10.3 Blind MCMC Receiver for Coded OFDM with Frequency-Selective Fading and Frequency Offset

In practical OFDM systems, the existence of frequency offset, which is caused by the mismatch between the oscillator in the transmitter and that in the receiver, destroys the orthogonality among OFDM subcarriers and leads to a performance degradation [374]. Several schemes of frequency offset estimation in OFDM systems have been investigated in [84, 89, 243, 295, 341, 440, 500, 507]. For OFDM applications over additive Gaussian white noise (AWGN) channels, the maximum- likelihood (ML) frequency offset estimates are derived in [89, 243, 295, 507]. Given that wireless channels typically exhibit frequency-selective fading, these methods designed for AWGN channels are not applicable in wireless OFDM systems. On the other hand, frequency offset estimators in frequency-selective fading channels are developed in [84, 341, 440], which require some particular form of data redundancy (e.g., data repetition [341] or pilot insertion [84, 440]). In [500], a blind subspace method for frequency offset estimation is proposed.

In wireless OFDM systems, in addition to the frequency offset, the frequency-selective fading channel states are also unknown to the receiver. The problem of channel estimation in OFDM systems has been studied in many previous works. The methods proposed in [260, 506] estimate the fading channel based on the pilot symbols, while blind estimation schemes based on second- or high-order statistics are proposed in [109, 345, 614]. Moreover, in [205, 367], subcarrier phase estimators are proposed by employing the expectation-maximization (EM) algorithm.

As an important remark, we note that the ultimate objective of the receiver is to recover the information- bearing data symbols from the received signals. Although the prevailing receiver-design paradigm is to estimate the unknown parameters first and then to use these estimated parameters in the detector, such an "estimate-then-plug-in" approach is ad hoc and bears no theoretical optimality. In this section we treat the problem of blind receiver design for coded OFDM systems in the presence of unknown frequency offset and frequency-selective fading, under the Markov chain Monte Carlo (MCMC) framework for Bayesian computation (cf. Chapter 8) and the principle of turbo processing (cf. Chapter 6). The techniques in this section were developed in [290].

10.3.1 System Description

Channel Model with Frequency Offset

When there is a carrier frequency offset in the OFDM channel, the received time-domain signal in (10.4) becomes [341]

Equation 10.11

where is the relative frequency offset of the channel (the ratio of the actual frequency offset to the intercarrier spacing). Note that for practical purposes, we assume that the absolute value of the frequency offset is no larger than half of the OFDM subcarrier spacing (i.e., < 0.5). That is, any large frequency offset has already been compensated (e.g., by an automatic frequency control circuit [120]) and what remains is the residual frequency offset. We next write the signal model (10.11) in matrix form. Denote

Note that W is the DFT matrix and (1/ Q ) W ^H is the inverse DFT matrix [i.e., WW ^H / Q = W ^H W / Q = I _Q ]. Hence H [ i ]= Wh [ i ] and N [ i ]= Wn [ i ].

Then upon applying a DFT to { y _m [ i ]} _m in (10.4), we obtain the signal model

Equation 10.12

For a better understanding of the effect of the frequency offset, we now take a closer look at the matrix in (10.12). Since < 0.5, after some simple algebra, the ( i,j )th element of the matrix can be expressed as

with

y ( i , j ) 1, i , j and y ( i , j ) y ( i' , j' ) if i - j i' - j' .

Hence I when 0; and the spillover to off-diagonal elements of , which corresponds to the ICI [341], increases as increases.

Bayesian Formulation of Optimal Demodulation

We consider a coded OFDM system with Q subcarriers, signaling through a frequency-selective fading channel in the presence of frequency offset. The system model, which has taken into account the frequency offset, is illustrated in Fig. 10.2. Each signal frame contains the information to be transmitted in one OFDM slot. The information bits of each signal frame are first encoded by a channel encoder; the encoded bits are then interleaved. After interleaving, the code bits { b _m } are mapped into MPSK symbols { c _k }. Finally, the differentially encoded MPSK symbols { X _k } are transmitted at the Q OFDM subcarriers. Note that the receiver processes each OFDM word independently, and hence in the remainder of this section, we drop the word index i in the signal model (10.12).

Figure 10.2. Block diagram of a coded OFDM system, including the transmitter, effect of frequency offset, frequency-selective fading channel, and receiver.

Since the receiver design problem addressed here is blind in nature, differential encoding is employed to resolve the phase ambiguity. For each signal frame, a block of MPSK symbols { c ₁ , ..., c _{Q “1} } is input to the differential encoder and the output MPSK symbols { X , ..., X _{Q “1} }, with

Equation 10.13

are then transmitted from the Q OFDM subcarriers. As will be seen in the following section, the decoding of the differentially encoded MPSK symbols is carried out implicitly in the Bayesian demodulator.

