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Performance limits of conventional and widely linear DFTprecodedOFDM receivers in wideband frequencyselective channels
EURASIP Journal on Wireless Communications and Networking volume 2014, Article number: 159 (2014)
Abstract
This paper describes the limiting behavior of linear and decision feedback equalizers (DFEs) in single/multiple antenna systems employing real/complexvalued modulation alphabets. The wideband frequencyselective channel is modeled using a Rayleigh fading channel model with infinite number of time domain channel taps. Using this model, we show that the considered equalizers offer a fixed post detection signaltonoise ratio (postSNR) at the equalizer output that is close to the matched filter bound (MFB). General expressions for the postSNR are obtained for zeroforcing (ZF)based conventional receivers as well as for the case of receivers employing widely linear (WL) processing. Simulation is used to study the bit error rate (BER) performance of both minimummeansquareerror (MMSE) and ZFbased receivers. Results show that the considered receivers advantageously exploit the rich frequencyselective channel to mitigate both fading and intersymbol interference (ISI) while offering a performance comparable to the MFB.
1 Introduction
Linear and decision feedback equalizers (DFEs) have been widely studied for the past 50 years. With the introduction of discrete Fourier transformprecodedorthogonal frequencydivision multiple access (DFTprecodedOFDMA) [1, 2] in the uplink of the longterm evolution (LTE) standard [3], there has been renewed interest in the design and analysis of these two receivers operating in wideband frequencyselective channels. DFTprecodedOFDM, also known as singlecarrier FDMA (SCFDMA), is a variant of OFDM in which the modulation data is precoded using the DFT before mapping the data on the subcarriers. The resultant modulation signal exhibits low peaktoaverage power ratio (PAPR). As the frequencyselective channel introduces intersymbol interference (ISI), this method requires sophisticated channel equalization at the receiver.
In broadband wireless systems employing high bandwidths, the propagation channel typically exhibits high frequency selectivity. For these systems, link performance measures such as the diversity order and bit error rate (BER) of a conventional minimum meansquare error (MMSE)based linear equalizers have not yet been fully characterized [4–9]. The noise enhancement phenomenon which is inherent in linear equalizers poses a difficulty in analyzing the receiver performance. The minimum meansquare error decision feedback equalizer (MMSEDFE) [10, 11], on the other hand, is an optimum canonical receiver for channels with ISI. In frequencyselective channels, it provides full diversity, and the performance is generally comparable to the optimum matched filter bound (MFB) [12]. Most of the prior works related to linear and decision feedback equalizers discuss the diversity order of the equalizers and do not quantify the exact performance of the equalizer. In many cases, simulation is typically used to determine the link performance.
The performance loss caused by the decision feedback section of the MMSEDFE can be minimized by using a receiver structure that uses the MMSEDFE feedforward filter (FFF) as a prefilter [13] which provides a minimum phase response followed by a reduced state sequence estimation (RSSE) [14] algorithm that uses set partitioning and state dependent decision feedback principles. Note that the maximum likelihood sequence estimator (MLSE) [15, 16] can be viewed as a special case of RSSE. In typical channels, RSSE with an appropriately chosen number of states performs close to MLSE [17]. In spite of the availability of a number of alternatives to MLSE, linear and decision feedback equalizers are generally preferred in wideband systems due to low implementation complexity.
In DFTprecodedOFDM systems, the MMSEDFE [18–20] equalizer can be implemented efficiently using a frequency domain FFF followed by a time domain DFE [21–31]. Computation of FFF and feedback filters (FBF) for DFTprecodedOFDM differs from conventional singlecarrier methods. Since DFTprecodedOFDM permits frequency domain equalization, it simplifies the computational requirements of both filter calculation and implementation. In [32], an iterative block DFE method is proposed. This method uses a linear equalizer in the first iteration and applies blocklevel soft decision feedback in subsequent iterations. In this paper, we are mainly concerned with the analysis of conventional DFEs based on hard decision feedback.
For realvalued data transmission (e.g., binary phaseshift keying (BPSK) or amplitudeshift keying (ASK)), widely linear (WL) equalizers which jointly filter the received signal and its complexconjugate [33] are known to outperform conventional receivers. This concept has been applied for numerous wireless applications [34–43] including equalization, interference suppression, multiuser detection, etc. Implementation WL equalizers is discussed in [39] for conventional time domain singlecarrier systems. WL receiver algorithms are widely employed in global system for mobile communication (GSM) for (a) lowcomplexity equalization of binary Gaussian minimum shift keying (GMSK) modulation in frequencyselective channels (b) cochannel interference suppression using a singlereceiver antenna. The latter feature is popularly known as single antenna interference cancelation (SAIC) [43, 44].
Throughout this paper, we assume that the receiver has multiple spatially separated antennas. However, the analysis, and the results of this paper hold for the case of single antenna as well. We consider a channel with v time domain taps where the individual taps are modeled as independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean with per tap variance of \frac{1}{v}. The postprocessing signaltonoise power ratio (postSNR) of the considered equalizers is analyzed in the limiting case as v → ∞. Using this model, Kuchi [45] has shown that the SNR at the output of a multiantenna zeroforcing linear equalizer (ZFLE) with N_{ r } antennas reaches a mean value of \frac{{N}_{r}1}{{\sigma}_{n}^{2}}, where {\sigma}_{n}^{2} denotes the noise variance and N_{ r }> 1. For the case of the singlereceiver antenna, both ZFLE and MMSELE are shown to perform poorly. Therefore, it is worthwhile to consider the DFE as an implementation alternative.
