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Waveform and Transceiver Design for Simultaneous Wireless Information and Power Transfer
arXiv:1607.05602v1 [cs.IT] 19 Jul 2016
Bruno Clerckx
Abstract—Simultaneous Wireless Information and Power Transfer (SWIPT) has attracted significant attention in the communication community. The problem of waveform design has however never been addressed so far. In this paper, we first investigate how a communication waveform (OFDM) and a power waveform (multisine) compare with each other in terms of harvested energy. We show that due to the non-linearity of the rectifier and the randomness of the information symbols, the OFDM waveform is less efficient than the multisine waveform for wireless power transfer. This observation motivates the design of a novel SWIPT transceiver architecture relying on the superposition of multisine and OFDM waveforms at the transmitter and a power-splitter receiver equipped with an energy harvester and an information decoder. The superposed SWIPT waveform is optimized so as to maximize the rate-energy region of the whole system. Its design is adaptive to the channel state information and result from a posynomial maximization problem that originates from the non-linearity of the energy harvester. Numerical results illustrate the performance of the derived waveforms and SWIPT architecture. Key (and refreshing) observations are that 1) a power waveform (superposed to a communication waveform) is useful to enlarge the rate-energy region of SWIPT, 2) a combination of power splitting and time sharing is in general the best strategy, 3) exploiting the nonlinearity of the rectifier is essential to design efficient SWIPT architecture, 4) a non-zero mean Gaussian input distribution outperforms the conventional capacity-achieving zero-mean Gaussian input distribution.
I. I NTRODUCTION Simultaneous Wireless Information and Power Transfer (SWIPT) has recently attracted significant attention in academia, with works addressing many scenarios, a.o. MIMO broadcasting [3], architecture [4], interference channel [5], [6], broadband system [7]–[9], relaying [10], [11]. Wireless Power Transfer (WPT) is a fundamental building block of SWIPT and the design of an efficient SWIPT architecture fundamentally relies on the ability to design efficient WPT. The major challenge with WPT, and therefore SWIPT, is to find ways to increase the DC power level at the output of the rectenna without increasing the transmit power. To that end, the vast majority of the technical efforts in the literature have been devoted to the design of efficient rectennas. The rectenna is made of a non-linear device followed by a low-pass filter to extract a DC power out of an RF input signal. The amount of DC power collected is a function of the input power level and the RF-to-DC conversion efficiency. Interestingly, the RF-toDC conversion efficiency is not only a function of the rectenna design but also of its input waveform [12]–[17]. Bruno Clerckx is with the EEE department at Imperial College London, London SW7 2AZ, United Kingdom (email:
[email protected]). This work has been partially supported by the EPSRC of UK, under grant EP/P003885/1. The material in this paper was presented in part at the ITG WSA 2016 [1].
This observation has triggered very recent interests on wireless power waveform design in the signal processing literature [15]–[17]. The objective is to understand how to make the best use of a given RF spectrum in order to deliver a maximum amount of DC power at the output of a rectenna. This problem can be formulated as a link optimization where transmit waveforms are adaptively designed as a function of the channel state information (CSI) so as to maximize the DC power at the output of the rectifier. In [16], the waveform design problem for WPT has been tackled by introducing a simple and tractable analytical model of the non-linearity of the diode through the second and higher order terms in the Taylor expansion of the diode characteristics. Comparisons were also made with a linear model of the rectifier, that only accounts for the second order term. Assuming perfect Channel State Information at the Transmitter (CSIT) can be attained, relying on both the linear and non-linear models, an optimization problem was formulated to adaptively change on each transmit antenna a multisine waveform as a function of the CSI so as to maximize the output DC current at the energy harvester. Important conclusions of [16] are that 1) multisine waveforms designed accounting for nonlinearity are spectrally more efficient that those designed based on a linear model of the rectifier, 2) the linear model does not characterize correctly the rectenna behaviour and leads to inefficient multisine waveform design, 3) rectifier nonlinearity is key to design efficient wireless powered systems. Interestingly, the SWIPT literature has so far entirely relied on the aforementioned linear model of the rectifier, e.g. [4] and subsequent works. The problem of SWIPT waveform and transceiver design that accounts for the nonlinearity of the rectifier has indeed never been addressed so far. In view of the recent results in [16], it is expected that accounting for the rectifier nonlinearity is key to efficient SWIPT design. In this paper, we address the important problem of waveform and transceiver design for SWIPT. The contributions of the paper are summarized as follows. First, we leverage the analytical model of the rectenna nonlinearity introduced in [16] and investigate how a communication waveform (OFDM) and a power waveform (multisine) compare with each other in terms of harvested energy. Comparison is also made with the linear model commonly used in the SWIPT literature [4]. Scaling laws of the harvested energy with multisine and OFDM waveforms are analytically derived as a function of the number of sinewaves and the propagation conditions. We show that with the nonlinear model of the rectifier, there is a clear benefit of using a deterministic multisine over an OFDM waveform for WPT. On the other hand, the benefits vanish when we consider the linear model.
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Second, we introduce a novel SWIPT transceiver architecture relying on the superposition of multisine and OFDM waveforms at the transmitter and a power-splitter receiver equipped with an energy harvester and an information decoder. Both cases where the receiver does and does not have the capability of cancelling the multisine waveform are considered. The SWIPT multisine/OFDM waveforms and the power splitter are jointly optimized so as to maximize the rateenergy region of the whole system. The superposed SWIPT waveforms are adaptive to the channel state information and result from a posynomial maximization problem that originates from the non-linearity of the energy harvester. Third, numerical results illustrate the performance of the derived waveforms and SWIPT architecture. Key observations are that 1) a power waveform (superposed to a communication waveform) is always useful to enlarge the rate-energy region of SWIPT, 2) a combination of power splitting and time sharing is in general the best strategy, 3) a non-zero mean Gaussian input distribution outperforms the conventional capacity-achieving zero-mean Gaussian input distribution. Those observations are consequences of the non-linearity of the rectifier and highlights that exploiting the nonlinearity of the rectifier is essential to design an efficient SWIPT architecture. Organization: Section II introduces the multisine and OFDM waveforms for WPT, section III introduces the rectenna model, section IV optimizes multisine and OFDM waveforms for WPT and discusses the impact of the waveform type (multisine vs OFDM) on the harvested energy. Section V introduces the SWIPT architecture, section VI addresses the SWIPT waveform design, section VII evaluates the performance and section VIII concludes the work. Notations: Bold lower case and upper case letters stand for vectors and matrices respectively whereas a symbol not in bold 2 font represents a scalar. k.kF refers to the Frobenius norm a matrix. A {.} refers to the DC component of a signal. EX {.} refers to the expectation operator taken over the distribution of the random variable X (X may be omitted for readability if the context is clear). .∗ refers to the conjugate of a scalar. T H (.) and (.) represent the transpose and conjugate transpose of a matrix or vector respectively.
