Blind and semi-blind ML detection for space-time block-coded OFDM wireless systems

The increasing demand for higher data rates in recent years has called for transmissions over a broadband wireless channel which is frequency selective. The wireless channel is thus prone to inter-symbol interference (ISI) which severely degrades the system performance and requires complex equalization techniques at the receiver. The orthogonal frequency-division multiplexing (OFDM) has emerged as the most promising scheme to combat ISI and improve system performance. OFDM essentially transforms a broadband channel into a number of parallel narrowband channels using a cyclic prefix (CP) of appropriate length and renders simple one-tap channel equalizer for each OFDM subcarrier [1, 2]. Due to these crucial advantages, OFDM is not only being used in many existing standards such as digital subscriber line (DSL), WLAN standards (IEEE 802.11 a/b/g), LAN standard (IEEE 802.11n) [3], MAN standards (IEEE 802.16e), digital audio broadcast (DAB), and digital video broadcast (DVB) [4] but also adopted for future wireless standards such as LTE and 4G+ [5]. Besides OFDM, the spatial dimensions in wireless communications are often exploited to further enhance the system capacity and /or improve the transmission reliability by employing multiple antennas at the transmitter and/or receiver. This offers many advantages over single-antenna systems including multiplexing gain and diversity gain [6]. Of several diversity schemes available in the literature, the Alamouti scheme [7] with two transmit and one receive antenna is the optimum in both the capacity and the diversity. Alamouti coding achieves full spatial diversity at full transmission rate for any signal (real or complex) constellation and offers very simple receiver structures. However, to decouple the signals at the receiver side via simple decoding, the Alamouti scheme requires the channel between each transmit-receive antenna to be constant over two consecutive OFDM symbols. Moreover, when dealing with frequency-selective channel, the Alamouti scheme has to be implemented over the block level.

In many wireless communications studies, it is often assumed that channel state information (CSI) is available at the receiver side for coherent data detection. This assumption is certainly not realistic. The current standards use pilot symbols to estimate the channel, thus sacrificing bandwidth which otherwise would have been available for data transmission. In high-mobility wireless systems, the channels may even change so rapidly that this approach will become infeasible. Blind or semi-blind detection over time-varying wireless channels has shown to enhance the system performance considerably [8, 9].

There are numerous blind estimation and equalization techniques available in the literature, namely, subspace-based methods [10, 11], second-order statistics [12], Cholesky factorization [13], and iterative methods [14]. These methods either suffer from slow convergence, higher computational costs or assume channel to be stationary over several OFDM symbols. These drawbacks make maximum likelihood (ML)-based approaches, e.g., [15, 16] more attractive due to their fast convergence despite having the higher computational cost. Usually, suboptimal techniques are employed to reduce the computational cost by restricting the search space of exhaustive ML search. Some of the suboptimal techniques are applicable to specific constant modulus constellations [17, 18]. Recently, in [19] and [20], the authors have proposed a low-complexity blind ML method for general constellations for single-input-multiple-output (SIMO) and single-input-single-output (SISO) systems, respectively, which form the basis of our paper.

Consider a single-user OFDM system with two transmit and one receive antenna as shown in Figure 1. The frequency-selective channels from two transmit antennas to the receive antenna are modelled as finite impulse response (FIR) filters. We assume that both channels are independent Rayleigh-fading channels having maximum length L and that OFDM CP length is at least L−1 to avoid ISI.

where$\mathcal{Y}$ and$\mathcal{N}$ are N-dimensional received OFDM symbol and noise vector, respectively, while h is the length L SISO channel vector. In either case, the task of receiver is to jointly estimate the channel h and the data vector$\mathcal{X}$ given only the received data symbol$\mathcal{Y}$.

where Ω^2N denotes all possible 2N-dimensional signal vectors. As seen from (12), the joint ML problem is a combinatorial problem involving |Ω|^2N hypothesis tests, and it is almost impossible to solve it exactly for sufficiently large Ω and N.