As the first step in the receiver, the code bits are demodulated (cf. Fig. 10.2). The optimal demodulator computes the a posteriori probabilities of the code bits as

Equation 10.14

where p ( Y b, h , ) is a complex Gaussian density function [cf. (10.12)]. The computation above is prohibitive and we therefore resort to the Markov chain Monte Carlo (MCMC) techniques introduced in Chapter 8 to calculate P ( b _m = +1 Y ) in (10.14).

10.3.2 Bayesian MCMC Demodulator

In this section we focus on the design of the MCMC demodulator for OFDM systems in the presence of frequency offset and frequency-selective fading. The receiver algorithm is summarized as follows .

Algorithm 10.1: [MCMC-based OFDM blind demodulator in the presence of frequency offset and frequency-selective fading] Given the initial samples { X ⁽⁰⁾ , h ⁽⁰⁾ , ⁽⁰⁾ } drawn from their prior distributions, proceed as follows. For n = 1, 2, ...:

• Draw a sample of h ^{( n )} from p ( h Y, X ^{( n -1)} , ^{( n -1)} ).

• For k = 0, 1, ..., Q “ 1, draw a sample of from P ( X _k Y, h ^{( n )} , ^{( n -1)} , ).

• Draw a sample of ^{( n )} from p ( Y , h ^{( n )} , X ^{( n )} ).

  We next elaborate on each step of the MCMC blind demodulator above.

Prior Distributions

The prior distributions of { X , h , } are assigned as follows.

1. The data sequence X = { X _k }, which is differentially encoded from c = { c _k }, forms a Markov chain. Its prior distribution can be expressed as

   Equation 10.15

   

   
   

   In (10.15), can be computed from the extrinsic information fed from the channel decoder; and we set P ( X ) = 1/ W to count for the phase ambiguity in X , where W represents the constellation of MPSK symbols.

2. For the unknown frequency-selective fading channel response h , a complex Gaussian prior distribution is assumed:

   Equation 10.16

   

   
   

   We set h = and S = a I _L , where a usually takes a large value corresponding to a noninformative prior of h .

3. For the unknown frequency offset , a uniform prior distribution is assumed:

   Equation 10.17

   

   
   

   where v denotes the prior range of , which is a real number that satisfies v > and v (0, 0.5).

Conditional Posterior Distributions

The following conditional posterior distributions are required in the MCMC blind demodulation algorithm.

1. The conditional posterior distribution of the channel response h given Y , X , and , as derived in the Appendix, is given by

   Equation 10.18

   

   
   

   with

   Equation 10.19

   

   
   

   Equation 10.20

   

   
   

   where the approximations in (10.19) and (10.20) follow from the fact that a in (10.16) is large and hence can be neglected. Moreover, it is seen that due to the orthogonality property of OFDM modulation, no matrix inversion is involved in generating the Monte Carlo samples of h; therefore, the computational complexity is low.

2. The conditional posterior distribution of the data symbol X _k is obtained by conditioning on Y , h , , and the samples of other data symbols { X , ..., X _{k -1} , X _{k +1} , ..., X _{Q -1} }. As shown in the Appendix, the conditional posterior distribution of X _k is given by

   Equation 10.21

   

   
   

   where a _j W and is the k th element of . The term in (10.21) is the a priori probability of the MPSK symbol c _k , which is delivered by the channel decoder. Through this term, the channel coding constraint embedded in { c _k } is exploited in the demodulator. Note that in the final step of (10.21), the term is dropped. This is because any random samples of the data sequence X must satisfy the differential coding constraint [cf. Eq. (10.15)]; since the conditioned data sequence in (10.21) may not satisfy this constraint, it is not a valid sample of the data sequence X . To avoid this problem, we propose to compute the conditional posterior probability of X _k by conditioning only on those data samples drawn in the current Gibbs iteration as { }, and correspondingly to drop the term related to the sample of the previous Gibbs iteration [i.e., ]. Our simulation results confirm that by neglecting this term, the Bayesian blind turbo receiver can yield much better performance through the turbo iterations.

3. The conditional posterior distribution of can be expressed as

   Equation 10.22

   

   
   

   Due to the nonlinear signal model of in (10.12), the conditional posterior distribution of in (10.22) is not a commonly used distribution (e.g., Gaussian, chi-square, etc.); hence generally there does not exist an efficient way to draw the random samples of directly from such a distribution function as we did above for h and X . As an important component in the Bayesian demodulator, we next discuss three methods for drawing samples of the frequency offset .

