- Open Access
Performance limits of conventional and widely linear DFT-precoded-OFDM receivers in wideband frequency-selective channels
© Kuchi; licensee Springer. 2014
- Received: 13 March 2014
- Accepted: 26 August 2014
- Published: 3 October 2014
This paper describes the limiting behavior of linear and decision feedback equalizers (DFEs) in single/multiple antenna systems employing real/complex-valued modulation alphabets. The wideband frequency-selective 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 signal-to-noise ratio (post-SNR) at the equalizer output that is close to the matched filter bound (MFB). General expressions for the post-SNR are obtained for zero-forcing (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 minimum-mean-square-error (MMSE) and ZF-based receivers. Results show that the considered receivers advantageously exploit the rich frequency-selective channel to mitigate both fading and inter-symbol interference (ISI) while offering a performance comparable to the MFB.
- Fading Channel
- Cyclic Prefix
- Decision Feedback
- Widely Linear
- Decision Feedback Equalizer
Linear and decision feedback equalizers (DFEs) have been widely studied for the past 50 years. With the introduction of discrete Fourier transform-precoded-orthogonal frequency-division multiple access (DFT-precoded-OFDMA) [1, 2] in the uplink of the long-term evolution (LTE) standard , there has been renewed interest in the design and analysis of these two receivers operating in wideband frequency-selective channels. DFT-precoded-OFDM, also known as single-carrier FDMA (SC-FDMA), 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 peak-to-average power ratio (PAPR). As the frequency-selective channel introduces inter-symbol 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 mean-square 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 mean-square error decision feedback equalizer (MMSE-DFE) [10, 11], on the other hand, is an optimum canonical receiver for channels with ISI. In frequency-selective channels, it provides full diversity, and the performance is generally comparable to the optimum matched filter bound (MFB) . 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 MMSE-DFE can be minimized by using a receiver structure that uses the MMSE-DFE feed-forward filter (FFF) as a pre-filter  which provides a minimum phase response followed by a reduced state sequence estimation (RSSE)  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 . 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 DFT-precoded-OFDM systems, the MMSE-DFE [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 DFT-precoded-OFDM differs from conventional single-carrier methods. Since DFT-precoded-OFDM permits frequency domain equalization, it simplifies the computational requirements of both filter calculation and implementation. In , an iterative block DFE method is proposed. This method uses a linear equalizer in the first iteration and applies block-level 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 real-valued data transmission (e.g., binary phase-shift keying (BPSK) or amplitude-shift keying (ASK)), widely linear (WL) equalizers which jointly filter the received signal and its complex-conjugate  are known to outperform conventional receivers. This concept has been applied for numerous wireless applications [34–43] including equalization, interference suppression, multi-user detection, etc. Implementation WL equalizers is discussed in  for conventional time domain single-carrier systems. WL receiver algorithms are widely employed in global system for mobile communication (GSM) for (a) low-complexity equalization of binary Gaussian minimum shift keying (GMSK) modulation in frequency-selective channels (b) co-channel interference suppression using a single-receiver 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 . The post-processing signal-to-noise power ratio (post-SNR) of the considered equalizers is analyzed in the limiting case as v → ∞. Using this model, Kuchi  has shown that the SNR at the output of a multi-antenna zero-forcing linear equalizer (ZF-LE) with N r antennas reaches a mean value of , where denotes the noise variance and N r > 1. For the case of the single-receiver antenna, both ZF-LE and MMSE-LE 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  and analyze the limiting performance of three receiver algorithms, namely (a) conventional ZF-DFE, (b) WL ZF-LE and (c) WL ZF-DFE. While ZF-based methods permit analytical evaluation of the post-SNR of the receiver, simulation is used to study the performance of MMSE-based receivers. The post-SNR 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 post-SNR 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 multi-user OFDMA systems, impairments such as frequency offsets, I/Q imbalance, and channel time variations affect the orthogonality of subcarriers and give rise to multi-user 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 frequency-selective channels without considering any of the aforementioned impairments.