as the partial joint ML metric up to the index i for$\mathcal{X}$, where ${\mathbf{X}}_{a(i)}=\left[\begin{array}{ll}\text{diag}\left({\mathcal{X}}_{1(i)}^{(k)}\right){\mathbf{A}}_{(i)}^H& \text{diag}\left({\mathcal{X}}_{2(i)}^{(k)}\right){\mathbf{A}}_{(i)}^H\\ -\text{diag}\left({\mathcal{X}}_{2(i)}^{\ast (k)}\right){\mathbf{A}}_{(i)}^H& \text{diag}\left({\mathcal{X}}_{1(i)}^{\ast (k)}\right){\mathbf{A}}_{(i)}^H\end{array}\right]$ is a partial Alamouti matrix of dimension 2i×2L,X _a (j), the 2×2L matrix is the same as X_a(j) with all ${\mathcal{X}}_{(j)}$ replaced by $\mathcal{X}(j)$, ${\mathcal{Y}}_{(i)}={\left[{\left[{\mathcal{Y}}_{(i)}^{(k)}\right]}^T{\left[{\mathcal{Y}}_{(i)}^{(k+1)}\right]}^T\right]}^T$ is the partial data vector of dimension 2i×1 and the partial matrix ${\mathbf{A}}_{(i)}^H$ consists of first i rows of A ^H . It should be noted that partial Alamouti matrix X_a(i) is the function of the first i data points, while X _a (i) is a function of the i th data point. Obviously, the solution that minimizes this partial joint ML metric is not the globally optimal. But we have the following lemma^a:

3.1 Recursive derivation of bound

For our blind search strategy, the calculation of the metric or bound $M_{{\widehat{\mathcal{X}}}_{(i)}}$ is needed at each tree node for comparison with the optimal value of objective function, R. This bound can be derived recursively by simply expressing $M_{{\widehat{\mathcal{X}}}_{(i)}}$ in terms of new observation and an additional regressor ${\widehat{\mathbf{X}}}_a(i)$ as follows:

$\begin{array}{l}{M}_{{\widehat{\mathcal{X}}}_{(i)}}=\underset{\mathbf{h}}{\text{min}}\left\{\parallel \mathbf{h}{\parallel }_{{\mathbf{R}}_h^{-1}}^2+\parallel {\mathcal{Y}}_{(i)}-\sqrt{\rho }{\widehat{\mathbf{X}}}_{a(i)}\mathbf{h}{\parallel }^2\right\}\\ =\underset{\mathbf{h}}{\text{min}}\left\{\parallel \mathbf{h}{\parallel }_{{\mathbf{R}}_h^{-1}}^2+{∥\left[\begin{array}{l}{\mathcal{Y}}_{(i-1)}\\ \mathcal{Y}(i)\end{array}\right]-\sqrt{\rho }\left[\begin{array}{l}{\widehat{\mathbf{X}}}_{a(i-1)}\\ {\widehat{\mathbf{X}}}_a(i)\end{array}\right]\mathbf{h}∥}^2\right\}\end{array}$

(19)

By invoking the block version of recursive least squares (RLS) algorithm to the cost function in (19) with the data vector of size 2×1 and the regressor matrix of dimension 2×2L, we get [23]

$M_{{\widehat{\mathcal{X}}}_{(i)}}=M_{{\widehat{\mathcal{X}}}_{(i-1)}}+{\mathbf{e}}_i^H{\mathit{\Gamma }}_i{\mathbf{e}}_i$

(20)

${\widehat{\mathbf{h}}}_i={\widehat{\mathbf{h}}}_{i-1}+{\mathbf{G}}_i{\mathbf{e}}_i,$

(21)

where

${\mathbf{e}}_i=\mathcal{Y}(i)-\sqrt{\rho }{\widehat{\mathbf{X}}}_a(i){\widehat{\mathbf{h}}}_{i-1}$

(22)

${\mathbf{\Gamma }}_i={\left[{\mathbf{I}}_2+\rho {\widehat{\mathbf{X}}}_a(i){\mathbf{P}}_{i-1}{\widehat{\mathbf{X}}}_a{(i)}^H\right]}^{-1}$

