‡ ‡ ‡ Femto 0.05 ‡ ‡ ‡ ‡⋄ remote radio head or remote radio unit (RRU) Since the model in (16) is obtained from empirical measurements, it gives a reasonable indication of powerconsumption; however, no accurate indication is given for the specific PA type used. Furthermore, as shown in[19], there is a nonlinear relationship between the loading factor and the actual power consumption, especiallyin high power transmission, e.g., at the macro BS. To address these limitations, a PA-dependent model is givenby [24], [25] P c = (1 + C PS )(1 + C CB )( P BB + P RF + P PA ) (17)where C PS is a PS coefficient (typically . ≤ C PS ≤ . ) and C CB is an CB coefficient (typically less than . ). We can modify the model in (17) according to the PA types and the power loading factor ξ because the PApower consumption is modeled explicitly and separately from the other power consumption factors. Note thatthe frequency loading factor ξ ′ in (16) can be interpreted as the power loading factor ξ in the time domain. B. Proposed PA-dependant Nonlinear Power Consumption Model
Though the PA power consumption P PA depends on many factors including the specific hardware implemen-tation, DC bias condition, load characteristics, operating frequency and PA output power, the component thatconsumes the majority of the power is given by the DC power fed to the PA [26]. Since the drain efficiency η depends on the PA types, we can express P PA for different types of PA as a function of ξ [27]. For the ℓ -wayDoherty PA, where ℓ is a fixed positive integer that depends on the implementation, the PA power consumptionis expressed as P PA ( ξ ) = 4 P maxout ℓπ × √ ξ, < ξ ≤ ℓ ( ℓ + 1) √ ξ − , ℓ < ξ ≤ . (18)Henceforth, we assume the use of the ℓ -way Doherty PA which has widespread use [28], [29]. The DohertyPA includes the special case of the class B PA with ℓ = 1 . The PA modeled in (18) can be considered to be anone-stage PA, which is relevant typically for low power transmission. We can obtain P PA ( ξ ) similarly for otherPA types, e.g., multi-stage PA combining class-A and Doherty, for high power transmission. It is straightforwardto generalize to a multi-stage PA, in which the PA efficiency will change and (18) will be slightly modifiedaccordingly with more levels. However, the EE analysis in the paper will remain without changes in the lowpower region.Substituting (18) to (17), we get a PA-dependant nonlinear power consumption model as P c ( ξ ) = P + c (cid:16) c + c p ξ (cid:17) P maxout (19)for < ξ ≤ , where P = (1 + C PS )(1 + C CB )( P BB + P RF ) , and ( c , c ) = 4 ℓπ × (0 , , < ξ ≤ ℓ , (20a) ( − , ℓ + 1) , ℓ < ξ ≤ . (20b)Comparing (19) with the model in (16), we also see that the new model in (19) reflects the PAs’ characteristics.However, since P RF is actually related to ξ , there are degrees of freedom to determine P and c . In this work,we set P = P fix and c = π c , so that (19) matches to (16) when ξ = 1 . In other words, this alignment allowsus to match the power consumption in (19) with that of (16) at the critical points of ξ , namely, ξ = 0 (idle), PSfrag replacements
Power loading factor, ξ P o w e r c on s u m p t i on , P c ( ξ ) , W idle mode in (16)linear model in (16) nonlinear model with class A PA in (19) nonlinear model with class B PA in (19)nonlinear model with -way Doherty PA in (19)model with ideal PA in (21) Fig. 6. Power consumption of microcell BS with P fix = 130 W and c = 4 . . ξ = ℓ , and ξ = 1 as shown in Fig. 6. In Fig. 6, we use a macrocell setup in Table I where P fix = 130 W and c = 4 . . Following the same procedure of modeling in this subsection, any PA can be reflected in (19).If a PA is ideal, namely, the PA is perfectly linear and efficient , then P out = gP in and P PA = P out − P in ,respectively. Thus, P PA = (1 − g − ) ξP maxout . From (17), we can model the PA power consumption with the idealPA as follows ( < ξ ≤ ): P idealc ( ξ ) = P fix + c (cid:0) − g − (cid:1) ξP maxout . (21)From Fig. 6, P idealc ( ξ ) gives a lower bound for the power consumption