On performance of cooperative network based on OFDM combined with TDM using MMSE-FDC in the presence of nonlinear HPA
- Amir Ligata^{1}Email author,
- Haris Gacanin^{2} and
- Fumiyuki Adachi^{3}
https://doi.org/10.1186/1687-1499-2013-185
© Ligata et al.; licensee Springer. 2013
Received: 20 June 2012
Accepted: 27 June 2013
Published: 8 July 2013
Abstract
In this paper, we analyze the impact of nonlinear high-power amplifier (HPA) on the performance of cooperative network based on orthogonal frequency division multiplexing combined with time-division multiplexing (OFDM/TDM) using minimum mean-square-error frequency-domain combining (MMSE-FDC) in a frequency-selective fading channel. We design a novel MMSE-FDC weights while taking into account the nonlinearity of HPA at source and relay. Closed-form symbol error rate and outage probability expressions are derived while approximating the residual inter-slot interference after the MMSE-FDC as a random Gaussian variable. We discuss and address the nonlinear OFDM/TDM system design issues in cooperative network using the obtained simulation and theoretical results. We show that the OFDM/TDM with MMSE-FDC can be used to reduce the impact of nonlinear HPA on overall performance of cooperative network in comparison to OFDM while providing the target quality-of-service for reduced required signal-to-noise ratio. This is because OFDM/TDM with MMSE-FDC achieves frequency diversity in addition to cooperative diversity, while reduced peak-to-average power ratio makes it more robust on nonlinear degradation due to HPA saturation in comparison to conventional OFDM.
Keywords
1 Introduction
Multi-antenna techniques are promising candidates to achieve broadband data services in a limited available bandwidth. However, their application often encounters practical implementation problem if a large number of antennas is to be deployed. In order to overcome this problem, a new mode of transmit diversity, called cooperative diversity, was proposed based on user cooperation [1, 2], where the antennas of the sender and the partners together form a multiple transmit antenna situation. A variety of algorithms have been developed to obtain cooperative diversity gain [3–9]. Cooperative network based on orthogonal frequency division multiplexing (OFDM) physical-layer access is an attractive solution to achieve cooperative diversity while overcoming the channel frequency selectivity. However, high peak-to-average power ratio (PAPR) of OFDM causes performance degradation due to the nonlinearity of high-power amplifiers (HPA) at the transmitters.
The importance of design choices for different relaying protocols and the implementation complexity as a result of a particular protocol has been discussed with regards to the achieved average system capacity in [10]. In [11], a trade-off between half-duplex (HD) and full-duplex (FD) mode in relay link was studied and a new scheme named hybrid HD/FD based on opportunistic switching between the two modes was presented. Cooperative dual-hop network based on OFDM using amplify-and-forward (AF) relaying with average power scaling (APS) and instantaneous power scaling (IPS) at HPA is analyzed in [12, 13], and it was shown that AF-IPS outperforms AF-APS. The authors in [14] showed that in dual-hop cooperative network based on OFDM with AF protocol relay’s HPA cause only minimal loss in performance, while the authors in [15] showed that the choice of the HPA input back-off is a trade-off between good performance at low signal-to-noise ratio (SNR) or low bit error rate (BER) at high SNR. Methods for compensating the effect of nonlinear HPA in cooperative network based on OFDM using AF and decode-and-forward protocol were presented in [16, 17], showing that the performance considerably approaches to the case with linear HPA. At receiver, a maximum ratio combiner is used in [16], while the authors in [17] used maximum a posteriori probability expectation maximization. Cooperative network using single-carrier frequency-domain equalization (SC-FDE) was presented in [18], while [19] gives a performance comparison of cooperative network based on SC-FDE and OFDM.
The OFDM combined with time-division multiplexing (OFDM/TDM) using minimum mean-square-error frequency-domain equalization (MMSE-FDE) can be used to reduce the PAPR of conventional OFDM [20]. Thus, the cooperative network using OFDM/TDM access based on MMSE frequency-domain combining (MMSE-FDC) in a frequency-selective fading channel may be an attractive solution to achieve both cooperative and frequency diversity gains with a lower PAPR in comparison with cooperative OFDM [21]. Still, the PAPR is not completely eliminated and thus, the nonlinear HPA may considerably degrade the performance of cooperative network based on OFDM/TDM with MMSE-FDC. It is worth mentioning that SC-FDE outperforms OFDM/TDM with MMSE-FDE in terms of BER performance but suffers from inferior channel capacity [22]. However, the benefit of OFDM/TDM over SC-FDE is related to its multi-carrier properties which can be exploited for bit, subcarrier, and power allocation.