In this paper, we further generalize the results of [45] and analyze the limiting performance of three receiver algorithms, namely (a) conventional ZFDFE, (b) WL ZFLE and (c) WL ZFDFE. While ZFbased methods permit analytical evaluation of the postSNR of the receiver, simulation is used to study the performance of MMSEbased receivers. The postSNR bounds developed in this paper provide new insights into the receiver performance. Specifically, we show that, in i.i.d. fading channels with infinitely high frequency selectivity, the postSNR at the output of all the considered receivers reach a fixed SNR. Using these results, we quantify the performance gap of a given receiver with respect to the MFB. In contrast to the previous works where the focus is restricted to diversity analysis, the results of this paper provide a framework to analyze the link performance in channels with high frequency selectivity.
We would like to remark here that in multiuser OFDMA systems, impairments such as frequency offsets, I/Q imbalance, and channel time variations affect the orthogonality of subcarriers and give rise to multiuser interference. Sophisticated equalization techniques are proposed in [46–49] to combat these impairments. In this paper, we restrict our attention to performance analysis in the presence of frequencyselective channels without considering any of the aforementioned impairments.
The organization of the paper is as follows: In section 3, we first generalize the finitelength ZF/MMSEDFE results to the infinitelength case. Then, we obtain a general expression for the postSNR of a ZFDFE for the case of infinite length i.i.d. fading channel under the assumption of errorfree decision feedback (ideal DFE). In section 4, we present the limiting analysis for receivers employing WL processing. Collection of complex and complexconjugated copies of the received signal effectively doubles the number of receiver branches. We show that these additional signal copies obtained through WL processing helps the receiver to obtain a substantially higher postSNR compared to conventional LEs. Analogous to the case of conventional ZFDFE, in section 5, we obtain filter settings for the WL ZF/MMSEDFE receiver. Then a general expression for the postSNR of the WL ZFDFE is obtained for the case of infinitelength i.i.d. fading channel. In section 6, we present simulation results. Finally, conclusions are drawn in section 7.
Notation
The following notation is adopted throughout the paper. Vectors are denoted using boldface lowercase letters, matrices are denoted using boldface uppercase letters. Time domain quantities are denoted using the subscript t. The Mpoint DFT of a vector h_{ t }(l) is defined as \mathbf{h}\left(k\right)=\sum _{l=0}^{M1}{\mathbf{h}}_{t}\left(l\right){e}^{\frac{j2\pi \mathrm{kl}}{M}}, where k = 0,1,..,M  1. The corresponding Mpoint IDFT is given by {\mathbf{h}}_{t}\left(l\right)=\frac{1}{M}\sum _{k=0}^{M1}\mathbf{h}\left(k\right){e}^{\frac{j2\pi \mathrm{kl}}{M}}. The squared Euclidean norm of a row/column vector h(k) = [h_{1}(k),h_{2}(k),..,h_{ n }(k)] is denoted as \left\right\mathbf{h}\left(k\right){}^{2}=\sum _{m=1}^{n}{h}_{m}\left(k\right){}^{2}. The circular convolution between two length N sequences is defined as {x}_{1}\left(n\right)\odot {x}_{2}\left(n\right)=\sum _{n=0}^{N1}{x}_{1}\left(n\right){x}_{2}{\left(\right(mn\left)\right)}_{N} where the subscript in x_{2}((mn))_{ N } denotes modulo N operation and ⊙ denotes circular convolution operation. The symbols †, ∗, Tr denote Hermitian, complexconjugate and transpose operations, respectively and E[.] denotes expectation operator.
2 System model
The DFTprecodedOFDMA transmitter sends a block of M i.i.d. real/complexvalued modulation alphabets with zeromean and variance {\sigma}_{x}^{2}. The DFT precoding of the data stream x_{ t }(l) is accomplished using a Mpoint DFT as
where l and k denote the discrete time and subcarrier indices, respectively. Throughout this paper, we consider a wideband allocation. Therefore, the precoded data is mapped to all the available M contiguous subcarriers. The time domain baseband signal s(t) is obtained using an inverse discrete time Fourier transform (IDTFT)
where T is the useful portion of OFDMA symbol, T_{ CP } is the duration of the cyclic prefix (CP) and \mathrm{\Delta f}=\frac{1}{T} is the subcarrier spacing.