The magnitudes and phases of the sinewaves can be collected into N ×M matrices SP and ΦP . The (n, m) entry of SP and ΦP write as sP,n,m and φP,n,m , respectively. The transmitter is subject to an average transmit power constraint PP = 2 1 signals, we can write 2 kSP kF ≤ P . Stacking up all transmit PN −1 the transmit signal vector as xP (t) = ℜ wP,n ejwn t n=0 T . where wP,n = wP,n,1 . . . wP,n,M The multi-antenna transmitted sinewaves propagate through a multipath channel, characterized by L paths whose delay, amplitude, phase and direction of departure (chosen with respect to the array axis) are respectively denoted as τl , αl , ξl and θl , l = 1, . . . , L. We assume transmit antennas are closely located so that τl , αl and ξl are the same for all transmit antennas (assumption of a narrowband balanced array) [19]. Denoting ζn,m,l = ξl + ∆n,m,l with ∆n,m,l the phase shift between the mth transmit antenna and the first one1 , the signal transmitted by antenna m and received at the single-antenna receiver after multipath propagation can be written as yP,m (t) =
N −1 X
sP,n,m An,m cos(wn t + ψP,n,m )
(2)
n=0
where the amplitude An,m and the phase ψP,n,m are such that ¯
An,m ejψP,n,m = An,m ej (φP,n,m +ψn,m ) = ejφP,n,m hn,m (3) PL−1 ¯ j(−wn τl +ζn,m,l ) the with hn,m = An,m ej ψn,m = l=0 αl e frequency response of the channel of antenna m at wn. The vector channel is defined as hn = hn,1 . . . hn,M . The total received signal comprises the sum of (2) over all transmit antennas, namely yP (t) =
N −1 X
XP,n cos(wn t + δP,n )
n=0
=ℜ where XP,n ejδP,n =
(N −1 X
hn wP,n e
jwn t
n=0
PM
m=1 sP,n,m An,m e
)
jψP,n,m
(4) = hn wP,n .
B. OFDM II. M ULTISINE AND OFDM- BASED WPT We consider two types of waveform: multisine and OFDM. A. Multisine Consider the transmitter made of M antennas and N sinewaves whose multisine transmit signal at time t on transmit antenna m = 1, . . . , M is given by (N −1 ) X jwn t xP,m (t) = ℜ wP,n,m e (1) n=0
jφP,n,m
where sP,n,m and φP,n,m with wP,n,m = sP,n,m e refer to the amplitude and phase of the nth sinewave at frequency wn on transmit antenna m, respectively. We assume for simplicity that the frequencies are evenly spaced, i.e. wn = w0 + n∆w with ∆w = 2π∆f the frequency spacing.
The baseband OFDM signal over one duration P symbol −1 j 2πt T n, T = 1/∆f can be written as xB,m (t) = N n=0 xn,m e 0 ≤ t ≤ T , where xn,m = w ˜I,n,m x ˜n refers to the precoded input symbol on frequency tone n and antenna m. We assume a capacity achieving Gaussian input distribution such that x˜n are i.i.d. circularly symmetric complex Gaussian distributed. We further write the precoder w ˜I,n,m = xn | ejφx˜n . |w ˜I,n,m | ejφI,n,m and the input symbol x˜n = |˜ After adding the cyclic prefix over duration Tg , it comes PN −1 j 2πt T n , −T to xB,m (t) = g ≤ t ≤ T . Vectorn=0 xn,m e wise, the baseband OFDM signal vector tone n writes as T on P N −1 j 2πt T n xB (t) = xB,1 (t) . . . xB,M (t) = n=0 xn e T ˜ I,n x˜n and w ˜ I,n = w ˜I,n,1 . . . w ˜I,n,M with xn = w d 1 For a Uniform Linear Array (ULA), ∆ n,m,l = 2π(m − 1) λn cos(θl ) where d is the inter-element spacing, λn the wavelength of the nth sinewave.