Unlike [21], in this work, we present extensive analysis of cooperative network based on OFDM/TDM with MMSE-FDC in frequency-selective fading channel in respect to impact of nonlinear HPA. We resolve to AF protocol at relay with time-domain amplification and equal time sharing for uplink/downlink due to its simple implementation while achieving satisfactory system performance. In addition, we adopt HD at relay (on the cost of reduced spectrum efficiency) since there is no self-interference at relay which is present in FD operation, and consequently, algorithms for mitigating this interference are not needed. We design a MMSE-FDC weights while taking into account nonlinearity of HPA at source and relay. Derivation of the closed-form symbol error rate (SER) and outage probability expressions is done while approximating the residual inter-slot interference (ISI) after MMSE-FDC as a zero-mean Gaussian variable. In addition, we present a closed-form average sum-rate expression for high SNR region. Theoretical average SER has been consisted with the simulation result validating the presented analysis. We discuss and address the nonlinear OFDM/TDM design issue and show that cooperative network based on OFDM/TDM with MMSE-FDC can be used to improve the robustness against nonlinear degradation due to HPA saturation in comparison to conventional OFDM. Simultaneously, the target quality-of-service (QoS) is achieved for reduced required SNR. The remainder of this paper is organized as follows: section 2 presents the network model; performance analysis is provided in section 3, while numerical results and discussions are presented in section 4; and section 5 concludes the paper.
2 Network model overview
In this section, we give an overview of the cooperative network based on OFDM/TDM with MMSE-FDC in the presence of nonlinear HPA. We present the transmit-and-receive signal representation, and then derive the decision variable. In this paper, T_{ c } - spaced discrete time signal representation is used, where T_{ c } represents a sampling interval of fast Fourier transform (FFT).
2.1 Transmit signal representation
The data-modulated symbols are divided into OFDM/TDM frames denoted as {d(i); i = 0 ∼ N_{ c } − 1} with $\mathbb{E}\left[{\left|d\left(i\right)\right|}^{2}\right]=1$ (where $\mathbb{E}[\xb7]$ denotes the ensemble average operation), and then each frame is decomposed into K blocks, each of which has N_{ m } ( = N_{ c }/K) data-modulated symbols [20]. The k th (k = 0, ..., K − 1) block of the OFDM/TDM frame is denoted as {d(k, i); i = 0 ∼N_{ m } − 1}, where d(k, i) = d(k N_{ m } + i). Then, an N_{ m }- point inverse FFT (IFFT) is applied to each of the K blocks to generate a sequence of K OFDM signals with N_{ m } subcarriers. Unlike conventional OFDM, the guard interval (GI) with cycle prefix (CP) is inserted over K slots which constitute the OFDM/TDM frame [20]. The resulting OFDM/TDM signal can be expressed using the equivalent low-pass representation as $s\left(t\right)=\sqrt{P}\sum _{i=0}^{{N}_{m}-1}d(\lfloor t/{N}_{m}\rfloor ,i)exp\left[j2\pi t\right(i/{N}_{m}\left)\right],$ for t = 0 ∼N_{ c } − 1, where P denotes the power coefficient given by P = 2E_{ s } / T_{ c }N_{ c } with E_{ s } being the data-modulated symbol energy and T_{ c } the sampling period. We note here that OFDM/TDM signal with K = 1 (i.e., N_{ m } = N_{ c }) represents the conventional OFDM system with N_{ c } subcarriers, while for K = N_{ c } (i.e., N_{ m } = 1), it becomes the SC system.
We assume a soft envelope limiter model for HPA [23], where the output of HPA represents a linearly amplified input signal for the input amplitude below the HPA saturation level φ_{ s }, while for the input signal amplitudes beyond φ_{ s }, the output signal amplitude is limited to φ_{ s }. Due to the nonlinear input-output characteristic of HPA, the output signal can be approximated as a sum of attenuated input replica and nonlinear noise as [24]$\u015d\left(t\right)={\alpha}_{\mathrm{s}\left(\text{or}\phantom{\rule{0.3em}{0ex}}\mathrm{r}\right)}s\left(t\right)+{c}_{\mathrm{s}\left(\text{or}\phantom{\rule{0.3em}{0ex}}\mathrm{r}\right)}\left(t\right)$, where α_{s(or r)}, s(t), and c_{s(or r)}(t), respectively, denote the attenuation constant, the transmitted signal, and the nonlinear noise due to HPA at source (s) or relay (r). Finally, the signal is multiplied with power coefficient as $\stackrel{~}{s}\left(t\right)=\sqrt{P}\u015d\left(t\right)$ and transmitted over a frequency-selective fading channel.