The organization of the paper is as follows: In section 3, we first generalize the finite-length ZF/MMSE-DFE results to the infinite-length case. Then, we obtain a general expression for the post-SNR of a ZF-DFE for the case of infinite length i.i.d. fading channel under the assumption of error-free decision feedback (ideal DFE). In section 4, we present the limiting analysis for receivers employing WL processing. Collection of complex and complex-conjugated 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 post-SNR compared to conventional LEs. Analogous to the case of conventional ZF-DFE, in section 5, we obtain filter settings for the WL ZF/MMSE-DFE receiver. Then a general expression for the post-SNR of the WL ZF-DFE is obtained for the case of infinite-length i.i.d. fading channel. In section 6, we present simulation results. Finally, conclusions are drawn in section 7.
The following notation is adopted throughout the paper. Vectors are denoted using bold-face lower-case letters, matrices are denoted using boldface upper-case letters. Time domain quantities are denoted using the subscript t. The M-point DFT of a vector h t (l) is defined as , where k = 0,1,..,M - 1. The corresponding M-point IDFT is given by . The squared Euclidean norm of a row/column vector h(k) = [h1(k),h2(k),..,h n (k)] is denoted as . The circular convolution between two length N sequences is defined as where the subscript in x2((m-n)) N denotes modulo N operation and ⊙ denotes circular convolution operation. The symbols †, ∗, Tr denote Hermitian, complex-conjugate and transpose operations, respectively and E[.] denotes expectation operator.
where T is the useful portion of OFDMA symbol, T CP is the duration of the cyclic prefix (CP) and is the subcarrier spacing.
where z t (l) is obtained after taking the IDFT of z(k) and is the ISI free time domain signal which is fed to the symbol demodulator. Here, the symbol ⊖ denotes right circular shift operation.
3.1 Limiting performance of ZF-DFE in wideband channels
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. zero-mean, complex-Gaussian vector with covariance . Note that per-tap power is set to so that the total power contained in the multi-path channel becomes unity. As v → ∞, we can express the covariance term as: . 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. . 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,..,M-1.
For comparison, post-SNR corresponding to the MFB is given by .
The above result suggests that highly dispersive nature of the frequency-selective channel can be exploited advantageously to obtain a performance comparable to the MFB. After evaluating the expression (13) for the case of a single-receiver antenna, the ZF-DFE provides a post-SNR of that is 2.5-dB less than the MFB. For this case, both ZF- and MMSE-based LEs perform poorly compared to the MFB . However, the ZF-DFE does not suffer from this limitation and provides a substantial gain over MMSE/ZF-LE. For N r = 2, the loss of ZF-DFE with respect to the MFB reduces to 1.19 dB whereas the ZF-LE has a higher loss of 3.0 dB.
3.2 DFE initialization
In the MMSE-DFE implementation considered in this paper, the feedback filter is implemented in the time domain. In (5), the ISI term 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 (N-L),..,x t (N-2),x t (N-1)]. As proposed in , 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  to address the DFE initialization problem. The results of this paper show that MMSE-LE-based initialization is sufficient to obtain near-ideal performance. An alternative receiver initialization method is also discussed in  for trellis-based receivers. Different iterative block DFE methods have been proposed in [23, 32] for DFE-precoded-OFDMA 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.
4.1 Liming performance of WL ZF-LE
Note that the variance of is bounded only for N r > 1.
In Equation 24, ||h(0)||2 and are sum of squares of N r i.i.d. complex Gaussian r.v.’s which give a chi-square random variable with 2N r DOF while [||h(k)||2+||h(M-k)||2] has chi-square random variable with 4N r DOF. For the special case of N r = 1, the expected value of or is unbounded since it has inverse chi-square distribution with two DOF. However, the mean of 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(M-k) become correlated random variables. Specifically for values of k = 0 and , these terms become equal while for values of k in the vicinity of 0 and they become highly correlated. Considering the first two terms of Equation 24, we see that the terms or contribute to an increase in the MSE. Similarly, since h(k) and h(M-k) can be highly correlated for certain subcarrier locations, the term 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 MMSE-LE over ZF case.