of the other models, as expected. Power-added efficiency (PAE) and overall efficiency are defined as P out − P in P PA and P out P in + P PA , respectively [26]. . Analytical Results on EE Using the practical SE in (9) and PA-dependent nonlinear power consumption P c ( ξ ) in (19), we obtain the practical EE given by (1b) as EE ( ξ ) = Ω SE ( ξ ) P c ( ξ ) . (22)An upper bound of EE ( ξ ) is obtained assuming an ideal PA with perfect linearity and efficiency as EE ideal ( ξ ) , Ω SE ideal ( ξ ) P idealc ( ξ ) . (23)However, the bound EE ideal ( ξ ) is not tight enough. Furthermore it does not reflect the PA types. Thus, we removethe perfect efficiency assumption from (23) and get a PA-dependant tighter bound as follows: EE linear ( ξ ) , Ω SE ideal ( ξ ) P c ( ξ ) (24)where we retain the assumption of a perfectly linear PA. Using EE linear ( ξ ) , we can obtain the following theorems,which allow us to obtain insights on the structured properties of the practical EE, EE ( ξ ) , at least for ξ ≪ . Theproofs are given in Appendix D. Theorem 4: EE linear ( ξ ) is a piecewise quasi-concave function over ξ ≥ ζ , (cid:16) v + √ v (cid:17) /γ , where v = P maxout c c / ( P + P maxout c c ) . Specifically, EE linear ( ξ ) is quasi-concave over ζ ≤ ξ ≤ /ℓ and also over /ℓ < ξ ≤ .We denote ξ ⋆ EE as the optimal power loading factor that maximizes EE linear ( ξ ) , which in general depends onthe PA parameters. Typically, ζ ≈ as γ , P maxout /σ z is large, see e.g., the numerical results in Section IV-D.Assuming ξ ⋆ EE ≥ ζ , Theorem 5 states the solution for ξ ⋆ EE . Theorem 5:
Assuming ξ ⋆ EE ≥ ζ , ξ ⋆ EE equals either [ ξ ⋆ ] /ℓ ζ or [ ξ ⋆ ] /ℓ , , where ξ ⋆ and ξ ⋆ are the solutions of ∂ EE linear ( ξ ) ∂ξ = 0 in (D.1) with ( c , c ) defined as (20a) and (20b), respectively. Here, notation [ x ] ba = a if x < a , [ x ] ba = b if x > b , and [ x ] ba = x otherwise.Assuming ξ ⋆ EE ≥ ζ , Theorem 5 shows that there are at most two candidates for ξ ⋆ EE . Hence, ξ ⋆ EE can be obtainedeasily by checking which candidate maximizes EE linear ( ξ ) . Moreover, Proposition 6 shows that an approximationof ξ ⋆ EE can be obtained in closed form. For the special case of class B PA (i.e., ℓ = 1 ), it follows from Theorem 5 that the optimal solution is given exactly by ξ ⋆ EE = ξ ⋆ , assuming ξ ⋆ EE ≥ ζ . Proposition 6:
A closed form approximation of ξ ⋆ EE , denoted by e ξ ⋆ EE , equals either [ e ξ ⋆ ] /ℓ ζ or [ e ξ ⋆ ] /ℓ , , where e ξ ⋆i ( i ∈ { , } ) approximates ξ ⋆i and is given by ξ ⋆i ≈ e ξ ⋆i = 1 γ exp (cid:18) W (cid:18) √ γev (cid:19)(cid:19) . (25) PSfrag replacements
Power loading factor, ξ EE ( ξ ) , M b / J (cid:3) EE linear ( e ξ ⋆ EE ) ; × EE ( e ξ ⋆ EE ) ; e ξ ⋆ EE in (25) EE ideal ( ξ ) in (23) EE linear ( ξ ) in (24) EE ( ξ ) in (22): class A PA EE ( ξ ) in (22): class B PA EE ( ξ ) in (22): -way Doherty PA Fig. 7. Energy efficiency evaluation with P fix = 130 W and c = 4 . . In (25), v is defined in Theorem 4, and W ( · ) > as √ γev > , so that W ( · ) is unique.Numerical results in next subsection show the tightness of the EE bound and that e ξ ⋆ EE is a near maximizer ofthe practical EE, EE ( ξ ) . D. Numerical Results on EE
To verify the analysis on EE, we evaluate the EE numerically. For power consumption parameters, the macrocellsetup in Section V-B is employed. Other parameters are the same as environment given in Section IV-D.Fig. 7 shows the EE for class B and -way Doherty PAs. Though the PA specifications, such as the maximumoutput power and gain, are identical, each of them has different efficiency resulting in different PA parameters in(20). From Fig. 7, we observe that the EE functions are concave ( Theorem 4 ), and that the Doherty PA achievesthe closest EE to the ideal PA’s. The EEs, EE ( e ξ ⋆ EE ) and EE linear ( e ξ ⋆ EE ) , are illustrated by ‘ × ’ and ‘ (cid:3) ,’ respectively.As shown in Theorem 5 , e ξ ⋆ EE yields the maximum EE linear ( e ξ ⋆ EE ) and it is almost identical to EE ( e ξ ⋆ EE ) (which areoverlapped in the figure). This is because the