We assume frequency-selective fading channel having a discrete-time channel impulse response given by ${h}_{\mathit{\text{xy}}}\left(\tau \right)=\sum _{l=0}^{L-1}{h}_{l,\mathit{\text{xy}}}\delta (\tau -{\tau}_{l}),$ where L, h_{l,x y}, τ_{ l }, and δ(τ), respectively, denote the number of paths, the path gain for xy link, and the delay of the l th path and the delta function. x y ∈ {sd, sr, rd} where sd, sr and rd, respectively, denote source-to-destination, source-to-relay, and relay-to-destination links. We assume Jake’s isotropic scattering model where incoming rays constituting each propagation path arrive at a user with uniformly distributed angles [25]. Thus, the normalized autocorrelation function of a Rayleigh faded channel with motion at a constant velocity is given by $R\left(\epsilon \right)=\mathbb{E}\left[{h}_{l,\mathit{\text{xy}}}{h}_{l,\mathit{\text{xy}}+\epsilon}^{\ast}\right]={J}_{0}\left(2\pi {f}_{D}\epsilon \right)$ at delay ε when the maximum Doppler shift is f_{ D }, and where ${J}_{0}\left(\varrho \right)=(1/\pi ){\int}_{0}^{\pi}exp(\mathrm{j\varrho}cos(\varphi \left)\mathrm{d\varphi}\right)$ is the 0th order Bessel function of the first kind.
The signal is transmitted in two orthogonal time stages. In the first time stage, the signal from source is received at destination and relay, while in the second time stage, the signal from relay is forwarded towards destination.
2.2 Received signal representation
2.2.1 Stage I
In Equation 1, α_{ s }, C_{ s }(n), d(k, i), N_{d,1}(n) and H_{sd}(n), respectively, denote the attenuation constant, nonlinear noise due to the source terminals’ HPA, data-modulated symbol, the additive white Gaussian noise (AWGN) having the variance 2N_{0}/T_{ c }N_{ c }, with N_{0} being the single-sided power spectrum density, and the channel gain between source and destination defined as ${H}_{\text{sd}}\left(n\right)=\sum _{t=0}^{{N}_{c}-1}{h}_{\text{sd}}\left(t\right)exp(-j2\pi nt/{N}_{c})$, for n = 0 ∼N_{c} − 1. We note here that the ISI during the first stage is embedded in s(t) ∗ h_{sd}(t), and it is defined after the data-demodulation in section 2.4.
for n = 0 ∼N_{ c } − 1, where ∗, s(t), h_{sr}(t), and n_{ r }(t), respectively, denote the convolution operator, the OFDM/TDM transmitted signal, the channel impulse response between source, and relay and the AWGN at relay.
2.2.2 Stage II
At relay, the AF-IPS protocol is implemented where the received signal is normalized with coefficient $\beta =\sqrt{P/\mathbb{E}\left[\right|{r}_{r}\left(t\right){|}^{2}]}$, and transmitted over the channel during the second time stage. The nonlinear output of HPA at relay can be represented as ${\stackrel{~}{r}}_{r}\left(t\right)=\beta \left[{\alpha}_{r}{r}_{r}\right(t)+{c}_{r}(t\left)\right]$, where α_{ r } and c_{ r }(t), respectively, denote the attenuation constant and nonlinear noise corresponding to relay’s HPA.
where ${\stackrel{~}{R}}_{r}\left(n\right)$, H_{rd}(n), and N_{d,2}(n), respectively, denote the frequency-domain representation of relay’s HPA output, the channel gain between relay and destination, and the AWGN at relay during the second stage. We underline that the ISI during the second stage can be clearly defined only after the data-demodulation in section 2.4.
2.3 MMSE-FDC
where w_{ j }(n) denotes the equalization weight for the j th stage. Below, we present the equalization weight based on MMSE criteria to capture the effect of nonlinear HPA. The equalization weight at the j th (j = 1, 2) stage is chosen to minimize the mean square error (MSE) e_{ j }(n) = R_{d,j}(n)w_{ j }(n) − S(n) as ${\text{MSE}}_{j}=\mathbb{E}\left[{\left|{e}_{j}\left(n\right)\right|}^{2}\right]$. We take into account the following assumptions: (1) the transmitted signal is not correlated with nonlinear noise due to HPA and AWGN at source and relay, (2) the nonlinear noise due to HPA at source/relay is not correlated with AWGN at source/relay, (3) the nonlinear noise due to HPA at source and relay are not correlated, and (4) the AWGN at source and relay are not correlated.