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is the precoder. After upconversion, the transmitted OFDM signal on antenna m is written as ) (N −1 X jw0 t jwn t (5) =ℜ xI,m (t) = ℜ xB,m (t)e xn,m e n=0
˜I,n,m jφ
with s˜I,n,m = |w ˜I,n,m | |˜ xn | where xn,m = s˜I,n,m e ˜I,n,m = φI,n,m + φx˜ . We also define wI,n,m = and φ n p PI,n w ˜I,n,m and sI,n,m = |wI,n,m | where PI,n = T 2 . E |˜ xn | . Vector-wise, wI,n = wI,n,1 . . . wI,n,M We define N × M matrices such that the (n, m) entry of ˜ I , SI , Φ ˜ I , ΦI write as s˜I,n,m , sI,n,m , φ˜I,n,m , φI,n,m , matrix S respectively. The OFDM is subject to the power 2waveform constraint PI = 12 E ˜ SI F = 21 kSI k2F ≤ P . After multipath, the received signal can be written as yI (t) =
N −1 X
XI,n cos(wn t + δI,n )
n=0
=ℜ
(N −1 X
˜ I,n x ˜n e hn w
jwn t
n=0
)
(6)
PM ˜ ˜ I,n x ˜n where XI,n ejδI,n = m=1 s˜I,n,m An,m ej ψI,n,m = hn w and ψ˜I,n,m = φ˜I,n,m + ψ¯n,m = φI,n,m + φx˜n + ψ¯n,m . Let us also define ψI,n,m = φI,n,m + ψ¯n,m such that ψ˜I,n,m0 − ψ˜I,n,m1 = ψI,n,m0 − ψI,n,m1 . III. A NALYTICAL M ODEL OF THE R ECTENNA In [15], [16], a simple and tractable model of the rectifier nonlinearity in the presence of multisine excitation was derived and its validity verified through circuit simulations. In this paper, we reuse the same model and further expand it to OFDM excitation. A multisine waveform is determinsitic while an OFDM waveform exhibits randomness due to information symbols x ˜n . This randomness has an impact on the amount of harvested energy and needs to be captured in the rectenna model. In the next two subsections, we introduce the antenna equivalent circuit and rectifier non-linearity2. A. Antenna Equivalent Circuit The antenna model reflects the power transfer from the antenna to the rectifier through the matching network. The signal impinging on the antenna is yP (t) for a multisine or yI (t) for an OFDM waveform. For readability, let us drop the index for a moment and simply refer to a received signal PN −1 of the form y(t) = n=0 Xn cos(wn t + δn ), i.e. applicable to both multisine and OFDM. y(t) has an average power Pin,av = E |y(t)|2 . A lossless antenna can be modelled as a voltage source vs (t) followed by a series resistance Rant (see Fig 1). Let Zin = Rin +jXin denote the input impedance of the rectifier with the matching network. Assuming perfect matching (Rin = Rant , Xin = 0), all the available RF power Pin,av is transferred to the rectifier and absorbed by Rin , 2 so that Pin,av = E |vin (t)| /R in and vin (t) = vs (t)/2. 2 Since√Pin,av = E |y(t)| , vs (t) can be formed as vs (t) = √ 2y(t) Rin = 2y(t) Rant . We also assume that the antenna noise is too small to be harvested so as no antenna noise term is added and vin (t) is delivered as such to the rectifier. 2 For
more details, the reader is invited to check [16].
Fig. 1.
Antenna equivalent circuit (left) and a single diode rectifier (right).
B. Rectifier and Diode Non-Linearity Consider a rectifier composed of a single series diode followed by a low-pass filter with load (see Fig 1). Denoting the voltage drop across the diode as vd (t) = vin (t) − vout (t) where vin (t) is the input voltage to the diode and vout (t) is the output voltage across the load resistor, a tractable behavioural diode model is obtained by Taylor series expansion of the vd (t) diode characteristic equation id (t) = is e nvt − 1 (with is the reverse bias saturation current, vt the thermal voltage, n the ideality factor assumed equal to 1.05) around a quiescent operating point vd = a. The diode current is then written as id (t) =
∞ X i=0
ki (vd (t) − a)i ,
(7)
a a e nvt where k0 = is e nvt − 1 and ki = is i!(nv i , i = 1, . . . , ∞. t) Assume a steady-state response and an ideal low pass filter such vout (t) is at constant DC level. If a is chosen as a = E {vd (t)} = −vout , (7) can be simplified as
id (t) =
∞ X
ki vin (t)i =
∞ X
i/2
ki Rant y(t)i .
(8)
i=0
i=0
The problem at hand will be the design of {xP,m (t)} or {xI,m (t)} such that the output DC current is maximized. Under the ideal rectifier assumption and a deterministic incoming waveform y(t) (i.e. with fixed weights Xn and phases δn ), the current delivered to the load in a steady-state response is constant and given by iout = A {id (t)}. In order to make the optimization tractable, we truncate the Taylor expansion to the nth o order. We consider two models: a non-linear model that truncates the Taylor expansion to the nth o order but retains the fundamental non-linear behaviour of the diode and a linear model that truncates to the second order term and ignores the non-linearity. C. A Non-Linear Model After truncation, the output DC current approximates as iout = A {id (t)} ≈
no X i=0
i/2 ki Rant A y(t)i .
(9)
Let us first consider a multisinePwaveform. Applying a N −1 received signal of the form y(t) = n=0 Xn cos(wn t + δn ) to (9) and averaging over time, we get an approximation of the DC component of the current at the output of the rectifier (and
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"N−1 # N−1 X X X hn wP,n 2 = 1 sP,n,m0 sP,n,m1 An,m0 An,m1 cos ψP,n,m0 − ψP,n,m1 , 2 n=0 m ,m n=0 n=0 0 1 X 3 ∗ ∗ hn3 wP,n3 , hn0 wP,n0 hn1 wP,n1 hn2 wP,n2 A yP (t)4 = ℜ 8 n0 ,n1 ,n2 ,n3 n0 +n1 =n2 +n3 # " 3 X X Y 3 sP,nj ,mj Anj ,mj cos(ψP,n0 ,m0 + ψP,n1 ,m1 − ψP,n2 ,m2 − ψP,n3 ,m3 ) . = 8 m0 ,m1 , n0 ,n1 ,n2 ,n3 j=0 1 A yP (t)2 = 2
"N−1 X
2 XP,n
#
1 = 2
n0 +n1 =n2 +n3
(12)
(13)
m2 ,m3
the low-pass filter) with a multisine excitation over a multipath channel as no X i/2 iout ≈ k0 + ki Rant A yP (t)i . (10) i even,i≥2
where A yP (t)2 and A yP (t)4 are detailed in (11) and (13), respectively3 (at the top of next page). There is no i odd (first, third, fifth, etc) order terms since A y(t) = E y(t)i = 0 for i odd. We note that the second order term (11) is linear, with the DC power being the sum of the power harvested on each frequency. On the other hand, even terms with i ≥ 4 such as (13) are responsible for the non-linear behavior of the diode since they are function of terms expressed as the product of contributions from different frequencies. Note that the non-linear model in (10) with no = 4 has been validated in [16] using a circuit simulator for multisine excitation. Let us now consider the OFDM waveform. It can be viewed as a multisine waveform for a fixed set of input symbols {˜ xn }. Hence, we can also write the DC component of the current at the output of the rectifier (and the low-pass filter) with an OFDM excitation and fixed set of input symbols over Pn i/2 a multipath channel as k0 + i oeven,i≥2 ki Rant A yI (t)i . Similar expressions as (11) and (13) can be written for A yI (t)2 and A yI (t)4 for a fixed set of input symbols {˜ xn }. However, contrary to the multisine waveform, the input symbols {˜ xn } of the OFDM waveform change randomly every symbol duration T . For a given channel impulse response, the proposed model for the DC current with an OFDM waveform is obtained by averaging out over the distribution of the input symbols {˜ xn } such that no X i/2 iout ≈ k0 + (14) ki Rant E{˜xn } A yI (t)i . i even,i≥2
For E A yI (t)i with i even, the DC component is first extracted for a given set of amplitudes {˜ sI,n,m } and phases φ˜I,n,m and then expectation is taken over the randomness of the input symbols x ˜n . Due to the i.i.d. circularly symmetric complex Gaussian distribution of the input symbols, |˜ xn |2 is 2 = PI,n and φx˜n is ˜n exponentially distributed with E x uniformly distributed. From the moments 4 of an exponential 2 = 2PI,n ˜n . This distribution, we also have that E x enables to express (15) and (16) as a function of s I,n,m = p PI,n |wI,n,m |. 3 Readers
(11)
interested in higher order terms can refer to [16].
