2.03.2.1.1 Tapped delay line model

We now consider a discrete-time channel model. If a linear modulation scheme is used, the baseband transmitted signal can be represented as

(3.3)

where is the information sequence and is the transmit (lowpass) filter (typically a root raised cosine filter). Therefore, the baseband signal at the receiver is given by

(3.4)

After filtering with a receive filter with impulse response , the received baseband signal is given by

(3.5)

where . If the continuous-time signal is sampled at once every sec., we obtain the discrete-time sequence

(3.6)

where is the (effective) channel response at time n to a unit impulse input at time and

(3.7)

Note that the noise sequence in (3.6) is no longer necessarily white; it can be whitened by further time-invariant linear filtering (see [1]). Henceforth, we assume that a whitening filter has been applied to , but with an abuse of notation, we will still use (3.6). For a causal system, for () and for a finite length channel of maximum length for (). In this case we modify (3.6) as (recall the noise whitening filter)

(3.8)

The model (3.8) represents a time- and frequency-selective linear channel. A tapped delay line structure for this model is shown in Figure 3.1. For a slowly (compared to the baud-rate) time-varying system, one often simplifies (3.8) to a time-invariant system as

(3.9)

where is the time-invariant channel response to a unit impulse input at time 0. The model (3.9) represents a frequency-selective linear channel with no time selectivity. It is a widely used model for receiver design.

Suppose that . Then we have the time-selective and frequency-nonselective channel whose output is given by

(3.10)

Finally, a time-nonselective and frequency-nonselective channel is modeled as

(3.11)

where h is a random variable (or a constant).

2.03.2.1.2 Autoregressive (AR) models

It is possible to accurately represent a wide-sense stationary uncorrelated scattering (WSSUS) channel by a large order AR model; see [3–5] and references therein. Let

(3.12)

where is vector. Then a pth order AR model, AR(p), for is given by

(3.13)

where s are the AR coefficient matrices, is also and independent and identically distributed (i.i.d.), driving noise is zero-mean with identity covariance matrix. Suppose that we know the correlation function for lags . The following Yule-Walker equation holds for (3.13) [6]:

(3.14)

Using (3.14) for , and the fact that , one can estimate s. Using the estimated s and (3.14) for , one can find , from which one can find (non-unique) by computing its “square root” [7, p. 358]. In [3] high values of p (several tens) have been used for channel simulation, whereas in [4,5] only AR(1) or AR(2) models have been used where the objective is channel estimation and related issues. An AR(1) model is given by

(3.15)

where is the AR coefficient, and the driving noise is zero-mean complex Gaussian with variance and statistically independent of . Assume that is also zero-mean, complex Gaussian with variance . Then [8]

(3.16)

2.03.2.1.3 Basis expansion models

Basis expansion models (BEMs) have also been widely investigated to represent doubly selective channels in wireless applications [9–13], where the time-varying taps are expressed as superpositions of time-varying basis functions in modeling Doppler effects, weighted by time-invariant coefficients. Candidate basis functions include complex exponential (Fourier) functions [10,11], polynomials [9], and discrete prolate spheroidal sequences [13], etc. In contrast to AR models that describe temporal variation on a symbol-by-symbol update basis, a BEM depicts the evolution of the channel over a period (block) of time. Intuitively, the coefficients of the BEM approximation should evolve much more slowly in time than the channel, and hence are more convenient to track in a fast fading environment.

Suppose that we include the effects of transmit and receive filters in the time-variant impulse response in (3.1). Suppose that this channel has a delay-spread and a Doppler spread . Consider the kth block of data consisting of an observation window of symbols where the baud-rate data samples in the block are indexed as . If (underspread channel), the complex exponential basis expansion model (CE-BEM) representation of in (3.8) is given by

(3.17)

where one chooses (, and K is an integer)

(3.18)

(3.19)

The BEM coefficients s remain invariant during this block, but are allowed to change at the next block, and the Fourier basis functions () are common for each block. If the delay spread and the Doppler spread of the channel (or at least their upper-bounds) are known, one can infer the basis functions of the CE-BEM [11]. Treating the basis functions as known, estimation of a time-varying process is reduced to estimating the invariant coefficients over a block of length symbols. Note that the BEM period is whereas the block size is symbols. If (e.g., or ), then the Doppler spectrum is said to be over-sampled [14] compared to the case where the Doppler spectrum is said to be critically sampled. In [10,11] only (henceforth called CE-BEM) is considered whereas [14] considers (henceforth called over-sampled CE-BEM).