The attenuation constant can be well approximated as [26]${\alpha}_{\mathrm{s}\left(\text{or}\phantom{\rule{0.3em}{0ex}}\mathrm{r}\right)}=1-exp(-\underset{\mathrm{s}\left(\text{or}\phantom{\rule{0.3em}{0ex}}\mathrm{r}\right)}{\overset{2}{\phi}})+\frac{\sqrt{\pi}}{2}{\phi}_{\mathrm{s}\left(\text{or}\phantom{\rule{0.3em}{0ex}}\mathrm{r}\right)}\text{erfc}\left({\phi}_{\mathrm{s}\left(\text{or}\phantom{\rule{0.3em}{0ex}}\mathrm{r}\right)}\right)$ and $\text{erfc}\left(x\right)=(2/\sqrt{\pi})\underset{x}{\overset{\infty}{\int}}exp(-{t}^{2})\mathit{\text{dt}}$ is the complementary error function.
We observe from Equations 6 and 8 that unlike previous works [19–21], the negative effect of HPA saturation at both source and relay has been taken into account when calculating the equalization weights.
2.4 Data demodulation
3 Performance analysis
Here, we first derive the instantaneous SINR expression and afterwards a closed-form SER and outage probability for cooperative network based OFDM/TDM with MMSE-FDC and nonlinear HPA is designed. Finally, the average sum rate is presented.
3.1 SINR
In this subsection, we derive the instantaneous SINR expression γ[P, φ_{ s }, φ_{ r }] while taking into account nonlinear HPA at source and relay.
Next, we present two special cases of OFDM/TDM: the conventional OFDM when K = 1 and SC-FDE when K = N_{ c }.
3.1.1 Special case of conventional OFDM (K = 1)
In the case of conventional OFDM, the expression for instantaneous SINR can be obtained by substituting Equation 13 into Equation 14 for N_{ m } = N_{ c } and Ψ(n, i) = δ(i − n), while the ISI is omitted since GI is inserted between two consecutive OFDM data symbols.
3.1.2 Special case of SC-FDE (K = N_{ c })
In the case of SC-FDE, the instantaneous SINR is obtained by substituting Equation 13 into Equation 14 for N_{ m }=1 and Ψ(n,i)=1, while nonlinear noise due to HPA can be neglected. This is because SC-FDE with quadrature phase shift keying (QPSK) data modulation has a low PAPR, and consequently, nonlinear HPA has no impact on its performance.
3.2 Closed-form SER
where ${\mathcal{M}}_{\gamma}(\xb7)$denotes the moment-generating function (MGF) as a function of the random variable γ which is the instantaneous SINR of OFDM/TDM with MMSE-FDC. The MGF can be calculated as a Laplace transform of the probability density function (PDF), while the PDF is obtained as a derivation of the corresponding SINR’s cumulative density function (CDF).
The SINR given by Equation 14 can be rewritten as γ = X/Y, where X and Y, respectively, denote the useful signal component and the composite noise (i.e., sum of the nonlinear noise due to HPA saturation, the residual ISI after the MMSE-FDE and AWGN). The random variables X and Y represent a sum of many random variables including $\left\{{\mathit{\u0124}}_{\text{sr}}\right(n);\phantom{\rule{0.3em}{0ex}}n=0\sim {N}_{c}-1\}$ and $\left\{{\mathit{\u0124}}_{\text{srd}}\right(n);\phantom{\rule{0.3em}{0ex}}n=0\sim {N}_{c}-1\}$. According to the central limit theorem, each of those variables X_{ i } (Y_{ i }) for i = 1, 2, can be approximated as exponentially distributed (chi-square distribution with 2 degrees of freedom) random variable (as we assume mobile terminals, i.e., Rayleigh fading channels [10, 11]). As a consequence, X and Y may follow a chi-square distribution with 4 degrees of freedom if both variables have the same average. However, the averages may not be the same since the path losses between the source-destination and the source-relay-destination are not the same. Moreover, it was also shown in [28] that in the case when there is a low number of exponentially distributed random variables in the sum, it is reasonable to approximate the resulting variable as exponentially distributed random variable [28]. Thus, to facilitate the analysis, we approximate X and Y as exponential distributed random variables. The random variables X and Y represent a combination of the same equalized channel gains, but one must know that besides those channel gains, each term contains also independent variables (i.e., C_{ s }(n), C_{ r }(n), N_{d,1}(n), N_{ r }(n), N_{d,2}(n)) which are not correlated to the useful signal. Therefore, the nonlinear noise due to HPA, AWGN term, and the residual ISI after the MMSE-FDE are not correlated with the useful signal [22, 24]. Moreover, the Rayleigh fading channel coefficients H_{sr}(n), H_{sd}(n), and H_{rd}(n) are assumed to be independent random variables. By taking into account the above mentioned, it is reasonable to assume as well that X and Y are independent random variables.