practical EE is maximized in the linear region, and the practical PSfrag replacements SE ( ξ ) , b/s/Hz EE ( ξ ) , M b / J SE-EE tradeoff with PA low SE-EE tradeoff with PA high ◦× : ( SE ( e ξ ⋆ SE ) , EE ( e ξ ⋆ SE ) ) from (15): ( SE ( e ξ ⋆ EE ) , EE ( e ξ ⋆ EE ) ) from (25) Fig. 9 Fig. 8. SE-EE tradeoff with -way Doherty PAs. PA low is a low power PA with P maxout = 25 W and g = 55 dB , and PA high is a highpower PA with P maxout = 100 W and g = 50 dB . SE is also maximized in linear region as shown numerically in the previous section. From the results, we cansurmise that the optimal e ξ ⋆ EE in (25) is a good approximation of the maximizer of EE ( ξ ) .VI. SE-EE T RADEOFF AND
PA S
WITCHING S TRATEGY
To obtain the practical SE-EE tradeoff, Fig. 8 is regenerated from the results of SE in Fig. 4 and EE in Fig. .In addition to the SE and EE of PA SM2122-44L in subsection IV-D, we include the results obtained from a PASM1720-50 ( P maxout = 50 dBm = 100 W and g = 50 dB ) in Table II. The former and the latter PAs are denotedby PA low and PA high , respectively. We use a -way Doherty PA for all results.In contrast to the SE-EE tradeoff for an ideal PA which is a decreasing convex function, the EE in the practicalSE-EE tradeoff drops rapidly when the SE exceeds beyond a threshold that corresponds to the maximum EE. Theclosed-form analysis of the SE-EE tradeoff appears intractable. Instead, we focus on the analysis of the tradeoffbased on the approximated SE and EE defined (13) and (24), respectively. Proposition 7:
The Pareto-optimality of the approximated SE-EE tradeoff is characterized as follows:) For min { e ξ ⋆ EE , e ξ ⋆ SE } ≤ ξ ≤ max { e ξ ⋆ EE , e ξ ⋆ SE } , the corresponding approximated SE-EE tradeoff is Pareto-optimal: to increase the approximated SE, the approximated EE must decrease, and vice versa.ii) For ξ < min { e ξ ⋆ EE , e ξ ⋆ SE } , both the approximated SE and EE increase as ξ increases.iii) For ξ > max { e ξ ⋆ EE , e ξ ⋆ SE } , both the approximated SE and EE decrease as ξ increases.From Proposition 7 , it is sufficient to consider only the region in i), because the remaining regions do not leadto the approximated Pareto-optimal SE-EE tradeoff. In Fig. 8, the approximated Pareto-optimal SE-EE tradeoffis narrow, which lies between the maximum SE and the maximum EE as indicated by ‘ ◦ ’ and ‘ × ,’ respectively.In cellular communications, however, a wide range of SE-EE tradeoff such as that illustrated by the dotted boxin Fig. 8 is desired. This motivates us to use multiple PAs, where one or more PAs are switched on at any time.We call this technique PA switching (PAS) . Although PAS incurs a switch insertion loss of G S and an overheadof switching time ǫ which decrease the SE and EE, we may obtain a better tradeoff of SE-EE from the degreeof freedom of choosing different PAs.For simplicity of description, we consider two PAs, PA- and PA- ; subsequent results are readily extendedto multiple PAs. Let the SE and EE of PA- i , i ∈ { , } , including the switch insertion loss G S , be SE ′ i ( ξ ) and EE ′ i ( ξ ) = Ω SE ′ i ( ξ ) P i c ( ξ ) , respectively, where P i c ( ξ ) is the total power consumption with PA- i . In the followingsubsections, we apply the PAS technique to two systems, namely, frequency division duplex (FDD) and timedivision duplex (TDD) systems, and derive their SEs and EEs. A. PA Switching for FDD Systems
Consider K FDD frames each with length of T . For PAS, we assume PA- is used for the first k frames, thenPA- is switched to PA- which consumes ǫ seconds, and finally PA- is used for the remaining K − k frames.Defining the time sharing factor as κ , kK , ≤ κ ≤ , the achievable SE and EE from PAS can be derived asfollows: SE FDDs ( ξ, κ ) = kT SE ′ ( ξ ) + ( K − k ) T SE ′ ( ξ ) KT + ǫ = KTKT + ǫ (cid:0) κ SE ′ ( ξ ) + (1 − κ ) SE ′ ( ξ ) (cid:1) (26)and EE FDDs ( ξ, κ ) = KT Ω SE FDDs ( ξ, κ ) kT P ( ξ ) + ( K − k ) T P ( ξ )= KTKT + ǫ EE ′ ( ξ ) EE ′ ( ξ ) (cid:0) κ SE ′ ( ξ ) + (1 − κ ) SE ′ ( ξ ) (cid:1) κ SE ′ ( ξ ) EE ′ ( ξ ) + (1 − κ ) SE ′ ( ξ ) EE ′ ( ξ ) ! , (27) PSfrag replacements G S = 0 dB , ǫ = 0 µ s , ideal G S = 1 dB , ǫ = 0 µ s , TDD G S = 1 dB , ǫ = 10 µ s , FDD G S = 1 dB , ǫ = 1 ms , FDD +210% EE − SE SE ( ξ ) , b/s/Hz EE ( ξ ) , M b / J ADBC E