D. A Linear Model The linear model was orginally introduced a few decades ago in [18] and is nowadays extensively used throughout the SWIPT literature, e.g. [4]. It could be argued that if yP/I (t) is very small (i.e. for a very low input power), the higher order terms would not contribute much to iout . Hence, the linear model truncates the Taylor expansion to 2 no = 2 such that iout ≈ k0 +k2 R antA yP2(t) for multisine and iout ≈ k0 + k2 Rant E{˜xn } A yI (t) for OFDM. It therefore completely omits the non-linearity behavior of the rectifier. The linear model is motivated by its simplicity rather than its accuracy. In [16], the linear model was shown to be inaccurate in predicting multisine waveform performance. Nevertheless, the loss incurred by using a linear vs a nonlinear model for the design of WPT based on other types of waveforms (i.e. others than deterministic multisine) and the design of SWIPT has never been addressed so far. In the next section, we study the design of multisine and OFDM waveforms under the assumption of a linear and nonlinear model. IV. M ULTISINE /OFDM WAVEFORM D ESIGN FOR WPT Assuming the CSI (in the form of frequency response hn,m ) is known to the transmitter, we aim at finding the optimal set of amplitudes and phases (SP , ΦP for multisine and SI , ΦI for OFDM) that maximizes iout , i.e. 1 2 max iout (Si , Φi ) subject to kSi kF ≤ P, (18) Si ,Φi 2 for i ∈ {P, I}. Following [16], problem (18) can equivalently be written as 1 2 kSi kF ≤ P, (19) max zDC (Si , Φi ) subject to Si ,Φi 2 where zDC (SP , ΦP ) =
no X
i even,i≥2
for the multisine waveform, and zDC (SI , ΦI ) =
no X
i even,i≥2
i/2 ki Rant A yP (t)i
i/2 ki Rant E{˜xn } A yI (t)i
(20)
(21)
for the OFDM waveform. In (20) and (21), we define ki = is (with a slight abuse of notation). Assuming is = 5µA, i!(nvt )i a diode ideality factor n = 1.05 and vt = 25.86mV , typical values of those parameters for second and fourth order are
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N−1 X X 1 E A yI (t)2 = E s˜I,n,m0 s˜I,n,m1 An,m0 An,m1 cos ψ˜I,n,m0 − ψ˜I,n,m1 2 n=0 m ,m 0 1 "N−1 # N−1 2 1 X 1 X X sI,n,m0 sI,n,m1 An,m0 An,m1 cos ψI,n,m0 − ψI,n,m1 = hn wI,n (15) = 2 n=0 m ,m 2 n=0 0 1 " 3 # X X Y 3 s˜I,nj ,mj Anj ,mj cos(ψ˜I,n0 ,m0 + ψ˜I,n1 ,m1 − ψ˜I,n2 ,m2 − ψ˜I,n3 ,m3 ) E A yI (t)4 = E 8 n0 ,n1 ,n2 ,n3 m0 ,m1 , j=0 n0 +n1 =n2 +n3 m2 ,m3 " " #" # # X Y Y 6 X = sI,n0 ,mj An0 ,mj sI,n1 ,mj An1 ,mj cos(ψI,n0 ,m0 + ψI,n1 ,m1 − ψI,n0 ,m2 − ψI,n1 ,m3 ) 8 n ,n m0 ,m1 , j=0,2 j=1,3 0
1
m2 ,m3
(16) "N−1 #2 2 6 X hn wI,n = 8 n=0
given by k2 = 0.0034 and k4 = 0.3829 (and will be used as such in any evaluation in the sequel).
A. Multisine Waveform Design The problem of (deterministic) multisine waveform design with a linear and non-linear rectenna model has been addressed in [15], [16]. linear model leads to the P equivalent problem PNThe N −1 2 −1 2 1 |h w | subject to maxwP,n n P,n n=0 kwP,n k ≤ n=0 2 P whose solution is the adaptive single-sinewave (ASS) strategy √ 2P hH ¯, ⋆ n / khn k , n = n wP,n = (22) 0, n 6= n ¯. The ASS performs a matched beamformer on a single sinewave, namely the one corresponding to the strongest channel n ¯ = arg maxn khn k2 . On the other hand, the nonlinear model leads to a posynomial maximization problem that can be formulated as a reverse geometric program and solved iteratively. Interestingly, for multisine waveforms, the linear and nonlinear models lead to radically different strategies. The former favours transmission on a single frequency while the latter favours transmission over multiple frequencies. Design based on the linear model was shown to be inefficient and lead to significant loss over the nonlinear-based design.
B. OFDM Waveform Design The design of OFDM waveform is rather different. From (15) and (17), both the P second and fourth order terms are N −1 2 exclusively function of n=0 |hn wI,n | . This shows that both the linear and nonlinear model-based design of OFDM waveforms for WPT lead to the ASS strategy and the optimum ⋆ wI,n should be designed according to (22). This is in sharp contrast with the deterministic multisine waveform design and originates from the fact that the OFDM waveform is subject to randomness due to the presence of input symbols x ˜n . Note that this ASS strategy has already appeared in the SWIPT literature with OFDM transmission, e.g. [7], [20].