Equation (3.17) applies to single-input single-output systems—one user and one receiver with symbol-rate sampling. It is easily modified to handle multiuser, multiple transmit and receive antennas, and higher than symbol rate sampling (multiple samples per symbol)—the basic representation remains essentially unchanged.

The representation in (3.17) is a special case of a more general representation

(3.20)

where are a set of orthogonal basis functions (over the time interval under consideration). Examples include wavelet-based expansions as in [15], polynomial bases as in [9] and other possibilities [16]. In discrete prolate spheroidal BEM (DPS-BEM), the ith DPS vector

(called Slepian sequence in [13], which is a time-windowed (infinite) DPS sequence) is the ith eigenvector of a matrix [17]:

is the th entry of and are the eigenvalues of . The Slepian sequences are orthonormal over the finite time interval . The modeling error of the CE-BEM can result in a noticeable floor in BER curves [18]. The polynomial basis functions are neither time-limited nor band-limited and their square bias varies heavily over the range of Doppler spread considered in [13]. DPS sequences are a good alternative as a basis set to approximate bandlimited channels alleviating the spectral leakage of CE-BEM [13]. The (infinite) DPS sequences have their maximum energy concentration in an interval with length T while being bandlimited to , where is the unique sequence that is bandlimited and most time-concentrated, is the next sequence having maximum energy concentration among the DPS sequences orthogonal to , and so on [17].

Figure 3.2 shows the channel modeling errors resulting from (critically sampled) CE-BEM, DPS-BEM and oversampled CE-BEM ( or 3) when the underlying channel is a one-tap time-selective channel following Jakes’ spectrum. The results are based on Monte Carlo averaging over 1000 runs with and varying Doppler spreads. (The results were obtained following the procedure in [13].) For a fixed value of Q, DPS-BEM provides the best fit whereas CE-BEM (no oversampling) yields minor improvements with increasing Q. On the other hand, the basis functions in oversampled CE-BEM are not mutually orthogonal leading to “analytical” difficulties. There exists a vast literature based on CE-BEM (no oversampling) where it is assumed that physical channel is accurately described by CE-BEM both for analysis and simulations; see e.g., [11,19–24].

2.03.3.1.1 Time-variant channels

In case of general time-varying channels represented by (3.6), a simple generalization of [32] (see also [11]) is to use a periodic Kronecker delta function sequence with period as training:

(3.27)

With (3.27) as input to model (3.6), one obtains

(3.28)

so that if , we have for ,

(3.29)

Therefore, one may take the estimate of as

(3.30)

For time samples between (k an integer), linear interpolation may be used to obtain channel estimates.

If we use the BEM representation (3.20), then we directly estimate the time-invariant parameters s. From (3.6) and (3.20) we have

(3.31)

(3.32)

Collecting samples of the observations we have the linear model

(3.33)

Now we have a model similar to (3.24) with a solution similar to (3.26).

2.03.3.2.1 Combined channel and symbol estimation

The simultaneous estimation of the input vector and the channel is general ill-posed; however, utilization of qualitative information about the channel and the input can help alleviate this deficiency. To this end, we consider two different types of maximum likelihood techniques based on different models of the input sequence.

Stochastic Maximum Likelihood Estimation. While the input vector is unknown, it may be modeled as a random vector with a known distribution. In such a case, the likelihood function of the unknown parameter (cf. (3.24)) can be obtained by

(3.34)

where is the marginal pdf of the input vector and is the likelihood function when the input is known. Assume, for example, that the input data symbol takes, with equal probability, a finite number () of values. Consequently, the input data vector also takes values from the signal set . The likelihood function of the channel parameters is then given by

(3.35)

where c is a constant, , and the stochastic maximum likelihood estimator is given by

(3.36)

The maximization of the likelihood function defined in (3.34) is in general difficult because is non-convex. The Expectation-Maximization (EM) algorithm can be applied to transform the complicated optimization to a sequence of quadratic optimizations. Kaleh and Vallet [33] first applied the EM algorithm to the equalization of communication channels with input sequence having finite alphabet property. By using a Hidden Markov Model (HMM), they developed a batch (off-line) procedure that includes the so-called forward and backward recursions. The complexity of this algorithm increases exponentially with the channel memory.