SE-EE tradeoff with PA high Fig. 9. SE-EE tradeoff with switching -way Doherty PAs, PA low and PA high , when ǫ = 10 µ s , L S = 1 dB , T = 10 ms , and K = 20 . where ǫ = 0 if κ = 0 or if κ = 1 (i.e., no switching), and ǫ > otherwise. Here, we ignore the switch powerconsumption as it is relatively negligible compared to P ic ( ξ ) . B. PA Switching for TDD Systems
In TDD systems, the downlink (DL) and uplink (UL) frames are transmitted alternately from BS to UE andfrom UE to BS. Here, we assume that ǫ is less than UL frame length which is typically true. For example, oneLTE frame consumes a time period of
10 ms [22], while the switching time is much less than (refer tothe PA turn-on time in Table II which consumes most of the switching time). We therefore can switch the PAsbetween consecutive DL frames while receiving UL frame, without switching time overhead. The correspondingSE and EE can be readily obtained from (26) and (27) by setting ǫ to be zero. Note that the switching insertionloss is still incurred. . Numerical Results and Discussion on PAS The PAS is useful for adaptive systems where the traffic and channel conditions change dynamically. Fig. 9shows the SE-EE tradeoff with PAS between PA low and PA high . For comparison, we include the results of asingle PA PA high and an ideal switching, namely, G S = 0 dB and ǫ = 0 . For practical switching, the switchinsertion loss G S is set to , and ǫ is set to µ s for TDD frame, while µ s and are used for FDDframes. We consider K = 20 frames with T = 10 ms for each frame length. From Fig. 9, we can verify that anSE-EE tradeoff is substantially improved by PAS. For example, let us consider the TDD system. The EE can beimproved by around ( ) if we reduce SE by ( ) from A to B ( C ), respectively, as marked inFig. 9. In contrast, if a single PA PA high is used instead, the EE is improved by only around ( ) withthe same reduction of SE from A to D ( E ). Next, consider the FDD system. Even with a switching time that is of the frame size, i.e., ǫ = 1 ms , a better SE-EE tradeoff is observed for most of the tradeoff region.To implement the PAS in practice, the network overhead to obtain full channel state information at thetransmitter can be significant, but it can be resolved by limiting the PA numbers with limited feedback information.Other issue is the increased form factor; however, this may not be significant issue in cellular networks wherethe BSs are already large in form factor due to other circuits. Furthermore, even with a small number of PAs, aswe show in our recent work [30], significant performance gain can be achieved. In the near future, advancementof semiconductor technology will help further reduce the related concerns with form factor and hardware cost,making the proposed PAS an even more convincing technology for any type of transmitters.VII. C ONCLUSION
In this paper, we provided a theoretical analysis of the spectral efficiency and energy efficiency (SE-EE) tradeoffof OFDM systems by taking into account the practical non-ideal effects of the power amplifiers (PAs). Optimalpower loading factors of PA are derived to achieve the maximum SE and EE. We identified the problem of anarrow SE-EE tradeoff region due to the nonlinearity and inefficiency of the practical PAs, and proposed a PAswitching that is a useful technique to achieve a wide SE-EE tradeoff. Future studies include the SE-EE analysisof multiuser communication systems, MIMO systems, and a more accurate PA model with a memory effect andnonlinearity at low power regime. A
PPENDIX
ASee Table II.
ABLE IIP