(17)
C. Scaling Laws In order to assess the performance benefits of a multisine waveform over an OFDM waveform for WPT, we quantify how zDC scales as a function of N . For simplicity we truncate the Taylor expansion to the fourth order (no = 4). We consider frequency-flat and frequency-selective channels. We assume that the complex channel gains αl ejξl are modeled as independent circularly symmetric complex Gaussian random variables. αl are therefore independent Rayleigh distributed such that α2l ∼ EXPO(λl ) with 1/λl = βl = E α2l . The impulse responses have a constant PL−1 average received power normalized to 1 such that l=0 βl = 1. In the frequency flat channel, ψ¯n = ψ¯ and An = A ∀n. This is met when the bandwidth of the multisine waveform (N − 1)∆f is much smaller than the channel coherence bandwidth. In the frequency selective channel, we assume that L >> 1 and frequencies wn are far apart from each other such that the frequency domain circularly symmetric complex Gaussian random channel gains hn,m fade independently (phase and amplitude-wise) across frequencies and antennas. Taking the expectation over the distribution of the channel, we denote z¯DC,UP = E {zDC,UP }. Leveraging the derivation of the scaling laws for multisine waveforms designed based on the ASS in [16], we can easily compute the scaling laws of OFDM waveform designed based on ASS. Table I summarizes the scaling laws for both multisine and OFDM waveforms based on the ASS strategy. It also compares with the UPMF strategy for multisine (introduced in [16]) that consists in uniformly allocating power to all sinewaves and matching the waveform phase to the channel phase. Such a UPMF srategy is suboptimal for multisine excitation and its scaling law is therefore a lower bound on what can be achieved with the optimal strategy of (19). Recall that the ASS strategy for OFDM is optimal for the maximization of zDC with the linear and nonlinear model. The ASS for multisine is only optimal for the linear model. As it can be seen from the scaling laws, the UPMF strategy leads to a linear increase of z¯DC with N while the ASS strategy for multisine and OFDM only lead to at most a logarithmic increase with N . Observation 1: The linear model highlights that there is no
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TABLE I S CALING L AWS OF M ULTISINE VS OFDM. Waveform OFDM z¯DC,ASS Multisine z¯DC,ASS z¯DC,U P M F
N, M
Frequency-Flat (FF)
Frequency-Selective (FS)
N >> 1, M = 1
k2 Rant P + 6k4 R2ant P 2
k2 Rant P log N + 3k4 R2ant P 2 log2 N
N >> 1, M = 1 N >> 1, M = 1
k2 Rant P + 3k4 R2ant P 2 k2 Rant P + 2k4 R2ant P 2 N
k2 Rant P log N + 32 k4 R2ant P 2 log2 N ≥ k2 Rant P + π 2 /16k4 R2ant P 2 N ≤ k2 Rant P + 2k4 R2ant P 2 N
as xm (t) = xP,m (t) + xI,m (t) =
N −1 X
sP,n,m cos(wn t + φP,n,m )
n=0
(a) Transmitter
+ s˜I,n,m cos(wn t + φ˜I,n,m ), ) (N −1 X jwn t =ℜ (wP,n,m + xn,m ) e
(23)
n=0
(b) Receiver Fig. 2.
A transceiver architecture for SWIPT.
difference in using a deterministic multisine waveform or an OFDM waveform for WPT, since according to this model the 2 PN −1 harvested energy is a function of n=0 hn wP/I,n . Hence OFDM and multisine waveforms are equally suitable. On the other hand, the nonlinear model highlights that there is a clear benefit of using a deterministic multisine over a (randomized) OFDM waveform in WPT, with the scaling law of multisine significantly outperforming that of OFDM. While z¯DC scales linearly with N with a multisine waveform, it scales at most logarithmically with N with an OFDM waveform. This loss in scaling law is inherently due to the randomness of information symbols.
˜
with wP,n,m = sP,n,m ejφP,n,m and xn,m = s˜I,n,m ej φI,n,m . This is illustrated in Figure 2(a). Due to the superposition of the two waveforms, the total average transmit power constraint now writes as PP + PI ≤ P . At the receiver, we can write the received signal as y(t) = yP (t)+yI (t), i.e. the sum of two contributions at the output of the channel, namely one from WPT yP (t) and the other from WIT yI (t), expressed as in (4) and (6), respectively. Using a power splitter with a power splitting ratio ρ and assuming perfect matching (as in Section III-A), the input voltage signal √ ρRant y(t)pis conveyed to the input of the energy harvester (EH) while (1 − ρ)Rant y(t) is conveyed to the information decoder (ID). Remark 1: It is worth noting the effect of the deterministic multisine waveform on the input distribution in (23). ˜ Recall that xn,m = s˜I,n,m ej φI,n,m ∼ CN (0, s2I,n,m ). Hence wP,n,m + xn,m ∼ CN (wP,n,m , s2I,n,m ) and the effective input distribution on a given frequency and antenna is not zero mean. The magnitude |wP,n,m + xn,m | is Ricean distributed with a K-factor on frequency n and antenna m given by Kn,m = s2P,n,m /s2I,n,m .
V. A SWIPT T RANSCEIVER A RCHITECTURE Observation 1 motivates a novel SWIPT architecture that is based on a superposition of multisine waveform for efficient WPT and OFDM waveform for efficient Wireless Information Transfer (WIT). In Figure 2, we introduce the proposed SWIPT architecture where power and information are transmitted simultaneously from one transmitter to one receiver equipped with a power splitter.
A. Transmitter and Receiver The SWIPT waveform on antenna m, xm (t), consists in the superposition of one multisine waveform xP,m (t) at frequencies wn = w0 + n∆w , n = 0, . . . , N − 1 for WPT and one OFDM waveform xI,m (t) at the same frequencies for WIT. The total transmitted SWIPT waveform on antenna m writes
B. Information Decoder Let us consider two types of information decoders: the perfect cancellation (PC) receiver and the no-cancellation (NC) receiver. 1) PC-ID receiver: : Since xP,m (t) does not contain any information, it is deterministic and can be cancelled at the ID receiver. Therefore, after down-conversion and ADC, the contribution of the WPT waveform is subtracted from the received signal (Fig 2(b)). Conventional OFDM processing is then conducted, namely removing the cyclic prefix and performing FFT. We can write the equivalent baseband system model of the ID receiver as p ˜ I,n x ˜n + vn (24) yP C−ID,n = 1 − ρhn w
where vn is the AWGN noise on tone n (with variance σn2 ) originating from the antenna and the RF to baseband downconversion.