To relax the memory requirements and facilitate channel tracking, “on-line” sequential approaches have been proposed in [34] for input with finite alphabet properties under a HMM formulation. Given the appropriate regularity conditions and a good initialization guess, it can be shown that these algorithms converge to the true channel value.

Deterministic Maximum Likelihood Estimation. The deterministic ML approach assumes no statistical model for the input sequence . In other words, both the channel vector and the input source vector are parameters to be estimated. When the noise is zero-mean Gaussian with covariance , the ML estimates can be obtained by the nonlinear least squares optimization

(3.37)

The joint minimization of the likelihood function with respect to both the channel and the source parameter spaces is difficult. Fortunately, the observation vector is linear in both the channel and the input parameters individually. In particular, we have

(3.38)

where

(3.39)

is the so-called filtering matrix. We therefore have a separable nonlinear least squares problem that can be solved sequentially

(3.40)

If we are only interested in estimating the channel, the above minimization can be rewritten as

(3.41)

where is a projection transform of into the orthogonal complement of the range space of , or the noise subspace of the observation, and denotes the pseudo-inverse of . Discussions of algorithms of this type can be found in [35].

Similar to the HMM for statistical maximum likelihood approach, the finite alphabet properties of the input sequence can also be incorporated into the deterministic maximum likelihood methods. These algorithms, first proposed by Seshadri [36] and Ghosh and Weber [37], iterate between estimates of the channel and the input. At iteration k, with an initial guess of the channel , the algorithm estimates the input sequence and the channel for the next iteration by

(3.42)

(3.43)

where is the (discrete) domain of . The optimization in (3.43) is a linear least squares problem whereas the optimization in (3.42) can be achieved by using the Viterbi algorithm [1]. Seshadri [36] presented blind trellis search techniques. Reduced-state sequence estimation was proposed in [37]. Raheli et al. proposed a per-survivor processing technique in [38]. The convergence of such approaches is not guaranteed in general. Interesting examples have been provided in [39] where two different combinations of and lead to the same cost .

2.03.3.2.2 The methods of moments

Although the ML channel estimator discussed in Section 2.03.3.2.1 usually provides better performance, the computation complexity and the existence of local optima are the two major difficulties. Therefore, “simpler” approaches have also been investigated.

SISO Channel Estimation. For baud-rate data, second-order statistics of the data do not carry enough information to allow estimation of the channel impulse response as a typical channel is nonminimum-phase. On the other hand, higher order statistics (in particular, fourth-order cumulants) of the baud-rate (or fractional rate) data can be exploited to yield the channel estimates to within a scale factor. Given the mathematical model (3.9), there are two broad classes of direct approaches to channel estimation, the distinguishing feature among them being the choice of the optimization criterion. All of the approaches involve (more or less) a least-squares error measure. The error definition differs, however, as follows:

Further details may be found in [44] and references therein.

SIMO Channel Estimation. Here we will concentrate upon second-order statistical methods. For single-input multiple-output vector channels the autocorrelation function of the observation is sufficient for the identification of the channel impulse response up to an unknown constant [45,46], provided that the various subchannels have no common zeros. This observation led to a number of techniques under both statistical and deterministic assumptions of the input sequence [35]. By exploiting the multichannel aspects of the channel (e.g., crosscorrelation among the outputs of various subchannels), many of these techniques lead to a constrained quadratic optimization

(3.44)

where is a positive definite matrix constructed from the observation. Asymptotically (either as the sample size increases to infinity or the noise variance approaches to zero), these estimates converge to true channel parameters.

2.03.4.1.1 Kalman detector (KD)

The Kalman filter, together with a quantizer, acts as the symbol detector at the receiver end. The state and the measurement equations are given by

(3.50)

(3.51)

with the following definitions

where is K-column vector of symbols (data or training), is the known or estimated channel matrix and integer (it will also be the equalization delay). Assume data symbols are zero-mean and white. If is a data symbol, we have and ; if is a training symbol, and .

Kalman filtering for the system described by (3.50) and (3.51) is initialized with

where denotes the estimate of given the observations , and denotes the error covariance matrix of , defined as

Then recursive filtering (for ) is applied via the following steps:

The estimated state vector is given by

and we extract its last (K-column vector) term as the desired equalized output for K-users with equalization delay d. Finally, we hard-quantize to acquire the detected symbols.