3 MMSEDFE receiver
The receiver front end operations such as sampling, synchronization, CP removal and channel estimation operations are similar to a conventional system. Further, the memory introduced by the propagation channel is assumed to be less than that of the CP duration. Throughout this paper, ideal knowledge of channel state information is assumed at the receiver. We consider a receiver equipped with N_{ r } antennas. Stacking up the time domain sample outputs of multiplereceiver antennas in a column vector format, we get
where
denote the received signal, channel, and noise vectors of size N_{ r }× 1. Here, s_{ t }(l) corresponds to the sampled version of the analog signal s(t). The noise vector n_{ t }(l) is composed of N_{ r } i.i.d. complexGaussian noise random variables each with zeromean and variance \frac{{\sigma}_{n}^{2}}{2} per dimension. Note that h_{ t }(n) is assumed to be a timelimited channel vector where each element of h_{ t }(n) has a duration v samples and M > > v. Taking the Mpoint DFT of y_{ t }(l), we get
where y(k) = DFT[y_{ t }(l)], h(k) = DFT[h_{ t }(l)], x(k) = DFT[s_{ t }(l)], n(k) = DFT[n_{ t }(l)]. In the MMSEDFE receiver (see Figure 1), the received signal is filtered using a vectorvalued feedforward filter to obtain: z(k) = w(k)y(k). Let \overrightarrow{z}\left(k\right)=z\left(k\right)b\left(k\right)x\left(k\right) is the ISI free signal where b(k) is the frequency domain FBF. Here, 1\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}b\left(k\right)=1\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\sum _{l=1}^{L}{b}_{t}\left(l\right){e}^{\frac{j2\pi \mathrm{kl}}{M}} where the FBF is constrained to have L time domain taps. Note that L is a receiver design parameter and its value can be chosen to be equal to the channel memory. As shown in Figure 1, the FBF is implemented in time domain to obtain a decision variable
where z_{ t }(l) is obtained after taking the IDFT of z(k) and {\overrightarrow{z}}_{t}\left(l\right) is the ISI free time domain signal which is fed to the symbol demodulator. Here, the symbol ⊖ denotes right circular shift operation.
MMSEDFE filter expressions are given in [21] for the singlereceiver antenna case and results for multipleinputmultipleoutput (MIMO) systems are available in [31]. In order to obtain an expression for the meansquare error (MSE) which enables closedform analysis in i.i.d. fading channels, in Appendix 1, we first provide expressions for the MMSEDFE filter for the finitelength case, then we generalize the results for the infinitelength scenario. Using these results, for M → ∞, the postSNR defined as the SNR at the output of an unbiased MMSEDFE receiver is given by
3.1 Limiting performance of ZFDFE in wideband channels
Recognizing that the ZFDFE enables one to obtain a closedform expression for the postSNR, we analyze its behavior for the proposed i.i.d. fading channel with infinite length. To this end, let
where v is the effective channel length. We are interested in determining the performance of the link for the limiting case where v → ∞. Note that as v tends to ∞, since M > > v, M also tends to ∞. Therefore,
Note that the variable v is replaced with M in (8) line 2 because as v → ∞, M → ∞, since v < < M. Next, we model h_{ t }(l) as an i.i.d. zeromean, complexGaussian vector with covariance E\left({\mathbf{h}}_{t}\left(l\right){\mathbf{h}}_{t}^{\u2020}\left(l\right)\right)=\frac{\mathbf{I}}{v}. Note that pertap power is set to \frac{1}{v} so that the total power contained in the multipath channel becomes unity. As v → ∞, we can express the covariance term as: {\text{lim}}_{v\to \infty}E\left({\mathbf{h}}_{t}\left(l\right){\mathbf{h}}_{t}^{\u2020}\left(l\right)\right)={\text{lim}}_{v\to \infty}\frac{\mathbf{I}}{v}={\text{lim}}_{M\to \infty}\frac{\mathbf{I}}{M}. Again here, v is replaced with M in the limit as v → ∞. We have an infinite number of taps with vanishingly small power. However, the sum total power of all the taps is equal to unity. Using (8), it can be shown that h(k) approaches an i.i.d. complex Gaussian vector with zero mean and the covariance tends to an identity matrix, i.e. {\text{lim}}_{v\to \infty}E\left(\mathbf{h}\left(k\right){\mathbf{h}}^{\u2020}\left(k\right)\right)\to \mathbf{I}. More specifically, the probability density function of the elements of the channel vector h(k) approaches an i.i.d. complex Gaussian distribution with zero mean and unit variance, and the vectors h(k) become statistically independent for k = 0,1,..,M1.
By setting {\sigma}_{n}^{2}=0 in the numerator of (6), we obtain the postSNR of a ZFDFE as
Applying the central limit theorem (CLT), the r.v., {\text{lim}}_{M\to \infty}\frac{1}{M}\sum _{k=0}^{M1}ln\left\right\mathbf{h}\left(k\right){}^{2} approaches Gaussian distribution with mean
and variance
The expected logarithm of a chisquare random variable with 2N_{ r } degreesoffreedom (DOF) is [50]: E\left[ln\left\right\mathbf{h}\left(k\right){}^{2}\right]=\left(\beta +\sum _{m=1}^{{N}_{r}1}\frac{1}{m}\right) where β = 0.577 is the Euler’s constant, and the variance is [50]: \mathtt{\text{Var}}\left[ln\left\right\mathbf{h}\left(k\right){}^{2}\right]=\sum _{p=1}^{\infty}\frac{1}{(p+{N}_{r}1)}. Since the variance term takes a finite value (the series is absolutely convergent), the variance of {\text{lim}}_{M\to \infty}\frac{1}{M}\sum _{k=0}^{M1}ln\left\right\mathbf{h}\left(k\right){}^{2} approaches zero. Therefore, the SNR at the output of the ZFDFE approaches a constant value of
Note that the ZFLE provides a fixed mean SNR of [45]
For comparison, postSNR corresponding to the MFB is given by {\mathtt{\text{SNR}}}_{\mathtt{\text{MFB}}}=\frac{{N}_{r}}{{\sigma}_{n}^{2}}.