(23)

${\mathbf{G}}_i=\sqrt{\rho }{\mathbf{P}}_{i-1}{\widehat{\mathbf{X}}}_a{(i)}^H{\mathit{\Gamma }}_i$

(24)

${\mathbf{P}}_i={\mathbf{P}}_{i-1}-{\mathbf{G}}_i{\mathit{\Gamma }}_i^{-1}{\mathbf{G}}_i^H$

(25)

The RLS recursions are initialized by

$M_{{\widehat{\mathcal{X}}}_{(i-1)}}=0,{\widehat{\mathbf{h}}}_{-1}=\mathbf{0}\text{and}{\mathbf{P}}_{-1}={\mathbf{R}}_h.$

Before introducing our algorithm, we first number the |Ω|² combinations of the constellation points from two antennas by 1,2,…,|Ω|² and treat them as a big constellation set Ψ, where the k th(1≤k≤|Ω|²) vector constellation point is denoted by Ψ(k). We then perform the depth-first search of signal tree for joint ML solution.

The algorithm essentially reduces the search space of exhaustive ML search by performing a trimmed search over the signal tree of N layers, where each tree node at the i th layer corresponds to a specific partial sequence ${\mathcal{X}}_{(i)}$ and each tree node at the intermediate layer has |Ω|² offsprings to the next layer.

The parameter ρ can be easily determined by estimating the noise variance, whereas for R _h , our simulation results indicate that we can replace it with an identity matrix with almost no effect on the performance via carrier reordering (see next section). To obtain the initial guess of search radius, we can use the strategy described in [20] to determine r that would guarantee a MAP solution with very high probability. Nevertheless, the algorithm itself takes care of the value of r, in that if it is too small such that the algorithm is not able to backtrack, then it doubles the value of r; if it is too large such that the algorithm reaches the last subcarrier too quickly, then it reduces r to the most recent value of objective function (see steps 4 and 6). Therefore, any choice of r would guarantee the MAP solution.

The complexity of the algorithm is mainly attributed to the calculation of the bound $M_{{\widehat{\mathcal{X}}}_{(i)}}$ (step 2) and the backtracking (step 3). The rest of the steps are simple additions and subtractions. From the RLS recursions, it can be seen that the calculation of the bound depends heavily on computation of 2 L×2L matrix P _i in (25). In Section 4, we shall see how computation of P _i can be avoided by exploiting the structure of the FFT matrix, while in Section 5 we shall deal with the issue of backtracking.

As mentioned earlier, the complexity of computing the metric $M_{{\widehat{\mathcal{X}}}_{(i)}}$ depends heavily on computation of matrix P _i . We show how we can completely avoid computing P _i and hence simply discard (25) from RLS recursions. This means that RLS algorithm will reduce to least mean square (LMS) in terms of complexity.

4.2 Reducing complexity by carrier reordering

In the above approximation, we assumed that P₁=I and a _i are orthogonal which allows us to use (27). However, a _i are columns of the partial FFT matrix A _i , so strictly speaking, they are not orthogonal. Hence, the successive regressor matrices would not be orthogonal, too. We will show that we can make them orthogonal or semi-orthogonal by carrier reordering. To understand the idea, we compute and plot the magnitude of autocorrelation of these partial vectors given by

$\left|{\mathbf{a}}_i^H{\mathbf{a}}_l\right|=\left\{\begin{array}{ll}L& \text{if}i=l\\ \frac{1}{L}\left|\right.\frac{sin(\pi (i-1)L/N)}{sin(\pi (i-1)/N)}& \text{if}i\ne l\end{array}\right.$

(30)

in Figure 2 for N=16 and L=4 and where we set l=1. It can be seen that columns 1, 5, 9, and 13 are orthogonal to each other and so are the columns 2, 6, 10, 14 and so on. If we visit the subcarriers in the order 1,5,9,13,2,6,10,14,⋯,4,8,12,16, we find that consecutive vectors will be orthogonal or approximately orthogonal. In general, it is found that with △=N/L, the vectors a _i ,a_i+△,a_i+2△,∀i are approximately orthogonal. Therefore, by simple reordering the carriers, we can achieve orthogonality among different subcarriers and use that fact to reduce the complexity of our algorithm as done in section 4.1.