When using the estimated channel, one may rewrite the received signal (3.48) as

(3.52)

where the “effective” noise is instead of . In order to compensate for this channel estimation error, as a first-order approximation, one may wish to take the variance of in (3.52) to be larger than , the variance of .

2.03.4.4.1 Principle of turbo equalization

Consider a simple transmitter where a sequence of data is encoded to the code symbols with a code rate , which is interleaved in a block and then mapped into binary phase shift keying (BPSK) symbols. Figure 3.4 depicts the receiver structure for turbo equalization [77], where both the soft-input soft-output (: we use subscript f to distinguish between single-input single-output abbreviated as SISO and soft-input soft-output abbreviated as ) equalizer and decoder are of the MAP type. The extrinsic log-likelihood ratios (LLRs), are transferred iteratively between the equalizer and decoder. [The subscript “e” for representing “extrinsic,” the superscript “E” and “D” for output of “Equalizer” and “Decoder” respectively]. The MAP equalizer computes the a posteriori probabilities (APPs) given received symbols, and generates the extrinsic LLR as (a posteriori LLR – a priori LLR)

(3.60)

The a priori LLR of the MAP equalizer, is provided by the interleaved output of the MAP decoder at the previous iteration but for the first iteration. The extrinsic LLRs produced by the MAP demodulator is sent to the MAP decoder as the a priori LLRs for channel decoding. Based on the a priori LLRs and the channel code constraints, the MAP decoder computes the APPs and generates the extrinsic LLR as

(3.61)

The extrinsic output of the MAP decoder is fed back to the MAP equalizer iteratively. Note that (3.60) and (3.61) are valid only if the a posteriori outputs are independent of the a priori inputs for the equalizer and the decoder. Assuming ideal interleaver between the equalizer and the decoder, we can apply the turbo principle and the correct ordering of the LLRs and , which are input to the MAP decoder and the MAP equalizer respectively. The MAP decoder also compute the a posteriori probabilities of the input data bit and then the data estimate as

(3.62)

By combining a MAP equalizer and a MAP decoder, and exchanging probabilistic information about data symbols iteratively, turbo equalization usually can achieve close-to-optimal performance but much lower complexity [77,78]. In [79], a turbo-equalization-like system using linear equalizers based on soft interference cancellation and linear minimum mean-square error (MMSE) filtering is proposed as part of a multiuser detector for CDMA. Based on this work, a variety of equalizers employing linear MMSE and decision feedback equalization (DFE) are proposed in [75,80,81].

2.03.4.4.2 Turbo equalization for doubly-selective channels

For doubly-selective channels, an adaptive equalizer has been presented in [5], using extended Kalman filter (EKF) to incorporate channel estimation into the equalization process. This adaptive soft nonlinear Kalman equalizer takes the soft decisions of data symbols from the decoder as its a priori information, and performs equalization process iteratively. With such an approach, the proposed scheme jointly optimizes the estimates of the channel and data symbols in each iteration. This avoids the common drawback in separate channel estimation and equalization/detection approach in that the correlation between channel estimate and data symbol decision is considered. The complexity of the method of [5] is comparable to that of the turbo equalizers using linear filters [82–84], and is usually much lower than that of the ML/MAP based joint channel estimation and data detection schemes.

Based on the turbo equalization approach proposed in [5] and CE-BEM, an adaptive turbo equalizer with nonlinear Kalman filtering has been proposed in [85]. It is discussed in more detail in later under adaptive channel estimation and equalization.

2.03.6.4.1 Turbo equalization using EKF and CE-BEM

Now we describe an extension of [5] as reported in [85] where a CE-BEM-based subblock tracking model (with one sample long subblock) is used. Kin and Tuguait [85] considers two overlapping blocks (each of symbols) that differ by just one symbol: the “past” block beginning at time and the “present” block beginning at time . Since the two blocks overlap so significantly, one would expect the BEM coefficients to vary only “a little” from the past block to the present overlapping one. One can track the BEM coefficients (rather than the channel tap gains) symbol-by-symbol using a first-order AR model for their variations. Kin and Tuguait [85] use (3.17) for all times n, not just the particular block of size symbols, by allowing the coefficients s to change with time. Note that model (3.17) is periodic with period whereas the channel is by no means periodic. So long as the effective “memory” of the Kalman filter used later is less than the model period , there are no deleterious effects due to the use of (3.17) for all time.