The above result suggests that highly dispersive nature of the frequencyselective channel can be exploited advantageously to obtain a performance comparable to the MFB. After evaluating the expression (13) for the case of a singlereceiver antenna, the ZFDFE provides a postSNR of \frac{0.5616}{{\sigma}_{n}^{2}} that is 2.5dB less than the MFB. For this case, both ZF and MMSEbased LEs perform poorly compared to the MFB [45]. However, the ZFDFE does not suffer from this limitation and provides a substantial gain over MMSE/ZFLE. For N_{ r }= 2, the loss of ZFDFE with respect to the MFB reduces to 1.19 dB whereas the ZFLE has a higher loss of 3.0 dB.
3.2 DFE initialization
In the MMSEDFE implementation considered in this paper, the feedback filter is implemented in the time domain. In (5), the ISI term \sum _{m=1}^{L}{b}_{t}\left(m\right){x}_{t}(l\ominus m) is obtained by circularly convolving the FBF b_{ t }(l) with the data sequence x_{ t }(l). For detecting the first data symbol x_{ t }(0), the receiver has to eliminate the ISI caused by the last L data symbols of the data sequence x_{ t }(l). Specifically, the DFE requires knowledge of the data symbols x_{ i }= [x_{ t }(NL),..,x_{ t }(N2),x_{ t }(N1)]. As proposed in [28], we use a linear equalizer to obtain hard decisions for the required elements contained in x_{ i }. These symbol estimates are then used to initialize the DFE. Simulation shows that this approach works quite well and the loss in the performance compared to the case of an ideal DFE is acceptable. We would like to remark here that an iterative receiver is presented in [23] to address the DFE initialization problem. The results of this paper show that MMSELEbased initialization is sufficient to obtain nearideal performance. An alternative receiver initialization method is also discussed in [31] for trellisbased receivers. Different iterative block DFE methods have been proposed in [23, 32] for DFEprecodedOFDMA systems. These methods use a linear equalizer in the first iteration, then applies block level decision feedback based on soft decisions in subsequent iterations. In this paper, we are mainly concerned with the analysis of conventional DFEs based on hard decision feedback.
4 Widely linear frequency domain MMSE equalizer
For the special case of real constellations, we consider a frequency domain widely linear equalizer which jointly filters the complexvalued received signal and its complexconjugated and frequency reversed copy in frequency domain (see Figure 2). Recall that the frequency domain signal model is given by (4)
Applying complex conjugation and frequency reversal operation on y(k), we get
where we use the fact that x^{∗}(Mk) = x(k) for realvalued modulation data. Combining (4) and (17) in vector form, we have
We note that two copies of the frequency domain modulation signal x(k) are obtained with distinct channel coefficients. Using compact vector notation, \stackrel{\u0304}{\mathbf{y}}\left(k\right)=\stackrel{\u0304}{\mathbf{h}}\left(k\right)x\left(k\right)+\stackrel{\u0304}{\mathbf{n}}\left(k\right). The WL filter \stackrel{\u0304}{\mathbf{w}}\left(k\right)=\left[\mathbf{w}\left(k\right),{\mathbf{w}}^{\ast}(Mk)\right] jointly filters the frequency domain signal y(k), and its complexconjugated and frequencyreversed copy y^{∗}(Mk) to obtain the scalar decision variable denoted as \stackrel{\u0304}{z}\left(k\right). Let \stackrel{\u0304}{z}\left(k\right)=\mathbf{w}\left(k\right)\mathbf{y}\left(k\right)+{\mathbf{w}}^{\ast}(Mk){\mathbf{y}}^{\ast}(Mk)=\stackrel{\u0304}{\mathbf{w}}\left(k\right)\stackrel{\u0304}{\mathbf{y}}\left(k\right). An estimate of the desired data is obtained as {\stackrel{\u0304}{z}}_{t}\left(l\right)=\text{IDFT}\left[\stackrel{\u0304}{z}\right(k\left)\right]. Using standard MMSE estimation [18], the vectorvalued WL MMSE filter is given by
Since \stackrel{\u0304}{\mathbf{w}}\left(k\right)=\left[\mathbf{w}\left(k\right),{\mathbf{w}}^{\ast}(Mk)\right], where \mathbf{w}\left(k\right)=\frac{{\mathbf{h}}^{\ast}\left(k\right)}{\frac{{\sigma}_{n}^{2}}{{\sigma}_{x}^{2}}+{\stackrel{\u0304}{\mathbf{h}}}^{\u2020}\left(k\right)\stackrel{\u0304}{\mathbf{h}}\left(k\right)}, it is computationally efficient to calculate the filter w(k) explicitly. The filter w^{∗}(Mk) can be obtained from w(k) with low computational complexity using complex conjugation and frequency reversal operations. The minimum MSE for this case is expressed as
Note that
Using this result, the MSE can be expressed as
The postSNR defined as the SNR at the output of the WL MMSE receiver is given by
where D=\frac{1}{M}\sum _{k=0}^{M1}\left[\frac{1}{\frac{{\sigma}_{n}^{2}}{{\sigma}_{x}^{2}}+\left(\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}\right)}\right].