We can make use of the third point to our advantage. Specifically, the presence of some pilots and slow variation in the channel allows us to get a tentative estimate of the data. If we are able to arrange the data according to its reliability, starting with the most reliable data, then there would be a less chance that we need to backtrack. Since earlier data is reliable, there is no need to backtrack for this part. The later data might not be reliable, but by the time we start processing this data, the algorithm would have been already converged. However, measuring data reliability requires tentative channel/data estimates. Thus, the blind algorithm can be turned into a semi-blind algorithm to reliably track the channel along with the data detection. The semi-blind algorithm would require a short training sequence of L symbols only at the start of transmission to get a tentative estimate of the data and its reliability and no further pilots or channel statistics would be required.

5.1 Measuring the reliability

For measuring the reliability of data carriers, we borrow the idea presented in [21] by the author of the current paper, where it was used in the context of non-linear distortion mitigation in OFDM. To minimize backtracking, the algorithm must devise a procedure to identify the reliable subcarriers from the tentative estimates of channel and the data. With receiver having an estimate of channel, the decoding process can be accomplished by rewriting (10) into the form as shown below:

$\tilde{\mathcal{Y}}=\sqrt{\rho }{\mathbf{H}}_a\mathcal{X}+\tilde{\mathcal{N}},$

(31)

where $\tilde{\mathcal{Y}}={\left[{\left[{\mathcal{Y}}^{(k)}\right]}^T{\left[{\mathcal{Y}}^{\ast (k+1)}\right]}^T\right]}^T,\tilde{\mathcal{N}}={\left[{\left[{\mathcal{N}}^{(k)}\right]}^T{\left[{\mathcal{N}}^{\ast (k+1)}\right]}^T\right]}^T$ and H _a is an Alamouti-like matrix defined as

${\mathbf{H}}_a\triangleq \left[\begin{array}{ll}{\mathbf{\Lambda }}_1& {\mathbf{\Lambda }}_2\\ {\mathbf{\Lambda }}_2^{\ast }& -{\mathbf{\Lambda }}_1^{\ast }\end{array}\right]$

(32)

By left multiplying both sides of (31) with $\frac{1}{\sqrt{\rho }}{\mathbf{H}}_a^{-1}$ and rearranging the terms, we get:

$\widehat{\mathcal{X}}=\mathcal{X}+\mathbf{D}$

(33)

with the difference vector; $\mathbf{D}\triangleq \frac{1}{\sqrt{\rho }}{\mathbf{H}}_a^{-1}\tilde{\mathcal{N}}$. The imperfect knowledge of the channel results in an estimation error Δ H _a and consequently the vector D represents the distortion due to channel estimation error and the effect of additive noise.

To assess the reliability, consider a data carrier $\widehat{\mathcal{X}}(k)$ (in scalar case) and its nearest constellation point $〈\widehat{\mathcal{X}}(k)〉$. Treating channel estimation error as noise, ML-based decoding would yield $\mathcal{X}(k)$ by mapping $\widehat{\mathcal{X}}(k)$ to the nearest constellation point $〈\widehat{\mathcal{X}}(k)〉$. Such a scheme would be very efficient at higher SNR if distortion was only due to AWGN. However, in our case, we have an additional perturbation due to channel estimation error that is independent of SNR, and therefore, we expect that part of data samples would be severely effected by the channel distortion and fall outside their actual decision regions. Clearly, there is a need to assess and identify these unreliable data coefficients for our algorithm to reduce the backtracking.