OWER A MPLIFIER C HARACTERISTICS ( ASCENDING ORDER OF P maxout ).(a) PA P maxout (dBm) g (dB) V PA (Volt) C PA (mA) P maxin (dBm) Frequency (GHz) turn-on time ( µ s ) Institution1 MAX2242 5 28.5 3.3 50 10 2.4 – 2.5 1.5 MAXIM2 FMPA2151 7 31 3.3 280 0 2.4 – 2.5 – FAIRCHILD Semiconductor3 FMPA2151 7 33 3.3 375 0 4.9 – 5.9 – FAIRCHILD Semiconductor4 PA1137 8 17 2 20 10 2 – 2.2 – tyco Electronics5 ADL5570 10 29 3.5 100 – 2.3 – 2.4 1.0 Analog Devices6 MAX2242 13 28.5 3.3 90 10 2.4 – 2.5 1.5 MAXIM7 MAX2840 15 22.8 3.3 155 – 5.15 – 5.35 1.5 MAXIM8 BGA6289 15 13 4.1 88 – 1.95 – 2.5 – NXP Semiconductors9 AWT6134 16 23.5 3.4 130 10 1.75 – 1.78 – ANADIGICS10 AWT6138 16 15 3.4 57 10 1.85 – 1.91 – ANADIGICS11 AWT6252 16 24.5 3.4 54 10 1.92 – 1.98 – ANADIGICS12 AWT6252 16 20.5 3.4 54 10 1.92 – 1.98 – ANADIGICS13 RF2192 16 22 3.4 150 10 0.824 – 0.849 40 EF MICRO-DEVICES14 RF2196 16 20 3.4 160 10 1.85 – 1.91 – EF MICRO-DEVICES15 RF3163 16 24 3.4 125 10 0.824 – 0.849 46 EF MICRO-DEVICES16 RF3164 16 28 3.4 130 10 1.85 – 1.91 46 EF MICRO-DEVICES17 RF3165 16 27 3.4 130 10 1.75 – 1.78 46 EF MICRO-DEVICES18 RF6100-1 16 26 3.4 135 10 0.824 – 0.849 46 EF MICRO-DEVICES19 AP172-317 17 33 3.3 140 20 1.8 – 2.5 – SKYWORKS20 SKY65006 17 30 3.3 110 10 2.4 – 2.5 – SKYWORKS21 RMPA5255 18 33 3.3 230 – 4.9 – 5.9 1.0 FAIRCHILD Semiconductor22 MAX2841 18 22.5 3.3 260 – 5.15 – 5.35 1.5 MAXIM23 FMPA2151 19 31 3.3 600 0 2.4 – 2.5 – FAIRCHILD Semiconductor24 FMPA2151 19 33 3.3 600 0 4.9 – 5.9 – FAIRCHILD Semiconductor25 RMPA2458 19 31.5 3.3 103 5 2.4 – 2.5 1.0 FAIRCHILD Semiconductor26 RF5117 21 26 3 200 10 1.8 – 2.8 – EF MICRO-DEVICES27 RF5189 21 25 3 220 10 2.4 – 2.5 – EF MICRO-DEVICES28 SST13LP01 21 34 3.3 340 – 4.9 – 5.8 0.2 Silicon Storage Technology, Inc.29 RF5117 22 26 3 500 10 1.8 – 2.8 – EF MICRO-DEVICES30 RMPA2455 22 30 5 195 10 2.4 – 2.5 1.0 FAIRCHILD Semiconductor31 MAX2242 22 28.5 3.3 300 10 2.4 – 2.5 1.5 MAXIM32 AP172-317 22.5 33 3.3 220 20 1.8 – 2.5 – SKYWORKS33 MAX2247 23 29.5 3 305 5 2.4 – 2.5 1.5 MAXIM34 SST12LP00 23 27 3.3 115 – 2.4 – 2.5 – Silicon Storage Technology, Inc.35 SST12LP14 23 31 3.3 290 – 2.4 – 2.5 0.1 Silicon Storage Technology, Inc.36 SST13LP01 23 28 3.3 260 – 2.4 – 2.485 0.1 Silicon Storage Technology, Inc.37 AP178-321 23.5 19 3.3 186 20 1.8 – 2.5 – SKYWORKS38 MAX2247 24 29.5 3.3 307 5 2.4 – 2.5 1.5 MAXIM39 AP172-317 24 33 3.3 240 20 1.8 – 2.5 – SKYWORKS40 CX65003 24.5 11.5 5 138 15 1.4 – 2.5 – SKYWORKS41 ADL5570 25 29 3.5 440 – 2.3 – 2.4 1.0 Analog Devices42 ADL5571 25 29 3.3 450 – 2.5 – 2.7 1.0 Analog Devices43 RF2163 25 19 3.3 378 15 1.8 – 2.5 – EF MICRO-DEVICES44 MAX2247 25 30.5 4.2 345 5 2.4 – 2.5 1.5 MAXIM45 SST12LP14 25 31 3.3 340 – 2.4 – 2.5 0.1 Silicon Storage Technology, Inc.46 NE552R479A 26 11 3 217 19 2.45 – 2.45 – CEL California Eastern Laboratories47 PA1153 26.4 28.5 15 250 15 1.8 – 2 – tyco Electronics48 PA1133 26.5 29 15 200 15 1.85 – 1.91 – tyco Electronics49 ADL5571 27 27.5 5 620 – 2.5 – 2.7 1.0 Analog Devices50 RF2114 27 36 6.5 300 12 0.001 – 0.6 0.1 EF MICRO-DEVICES51 RF2161 27 30 3 477 6 1.85 – 2 – EF MICRO-DEVICES52 RF2186 27 31 3 668 6 1.85 – 2 – EF MICRO-DEVICES53 RF5117 27 26 5 500 10 1.8 – 2.8 – EF MICRO-DEVICES54 RF5176 27 26 3 476 6 1.85 – 2 – EF MICRO-DEVICES55 AWT6252 27.5 26.5 3.4 423 10 1.92 – 1.98 – ANADIGICS56 AWT6134 28 26 3.4 475 10 1.75 – 1.78 – ANADIGICS57 AWT6134 28 25 3.4 462 10 1.75 – 1.78 – ANADIGICS58 AWT6138 28 26 3.4 487 10 1.85 – 1.91 – ANADIGICS59 RF3163 28 28.5 3.4 455 10 0.824 – 0.849 46 EF MICRO-DEVICES60 RF3164 28 28 3.4 460 10 1.85 – 1.91 46 EF MICRO-DEVICES61 RF3165 28 28 3.4 460 10 1.75 – 1.78 46 EF MICRO-DEVICES62 RF6100-1 28 29 3.4 465 10 0.824 – 0.849 46 EF MICRO-DEVICES63 RF2132 28.5 29 4.8 327 12 0.824 – 0. 