4.1 Liming performance of WL ZFLE
To obtain a closedform expression for the postSNR at the output of the equalizer, we analyze the performance of a ZF WLLE. Letting {\sigma}_{n}^{2}=0 in the denominator of D, for v → ∞, we let M → ∞. In this case, we get
Let Q(k) = h(k)^{2} + h(Mk)^{2}. Then we have Q(k) = Q(Mk), Q(0) = 2h(0)^{2}, and Q\left(\frac{M}{2}\right)=2{\left\left\mathbf{h}\left(\frac{M}{2}\right)\right\right}^{2}. Using this,
For an i.i.d. channel with infinite frequency selectivity, the entries of h(k) are i.i.d. complex Gaussian r.v.’s with zero mean and unit variance. Therefore, h(k)^{2} has chisquare distribution. Since h(k)^{2} always takes positive values, in the limiting case as M → ∞, the first two terms of D become vanishingly small. Then we end up with
The expected value of D is given by
Since h(k) and h(Mk) are i.i.d r.v’s, [h(k)^{2} + h(Mk)^{2}] is a sum of squares of 2N_{ r } i.i.d. complex Gaussian r.v’s. The term \left[\frac{1}{\left[\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}\right]}\right] has inverse chisquare distribution with 4N_{ r } realvalued DOF. Applying the result of [51], we have E\left[\frac{1}{\left[\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}\right]}\right]=\frac{1}{2{N}_{r}1} and \mathtt{\text{Var}}\left[\frac{1}{\left[\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}\right]}\right]=\frac{1}{2(2{N}_{r}1)({N}_{r}1)} for N_{ r }> 1. Therefore, the variance of D is given by
The postSNR of WL ZFLE reaches a constant value of
Note that the variance of \left[\frac{1}{\left[\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}\right]}\right] is bounded only for N_{ r }> 1.
4.1.1 Remark
In Equation 24, h(0)^{2} and \left\right\mathbf{h}\left(\frac{M}{2}\right){}^{2} are sum of squares of N_{ r } i.i.d. complex Gaussian r.v.’s which give a chisquare random variable with 2N_{ r } DOF while [h(k)^{2}+h(Mk)^{2}] has chisquare random variable with 4N_{ r } DOF. For the special case of N_{ r }= 1, the expected value of \frac{1}{\left\right\mathbf{h}\left(0\right){}^{2}} or \frac{1}{\left\right\mathbf{h}\left(\frac{M}{2}\right){}^{2}} is unbounded since it has inverse chisquare distribution with two DOF. However, the mean of \frac{1}{\left[\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}\right]} is bounded for any value of N_{ r }. In the limiting case as M → ∞, the contribution of the first two terms in (24) vanishes. However, for the special case of N_{ r }= 1, and for finite values of v, h(k) and h(Mk) become correlated random variables. Specifically for values of k = 0 and k=\frac{M}{2}, these terms become equal while for values of k in the vicinity of 0 and \frac{M}{2} they become highly correlated. Considering the first two terms of Equation 24, we see that the terms \frac{1}{\left\right\mathbf{h}\left(0\right){}^{2}} or \frac{1}{\left\right\mathbf{h}\left(\frac{M}{2}\right){}^{2}} contribute to an increase in the MSE. Similarly, since h(k) and h(Mk) can be highly correlated for certain subcarrier locations, the term \frac{1}{\left[\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}\right]} contributes to an increase in MSE for those subcarrier locations. The overall increase in the MSE can be controlled by considering a WL MMSE which regularizes the denominator terms. Simulation is used to quantify the gain of WL MMSELE over ZF case.
For detection of realvalued symbols, only the real part of the noise at the output of the equalizer contributes to the error rate. Taking this into account, the postSNR of a conventional ZFLE should be modified as
5 WL MMSEDFE
In the WL MMSEDFE receiver (see Figure 3), the received signal and its conjugated timereversed replicas are filtered as
where \stackrel{\u0304}{\mathbf{w}}\left(k\right)=\left[\mathbf{w}\left(k\right),{\mathbf{w}}^{\ast}(Mk)\right] is composed of two vectorvalued filters. Next, the ISI is eliminated using a feedback filter as
Note that \stackrel{\u0304}{b}\left(k\right) is the frequency domain feedback filter where 1+\stackrel{\u0304}{b}\left(k\right)=1+\sum _{l=1}^{L}{\stackrel{\u0304}{b}}_{t}\left(l\right){e}^{\frac{j2\pi \mathrm{kl}}{M}}. The coefficients of the FBF take real values only. The FBF is implemented in the time domain to obtain a decision variable
which is fed to the symbol demodulator. Here, {\stackrel{\u0304}{z}}_{t}\left(l\right)=\text{IDFT}\left[\stackrel{\u0304}{z}\right(k\left)\right] and {\widehat{z}}_{t}\left(l\right)=\text{IDFT}\left[\widehat{z}\right(k\left)\right]. In Appendix 2, we generalize the MMSEDFE results for the WL case and provide expressions for the FFF, FBF, and the MSE. We note here that the FFF and FBF expressions are distinct from the ones reported in the literature [30]. In addition, the receiver design presented in the Appendix has low implementation complexity. Using the results in the Appendix 2, the postSNR of the WL MMSEDFE, for M → ∞, is given by (60)
5.1 Performance of WL ZFDFE in wideband channels
Setting {\sigma}_{n}^{2}=0 in (33), the postSNR of a WL ZFDFE can be expressed as
Inside the logarithm, we have a sum of squares of 2N_{ r } i.i.d. complex Gaussian r.v’s. In the limiting case as v → ∞, generalizing the analysis used for the conventional ZFDFE, we can show that the SNR of WL ZFDFE reaches a fixed value of
For realvalued modulation, since only the real part of the noise is relevant, the conventional ZFDFE provides a fixed SNR of
For N_{ r }= 1, the ideal WL ZFDFE offers a postSNR of \frac{1.5265}{{\sigma}_{n}^{2}} that is 1.17 dB away from the MFB. The actual performance gap with practical FBF is determined using BER simulation.