Bit-Interleaved Coded Modulation (BICM). We consider a BICM transmitter (as in [96]) for a doubly-selective fading channel as shown in Figure 3.6. A sequence of independent data vector are fed into a convolutional encoder with a code rate . The coded output is passed through a bit-wise random interleaver , generating the interleaved coded bit sequence . The binary coded bits are then mapped to a signal sequence over a 2-dimensional signal constellation of cardinality by a -ary modulator with an one-to-one binary map . In this section, we only consider the case of phase-shift keying (PSK) or quadrature amplitude modulation (QAM) with the average energy of the constellation to be unity. That is, the signal drawn from has mean and variance . After modulation, we periodically insert short training sequences into the data symbol sequence. The training symbols , which are known to the receiver, are randomly drawn from the signal constellation with equal probabilities. The symbol will be used to denote the symbol sequence after training insertion into data symbol sequence .

Receiver structure. A turbo equalization structure, as depicted in Figure 3.7, is employed in the receiver, as in [5] except that [5] uses symbol-wise AR models. The adaptive equalizer is embedded into the iterative decoding (ID) process of the BICM transmission system (BICM-ID) [96]. In each decoding iteration, the equalizer takes the training symbols and the soft decision information about data symbols supplied by the decoder from the previous iteration as its a priori information to perform joint adaptive channel estimation and equalization. The equalizer produces the soft-valued extrinsic estimate of the data symbols, which are independent of their a priori information. The output of the equalizer is an updated sequence of soft estimates and its error variance . Using the adaptive equalizer described later, we have extrinsic information for the data symbols . The training symbols are removed at the equalizer output and the iterative process that follows is only for data symbols. The equalizer based on the CE-BEM is described later. The demodulator follows [96] whereas the decoder follows the MAP decoding algorithm (“BCJR”) [99] Section 6.2.

Adaptivenonlinear Kalman equalizer. Using a symbol-wise AR-model for channel variations, an adaptive equalizer using fixed-lag EKF was presented in [5] for joint channel estimation and equalization where their correlation was (implicitly) considered. The CE-BEM model-based nonlinear Kalman equalizer for turbo equalization follows a similar approach. We will perform equalization with a delay . Define a parameter

(3.79)

and the data vector

(3.80)

In order to apply (extended) Kalman filtering to joint channel estimation and equalization, we stack (which is given by (3.77) after replacing p therein with n to reflect the fact our subblock size now is one sample) and data vector together into a state vector at time n as

(3.81)

As in [5] (and others), we consider the symbol sequence as a stochastic process so as to utilize the soft decisions on the data symbols generated in the iterative decoding process as its a priori information. We can express as where and is approximated as a zero-mean uncorrelated sequence such that , assuming an ideal interleaver. Note that and are provided via the a priori information. We have and for a data symbol (where and are obtained from the extrinsic LLRs of the decoder), while and for a training symbol . Using , the state equation turns out to be

(3.82)

where

(3.83)

(3.84)

the vector

(3.85)

is zero-mean uncorrelated process noise where is given in (3.77) and

(3.86)

The channel output in (3.6) can be rewritten by CE-BEM given in (3.17) as

(3.87)

where and . Using the state vector that comprises the information symbols and channel coefficients, the measurement equation can be given as

(3.88)

where

(3.89)

With (3.82) and (3.88) as the state and measurement equations, respectively, nonlinear Kalman filtering via EKF is applied to track for joint channel estimation and equalization.

Structure of adaptive soft-input soft-output equalizer. The fixed-lag EKF takes soft inputs and generates a delayed a posteriori estimate for . In order to generate extrinsic estimate independent of the a priori information , a “comb” structure in conjunction with the EKF in Figure 3.8 is used for the equalization, just as in [5]. At each time n, the vertical branch composed of EKFs produce the extrinsic estimate , while the horizontal branch keeps updating the a posteriori estimate and its error covariance ; here denotes the estimate of given the observations , and denotes the error covariance matrix of . The first vertical EKF has an input in place of to exclude the effect of the a priori information. Let and denote the state estimate and its error covariance matrix, respectively, generated by the th vertical filtering branch. Then the extrinsic estimate of and its error variance are given by

(3.90)

(3.91)

Note that the extrinsic outputs and are computed for data symbol , not for training symbol , and then used in the later parts of the turbo-equalization receiver (see Figure 3.7). Further details regarding generation of extrinsic estimates can be found in [5].