7
Assuming perfect cancellation (PC), the rate writes as ! N −1 2 X (1 − ρ) |hn wI,n | IP C (SI , ΦI , ρ) = log2 1 + . σn2 n=0 (25) Naturally, I(SI , ΦI , ρ) can never be larger than the maximum rate achievable when ρ = 0, i.e. I(S⋆I , Φ⋆I , 0), which is obtained by performing matched filtering on each subcarrier and water-filling power allocation across subcarrier. 2) NC-ID receiver: : In the event that the receiver is not equipped with a WPT multisine waveform canceller (NC), the equivalent baseband system model of the ID receiver is p p ˜ I,n x ˜n + 1 − ρhn wP,n +vn (26) yN C−ID,n = 1 − ρhn w The ID receiver treats the contribution from the WPT as noise and the rate writes as IN C (SP , SI , ΦP , ΦI , ρ) =
N −1 X
log2 1 +
n=0
(1 − ρ) |hn wI,n |
2
σn2 + (1 − ρ) |hn wP,n |
2
!
. (27)
C. Energy Harvester
and guarantees all arguments of the cosine functions in A yP (t)i i=2,4 (expressions (11) and (13)) and in E A yI (t)i (expressions (15) and (16)) to be i=2,4 equal to 0. Φ⋆P and Φ⋆I are obtained by collecting φ⋆P,n,m and φ⋆I,n,m ∀n, m into a matrix, respectively. With such phases Φ⋆P and Φ⋆I , zDC (SP , SI , Φ⋆P , Φ⋆I , ρ) can be finally written as (32). Similarly we can write IP C (SI , Φ⋆I , ρ)
= log2
n=0
max
SP ,SI ,ρ
subject to
i even,i≥2
Assuming no = 4, we can compute zDC as in (29), where {A {yP (t)yI (t)}} = 0, we use the fact that E 3 E A yP (t)3 yI (t) = 0, E A y P (t)y (t) = I 0 and E A yP (t)2 yI (t)2 = A yP (t)2 E A yI (t)2 . VI. SWIPT WAVEFORM O PTIMIZATION
We can now define the achievable rate-harvested energy (or more accurately rate-DC current) region as n CR−IDC (P ) , (R, IDC ) : R ≤ IP C/N C , o 1 2 2 IDC ≤ zDC , kSI kF + kSP kF ≤ P . (30) 2 Optimal values S⋆P ,S⋆I ,Φ⋆P ,Φ⋆I , ρ⋆ are to be found in order to enlarge as much as possible the rate-harvested energy region. We derive a methodology that is general to cope with any Taylor expansion order no 4 . A. PC-ID receiver: A General Approach Looking at (25) and (29), it is easy to conclude that matched filtering w.r.t. the phases of the channel is optimal from both rate and harvested energy maximization perspective. This leads to the same phase decisions as for WPT in [15], [16], namely φ⋆P,n,m = φ⋆I,n,m = −ψ¯n,m 4 We
(31)
display terms for no ≤ 4 but the derived algorithm works for any no .
(1 − ρ) Cn 1+ σn2
!
(33)
Q1 P where Cn = m0 ,m1 j=0 sI,n,mj An,mj . Recall from [21] that a monomial is defined as the function a1 a2 aN g : RN ++ → R : g(x) = cx1 x2 . . . xN where c > 0 and ai ∈ R. A sum of K monomials PKis called a posynomial and can be written as f (x) = k=1 gk (x) with gk (x) = ck xa1 1k xa2 2k . . . xaNN k where ck > 0. As we can see from (32), zDC (SP , SI , Φ⋆P , Φ⋆I , ρ) is a posynomial. In order to identify the achievable rate-energy region, we formulate the optimization problem as an energy maximization problem subject to transmit power and rate constraints
p Contrary to WPT, in SWIPT, ρRant y(t) is conveyed to the input of the energy harvester. Hence both WPT yP (t) and WIT yI (t) now contribute to the DC component zDC (SP , SI , ΦP , ΦI , ρ) no X i/2 = ki ρi/2 Rant E{˜xn } A y(t)i . (28)
N −1 Y
zDC (SP , SI , Φ⋆P , Φ⋆I , ρ) 1 2 2 kSI kF + kSP kF ≤ P, 2 ¯ IP C (SI , Φ⋆I , ρ) ≥ R.
(34) (35) (36)
It therefore consists in maximizing a posynomial subject to constraints. Unfortunately this problem is not a standard Geometric Program (GP) but it can be transformed to an equivalent problem by introducing an auxiliary variable t0 min
SP ,SI ,ρ,t0
subject to
1/t0 1 2 2 kSI kF + kSP kF ≤ P, 2 t0 /zDC (SP , SI , Φ⋆P , Φ⋆I , ρ) ≤ 1, "N −1 # Y (1 − ρ) ¯ R ≤ 1. Cn 1+ 2 / σn2 n=0
(37) (38) (39) (40)
This is known as a Reverse Geometric Program [21], [22]. A similar problem also appeared in the WPT waveform ⋆ ⋆ optimization Note that QN −1 [15].(1−ρ) 1/zDC (SP , SI , ΦP , ΦI , ρ) and 1/ are not posynomials, therefore Cn 2 n=0 1 + σn preventing the use of standard GP tools. The idea is to replace the last two inequalities (in a conservative way) by making use of the arithmetic mean-geometric mean (AM-GM) inequality. Let {gk (SP , SI , Φ⋆P , Φ⋆I , ρ)} be the monomial terms in the posynomial zDC (SP , SI , Φ⋆P , Φ⋆I , ρ) = PK ⋆ ⋆ Similarly we define k=1 gk (SP , SI , ΦP , ΦI , ρ). {gnk (SI , ρ¯)} asPthe set of monomials of the posynomial Kn ¯) with ρ¯ = 1 − ρ. For a 1 + σρ¯2 Cn = k=1 gnk (SI , ρ n given choice of {γ } and {γ k nk } with γk , γnk ≥ 0 and PK PKn γ = γ = 1, we perform single condensations k nk k=1 k=1
8
zDC (SP , SI , ΦP , ΦI , ρ) = k2 ρRant A yP (t)2 + k4 ρ2 R2ant A yP (t)4 + k2 ρRant E A yI (t)2 + k4 ρ2 R2ant E A yI (t)4 + 6k4 ρ2 R2ant A yP (t)2 E A yI (t)2 .