849 0.1 EF MICRO-DEVICES64 RF2146 28.5 18.5 4.8 393 12 1.5 – 2 0.55 EF MICRO-DEVICES65 RF6100-4 28.5 28 3.4 535 10 1.85 – 1.91 46 EF MICRO-DEVICES66 CX65105 28.5 25 5 470 7 1.7 – 2.2 – SKYWORKS67 SKY65162-70LF 28.8 20 5 306 – 0.869 – 0.96 – SKYWORKS68 RF2192 29 30 3 715 10 0.824 – 0.849 40 EF MICRO-DEVICES69 RF2196 29 27 3 755 10 1.85 – 1.91 – EF MICRO-DEVICES70 MGA-43228 29.2 38.5 5 500 – 2.3 – 2.5 – Avago Technologies71 MGA-43328 29.3 37.3 5 470 – 2.5 – 2.7 – Avago Technologies72 AWT6104M5 30 30 3.5 714 10 1.85 – 1.91 – ANADIGICS73 HMC457QS16G/E 30.5 25 5 500 15 2.01 – 2.17 – Hittite Microwave corporation74 HMC453QS16G/E 33 8 6.5 725 – 2.01 – 2.17 – Hittite Microwave corporation75 SM0825-33/33H 33 20 12 1100 4 0.8 – 2.5 – Stealth Microwave76 SM1025-36DMQ2 33 10 12 800 – 1 – 2.5 – Stealth Microwave77 SM1025-37MQ2 33 10 12 1600 – 1 – 2.5 – Stealth Microwave78 PA1110 33 10 10 725 28 1.8 – 2 – tyco Electronics79 PA1132 33 22 12 725 15 1.8 – 2 – tyco Electronics80 SM1727-34HS 34 33 12 1200 1 1.7 – 2.7 – Stealth Microwave81 SM1727-34HSQ 34 36.5 12 1200 1 1.7 – 2.7 – Stealth Microwave82 SM2023-34HS 34 33 12 1200 1 2 – 2.3 – Stealth Microwave83 PA1157 36 24.5 10 1350 15 2 – 2.2 – tyco Electronics84 PA1159 36.2 23.5 10 1700 28 2.3 – 2.4 – tyco Electronics A PPENDIX BD ERIVATION FOR (7)
Joint pdf f Y ( y, S = 0) : Given S = 0 , i.e., A ≤ a max , we have W = L PA ( A ) e jθ = √ gAe jθ . From assumption A , the conditional pdf of W is a truncated complex Gaussian as f W ( w | S = 0) = S =0) 1 πgP in exp (cid:16) − | w | gP in (cid:17) , | w | < b max , | w | ≥ b max . (B.1) b) Continued
PA P maxout (dBm) g (dB) V PA (Volt) C PA (mA) P maxin (dBm) Frequency (GHz) turn-on time ( µ s ) Institution85 PA1162 36.2 30 10 1450 11 0.8 – 0.96 – tyco Electronics86 SM04060-37HS 37 36 12 1800 1 0.4 – 0. 6 – Stealth Microwave87 SM04093-36HS 37 34 12 1600 1 0.4 – 0. 925 – Stealth Microwave88 SM5659-37S 37 20 12 2300 20 5.6 – 5.9 1.0 Stealth Microwave89 SM5759-37HS 37 39 12 2300 2 5.7 – 5.9 1.0 Stealth Microwave90 PA1182 37.5 23 28 1000 15 2.3 – 2.4 – tyco Electronics91 PA1223 37.5 25 28 1000 15 2.11 – 2.17 – tyco Electronics92 PA1224 37.5 25 28 1000 15 2 – 2.2 – tyco Electronics93 PA1186 38 29 28 1000 15 0.8 – 0.96 – tyco Electronics94 XD010-42S-D4F/Y 39 30 28 930 20 0.869 – 0.894 – Sirenza Micro Devices95 SM0822-39 39 45 12 3500 -4 0.8 – 2.2 1.0 Stealth Microwave96 SM0825-40Q 40 39 12 5500 1 0.8 – 2.5 – Stealth Microwave97 SM2023-41 41 55 12 4500 -13 2 – 2.3 – Stealth Microwave98 SM2027-41LS 41 51 12 6000 -7 2 – 2.7 – Stealth Microwave99 SM4450-41L 41 55 12 5000 -13 4.4 – 5 1.0 Stealth Microwave100 SM1822-42LS 42 52 12 5500 -8 1.8 – 2.2 – Stealth Microwave101 SM3338-43 43 50 12 8500 -6 3.3 – 3.8 1.0 Stealth Microwave102 SM5053-43L 43 55 12 9200 -7 5 – 5.3 1.0 Stealth Microwave103 SM7785-43A 43 48 12 9500 – 7.725 – 8.5 – Stealth Microwave104 SM1923-44L 44 55 12 8200 -8 1.9 – 2.3 – Stealth Microwave105 SM2025-44L 44 55 12 8500 -10 2 – 2.5 – Stealth Microwave
106 SM2122-44L 44 55 12 8200 -9 2.1 –
107 SM2325-44 44 55 12 8000 -10 2.3 – 2.5 – Stealth Microwave108 SM2025-46L 46.3 52 12 15000 -7 2 – 2.5 – Stealth Microwave109 SM04548-47L 47 55 12 14000 -8 0.45 – 0.48 – Stealth Microwave110 SM2023-47L 47 55 12 15000 -7 2 – 2.3 – Stealth Microwave111 SM3134-47L 47 55 12 15000 -6 3.1 – 3.4 – Stealth Microwave112 SM3436-47L 47 56 12 15000 -6 3.4 – 3.6 – Stealth Microwave
113 SM1720-50 50 50 12 27000 2 1.7 – 2 – Stealth Microwave
113 SM2325-50L 50 59 12 31000 -9 2.3 – 2.5 – Stealth Microwave115 SM1819-52LD 52 45 30 11000 – 1.8 – 1.9 – Stealth Microwave
The pdf of the AWGN Z is f Z ( z | S = 0) = f Z ( z | S = 1) = f Z ( z ) = πσ z exp (cid:16) − | z | σ z (cid:17) . (B.2)From (3), we can express the joint pdf as follows: f Y ( y, S = 0) = Pr( S = 0) ( f W ( w | S = 0) ∗ f Z ( z | S = 0))= Pr( S = 0) Z τ ∈ C f W ( τ | S = 0) f Z ( y − τ ) dτ = Pr( S = 0) Z ∞ Z π f W (cid:16) re jφ (cid:12)(cid:12) S = 0 (cid:17) f Z (cid:16) y − re jφ (cid:17) | J ( φ, r ) | dφdr (B.3)where ‘ ∗ ’ is the linear convolution operator and | J ( φ, r ) | = r is the Jacobian [17]. Noting ( f W | S = 0) = 0 if | w | ≥ b max , and using (B.1) and (B.2) to (B.3), we further derive f Y ( y, S = 0) = 1 π gP in σ z Z b max Z π r exp (cid:18) − r gP in − | y − re jφ | σ z (cid:19) dφdr = 1 π gP in σ z Z b max r exp (cid:18) − r gP in − r + | y | σ z (cid:19) Z π exp (cid:18) r | y | cos( θ − φ ) σ z (cid:19) dφdr = 2 πgP in σ z Z b max r exp (cid:18) − r gP in − r + | y | σ z (cid:19) I (cid:18) r | y | σ z (cid:19) dr = 1 π ( gP in + σ z ) exp (cid:18) − | y | gP in + σ z (cid:19) Z ρ max (cid:20)
