This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
Nonlinear Amplifier Distortion in Cooperative Amplify-and-Forward OFDM Systems Victor del Razo,1 Taneli Riihonen,1 Fernando Gregorio,2 Stefan Werner1 and Risto Wichman1 1
Helsinki University of Technology P.O. Box 3000, FIN-02015 TKK, FINLAND 2 CONICET-Department of Electrical and Computer Eng. Universidad Nacional del Sur, Av. Alem 1253, Bah´ıa Blanca, 8000 Argentina
Abstract—This paper studies receiver techniques for nonlinear amplifier distortion compensation in an OFDM relay-assisted cooperative communication system. The system model includes a nonlinear amplifier at the amplify-and-forward relay, modeled as a solid state power amplifier (SSPA). A maximum ratio combiner (MRC) is introduced that includes the effects of the amplifier distortions. The MRC is obtained by a proper modeling of the nonlinear distortion noise. Furthermore, we introduce a power amplifier nonlinearity cancellation (PANC) technique that is suitable for cooperative systems. Our simulation results confirm that the new MRC and PANC techniques offer substantial performance improvements if the relayed signal is subject to nonlinear distortion.
I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) is a multi-carrier transmission technique that has been recently used in different communication applications, e.g., digital radio and television, wireless networking systems, high datarates over the fixed phone line, and the Long Term Evolution (LTE) mobile systems. OFDM modulation has proved to operate efficiently in multipath environments at a low cost due to its spectral efficiency and simplicity of implementation and equalization [1]. Cooperative communications origins date back to the relay channel model, containing a source, a destination and a relay. The goal of the relay is to facilitate information transfer from the source to the destination. The relay channel was introduced by Van der Meulen in [2] and investigated extensively by Cover and El Gamal in [3]. These capacity studies considered the relay to work as a helper for the main channel. Cooperative communications evolved from this first approach to become a source of diversity. First proposed by Laneman in [4], cooperative communications is currently a topic under intensive research. In cooperative communications, neighbor devices are used as relays that cooperate with the transmitter and the receiver to provide diversity. The type of cooperation depends on the operation of the relay, e.g., amplify-and-forward (AF) and decode-and-forward (DF). These systems are expected to offer capacity gains similar to those of multiantenna systems [5] without the cost and complexity of increasing the number of antennas in mobile devices. This is achieved by using neighbor devices as virtual antennas.
When AF relays repeat OFDM signals the effect of relay power amplifiers should be taken into account. This is because OFDM systems are known to be sensitive to amplifier nonlinearities due to the high peak-to-average power ratio (PAPR) of OFDM signal. The resulting nonlinear distortion at the amplifier output causes signal waveform distortion and adjacent channel interference. This problem has been previously studied and some techniques for combating the nonlinear distortion effects are already available, see, e.g., [6]–[10]. However, most of these solutions are complex and expensive to implement in mobile devices, where resources are more limited than in equipments connected to the network backbone. This means that the nonlinearity problem in cooperative communication systems has to be approached in a different way. So far, information available on this issue is limited. In [11], the effect of the amplifier nonlinearity at the relay is minimized by limiting the gain to the linear region. However, limiting the relay gain to the linear region or increasing the output backoff (OBO) in OFDM reduces significantly the efficiency of the amplifier because of the high PAPR. In this paper we analyze the effects of nonlinear amplifier distortion (NLD) in OFDM cooperative systems with a single half-duplex AF relay. The objective is to include the NLD effects into the receiver processing to improve the power efficiency of the system. Taking into account NLD gives information on the performance of AF relays in practical systems. By using a similar approach as presented in [12], we model the NLD distortion as additive Gaussian noise. This noise term is used to implement an NLD-aware maximal ratio combiner (MRC). In addition to the NLD-aware MRC approach, we consider a power amplifier nonlinearity cancellation (PANC) technique [13] for mitigating the nonlinear distortion. Considering the more general case of cooperative communication, the basic technique in [13] will not yield an adequate performance because the memory effects due to the source-relay link are not taken into account. The rest of the paper is structured as follows: In Section II, the system model consisting of a single AF relay and independent multipath fading channels is presented. Section III describes the proposed techniques for mitigating the NLD distortion, including an NLD-aware MRC and a PANC for cooperative systems that minimize the detriment in the power
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
nSD
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S
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Fig. 1.
