Detection of Channel Variations to Improve Channel Estimation Methods

Abstract

In current digital communication systems, channel information is typically acquired by supervised approaches that use pilot symbols included in the transmit frames. Given that pilot symbols do not convey user data, they penalize throughput spectral efficiency, and transmit energy consumption of the system. Unsupervised channel estimation algorithms could be used to mitigate the aforementioned drawbacks although they present higher computational complexity than that offered by supervised ones. This paper proposes a simple decision method suitable for slowly varying channels to determine whether the channel has suffered a significant variation, which requires to estimate the matrix of the recently changed channel. Otherwise, a previous estimate is used to recover the transmitted symbols. The main advantage of this method is that the decision criterion is only based on information acquired during the time frame synchronization, which is carried out at the receiver. We show that the proposed criterion provides a considerable improvement of computational complexity for both supervised and unsupervised methods, without incurring in a penalization in terms of symbol error ratio. Specifically, we consider systems that make use of the popular Alamouti code. Performance evaluation is accomplished by means of simulated channels as well as making use of indoor wireless channels measured using a testbed.

Appendix

In this appendix we will derive the maximum log-likelihood expression for channel coefficients used in the paper to estimate such channel coefficients from pilot sequences.

Under the assumptions A1–A3, the expected log-likelihood of the observations is given, up to an irrelevant constant factor, by:

$$\begin{aligned} L(\mathbf{h}) = -\frac{1}{2\sigma _n^2N}\sum _{k=0}^{N-1} \Vert \mathbf{x}_k - \mathbf{S}_k\mathbf{h}\Vert ^2, \end{aligned}$$

(12)

where $N$ is the number of samples of the observed signal. The gradient of Eq. (12) with respect to the unknown vector $\mathbf{h}$ yields the score function:

$$\begin{aligned} \nabla L(\mathbf{h}) = \frac{1}{\sigma _n^2N}\sum _{k=0}^{N-1} \mathbf{S}_k^\mathrm {H}(\mathbf{x}_k - \mathbf{S}_k\mathbf{h}). \end{aligned}$$

(13)

Setting the score to zero leads to the ML estimate:

$$\begin{aligned} \hat{ \mathbf{h}}_\mathrm {ML}= \frac{1}{2N\hat{\sigma }_s^2} \sum _{k=0}^{N-1}\mathbf{S}^\mathrm {H}_k\mathbf{x}_k, \end{aligned}$$

(14)

where $\hat{\sigma }_s^2= \frac{1}{2N}\sum _{t=0}^{2N-1} |s(t)|^2$ is a sample estimate of the source variance $\sigma _s^2$. To derive the expression of Eq. (14), we have exploited the orthogonality of the source symbol matrix $\mathbf{S}_k$ given by Eq. (3), which means in particular that

$$\begin{aligned} \sum _{k=0}^{N-1}\mathbf{S}_k^\mathrm {H}\mathbf{S}_k = \sum _{k=0}^{N-1} \bigl ( |s_1(k)|^2 + |s_2(k)|^2 \bigr )\mathbf{I}_2 = \left( \sum _{t=0}^{2N-1} |s(t)|^2\right) \mathbf{I}_2 = 2N\hat{\sigma }_s^2\mathbf{I}_2, \end{aligned}$$

(15)

where $\mathbf{I}_2$ is the ($2\times 2$) identity matrix. According to Eq. (14), the ML estimates of the channel coefficients can be expressed as follows:

$$\begin{aligned} \hat{h}_{1,\mathrm {ML}}&= \frac{1}{2N\hat{\sigma }_s^2}\sum _{k=0}^{N-1} \left( s_1(k)^*x_1(k) - s_ 2(k)x_ 2(k)\right) \nonumber \\ \hat{h}_{2,\mathrm {ML}}&= \frac{1}{2N\hat{\sigma }_s^2}\sum _{k=0}^{N-1} \left( s_ 2(k)^*x_1(k) + s_1(k)x_2(k)\right) . \end{aligned}$$

(16)