We observe that HPA saturation level and OFDM/TDM parameter K affects the SER. In particular, lower saturation level (greater parameter K) will cause higher nonlinear noise (lower residual ISI after MMSE-FDC), and the average SINR will decrease (increase).
3.3 Outage probability
where γ_{th} denotes the SINR threshold. We observe that the outage probability of cooperative network is a function of the OFDM/TDM design parameter K and HPA saturation level. As the parameter K increases, so does the residual ISI after the MMSE-FDC and consequently the outage probability as well. However, if the HPA saturation level increases, corresponding total noise due to HPA saturation level will decrease as will the outage probability. Thus, to achieve the target QoS, these parameters have to be designed properly.
3.4 Average sum-rate
Here, we give a brief overview of the average sum rate of cooperative network based on OFDM/TDM access with MMSE-FDC in the presence of nonlinear HPA at source and relay. We provide an analytical description of the impact of nonlinear degradation on average sum rate performance of cooperative network.
We note here that the average sum rate upper bound which can be easily derived from Equation 24 converges to Equation 26 in the high SNR region.
4 Simulation results and discussion
Numerical parameters
Parameter | Value | |
---|---|---|
Data modulation | QPSK | |
IFFT/FFT size | N_{ m }=256/K | |
Transmitter | No. of | K=1, 16 |
slots | AND 256 | |
GI | N_{ g }=32 | |
Channel | L | 16-path frequency-selective |
block Rayleigh fading | ||
FDE | MMSE | |
Receiver | No. of FFT | N_{ c }=256 |
points | N_{ m }=256/K | |
Channel estimation | Ideal |
4.1 SER
In this subsection, we investigate the SER performance of cooperative network based on OFDM/TDM with MMSE-FDC in the case of ideal and nonlinear HPA.
4.1.1 Ideal HPA
4.1.2 Nonlinear HPA
4.2 Outage probability
4.3 Average sum rate
5 Conclusion
In this paper, we investigate the impact of nonlinear HPA using AF-IPS at relay on the performance of cooperative OFDM/TDM with MMSE-FDC in the frequency-selective fading channel. We design the MMSE-FDC weights while taking into account the nonlinear HPA at source and relay. In addition, we design a closed-form SER and outage probability expressions while approximating the residual ISI after the MMSE-FDC and noise due to HPA saturation as Gaussian random variables. It has been showed that by appropriate design, OFDM/TDM with MMSE-FDC can be used to reduce the impact of nonlinear HPA in cooperative network in comparison to OFDM access while achieving the target QoS for reduced required SNR. This is due to the reduced PAPR of OFDM/TDM and frequency diversity obtained through MMSE-FDC. Furthermore, we show that the nonlinear HPA at relay has limited impact on overall performance of cooperative OFDM/TDM.
Appendix
Here, we present a derivation of the MSE expressions for both stages.
By taking into account that; (1) the transmitted signal S(n) is not correlated with nonlinear noise due to HPA at source (i.e., C_{ s }(n)) and AWGN at destination (i.e., N_{d,2}(n)) and (2) the nonlinear noise due to HPA at source (i.e., C_{ s }(n)) is not correlated with AWGN at destination (i.e., N_{d,2}(n)), we come to Equation 5 in the main text.
Now, by taking into account that (1) the transmitted signal S(n) in not correlated with nonlinear noise due to HPA at source/relay (i.e., C_{ s }(n) and C_{ r }(n)) and AWGN at relay/destination (i.e., N_{ r }(n) and N_{d,2}(n)), (2) the nonlinear noise due to HPA at source/relay (i.e., C_{ s }(n) and C_{ r }(n)) is not correlated with AWGN at relay/destination (i.e., N_{ r }(n) and N_{d,2}(n)), (3) the nonlinear noises due to HPA at source and at destination are not correlated (C_{ s }(n) and C_{ r }(n)), and (4) the AWGN at relay and destination are not correlated (N_{ r }(n) and N_{d,2}(n)), we come to Equation 7 in the main text.
Declarations
Acknowledgements
This study was supported in part by the 2010 KDDI Foundation Research Grant Program.
Authors’ Affiliations
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