Sampling the Frequency Offset

We consider three methods for drawing samples of the frequency offset . The first two methods are within the Bayesian MCMC framework: the Metropolis “Hastings algorithm and the Gibbs sampler with local linearization. The third method simply ignores the frequency offset (i.e., it sets ^{( n )} = 0, n ).

Method I: Metropolis “Hastings Algorithm The Metropolis “Hastings algorithm was introduced briefly in Section 9.3.1. For the problem considered here, the target distribution is p ( Y , h , X ); and the transition function is chosen as T ( , ') = 1. Note that such a transition function is by no means the optimal choice, but it has been widely adopted in practice due to its simplicity [424]. Following the Metropolis “Hastings algorithm, and by using the prior distribution (10.17) as well as the posterior distribution (10.22), the procedure for drawing random samples of the frequency offset is as follows:

Algorithm 10.2: [Metropolis “Hasting sampling of frequency offset]

• Draw a sample ' ~ uniform [ ].

• Compute the Metropolis ratio:

  Equation 10.23

  

  
  

• Generate u ~ uniform (0, 1), and let

  Equation 10.24

  

  
  

Method II: Gibbs Sampler with Local Linearization As seen in (10.22), due to the nonlinear signal model of , we cannot directly apply the Gibbs sampler to draw samples of . However, following an idea that appeared in [104], we can first linearize the received signal model at its mode (the maximum value) with respect to ; then based on the linearized signal model, we obtain a locally linear conditional posteriori distribution of . Specifically, applying an inverse DFT on both sides of (10.12), we obtain

Equation 10.25

A Taylor series expansion is applied around the mode of the signal model in (10.25) and the linearized signal model is given by

Equation 10.26

with

Based on the locally linearized signal model (10.26), as derived in the Appendix, the conditional posterior distribution of is Gaussian, given by

Equation 10.27

with

Note that m and are real. Using (10.22) and (10.27), the procedure for frequency offset sampling is as follows:

Algorithm 10.3: [Gibbs sampling of frequency offset]

• Search for the mode of p ( Y , h , X ) from [ ].

• Draw a sample of from its linearized conditional posterior distribution: p ( Y , h , X ) ~ .

Method III: Null Sampling In this method, we simply ignore the frequency offset by setting ^{( n )} = 0, n . Although bearing no theoretical optimality, this method can be used to test the robustness of the Bayesian demodulator against a modeling mismatch. That is, when = 0 is assumed, we essentially ask the Gibbs sampler to fit an OFDM model with no frequency offset into an actual OFDM system with a certain frequency offset.

We consider a special case and see how the blind receiver behaves in the presence of a modeling mismatch. Let us revisit the system model in (10.12) and define

Equation 10.28

The vector can be seen as the time response of a "composite" channel with zero frequency offset, which incorporates the effect of the frequency offset, the data symbols, and the original frequency-selective fading channel. It is easy to see that when X = I , preserves the same statistics as h . In other words, no matter how large the frequency offset is, the blind receiver derived based on the statistics of h can also adapt to that zero-frequency-offset composite channel with response , by setting ^{( n )} = 0, n , in the Gibbs sampler.

When X I , the statistics of are usually different from those of h and the receiver will suffer from a performance loss. A quantitative evaluation of such a performance loss due to the modeling mismatch is given by computer simulations later in this section.

Computing Data Posterior Probabilities

After collecting J random samples of the differentially encoded MPSK symbols { X ^{( n )} }, the a posteriori probabilities of the MPSK symbols c and the code bits b are computed as follows. First, the a posteriori probabilities of c are computed from J random samples of { X ^{( n )} } as

Equation 10.29

where is an indicator such that

Equation 10.30

Furthermore, the a posteriori probabilities of the code bits b can easily be obtained from the a posteriori probabilities of the MPSK symbols c in (10.29). Assume that the symbol c _k is modulated from the code bits , with S = log ₂ W . Then the a posteriori probability of code bit is given by

Equation 10.31

where denotes all symbols in W that are modulated from the code bits with the i th bit as "+1".

So far, the Bayesian demodulator performs the soft demodulation of the code bits b . To further exploit the channel coding constraints embedded in the code bits b , we adopt a turbo receiver structure discussed in Chapter 6, which iteratively exchanges information about b between the Bayesian demodulator developed above and the channel decoder, to achieve successively improved receiver performance.

Bayesian Blind Turbo Receiver

The Bayesian blind turbo receiver consists of two stages ”the Bayesian demodulator as developed in Section 10.2, followed by a MAP channel decoder ”and these two stages are separated by an interleaver and a deinterleaver. The a posteriori log-likelihood ratios (LLRs) of the channel code bits b are iteratively exchanged between these two stages, to successively refine the receiver performance.