+
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In the case of the relay, the time-domain input of the amplifier can be expressed as
OFDM receiver
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ySR (t) = hSR (t) ∗ xSR (t) + nSR (t),
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and the output of the amplifier depends on the received signal which can be expressed in time domain as xRD (t) = F ySR (t) . (4)
xRD hRD
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System model of the cooperative communication system.
amplifier’s efficiency. In Section IV, we present the results of these techniques in simulations. II. S YSTEM M ODEL The cooperative system to be considered consists of one source (S), one relay (R), and one destination (D); all of them with single antennas as shown in Fig. 1. The three channels involved in this system (hSD , hSR , hRD ) are assumed to be multipath channels and each link is assumed to be distorted by additive white Gaussian noise (AWGN). The signals received from the source and from the relay are assumed to be orthogonal, e.g., they are transmitted on different time slots. This facilitates the use of half-duplex relay that is simpler to implement than full-duplex relay. Full-duplex relays that transmit and receive concurrently in the same frequency are virtually inviable in small devices, because they require separate transmit and receive antennas. The source is an OFDM transmitter with Nc subcarriers. In addition, the transmitter amplification is assumed to be linear and with unit gain. This assumption can be justified by considering that the source is a base station with enough processing and power resources to apply some of the existing techniques [6]–[10] to compensate the amplifier nonlinearities. In the source, a random bit sequence is modulated into an MQAM constellation. After that, inverse fast Fourier transform (IFFT) is applied and the cyclic prefix (CP) is added to generate the OFDM symbol. The relay is an amplify-and-forward (AF) relay with fixed gain. The maximum output power is limited by the saturation voltage. It consists of a nonlinear power amplifier and an adjustable output backoff (OBO) factor. The OBO is the ratio between the saturation power and the actual output power, which can be expressed as 2 Vsat Psat , = OBO = Pout E{|u(t)|2 }
(3)
(1)
where Vsat is the saturation voltage, u(t) is the output of the amplifier and E{|u(t)|2 } is the average output power of the amplifier. The nonlinear amplification is defined by FA [·] and FP [·] that are the AM/AM and AM/PM conversion functions, respectively. The output of the amplifier for a given input x(t) = ρ(t)ejφ(t) can, therefore, be expressed as [12] (2) y(t) = FA [ρ(t)]eFP [ρ(t)] ejφ(t) = F ρ(t) ejφ(t) .
At the destination, the signal is received, transformed, combined, and decoded. Let us first consider the ideal case when the amplifier at the relay is linear with unit gain, i.e., no nonlinear distortion is present in the received signal. The signals received from the source and the relay after removing the CP and applying the FFT are then given by YSD (k) = HSD (k)X(k) + NSD (k), YSRD (k) = HSRD (k)X(k) + NSRD (k).
(5) (6)
where for a given subcarrier k, X(k) is the transmitted symbol, NSD (k) and HSD (k) are the respective additive noise term and channel frequency response of the sourcedestination link, and NSRD (k) = HRD (k)NSR (k) + NRD (k) and HSRD (k) = HRD (k)HSR (k) are the respective additive noise term and channel frequency response of the source-relaydestination link. The additive noise term NSRD (k) will be Gaussian with variance 2 2 2 (k) = |HRD (k)|2 σSR + σRD . σSRD
(7)
The two signals, YSD (k) and YSRD (k), are then combined using an MRC weighting based on the SNR on each link YD (k) =
∗ HSD H ∗ (k) (k) YSRD (k). YSD (k) + 2SRD 2 σSD σSRD (k)
(8)
2 (k) can be estimated in the receiver by The noise term σSRD shutting down the source and transmitting a training signal only from the relay. After the combination, the signal is passed through the detector and the M-QAM demodulator. Note that if nonlinear distortion is generated by the relay, the MRC in (8) is no longer optimal and we can expect an SNR degradation. In next section, we derive an NLD-aware MRC that includes the nonlinear distortion effects.
III. M ITIGATION OF N ONLINEAR A MPLIFIER D ISTORTION Given the random nature of the input QAM symbols, the transmitted OFDM signal x(t) has approximately a Gaussian distribution. For sufficiently long OFDM symbols and relatively slow fading channels, this behavior can be extended to the time-domain signal ySR (t) received by the relay. In principle, if the channel varies slowly compared with the OFDM symbol variations, the channel effect on the received signal becomes quasi-static and the distribution of ySR (t) is similar to that of the transmitted signal. When the input to a nonlinear amplifier is Gaussian, the output of the amplifier can be expressed in terms of a scaled
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
version of the input x(t) (i.e., our time-domain OFDM signal) and an additive uncorrelated noise term d(t) [12]: y(t) = Kx(t) + d(t)
(9)
where K ≤ 1 is the gain of the linear part and d(t) is the nonlinear distortion which is a function of the transmitted symc bols {X(k)}N k=1 and the nonlinear power amplifier transfer function F [·], Nc being the number of OFDM subcarriers. By modeling the NLD as an additive noise term we can consider two options for the receiver processing. The first option is to treat the nonlinear distortion term as purely random and incorporate it in the MRC. Alternatively, we can exploit a known (or estimated) model of the power amplifier at the relay and estimate the NLD term, and thereafter, cancel it from the received signal. The second option leads to an extension of the PANC technique in [13] to the case of cooperative communications.