zDC (SP , SI , Φ⋆P , Φ⋆I , ρ) =
N−1 X
k2 ρ Rant 2 n=0 m
X
0 ,m1
"
# 2 3k4 ρ 2 Rant sP,n,mj An,mj + 8 j=0 1 Y
X
X
m0 ,m1 , n0 ,n1 ,n2 ,n3 n0 +n1 =n2 +n3 m2 ,m3
"
3 Y
j=0
# sP,nj ,mj Anj ,mj
" 1 " 1 #2 # N−1 N−1 2 X X Y X X Y k2 ρ 3k ρ 4 2 + Rant Rant sI,n,mj An,mj sI,n,mj An,mj + 2 4 n=0 m0 ,m1 j=0 n=0 m0 ,m1 j=0 " 1 # N−1 " 1 # N−1 Y X X Y 3k4 ρ2 2 X X Rant sP,n,mj An,mj sI,n,mj An,mj + 2 n=0 m ,m n=0 m ,m j=0 j=0
0
1
0
and write the standard GP as min
SP ,SI ,ρ,ρ,t ¯ 0
subject to
1/t0
(41)
1 2 2 kSI kF + kSP kF ≤ P, (42) 2 K −γk Y gk (SP , SI , Φ⋆P , Φ⋆I , ρ) ≤ 1, (43) t0 γk k=1 −γnk Kn N −1 Y Y gnk (SI , ρ¯) ¯ R 2 ≤ 1, (44) γnk n=0 k=1
ρ + ρ¯ ≤ 1.
(45)
It is important to note that the choice of {γk , γnk } plays a great role in the tightness of the AM-GM inequality. An iterative procedure can be used where at each iteration the standard GP (41)-(45) is solved for an updated set of {γk , γnk }. (i−1) (i−1) Assuming a feasible set of magnitude SP and SI and power splitting ratio ρ(i−1) at iteration i − 1, compute at
(29)
(32)
1
Algorithm 1 is also known as a sequential convex optimization or inner approximation method [24]. It cannot guarantee to converge to the global solution of the original problem, but only to yield a point fulfilling the KKT conditions [24], [25]. B. PC-ID receiver: Decoupling Space and Frequency When M > 1, previous section derives a general methodology to optimize the superposed waveform weights jointly across space and frequency. It is worth wondering whether we can decouple the design of the spatial and frequency domain weights without impacting performance. The optimal phases in (31) are those of a matched beamformer. Looking at (11), (12), (15), (17) and (25), the optimum weight vectors wP,n and wI,n that maximize the 2nd and 4th order terms and the rate, respectively, are matched beamformers of the form wP,n = sP,n hH n / khn k ,
wI,n = sI,n hH (46) n / khn k , PN −1 (i−1) (i−1) such from (4), yP (t) = g (S ,S ,Φ⋆ ,Φ⋆ ,ρ(i−1) ) n=0 khn k sP,n cos (wn t) = Pthat, N −1 iteration i γk = k P(i−1) I (i−1) P⋆ I ⋆ (i−1) k = 1, . . . , K jwn t ˜ I,n and from (6), w = ℜ zDC (SP ,SI ,ΦP ,ΦI ,ρ ) n=0 khn k sP,n e PN −1 (i−1) (i−1) (i−1) (i−1) kh k x ˜ cos (w t) = / kh k and y (t) = ) , n = hH and γnk = gnk (SI , ρ¯ )/ 1 + ρ¯ σ2 Cn (SI n n n n I n n=0 PN −1 n ˜n ejwn t . Hence, with (46), the multi0, . . . , N − 1, k = 1, . . . , Kn and then solve problem (41)- ℜ n=0 khn k x (i) (i) (45) to obtain SP , SI and ρ(i) . Repeat the iterations till antenna multisine SWIPT weight optimization is converted convergence. The whole optimization procedure for PC-ID into an effective single antenna multisine SWIPT optimization with the effective channel gain on frequency n given by receiver is summarized in Algorithm 1. khn k and the amplitude of the nth sinewave given by sP,n and sI,n for the multisine and OFDM waveform, Algorithm 1 SWIPT Waveform for PC-ID Receiver PN −1 2 2 ⋆ ⋆ respectively (subject to ¯ 1: Initialize: i ← 0, R, ΦP and ΦI , SP , SI , ρ, ρ ¯ = 1 − ρ, n=0 sP,n + sI,n = 2P ). The (0) optimum magnitude sP,n and sI,n in (46) can now be zDC = 0 obtained by using the posynomial maximization methodology 2: repeat of Section VI-A. Namely, focusing on no = 4 for simplicity, ¨ P ← SP , S ¨ I ← SI , ρ¨ ← ρ, ρ¨ 3: i ← i + 1, S ¯ ← ρ¯ plugging (46) into (11), (12), (15) and (17), we get (47) ⋆ ⋆ ⋆ ⋆ ¨P , S ¨ I , Φ , Φ , ρ¨)/zDC (S ¨P , S ¨ I , Φ , Φ , ρ¨), 4: γk ← gk (S P I P I and (48). The DC component zDC as defined in (29) k = 1, . . . , K ¨ ρ ¯ can simply be written as zDC (sP , sI , ρ), expressing that ¨ I , ρ¨ ¨ I ) , n = 0, . . . , N −1, 5: γnk ← gnk (S ¯)/ 1+ σ2 Cn (S n it is now only a function of the N -dimensional vectors k = 1, . . . , Kn s , . . . , s and sI = sI,0 , . . . , sI,N −1 . s = P,0 P,N −1 P 6: SP , SI , ρ, ρ¯ ← arg min (41) − (45) QN −1 2 (1−ρ) 2 (i) ⋆ ⋆ sI,n khn k Similarly, IP C (sI , ρ) = log2 2 n=0 1 + σn 7: zDC ← zDC (SP , SI , ΦP , ΦI , ρ) writes as a function of sI . (i) (i−1) 8: until zDC − zDC < ǫ or i = imax Following the posynomial maximization methodology, we PK gk (sP , sI , ρ) and 1 + can write zDC (sP , sI , ρ) = k=1 P Kn 2 ρ¯ 2 Note that the successive approximation method used in the σn2 Cn = k=1 gnk (sI , ρ¯) with Cn = sI,n khn k , apply the
9
1 A yP (t)2 = 2
"N−1 X
1 E A yI (t)2 = 2
"N−1 X
khn k2 s2P,n
n=0
#
3 4 , A yP (t) = 8
min
1/t0 i 1h 2 2 subject to ksP k + ksI k ≤ P, 2 −γk K Y gk (sP , sI , ρ) t0 ≤ 1, γk k=1 −γnk Kn N −1 Y Y gnk (sI , ρ¯) ¯ R ≤ 1, 2 γnk n=0
(49) (50) (51)
(52)
k=1
ρ + ρ¯ ≤ 1.