The Bayesian demodulator takes as input the interleaved a priori LLRs of code bits { l ₂ [ b _{p ( m )} ]} from the MAP channel decoder in the previous turbo iteration as well as the received signals Y , where p ( ·) denotes the interleaving function. It computes as output the a posteriori LLRs of the code bits:

Equation 10.32

where P ( b _{p ( m )} = +1 Y ) is the a posteriori probability of code bit b _{p ( m )} as computed in (10.31). Note that according to the original form of the turbo principle, the a priori LLRs { l ₂ [ b _{p ( m )} ]} are supposed to be deducted from the a priori LLRs { L ₁ [ b _{p ( m )} ]} to yield extrinsic information. However, the posterior distribution delivered by the Gibbs-sampler-based Bayesian demodulator takes on a quantized value as P ( c _k = a _j Y ) {0, 1/ J , 2/ J , ..., 1}, due to the finite samples of X [cf. Eq. (10.29)]. Hence, to enhance numerical stability and the iterative receiver performance, for this particular receiver structure, we feed the complete posterior information { L ₁ [ b _{p ( m )} ]} to the MAP channel decoder.

The MAP channel decoder employs the standard MAP decoding algorithm to compute the a posteriori LLRs of code bits:

Equation 10.33

In (10.33), the extrinsic information { l ₂ ( b _m )} is obtained by subtracting the prior information { L ₁ ( b _m )} from the posterior information { L ₂ ( b _m )}. After being interleaved, this extrinsic information is fed back to the Bayesian demodulator as a priori information for the next iteration; and thus we complete one turbo iteration. At the last turbo iteration, the LLRs and hard decisions of information bits are computed.

In addition to exchanging extrinsic information with the Bayesian demodulator, the channel decoder can also help the Gibbs-sampler-based Bayesian demodulator to assess its convergence, as discussed in Section 8.4.4. The number of bit corrections made by the MAP channel decoder is monitored , where the number of corrections is counted by comparing the signs of the code-bit LLRs at the input and output of the MAP channel decoder. If this number exceeds some predetermined threshold, we decide that the convergence of the Gibbs-sampler-based Bayesian demodulator is not achieved. In this case, the Bayesian demodulator will again be applied to the same received data.

Simulation Examples

In this section we provide computer simulation results to illustrate the performance of the MCMC blind turbo receiver for coded OFDM systems with frequency offset and frequency-selective fading. In simulations the available bandwidth is 1 MHz and Q = 256 subcarriers are used for OFDM modulation. These correspond to a subcarrier symbol rate of 3.9 KHz and an OFDM word duration of 1/ D _f = 256 m s. For each OFDM word, a guard interval of 40 m s is added to combat the effect of intersymbol interference. Simulations are carried out through an equal-power four-tap frequency-selective fading channel, where the delays of these taps are t _l = l/Q D _f , l = 0, ..., 3. The modulator employs a QPSK constellation. A four-state, rate- ½ convolutional code with generator (5,7) in octal notation is chosen as the channel code. For each OFDM slot, J + J = 100 samples are drawn by the MCMC demodulator, with the first J = 50 samples discarded. After completing 100 MCMC iterations, the convergence is tested by counting the number of corrections made by the decoders. In a few cases, when convergence is not reached, it gets restarted for another round of 100 MCMC iterations. In the following, the performance is demonstrated in terms of the bit-error rate (BER) and OFDM word-error rate (WER) versus the signal-to-noise ratio (SNR), defined as SNR h ² / s ² .

Performance Degradation Due to Frequency Offset First, we demonstrate the performance degradation due to the frequency offset in the coded OFDM system simulated here. The channel state information (CSI) (i.e., the channel response h ) is assumed to be known at the receiver. In Fig. 10.3 the performance of the turbo receiver under perfect CSI is shown for the coded OFDM system with different frequency offsets, = {0.00, 0.09, 0.18}. These results confirm the analysis in previous works (i.e., [374]) that the receiver performance degrades rapidly as the frequency offset increases. Hence, appropriate measures should be taken to combat the frequency offset.

Figure 10.3. BER and WER in a coded OFDM system with frequency offset = {0.00, 0.09, 0.18}. Perfect CSI (i.e., the channel response h ) is assumed at the receiver.