Fig. 2.
PANC model for a cooperative system.
TABLE I T HE MODIFIED PANC ALGORITHM FOR COOPERATIVE SYSTEMS
For i = 1 to number of iterations. ˆ 1. Demap received symbol: X(k) 2. Transform to time domain: x ˆ(t) 3. Convolve with channel: yˆSR (t) = hSR (t) ∗ x ˆ(t) 4. Amplify using the amplifier model: x ˆRD (t) = F [ˆ ySR (t)] ˆ SR (t)] = x 5. Compute the distortion term: d[y ˆRD (t) − KySR (t) 6. Subtract the distortion term from yRD 7. Transform to frequency domain and combine End
A. NLD-aware MRC The NLD term is introduced by the amplifier at the relay. From (9), the output of the amplifier in time-domain can be modeled as the scaled version of the input plus a noise term: xRD (t) = F [ySR (t)] = KySR (t) + d[ySR (t)] = KhSR (t) ∗ x(t) + KnSR (t) + d[ySR (t)].
(10)
At the destination, the received frequency-domain signal from the relay is now given by YSRD (k) = KHSRD (k)X(k)+ KHRD (k) NSR (k) + K −1 D(k) + NRD (k) (11) where D(k) is the nonlinear distortion at subcarrier k which is related time-domain distortion sequence d[ySR (t)] via the FFT. The main observation here is that if the input saturation voltage of the amplifier is fixed at a low level, several clipping events will be created during an OFDM symbol. In this case, the resulting nonlinear distortion in frequency domain, D(k), 2 can be considered Gaussian [12] with variance σN LD . Employing the Gaussian assumption on D(k), the MRC in (8) is modified to YD (k) =
∗ ∗ HSD KHSRD (k) (k) YSRD (k), Y (k) + SD 2 2 σSD σ ˆSRD (k)
(12)
where 2 2 2 (k) = |KHRD (k)|2 σ ˆSR + σRD , σ ˆSRD 2 σ ˆSR
=
2 σSR
+
2 K −2 σN LD .
(13) (14)
2 Unlike the linear case, σ ˆSRD (k) cannot be estimated by shutting off the source, because there would not be clipping events to cause NLD. Instead, the source should send a training 2 (k) when receiving only from the sequence to estimate σ ˆSRD S-R-D link.
B. PANC for Cooperative Communications The Power Amplifier Nonlinearity Cancellation (PANC) is an NLD mitigation technique applied at the receiver where the NLD is estimated and canceled as an additive term. In [13], [14] a PANC for SISO and MIMO systems are presented, respectively. In a cooperative system the destination receives at least two signals. Under the assumption that nonlinear distortion caused by the source can be mitigated using, e.g., predistortion, NLD can be considered to be present only on the S-R-D branch. In addition, the input of the amplifier is a channel-distorted version of the transmitted symbol. This also has to be considered when implementing the PANC. Figure 2 presents a block diagram of the modified PANC for this system. The algorithm of this modification of the PANC is presented in Table I. ˆ SR (t)] in Fig. 2 The estimate of the distortion term d = d[y is subtracted only from yRD . After taking the IFFT of the deˆ we can generate an estimate of the coded symbols X = X(k), ˆ(t)∗hSR (t). The signal received signal at the relay yˆSR (t) = x yˆSR (t) is then passed through our power amplifier model F [·]. Thereafter, an estimate of the NLD term is obtained from (10) ˆ SR (t)] = x ˆRD (t) − KySR (t). Like in [14], the channels as d[y and the amplifier model F [·] are assumed to be known at ˆ SR (t)] from the destination. After subtracting hRD (t) ∗ d[y yRD (or more precise, HRD (k)D(k) from YRD (k)) a better estimation of the transmitted symbol is obtained. This process can be applied iteratively to improve the symbol estimates ˆ X(k) further. C. NLD-aware MRC and PANC The proposed MRC and PANC cannot be combined directly, because the NLD-aware MRC weights the received signals according to NLD while the PANC’s objective is to remove the NLD. The NLD-aware MRC is optimal for the reception
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
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of the signal yRD but not for yRD − hRD (t) ∗ d[ySR (t)]. In order to combine these two techniques successfully, two steps must be considered: 1) The reception of the signal using NLD-aware MRC 2) The PANC process using regular MRC In the first step, MRC (12) is used for the first estimate of the transmitted symbol, because NLD has not been canceled yet. In the second step PANC has removed the estimated distortion from the received signal, and therefore the MRC in (8) should be used instead.