12 exp (cid:18) − µ ( y ) + ρ (cid:19) I (cid:16)p µ ( y ) ρ (cid:17)(cid:21) dρ where θ denotes the angle of y and I ( · ) is a modified Bessel function of first kind. The last equality is obtainedafter some mathematical manipulations, for which we observe the function outside the integral is a pdf of aormal distribution with a zero mean and variance ( gP in + σ z ) and the function inside the integral is a pdf of anoncentral chi-squared random variable with one degree of freedom. Since the cdf of a noncentral chi-squaredrandom variable is obtained as − Q ( p µ ( y ) , √ ρ max ) , we can readily arrive at (7a). Joint pdf f Y ( y, S = 1) : Given S = 1 , i.e., A > a max , we have W = L PA ( A ) e jθ = √ ga max e jθ = b max e jθ .Thus, the amplitude is a constant while the phase θ is uniformly distributed. Therefore, we can express theconditional pdf of W given S = 1 as f W ( w | S = 1) = cδ ( | w | − b max ) (B.4)where δ ( · ) is the Dirac delta function and c = (2 πb max ) − is a constant obtained from the normalization R w ∈ C f W ( w | S = 1) dw = 1 . Then the joint pdf is similarly to give f Y ( y, S = 1) = Pr( S = 1) Z ∞ Z π rf W (cid:16) re jφ (cid:12)(cid:12) S = 1 (cid:17) f Z (cid:16) y − re jφ (cid:17) dφdr. (B.5)Substituting (B.4) and f Z ( z | S = 1) = f Z ( z ) in (B.2) into (B.5), and integration over r , we get f Y ( y, S = 1) = Pr( S = 1)2 π σ z Z π exp − (cid:12)(cid:12) y − b max e jφ (cid:12)(cid:12) σ z ! dφ = Pr( S = 1)2 π σ z exp (cid:18) − | y | + b σ z (cid:19) π I (cid:18) b max | y | σ z (cid:19) = 1 πσ z exp (cid:18) − | y − b max | σ z (cid:19) (cid:20) Pr( S = 1) exp (cid:18) − b max y Re σ z (cid:19) I (cid:18) b max | y | σ z (cid:19)(cid:21) where the function outside [ · ] is a pdf of a normal distribution CN ( b max , σ z ) . Thus, we get (7b).A PPENDIX CP ROOF OF (8)Henceforth, we take the indices in the subscript to be modulo N , e.g., X N + i = X ( N + i ) mod N = X i . Weexpress the mutual information as I ( f X ; e Y ) = I ( X ; Y ) ( a ) = N − X t =0 I ( X t ; Y , · · · , Y N − | X , · · · , X t − ) ( b ) ≥ N − X t =0 I ( X t ; Y , · · · , Y t + L − | X , · · · , X t − ) (C.1)where (a) follows from the chain rule of mutual information and (b) follows from the data processing inequality (bydiscarding the received signals Y t + L , · · · , Y N ). The inequality in (b) is typically tight from numerical experiments,because X t is only present in Y t , · · · , Y t + L − ; therefore, intuitively the discarded signals do not directly contributeo the information on X t (although they contribute to the information on the interfering terms in Y t , · · · , Y t + L − ).The summand in (C.1) for t ≥ L can be lower bounded as follows: I ( X t ; Y , · · · , Y t + L − | X , · · · , X t − ) ( a ) = I ( X t ; Y t , · · · , Y t + L − | X , · · · , X t − ) ( b ) = I ( X t ; Y ′ t , · · · , Y ′ t + L − | X , · · · , X t − ) ( c ) = I ( X t ; Y ′ t , · · · , Y ′ t + L − ) ( d ) ≥ I ( X t ; Y ′′ t ) , I LB t (C.2)where (a) follows from the independence of X t and { Y , · · · , Y t − } given { X , · · · , X t − } ; (b) follows from thedefinition Y ′ t + i = Y t − P L − j = i +1 h j X t + i − j for i = 0 , · · · , L − ; (c) follows from the fact that Y ′ t , · · · , Y ′ t + L − consist only of the signal terms X t , · · · , X N − and noise; and (d) follows from the data processing inequalitywhere Y ′′ t is the maximum ration combining (MRC) of Y ′ t , · · · , Y ′ t + L − . For additive independent interferencewith variance σ , the mutual information is lower bounded by the channel where the noise is treated as AWGNwith the same variance σ [16]. Hence, a lower bound of I ( X t ; Y ′′ t ) , denoted by I LB t , is given by the mutualinformation of the following channel: Y ′′′ t = L PA ( A t ) e θ t + Z ′ t (C.3)where Z ′ t ∼ CN (0 , σ z / | h ′ t | ) and h ′ t is derived by h ′ t = vuut gP in | h | σ z + L − X i =1 gP in | h i | σ z + gP in P i − j =0 | h j | . (C.4)The channel in (C.4) is the equivalent channel after MRC of the current and ( L − future received signals,where we take yet-to-be decoded, transmitted signals as interferences. The channel in (C.3) is the flat fadingchannel (3) with equivalent noise variance given by the original noise variance divided by the equivalent channelgain. Hence, we can obtain I LB t from (4) directly. In summary, from (C.1) and (C.2), the mutual information islower bounded by (8). A PPENDIX DP ROOF OF T HEOREMS AND P ROPOSITIONS