n0 ,n1 ,n2 ,n3 n0 +n1 =n2 +n3
"
3 Y
sP,nj
j=0
"N−1 # #2 6 X khn k2 s2I,n , E A yI (t)4 khn k2 s2I,n . = 8 n=0 n=0
AM-GM inequality and write the standard GP problem sP ,sI ,ρ,ρ,t ¯ 0
X
(53)
Algorithm 2 summarizes the design methodology with spatial and frequency domain decoupling. Such an approach
#
hn , j
(47)
(48)
choice of phases, zDC (SP , SI , Φ⋆P , Φ⋆I , ρ) also writes as (32), while the achievable rate is written as ! N −1 Y (1 − ρ)C n 1+ 2 IN C (SI , Φ⋆I , ρ) = log2 σn + (1 − ρ)Dn n=0 (54) Q1 P and D = A s where Cn = n n,m I,n,m j j j=0 m ,m 0 1 Q1 P . A s n,m P,n,m j j j=0 m0 ,m1 The optimization problem can now be written as (37)-(40) with (40) replaced by QN −1 ρ¯ 1 + D 2 n n=0 σn ¯ ≤ 1. 2R Q (55) N −1 ρ¯ 1 + (D + C ) 2 n n n=0 σ n
Defining the setPof monomials of the posynomial 1 + Jn (Dn + Cn ) = j=1 fnj (SP , SI , ρ). For a given choice P Jn γnj = 1, we perform single of {γnj } with γnj ≥ 0 and j=1 condensations and write the standard GP for NC-ID receiver as ρ¯ 2 σn
Algorithm 2 SWIPT Waveform for PC-ID Receiver with Decoupling ¯ wP/I,n and in (46), sP , sI , ρ, ρ¯ = 1: Initialize: i ← 0, R, (0) 1 − ρ, zDC = 0 2: repeat 3: i ← i + 1, ¨sP ← sP , ¨sI ← sI , ρ¨ ← ρ, ρ¨ ¯ ← ρ¯ 4: γk ← gk (¨sP , ¨sI , ρ¨)/zDC (¨sP , ¨sI , ρ¨), k = 1, . . . , K ¨ 5: γnk ← gnk (¨sI , ρ¨ ¯)/ 1 + σρ¯2 Cn (¨sI ) , n = 0, . . . , N − 1, n k = 1, . . . , Kn 6: sP , sI , ρ, ρ¯ ← arg min (49) − (53) (i) 7: zDC ← zDC (sP , sI ρ) (i) (i−1) 8: until zDC − zDC < ǫ or i = imax
would lead to the same performance as the joint spacefrequency design of Algorithm 1 but would significantly reduce the computational complexity since only N -dimensional vectors sP and sI are to be optimized numerically, compared to the N × M matrices SP and SI of Algorithm 2. C. NC-ID receiver With NC-ID receiver, the multisine creates interference to the information decoder, which is reflected in the denominator of the SINR in the rate expression (27). This implies that decoupling the space frequency design by choosing the weights vectors as in (46) is not guaranteed to be optimal. We will therefore resort to a more general approach where space and frequency domain weights are jointly designed, similarly to the one used in Section VI-A. Let us assume the same phases Φ⋆P and Φ⋆I as in the PC-ID receiver in (31). Such a choice is optimal for M = 1 (even though there is no claim of optimality for M > 1). With such a
min
SP ,SI ,ρ,ρ,t ¯ 0
subject to
1/t0
(56)
1 2 2 kSI kF + kSP kF ≤ P, (57) 2 K Y gk (SP , SI , Φ⋆ , Φ⋆ , ρ) −γk P I ≤ 1, (58) t0 γk k=1 N −1 Y ρ¯ ¯ 1 + 2 Dn (SP ) 2R σn n=0 −γnj J n Y fnj (SP , SI , ρ) ≤ 1, (59) γnj j=1
ρ + ρ¯ ≤ 1.
(60)
The whole optimization procedure for NC-ID receiver is summarized in Algorithm 3. Note that Algorithm 3 boils down to Algorithm 1 for Dn = 0 ∀n. VII. S IMULATION R ESULTS We now illustrate the performance of the optimized SWIPT architecture. Parameters are taken as k2 = 0.0034, k4 = 0.3829 and Rant = 50Ω. We assume a WiFi-like environment at a center frequency of 5.18GHz with a 36dBm EIRP, 2dBi receive antenna gain and 58dB path loss. This leads to an average received power of about -20dBm. We assume a large open space environment with a NLOS channel power delay profile with 18 taps obtained from model B [26]. Taps are modeled as i.i.d. circularly symmetric complex Gaussian random variables, each with an average power βl . The multipath
10
3
30 N=2 N=4 N=8 N=16
WPT, ρ=1
25
2.5 20 IDC [µ A]
Frequency response
2.5 MHz 1 MHz
2
Time sharing
15 A, 0