5.1.1 Remarks

For the case of WL ZF/MMSEDFE, we ignore the potential MSE increase contributed by the terms located at k = 0 and k=\frac{M}{2}. Since at these locations, the exponent in (34) involves the terms E[lnh(0)^{2}], E\left[ln{\left\left\mathbf{h}\left(\frac{M}{2}\right)\right\right}^{2}\right] which take a finite value, the overall increase in the MSE can be neglected for finite values of M.

We note here that our main goal of the paper is to expose the basic properties of conventional and WL equalizers in wideband channels. Our aim is not to promote the use of real constellations over typically used complex modulation methods. However, the analysis and results related to WL equalizers are useful in systems where real constellations are employed. One such application is discussed in [52] where binary modulation along with duobinary precoding is employed in the uplink of DFTprecodedOFDM to reduce the PAPR.
6 Results
We present BER simulation results for BPSK and 8PSK and 16QAM (quadrature amplitude modulation) systems. In all cases, the FBF length is set equal to the channel memory. Throughout the paper, we present results for a 20tap i.i.d. Rayleigh fading channel with M=512 in all cases.
6.1 BER results for conventional equalizers
In the following, we consider the BER performance of the singleantenna receivers. In Figure 4, the results are shown for BPSK modulation using ZF receivers. The BER of ideal DFE is close to the conventional DFE up to BER = 10^{4} and shows a degradation at low error rates. This loss is mainly caused by the imperfect initialization of the DFE. Note that the ZFLE performs poorly due to high noise enhancement. Therefore, initialization of the ZFDFE with ZELE decisions leads to severe error propagation. In Figure 5, BER results are given for 16QAM system using ZF receivers. The results show that when the ZFDFE is initialized using known data, the BER follows the ideal DFE case while initialization using the ZFLE leads to an error floor. In Figure 6, BER is given for 16QAM employing MMSEbased receivers. Unlike the ZF case, initialization of the MMSEDFE using MMSELE does not cause severe error propagation and the BER is within 2.0 dB of ideal MMSEDFE. With lower modulation alphabets, like BPSK, the difference between the BER of MMSEDFE and ideal MMSEDFE is small (see Figure 7) and the difference increases for higher order constellations. This loss may be reduced using lowcomplexity sequence estimation techniques such as RSSE.Next, we consider the BER performance of BPSK and 8PSK system with two antennas (see Figures 8 and 9). We observe that in the presence of multiplereceiver antennas, the BER of LE improves considerably compared to the single antenna case. As a result, initialization of DFE using the LE does not cause significant degradation.
The theoretical performance gap between the postSNR of the considered receivers and the MFB is tabulated in Table 1 for an i.i.d. channel with infinite length. In Tables 2 and 3, we report the gap measured at BERs of 0.01 and 0.001, respectively. For ZFbased receivers, the gap measured using simulation is in good agreement with the analytically obtained results.
6.2 BER of WL equalizers
In Figure 10, we show the BER for WL MMSEbased receivers employing BPSK modulation for the case of N_{ r }= 1. Comparing with the results of Figure 7, we see that WL processing provides a gain over conventional receivers. In Figure 11, the results are given for WL ZF receiver with N_{ r }= 1. In section 4, it is shown that the postSNR of WL ZFLE approaches \frac{(2{N}_{r}1)}{{\sigma}_{n}^{2}} when v → ∞. For the case of N_{ r }= 1 and for finite values of v, we argued that certain SNR penalty is expected. For the single antenna case, while we expect a 3.0dB SNR gap between the postSNR of WL ZFLE and MFB, Tables 2 and 3 show a gap of 3.2 and 3.8 dB, respectively. However, for the dual antenna case, the gap reported in Tables 2 and 3 is in good agreement with the analytically obtained results given in Table 1.
Referring to Figures 10 and 11, we note that the SNR difference between ideal DFE and actual DFE with LEbased initialization is approximately 0.4 dB for both ZF and MMSE cases for N_{ r }= 1. In Figures 12 and 13, results are given for the case of N_{ r }= 2. We see that the performance of actual DFE is very close to that of the ideal DFE. The additional DOF obtained by WL processing aid the WL DFEs to mitigate the error propagation.
7 Conclusions
This paper describes the limiting behavior of conventional and WL equalizers in wideband frequencyselective channels. For systems employing DFTprecodedOFDM modulation, closedform expressions are obtained for the postSNR of conventional and WL receivers employing ZFLE and ZFDFE; simulation is used to assess the performance of MMSEbased receivers. In i.i.d. fading channels with infinite channel memory, the postSNR reaches a fixed value that is comparable to the MFB in most cases.
Both conventional MMSELE and ZFLE offer near optimal performance only when the receiver has multiple antennas, whereas ideal ZFDFE and ideal MMSEDFE perform close to the MFB with a fixed SNR penalty even when the receiver has a single antenna. For singleantenna MMSEDFE with decision feedback, the penalty compared to the ideal DFE is approximately 2.0 dB for 16QAM systems at high SNRs. The total gap compared to MFB is 4.5 dB. Lowcomplexity receiver algorithms that further reduce this gap need to be developed. Unlike the single antenna case, the presence of multiple antennas helps the DFEs to reach a performance close to the MFB. Multiplereceiver antennas are also shown to reduce the error propagation of the DFEs.