5.1 Performance of WL ZF-DFE in wideband channels
For N r = 1, the ideal WL ZF-DFE offers a post-SNR of that is 1.17 dB away from the MFB. The actual performance gap with practical FBF is determined using BER simulation.
For the case of WL ZF/MMSE-DFE, we ignore the potential MSE increase contributed by the terms located at k = 0 and . Since at these locations, the exponent in (34) involves the terms E[ln||h(0)||2], 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  where binary modulation along with duobinary precoding is employed in the uplink of DFT-precoded-OFDM to reduce the PAPR.
We present BER simulation results for BPSK and 8-PSK and 16-QAM (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 20-tap i.i.d. Rayleigh fading channel with M=512 in all cases.
6.1 BER results for conventional equalizers
Theoretically expected SNR gap of the receiver with respect to the MFB in decibels (dB)
Gap for N r = 1
Gap for N r = 2
SNR gap of the receiver with respect to MFB in decibels (dB) at BER = 0.01 for BPSK
Gap for N r = 1
Gap for N r = 2
SNR gap of the receiver with respect to MFB in decibels (dB) at BER = 0.001 for BPSK
Gap for N r = 1
Gap for N r = 2
6.2 BER of WL equalizers
This paper describes the limiting behavior of conventional and WL equalizers in wideband frequency-selective channels. For systems employing DFT-precoded-OFDM modulation, closed-form expressions are obtained for the post-SNR of conventional and WL receivers employing ZF-LE and ZF-DFE; simulation is used to assess the performance of MMSE-based receivers. In i.i.d. fading channels with infinite channel memory, the post-SNR reaches a fixed value that is comparable to the MFB in most cases.
Both conventional MMSE-LE and ZF-LE offer near optimal performance only when the receiver has multiple antennas, whereas ideal ZF-DFE and ideal MMSE-DFE perform close to the MFB with a fixed SNR penalty even when the receiver has a single antenna. For single-antenna MMSE-DFE with decision feedback, the penalty compared to the ideal DFE is approximately 2.0 dB for 16-QAM systems at high SNRs. The total gap compared to MFB is 4.5 dB. Low-complexity 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. Multiple-receiver antennas are also shown to reduce the error propagation of the DFEs.
For single-antenna systems employing real-valued modulation alphabets, WL receiver processing can be used to obtain a performance advantage over conventional receivers. In particular, the WL MMSE-LE performs within 3.2 to 3.8 dB of the MFB while the WL MMSE-DFE reduces the gap with respect to the MFB to 1.0 dB. Results show that the multi-antenna 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  for the case of MIMO ZF-LE where it is shown that the post-SNR of the receiver reaches a constant value of for N r > N t , where N t is the SM rate. Extension to the general case of SM employing ZF/MMSE-DFEs is yet to be considered.
a (A + B C D)-1 = A-1 + A-1B(C-1 + D A-1B)-1D A-1
Derivation of MMSE-DFE filter settings
where the (l,m)th element of the matrix A is given by A(l,m) = q(m-l), b = [b t (1),b t (2),..,b t (L)] T r , and q = [q(1),q(2),..,q(L + 1)] T r . 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 MMSE-LE. The FBF coefficients can be calculated efficiently using the Levinson-Durbin recursion. The minimum MSE can be obtained by substituting the values of the FBF coefficients in the MSE expression (44). Next, we characterize the MMSE-DFE for the case of M→∞.
Derivation of WL MMSE-DFE 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.
Let . This is a real-valued function which exhibits even symmetry, i.e. P(k) = P(M-k) for k = 0,1,..,M-1. Similar to the WL MMSE-LE 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 last term involves point type-1 DCT of P(k). Note that needs to be calculated only for the first terms since the rest of the coefficients can be obtained exploiting the even symmetry of i.e., .
where the (l,m)th element of the matrix denotes as is given by , , and . Note that FBF can be calculated with low complexity using Levinson-Durbin recursion which involves real-valued quantities whereas the FBF for the conventional case involves complex values. Now we consider the infinite length filter case.
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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