Authors in [21] have developed a rigorous method for assessing the reliability of estimated data coefficients. Intuitively, for the data carrier $\widehat{\mathcal{X}}(k)$, we can measure its reliability based on relative posterior probability that the difference term D(k) equals $\widehat{\mathcal{X}}(k)-〈\widehat{\mathcal{X}}(k)〉$ to the probability that it equals some other vector $\widehat{\mathcal{X}}(k)-{\Omega }_m|{\Omega }_m\ne 〈\widehat{\mathcal{X}}(k)〉$. For example,

$\begin{array}{l}\Re (k)=\text{log}\frac{\mathit{\text{Pr}}\left(〈\widehat{\mathcal{X}}(k)〉=\mathcal{X}(k)\mid \widehat{\mathcal{X}}(k)\right)}{\mathit{\text{Pr}}\left(〈\widehat{\mathcal{X}}(k)〉={\mathcal{X}}_{\mathit{\text{NN}}}(k)\mid \widehat{\mathcal{X}}(k)\right)}\\ =\text{log}\frac{\mathit{\text{Pr}}\left(D(k)=\widehat{\mathcal{X}}(k)-〈\widehat{\mathcal{X}}(k)〉\right)}{\mathit{\text{Pr}}\left(D(k)=\widehat{\mathcal{X}}(k)-{\mathcal{X}}_{\mathit{\text{NN}}}(k)\right)}\end{array}$

(34)

defines the reliability in decoding $\widehat{\mathcal{X}(k)}$ to the closest constellation point relative to decoding to the nearest neighbour ${\mathcal{X}}_{\mathit{\text{NN}}}(k)$. Figure 3 illustrates this concept, such that for instance even though $\widehat{\mathcal{X}}(1)$ and $\widehat{\mathcal{X}}(2)$ have the same distance from $〈\widehat{\mathcal{X}}(1)〉=〈\widehat{\mathcal{X}}(2)〉$, $\widehat{\mathcal{X}}(2)$ has a higher reliability than $\widehat{\mathcal{X}}(1)$ as it is farther from nearest neighbour. This suggests the dependence of phase ${\theta }_{\widehat{\mathcal{X}}(k)-〈\widehat{\mathcal{X}}(k)〉}$ on the reliability in addition to the magnitude $|\widehat{\mathcal{X}}(k)-〈\widehat{\mathcal{X}}(k)〉|$. In essence, the reliability of a measurement at each tone is a function that maps a triple (magnitude, phase and channel gain) into ${\mathcal{R}}_{[1,\infty ]}$. The exact expression for reliability is the generalization of (34) to the vector-wise likelihood ratio defined as

${\Re }^{\text{exact}}=\text{log}\frac{f_D\left(\widehat{\mathcal{X}}-〈\widehat{\mathcal{X}}〉\right)}{{\sum }_{m=0,{\Omega }_m\ne 〈\widehat{\mathcal{X}}〉}^{\left|\Omega \right|-1}{f}_D\left(\widehat{\mathcal{X}}-{\Omega }_m\right)},$

(35)

where f _D (.) is the pdf of the distortion vector D, which by definition, can be easily seen to be Gaussian circularly symmetric with variance ${\sigma }_D^2=\frac{1}{\rho }\left({\mathbf{Ha}}^{-1}\right){\left({\mathbf{Ha}}^{-1}\right)}^H$. The above computation for exact reliability is however inefficient since it would require O(N|Ω|) evaluations of f _D (.) which grows with constellation size |Ω|. Alternately, the geometric-based approximations for assessing reliability as derived in [21] may be employed with marginal loss in the performance. Ultimately, once the vector $\Re$ is computed, we can proceed to select the most reliable data tones. These reliable data tones can then be supplied to our algorithm for initial search of the ML solution. Based on the above developments, we now introduce the semi-blind algorithm.

where $\widetilde{\overline{\mathbf{B}}}=\left(\mathbf{A}\text{diag}\left(|{\mathcal{X}}_1^{(k)}{|}^2+|{\mathcal{X}}_2^{(k)}{|}^2\right){\mathbf{A}}^H\right)$. It can be seen that the above equations correspond to the SISO systems for which the joint ML solution can be derived easily. It is worth mentioning that the search space for blind ML is not reduced due to dependence of observations and the matrix $\overline{\mathbf{B}}$ on ${\mathcal{X}}_1^{(k)}$ and ${\mathcal{X}}_2^{(k)}$. However, the number of channel parameters is reduced to half, which results in an improved estimation performance. In short, the Alamouti structure enables us to either improve upon the estimation performance or the complexity reduction but not the both.