Performance of Various Frequency Offset Sampling Methods In Figs. 10.4 “10.11 the impact of different methods for drawing samples of the frequency offset on the overall Bayesian blind turbo receiver performance is examined. For methods I and II, the impact of the prior range ( v = {0.1, 0.5}) on the receiver performance is examined as well. In particular, in method II the mode of the conditional posterior distribution of the frequency offset is found by a global search with a step size of d = 0.05. Hence, the computational complexity of method II is still acceptable; and simulations show that only marginal performance improvement is obtained by using smaller step sizes.

Figure 10.4. BER and WER in a coded OFDM system with frequency offset = 0.09. The Metropolis “Hastings algorithm is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.5.

Figure 10.5. BER and WER in a coded OFDM system with frequency offset = 0.09. The Metropolis “Hastings algorithm is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.1.

Figure 10.6. BER and WER in a coded OFDM system with frequency offset = 0.09. The Gibbs sampler with local linearization is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.5 and the search step size d = 0.05.

Figure 10.7. BER and WER in a coded OFDM system with frequency offset = 0.09. The Gibbs sampler with local linearization is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.1 and the search step size d = 0.05.

Figure 10.8. BER and WER in a coded OFDM system with frequency offset = 0.18. The Metropolis “Hastings algorithm is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.5.

Figure 10.9. BER and WER in a coded OFDM system with frequency offset = 0.18. The Metropolis “Hastings algorithm is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.2.

Figure 10.10. BER and WER in a coded OFDM system with frequency offset = 0.18. The Gibbs sampler with local linearization is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.5 and the search step size d = 0.05.

Figure 10.11. BER and WER in a coded OFDM system with frequency offset = 0.18. The Gibbs sampler with local linearization is employed to generate Monte Carlo sampling of the frequency offset, where the prior range v = 0.2 and the search step size d = 0.05.

The performance of the receiver, when it employs method I or method II, is demonstrated through the BER and WER after the first, third, and fifth turbo iterations; whereas when method III is employed, for clarity, only the performance after the fifth turbo iteration is shown, denoted " WER,Iter#5,3 ^rd " and "BER,Iter#5,3 ^rd " in the figures. Moreover, for comparison, the ideal CSI performance of the system with zero frequency offset, which is approximately the best performance we can achieve in this system, is shown again in these figures and denoted " WER,CSI, =0 " and " BER,CSI, =0 ."

Example 1: Small Frequency Offset In Figs. 10.4 “10.7 we present the performance of the MCMC blind turbo receiver in a coded OFDM system with frequency offset = 0.09. From the simulation results, several conclusions can be drawn. First, the receiver performance is improved significantly through turbo iterations and can approach the performance under perfect CSI at the BER of 10 ^-3 and WER of 2 x 10 ^-2 , as can be seen from the performance curves when the receiver employs either method I or method II for frequency offset sampling. Second, the receiver performance is not sensitive to the prior range v of method I or method II, which can be seen by comparing Figs. 10.4 and 10.5 or Figs. 10.6 and 10.7. This is a favorable fact for receiver design, as we can always set the largest prior range of as [ “0.5, 0.5]. Third, the robustness of the receiver is tested by employing method III in the Bayesian demodulator. Compared with the performance of the receiver that explicitly samples the frequency offset (i.e., by method I or method II), we do not see any performance loss when the receiver samples null frequency offset (i.e., by method III). In other words, the MCMC blind turbo receiver is robust against a modeling mismatch.

Example 2: Large Frequency Offset In Figs. 10.8-10.11, in the same form as in Example 1, we present the receiver performance in a coded OFDM system with a larger frequency offset, = 0.18. Recall that from Fig. 10.3, when no proper methods are employed to combat the frequency offset, the receiver assuming perfect CSI completely fails in the presence of such a large frequency offset.

From Figs. 10.8-10.11, in addition to the conclusions drawn in Example 1, some new observations are made. When method I or method II is employed, it is seen from the figures that the receiver still performs very well and can approach the performance under perfect CSI after three to five turbo iterations, in the presence of such a large frequency offset. However, when method III is employed, the receiver performance mildly degrades by about 1.5 dB, due to the modeling mismatch, compared to the performance in a smaller frequency offset system (i.e., the performance shown in Figs. 10.4-10.7).

Finally, based on all the simulation results shown above, we compare the efficiency of all three methods for frequency offset sampling in terms of both the performance and complexity. Method III has the lowest complexity by ignoring the frequency offset, but it leads to a noticeable receiver performance degradation as the frequency offset becomes large. Method I has lower complexity than method II, and it can yield almost the same receiver performance as method II. Moreover, since no approximation has been made in deriving method I, its convergence is guaranteed by the theory of MCMC. Therefore, we advocate the use of method I, the Metropolis “Hastings algorithm, to draw samples of the frequency offset in the MCMC blind turbo receiver.