For singleantenna systems employing realvalued modulation alphabets, WL receiver processing can be used to obtain a performance advantage over conventional receivers. In particular, the WL MMSELE performs within 3.2 to 3.8 dB of the MFB while the WL MMSEDFE reduces the gap with respect to the MFB to 1.0 dB. Results show that the multiantenna WL receivers (both LEs and DFEs) perform very close to the MFB as predicted by the infinite length i.i.d. fading channel model.
We note here that the proposed infinite length i.i.d fading channel model can be used to obtain the limiting performance of MIMO systems employing spatial multiplexing (SM). The analysis has been carried out in [53] for the case of MIMO ZFLE where it is shown that the postSNR of the receiver reaches a constant value of \frac{{N}_{r}{N}_{t}}{{\sigma}_{n}^{2}} for N_{ r }> N_{ t }, where N_{ t } is the SM rate. Extension to the general case of SM employing ZF/MMSEDFEs is yet to be considered.
Endnote
^{a} (A + B C D)^{1} = A^{1} + A^{1}B(C^{1} + D A^{1}B)^{1}D A^{1}
Appendices
Appendix 1
Derivation of MMSEDFE filter settings
We obtain closedform expressions for the FFF, FBF and MSE for the case when the FBF is restricted to have finite length. First, we show that the MSE minimizing solution for the FBF becomes a finite length prediction error filter that whitens the error covariance at the output of the MMSELE. The solution obtained in our case becomes a multiplereceiver antenna generalization of the results presented in [21]. Similarly, Gerstacker et al. [31] presented an alternative approach for MIMO systems where the problem of designing the FBF is formulated as one of finite length prediction error filter design. This alternative approach results in a solution that agrees with our results for the multiplereceiver antenna case. We further generalize our results to the infinite length filter case which facilitates performance analysis in i.i.d. fading channels. The derivations presented in this section follow the approach presented in [13, 18]. We define an error signal
This is written in time domain as
Define: r_{ ee }(k) = E(e(k)^{2}). Using Parseval’s theorem: \frac{1}{M}\sum _{k=0}^{M1}\left\righte\left(k\right){}^{2}=\sum _{l=0}^{M1}\left{e}_{t}\right(l\left)\right{}^{2}. Taking expectation on both sides, we get
The MSE is defined as MSE=E(e_{ t }(l)^{2}) which is independent of the time index l. It can be written as
Next, we obtain an expression for the FFF in frequency domain. Applying orthogonality principle [18]
Substituting (37), in (39), and evaluating the expectation, the FFF can be expressed as
where R_{ xy }(k) = E(x(k)y^{†}(k)) = r_{ xx }(k)h^{†}(k) and R_{ yy }(k) = E(y(k)y^{†}(k)) = [h(k)r_{ xx }(k)h^{†}(k) + R_{ nn }(k)]. Here {r}_{\mathit{\text{xx}}}\left(k\right)=E\left(\left\rightx\left(k\right){}^{2}\right)=M{\sigma}_{x}^{2} and R_{ nn }(k) = E\left(\mathbf{n}\left(k\right){\mathbf{n}}^{\u2020}\left(k\right)\right)=M{\sigma}_{n}^{2}\mathbf{I}. The FFF can be expressed in alternative form as
Note that (43) follows from applying matrix inversion lemma^{a}. With this choice of FFF, the minimum MSE can be shown to be [13, 18]
To obtain the coefficients b_{ t }(l), we take the partial derivatives
Let
Treating b_{ t }(l) and {b}_{t}^{\ast}\left(l\right) as independent variables, we get \frac{\partial}{\partial {b}_{t}\left(l\right)}\left[{\left\left\left(1+\sum _{l=1}^{L}{b}_{t}\left(l\right){e}^{\frac{j2\mathrm{\pi kl}}{M}}\right)\right\right}^{2}\right]=\left[{e}^{\frac{j2\mathrm{\pi kl}}{M}}+\left(\right)close=")">\sum _{m=1}^{L}{b}_{t}^{\ast}\left(m\right){e}^{\frac{j2\mathrm{\pi k}(lm)}{M}}\right]\n Substituting this result in (45) and setting the partial derivatives to zero, we obtain the MSE minimizing condition
We define the following IDFT pair
Note that\frac{{\sigma}_{n}^{2}}{\left\right\mathbf{h}\left(k\right){}^{2}+\frac{{\sigma}_{n}^{2}}{{\sigma}_{x}^{2}}} is the frequency domain error covariance at the output of a MMSELE [18] where q(l) is the corresponding time domain error covariance. Now, we can express (46) in compact form as
where the (l,m)th element of the matrix A is given by A(l,m) = q(ml), b = [b_{ t }(1),b_{ t }(2),..,b_{ t }(L)]^{Tr}, and q = [q(1),q(2),..,q(L + 1)]^{Tr}. The elements of the FBF can be obtained by solving (48). It can be seen that the MSE minimizing solution for the FBF becomes a finite length prediction error filter of order L that whitens the error covariance at the output of the MMSELE. The FBF coefficients can be calculated efficiently using the LevinsonDurbin recursion. The minimum MSE can be obtained by substituting the values of the FBF coefficients in the MSE expression (44). Next, we characterize the MMSEDFE for the case of M→∞.