Simulation examples. A random time- and frequency-selective Rayleigh fading channel is considered. We assume is zero-mean, complex Gaussian, and white with autocorrelation . We take (3 taps) and . For different ls, s are mutually independent and satisfy Jakes’ model. To this end, we simulate each single tap following [100] (with a correction in the Appendix of [13]). We consider a normalized Doppler spread from 0.001 to 0.01. The additive noise is zero-mean complex white Gaussian. The (receiver) SNR refers to the average energy per symbol over one-sided noise spectral density. In the simulations, we use a four-state convolutional code of rate with octal generators . The information block size is set to 3000 bits () leading to a coded block size of 6000 bits, and the interleaver size is equal to the coded block size. In the modulator, the QPSK constellation with Gray mapping is used, which gives and a block size of 3000 symbols. After modulation, training symbol sequences of length are inserted in front of every data symbols, leading to a sequence of length when and (20% training overhead).

We compared the following schemes:

We evaluate the performances of various schemes by considering their normalized channel mean square error (NCMSE) and their bit error rates (BER). The NCMSE is defined as

where is the true channel and is the estimated channel at the ith Monte Carlo run, among total runs. The BERs are evaluated by employing the equalization delay , using the decoded information symbol sequences at the turbo-equalization receiver. All the simulation results are based on 1000 runs.

In Figures 3.9 and 3.10, the performance of all the above schemes, under normalized Doppler spread , are compared for different SNRs. In Figures 3.11 and 3.12 those schemes are compared over varying Doppler spread s, under SNR = 10 dB. It is clear from these figures that since the channel variations are well captured by the BEM coefficients, TE-BEM-based approach yields good performance even for “low” SNRs and over a wide range of Doppler spreads. Note that TE-BEM with larger block parameter has a (slightly) better performance than with the smaller parameter . The BER of TE-BEM varies only “slightly” with increasing normalized Doppler spread implying that its performance is not sensitive to the actual Doppler spread. Therefore, we do not have to know the exact Doppler spread of the channel—an upper bound on it is sufficient in practice. The performance of TE-AR5 is significantly worse than that of TE-BEM(200) (the two approaches have comparable computational complexity) in Figure 3.10 with increasing SNR for a fixed , and is slightly worse in Figure 3.12 for a fixed SNR of 10 dB and varying Doppler spreads. On the other hand, while the performance of TE-AR9 is slightly better than that of TE-BEM(400) (the two approaches have comparable computational complexity) in Figure 3.10 with increasing SNR for a fixed , it is significantly worse in Figure 3.12 for a fixed SNR of 10 dB and varying Doppler spreads. While increasing the BEM period improves performance, increasing the AR model order does not necessarily do so: we get inconsistent performance. A possible reason is that, as noted in [3], AR model fitting to a given correlation function can be numerically ill-conditioned for “large” model orders.

In Figures 3.10 and 3.12 the scheme TE-LE refers to the approach of [83] that uses the linear MMSE equalizer (e.g., [80]) coupled with modified RLS channel estimation. It is seen that this approach only works for normalized Doppler spread values of 0.002. In Figure 3.10 we also present the performance of the turbo equalizer based on the fixed-lag Kalman filter with knowledge of the true channel (curves with plus sign marker and labeled “TrueCH”) in order to illustrate the effectiveness of the proposed channel estimation approach; as there was little improvement beyond the second iteration, we only show the second iterative result with dotted curve labeled “TrueCH.” It is seen that there is a slightly more than 2 dB SNR penalty due to channel estimation. As has been noted in the literature, the Kalman filter based equalization is a sub-optimum equalizer compared to the trellis-based MAP (BCJR) equalizer [101]. In Figure 3.10 we present the performance of the turbo equalizer based on the optimum BCJR method with knowledge of the true channel (curves with asterisk marker and labeled “Opt-MAP-TrueCH”) in order to illustrate loss in performance due to suboptimality of the Kalman equalizer; as there was little improvement beyond the second iteration, we only show the second iterative result with dotted curve labeled “Opt-MAP-TrueCH”. It is seen that while there is a large difference in performance initially (see 1st iteration results for “TrueCH” and “Opt-MAP-TrueCH” where both are dashed curves with plus sign and asterisk markers, respectively), just one turbo iteration yields very close performance (see the two dotted curves). That is, at least for this example, performance loss in using Kalman equalizer instead of the BCJR equalizer is quite negligible.