For simulations, we first consider OFDM system with N=16 subcarriers and channel length L=4 for each transmit-receive channel and CP length of at least L−1. For blind algorithm, both channels are independent Rayleigh fading, assumed stationary over two consecutive OFDM blocks and each having an exponential power decay profile, i.e. E{|h _i (τ)|²}=e^−0.2τ. Information symbols are modulated using binary phase shift keying (BPSK) or quadrature amplitude modulation (QAM).

In Figure 4, we plot the results for N=16, BPSK data symbols using perfectly known channel and our exact blind algorithm, together with low-complexity variants, i.e. blind algorithm with (a) P _i =I and (b) P _i =I with subcarrier reordering. In the first case, the performance degrades and bit error rate (BER) reaches an error floor. However, with subcarrier reordering approach, we almost get the same performance as that of exact blind algorithm without requiring the channel statistics. A similar trend is observed in Figure 5, when 4-QAM signal modulation is considered.

For semi-blind algorithm, we adopt the AR(1) process to model the slow rayleigh fading channels, where the channel weight vector varies as [23]: h(n)=α h(n−1)+q(n) and where α=J₀(2π f_dT_s) and q is a complex normal vector with covariance matrix (1−α²)I. The product of maximum Doppler frequency f_d and sampling time T_s referred to as normalized Doppler frequency F_D, controls the amount of variations in the channel statistics. Two different values of normalized Doppler frequency, i.e., 0.1 and 0.001, are considered in the simulations. Results for semi-blind algorithm are presented against perfect coherent detection in Figures 6 and 7 for BPSK and 4−QAM modulations respectively which show favourable performance of algorithm under different fading conditions. For higher modulations and number of subcarriers, we use the SISO OFDM systems obtained after decoupling by Alamouti structure and results are presented in Figure 8 for 16−QAM constellation with N=64.To assess the computational complexity of proposed algorithm, we compare the average number of nodes visited by the algorithm with various reliability measures in Figure 9. It is clearly observed that proposed reliability scheme offers significantly lower complexity at lower SNR values. At higher SNR, the complexity is constant, confirming the fact that there is almost no backtracking. Figure 10 shows that the performance for various degrees of reliability measures is almost identical which means that computational advantages are attained without degrading the performance. Through simulations, it has also been observed that the reliability of around 50% to 60% is enough for a good performance, although more importantly, the algorithm does not disfavour the usage of more reliable carriers.Finally, in Figure 11, the complexity of comparison for different modulation schemes such as BPSK, 4-QAM and 16-QAM is presented which clearly indicates the computational advantages of proposed reliability-based method.

In this paper, we presented a blind ML algorithm for joint channel estimation and data detection in OFDM wireless systems using Alamouti STBC coding. The simulation results show favourable performance of algorithm. As evident from simulations, our low-complexity blind algorithm performs equally well as exact blind algorithm. Moreover, the new algorithm does not need any prior information about channel statistics as it avoids calculating matrix P _i with subcarrier reordering. Another major source of complexity in blind algorithm is the issue of backtracking which becomes more prominent in the low-SNR regime. This issue was not tackled in the previous studies. We proposed a semi-blind algorithm which minimizes the probability of backtracking by supplying the blind algorithm with reordered subcarriers based on their reliability computations using a sophisticated reliability criterion. By minimizing the backtracking, significant improvement is achieved in terms of complexity without compromising the performance. Moreover, the orthogonal structure of Alamouti coding was also exploited to yield complexity reduction or improved estimation performance. The reduced complexity variant turns out to be very handy when large number of subcarriers and/or large size constellation modulations are employed.

^a This lemma was proved in [20] for SISO case; we simply extend it here to the multi-antenna case.

^b In fact, a weaker condition that three consecutive vectors a _i , a_i+1 and a_i+2 are orthogonal would suffice.