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To evaluate the performance of the cooperative system affected by the NLD, simulations for evaluating the bit-error rate (BER) are performed, using the same cooperative system without NLD and a non-cooperative (consisting of a source and a destination only) system as references. Note that the transmitted SNR is calculated as the SNR of the S-D link. In other words, the results presented here show the performance of the different implementations for the same transmitted power. The simulations are performed using 16-QAM modulation, equal noise power at the three links, and three independent Rayleigh fading multipath channels with a Doppler spread fc ≈ 10Hz, with the following parameters: • delay profile: 1, 2, 3, 4 subsymbols • power profile (dB): 0, -1, -3, -9 • terminal velocity: 5 km/h • carrier frequency: 2 400 MHz • bandwidth: 6 000 kHz The OFDM symbols contain 512 data subcarriers plus a 16 subsymbols long CP. For simplicity, channel coding and interleaving are not considered and all subcarriers are modulated. The simulations comprise ten thousand OFDM 2 symbols, and the training sequence for the estimation of σN LD is 16 OFDM symbols long. The PANC is iterated three times.
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The nonlinear amplifier is a solid-state power amplifier (SSPA) implemented using the Rapp model [15]: vρ (15) FA [ρ] = [1 + (vρ/Vsat )2r ]1/(2r) FP [ρ] = 0 (16) FA [·] and FP [·] are the AM/AM and AM/PM conversion functions respectively, v = 1 is the small signal gain of the amplifier, r = 3 is the smoothing factor of the transition between the linear operation and the saturation level and Vsat = 1.4V is the saturation voltage of the amplifier. As reference, Fig. 3 shows the performance of the cooperative system for different OBO in the relay when the NLD is not considered, i.e., the amplifier is assumed to be linear even if it is not. From this result it is clear that NLD at the relay impacts the performance of the system significantly. The transmitted
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.
signal has normalized average power and maximum voltage Vmax ≈ 1.3V. Therefore, OBO = 1dB and Vsat = 1.4V should be enough to ensure acceptable performance. This is not the case here because the transmitted signal is distorted by a Rayleigh fading channel before being amplified by the relay, which causes higher NLD in the system. Figure 4 shows the results for the NLD-aware MRC for OBO = 1dB. The performance of the NLD-aware MRC is significantly enhanced when compared to the regular MRC. Since the NLD noise is considered additive, the MRC uses this additional term to define the weight of the S-R-D branch in a more realistic way by giving less weight to this path. NLD-aware MRC outperforms the noncooperative scheme but its performance is still inferior to the linear amplifier. The results for the PANC are presented in Fig 5. When the NLD-aware MRC and PANC are combined properly (NLDaware MRC + PANC + regular MRC), the performance is improved significantly. The BER curve is not just close to the linear case but it also shows a similar behavior. This means that both elements, PANC and MRC, are doing their part. The NLD-aware MRC (12), gives a first estimate of the transmitted symbol. This estimate is good enough for the PANC to accurately estimate the NLD term and subtract it from the received symbol. Once this is done, the symbol can be considered almost distortion-free and the regular MRC (8) combines the signals accordingly. When the NLD-aware MRC and PANC are combined directly (not considering the two steps presented in Section III), the results are better for higher SNR compared with the case where PANC is used with regular MRC. For low SNR, the results are better in the latter case. It is evident from these results that when PANC and MRC are used together, they must be combined as described in Section III. V. C ONCLUSION When using OFDM in cooperative systems, the nonlinear distortion (NLD) becomes an issue that has to be considered carefully. The source-relay channel plays an important role in the resulting NLD of the system. Increasing the output backoff of the amplifier improves the performance in terms of bit-error rate but it makes the system inefficient from the power resources point of view. This paper introduced the maximal ratio combiner (MRC) that properly includes the nonlinear distortion effects. In addition, a modification to the power amplifier nonlinearity cancellation (PANC) technique for cooperative systems was introduced. Details were provided on how to combine the MRC with PANC to yield results close to those observed when employing perfectly linear amplifiers. VI. ACKNOWLEDGMENTS This work was partially supported by the Academy of Finland, Smart and Novel Radios (SMARAD) Center of Excellence, Agencia Nacional de Promocion Cientifica y Tecnologica PICT - 21723 and Univ. Nacional del Sur, Argentina, Project 24/K035.
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