Expanding the log spectrum using the DFT
where
with c\left(0\right)=\frac{1}{M}\sum _{k=0}^{M1}ln\left[\left\right\mathbf{h}\left(k\right){}^{2}+\frac{{\sigma}_{n}^{2}}{{\sigma}_{x}^{2}}\right] and c(l) = c^{∗}(l) = c^{∗}(M ⊖ l). For odd values of M, let us rewrite (49) as
where γ = e^{c(0)}, g\left(k\right)={e}^{\sum _{l=1}^{\frac{M1}{2}}c\left(l\right){e}^{\frac{j2\mathrm{\pi lk}}{M}}}. In the limiting case as M → ∞, the DFTs approach discretetime Fourier transforms (DTFTs) i.e.,
where
The result on line 2 is obtained by expanding the exponential function into an infinite series. The coefficients g_{ t }(l) are obtained by collecting appropriate terms in the summation on line 2. Note that g(f) is a causal, monic, and minimumphase filter with all its poles and zeros inside the unit circle. Similarly, g^{∗}(f) is a noncausal, monic, and maximum phase filter. For M → ∞, (50) can be written as
where \mathbf{h}\left(f\right)=\sum _{l=0}^{\infty}{\mathbf{h}}_{t}\left(l\right){e}^{j2\mathrm{\pi lf}} is the DTFT of h_{ t }(l) which is a periodic in f with period 1. Further,
Now, we write the spectrum factorization for the continuous case as [13, 18]
The MSE given by (44) becomes
where
The optimum choice which minimizes the MSE given in (53) is given by 1+b(f) = g(f) [13, 18]. Using this, and substituting (51) in the MSE expression (53), we obtain the minimum MSE as
Assuming ideal decision feedback, the SNR at the output of the MMSEDFE is given by
Appendix 2
Derivation of WL MMSEDFE filter settings
In this section, we discuss the design aspects of WL MMSE DFE. The key implementation differences between conventional and WL equalizers are highlighted. Specifically, we notice that the noise covariance term at the output of the WL MMSE section exhibits even symmetry in frequency domain. This property is exploited to reduce the computational complexity of FFF and FBF filter calculation.
The error signal for WL case is defined as
In time domain
where {\stackrel{\u0304}{b}}_{t}\left(l\right)=\text{IDFT}\left(\stackrel{\u0304}{b}\right(k\left)\right). Let {\stackrel{\u0304}{r}}_{\mathit{\text{ee}}}\left(k\right)=E\left(\right\u0113(k\left)\right{}^{2}). The total MSE is given by
Following the conventional MMSEDFE case, the MSE minimizing solution for the WL FFF can be obtained as
Let P\left(k\right)=\left\right\mathbf{h}\left(k\right){}^{2}+\left\mathbf{h}\right(Mk\left)\right{}^{2}+\frac{{\sigma}_{n}^{2}}{{\sigma}_{x}^{2}}. This is a realvalued function which exhibits even symmetry, i.e. P(k) = P(Mk) for k = 0,1,..,M1. Similar to the WL MMSELE case, it is computationally efficient to calculate the filter w(k) explicitly. The second filter can be obtained from the first by applying complex conjugation and frequency reversal operations.
The minimum MSE can be expressed as
Consider the partial derivatives
Let
Substituting this result in (54) and setting the partial derivatives to zero, we get
Let us define the following transform pair
It can be implemented with low complexity using standard inverse fast Fourier transform (IFFT) algorithm. Alternatively, noting that P(k) = P(Mk), we can write (56) as
The last term involves \frac{M}{2} point type1 DCT of P(k). Note that \stackrel{\u0304}{q}\left(l\right) needs to be calculated only for the first \frac{M}{2} terms since the rest of the coefficients can be obtained exploiting the even symmetry of \stackrel{\u0304}{q}\left(l\right) i.e., \stackrel{\u0304}{q}\left(l\right)=\stackrel{\u0304}{q}(Ml).
The MSE minimizing condition (55) can be expressed in vectormatrix form as
where the (l,m)th element of the matrix \stackrel{\u0304}{\mathbf{A}} denotes as \u0100(l,m) is given by \u0100(l,m)=\stackrel{\u0304}{q}(ml), \stackrel{\u0304}{\mathbf{b}}={\left[{\stackrel{\u0304}{b}}_{t}\right(1),{\stackrel{\u0304}{b}}_{t}(2),\mathrm{..},{\stackrel{\u0304}{b}}_{t}(L\left)\right]}^{\mathit{\text{Tr}}}, and \stackrel{\u0304}{\mathbf{q}}={\left[\stackrel{\u0304}{q}\right(1),\stackrel{\u0304}{q}(2),\mathrm{..},\stackrel{\u0304}{q}(L\left)\right]}^{\mathit{\text{Tr}}}. Note that FBF can be calculated with low complexity using LevinsonDurbin recursion which involves realvalued quantities whereas the FBF for the conventional case involves complex values. Now we consider the infinite length filter case.
Following the derivations for the conventional case, we can show that
and
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Acknowledgements
This work was carried out as part of Converged Cloud Communication Technologies project sponsored by the Department of Electronics and Information Technology (DeitY), Government of India.
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Kuchi, K. Performance limits of conventional and widely linear DFTprecodedOFDM receivers in wideband frequencyselective channels. J Wireless Com Network 2014, 159 (2014). https://doi.org/10.1186/168714992014159
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DOI: https://doi.org/10.1186/168714992014159
Keywords
 Fading Channel
 Cyclic Prefix
 Decision Feedback
 Widely Linear
 Decision Feedback Equalizer