Proof of Theorem 1:
After substituting (12b) into (13), we can derive the second derivative of SE IBO ( ξ ) with respect to ξ as follows: ∂ SE IBO ( ξ ) ∂ ξ = − γ (1 + γξ ) + e − ξ − ξ − + e − ξ − ξ − (cid:0) ξ − − (cid:1) (cid:18) ln 1 πσ z − ξ (cid:19) ! E IBO ( ξ ) is concave over ξ if max (cid:16) , − πσ z ) (cid:17) ≤ ξ ≤ as the second derivative is negative. Proof of Theorem 2:
Since SE IBO ( ξ ) is concave assuming max (cid:16) , − πσ z ) (cid:17) ≤ ξ ≤ from Theorem 1 , wecan find the maximizer of SE IBO ( ξ ) by solving ∂ SE IBO ( ξ ) ∂ξ = 0 , which gives (14). Proof of Proposition 3:
Since γξ ≫ with practical value of γ , P maxout /σ z and typical value of ξ , we canapproximate γξ ≈ γξ in (14), and get ξe ξ = − ξ + 1 − ln (cid:0) πeσ z (cid:1) ≈ − ln (cid:0) πeσ z (cid:1) where we discard relativelysmall terms to obtain the closed form solution e ξ ⋆ SE in (15). Proof of Theorem 4:
First, assume ( c , c ) is fixed over all ξ (the dependence according to (20) will beconsidered shortly). The first derivative of EE with respect to ξ in (24) is given by ∂ EE linear ( ξ ) ∂ξ = Ω2 √ ξ (cid:0) v + v √ ξ (cid:1) (cid:18) γ γξ (cid:16) v p ξ + v ξ (cid:17) − v log (1 + γξ ) (cid:19) (D.1)where v = P + P maxout c c , and v = P maxout c c . Clearly v log (1 + γξ ) is increasing in ξ . It can be shown that γ γξ (cid:0) v √ ξ + v ξ (cid:1) is decreasing in ξ if ξ ≥ ζ , where ζ is defined in Theorem 4. Assuming ξ ≥ ζ , EE linear ( ξ ) then has at most one turning point and thus EE linear ( ξ ) is a quasi-concave function; it cannot be a quasi-convexfunction because EE linear ( ξ ) is decreasing in ξ for sufficiently large ξ . Now note that ( c , c ) is in fact constantfor ≤ ξ ≤ ℓ , and also constant for ℓ < ξ ≤ . Thus, EE linear ( ξ ) is a quasi-concave function for ζ ≤ ξ ≤ ℓ ,and also for ℓ < ξ ≤ . Proof of Theorem 5:
First, consider the optimal solution that maximizes EE linear ( ξ ) over ζ ≤ ξ ≤ ℓ .From Theorem 4 , EE linear ( ξ ) is a quasi-concave function. Thus the optimal solution is given by the turning point ξ ⋆ . In other words, the solution of ∂ EE linear ( ξ ) ∂ξ = 0 with ( c , c ) defined as (20a), if ξ ⋆ lies between ζ and ℓ . Since ξ ⋆ is a turning point, the optimal solution must be ζ if ξ ⋆ < ζ , while the optimal solution must be ℓ if ξ ⋆ > ℓ .More concisely, the optimal solution is [ ξ ⋆ ] /ℓ ζ . Similarly, the optimal solution that maximizes EE linear ( ξ ) over ℓ ≤ ξ ≤ can be shown to be [ ξ ⋆ ] /ℓ . Proof of Proposition 6:
The same approximation used in the proof of
Theorem 2 , i.e., γξ ≈ γξ , givesan equality ξ (cid:0) v √ ξ + v ξ (cid:1) = v log ( γξ ) to make (D.1) be a zero. After some mathematical manipulationsof the equality, we can find the unique solution given by (25). Proof of Proposition 7:
Suppose e ξ ⋆ EE ≤ e ξ ⋆ SE . From Theorems
Propositions ξ , and e ξ ⋆ SE and e ξ ⋆ EE are their maximizers, respectively. Thus,we can show the following. i) For ξ ≥ e ξ ⋆ EE , EE ( ξ ) decreases as ξ increases. For ξ ≤ e ξ ⋆ SE , SE ( ξ ) increases as ξ increases. Thus, for e ξ ⋆ EE ≤ ξ ≤ e ξ ⋆ SE , EE ( ξ ) decreases, while SE ( ξ ) increases as ξ increases. ii) For ξ < e ξ ⋆ EE ,both EE ( ξ ) and SE ( ξ ) increase as ξ increase. iii) For ξ > e ξ ⋆ SE , both EE ( ξ ) and SE ( ξ ) decrease as ξ increases.he proof is completed by showing the similar analysis for the case when e ξ ⋆ EE < e ξ ⋆ SE .R EFERENCES [1] J. Joung, C. K. Ho, and S. Sun, “Tradeoff of spectral and energy efficiencies: Impact of power amplifier on OFDM systems,” in
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