Adaptive-Slope Squashing-Function-Based ANN for CSI Estimation and Symbol Detection in SFBC-OFDM System

Abstract

This paper presents an adaptive-slope squashing-function (ASF)-based artificial neural network (ANN) for efficient estimation of smoothly time-varying multipath fading channels, in a $4 \times 1$ space-frequency-block-coded orthogonal-frequency-division-multiplexing (SFBC-OFDM) system using 64 subcarriers. The channel-state-information (CSI) estimated at first stage is further used for OFDM information symbol detection (through minimum mean square error criterion-based detection) at second stage. To combat the impact of smoothly time-varying environment, we emphasize on the utilization of ASF-ANN using backpropagation (BP) algorithm for the estimation of channel tap coefficients in frequency domain. The underlying ANN is modeled as feedforward multi-layered perceptron that updates the network weights. The major focus is on the gradient-descent algorithm-based adaptation of the slope of squashing-function (SF) along with other ANN parameters, which enhances the training efficiency of ASF-ANN in terms of the lower mean-squared channel estimation error in comparison with the traditional fixed-slope squashing-function (FSF) ANN technique. Simulation results corresponding to the underlying $4 \times 1$ SFBC-OFDM system are presented to depict that ASF-ANN-based approach outperforms the FSF-ANN technique by providing lower bit-error-rate (BER) due to the usage of well-estimated CSI. At 15 dB SNR and fade rate = 0.001, the average BER reduces to $2.85 \times 10^{ - 4}$ for the ASF-ANN based approach, due to improved CSI estimation, which accounts for approximately 5% improvement in the detection success rate as compared to the FSF-ANN-based approach.

Appendix

1.1 Brief Details About LMMSE-Based Channel Estimation for SFBC-OFDM System

Let the received symbol vector in underlying SFBC-OFDM system be

$${\mathbf{Y}} = {\mathbf{HX}} + {\mathbf{W}}$$

(34)

(subscripts and indices in Eq. (4) are not shown for mathematical convenience).

As per conventional LS channel estimation method, the following cost function is minimized to provide optimum LS solution

$$J({\hat{\mathbf{H}}}_{LS} ) = \left\| {{\mathbf{Y}} - {\mathbf{X\hat{H}}}_{LS} } \right\|^{2}$$

(35)

where ${\hat{\mathbf{H}}}_{LS}$ is the LS estimated channel vector. However, the above function can also be rewritten as

$$\begin{aligned} J({\hat{\mathbf{H}}}_{LS} ) & = ({\mathbf{Y}} - {\mathbf{X\hat{H}}}_{LS} )^{H} ({\mathbf{Y}} - {\mathbf{X\hat{H}}}_{LS} ) \\ & = {\mathbf{Y}}^{H} {\mathbf{Y}} - {\mathbf{Y}}^{H} {\mathbf{X\hat{H}}}_{LS} - {\hat{\mathbf{H}}}^{H}_{LS} {\mathbf{X}}^{H} {\mathbf{Y}} + {\hat{\mathbf{H}}}^{H}_{LS} {\mathbf{X}}^{H} {\mathbf{X\hat{H}}}_{LS} \\ \end{aligned}$$

(36)

In order to find the optimum value ${\hat{\mathbf{H}}}_{LS}$, for which the cost function in Eq. (35) gets minimized, the derivative of cost function with respect to the estimated channel vector needs to be calculated and equated to zero. It results in

$$\frac{{\partial J({\hat{\mathbf{H}}}_{LS} )}}{{\partial {\hat{\mathbf{H}}}_{LS} }} = - 2\left( {{\mathbf{X}}^{H} {\mathbf{Y}}} \right)^{*} + 2\left( {{\mathbf{X}}^{H} {\mathbf X}{\hat{\mathbf{H}}}_{LS} } \right)^{*} = 0$$

(37)

where $(.)^{*}$ is the complex conjugation operator. The mathematical simplification of above equation leads to

$${\hat{\mathbf{H}}}_{LS} = \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{H} {\mathbf{Y}}$$

(38)

This can also be expressed as

$$\begin{aligned} {\hat{\mathbf{H}}}_{LS} & = \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{H} \left[ {{\mathbf{XH}} + {\mathbf{W}}} \right] \\ & = {\mathbf{H}} + \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{H} {\mathbf{W}} \\ \end{aligned}$$

(39)

As the underlying system utilizes QO-codes, the matrix ${\mathbf{X}}^{H} {\mathbf{X}}$ is not purely diagonal, and therefore it contains some off-diagonal entries as well. Considering the LS estimate in (38), a better linear estimate in terms of weighted LS estimate can be applied by using the LMMSE channel estimation method, as suggested in [36]. This method finds a linear estimate to minimize the following error function

$$J\left( {{\hat{\mathbf{H}}}_{MM} } \right) = E\left[ {\left\|{\mathbf \psi} \right\|^{2} } \right] = E\left[ {\left\| {{\mathbf{H}} - {\hat{\mathbf{H}}}_{MM} } \right\|^{2} } \right]$$

(40)

where ${\hat{\mathbf{H}}}_{MM} = {\mathbf{W}}_{MM} {\hat{\mathbf{H}}}_{LS}$ is the LMMSE estimate. As the estimation error $\mathbf\psi$ is orthogonal to ${\hat{\mathbf{H}}}_{LS}$, therefore $E\left[ {{\mathbf{\psi \hat{H}}}^{H}_{LS} } \right] = 0$. It follows that

$$\begin{aligned} E\left[ {{\mathbf{\psi \hat{H}}}^{H}_{LS} } \right] & = E\left[ {\left( {{\mathbf{H}} - {\hat{\mathbf{H}}}_{MM} } \right){\hat{\mathbf{H}}}^{H}_{LS} } \right] \\ & = E\left[ {\left( {{\mathbf{H}} - {\mathbf W}_{MM} {\hat{\mathbf{H}}}_{LS} } \right){\hat{\mathbf{H}}}^{H}_{LS} } \right] \\ & = E\left[ {{\mathbf{H\hat{H}}}^{H}_{LS} } \right] - {\mathbf W}_{MM} E\left[ {{\hat{\mathbf{H}}}_{LS} {\hat{\mathbf{H}}}^{H}_{LS} } \right] \\ & = {\mathbf R}_{{{\mathbf{H\hat{H}}}_{LS} }} - {\mathbf W}_{MM} {\mathbf R}_{{{\hat{\mathbf{H}}}_{LS} {\hat{\mathbf{H}}}_{LS} }} \\ \end{aligned}$$

(41)

where ${\mathbf{R}}_{{{\mathbf{H\hat{H}}}_{LS} }}$ is the cross-correlation matrix between true channel vector and LS estimated channel vector, and ${\mathbf{R}}_{{{\hat{\mathbf{H}}}_{LS} {\hat{\mathbf{H}}}_{LS} }}$ is the auto-correlation matrix of LS estimated channel vector. Further solution of Eq. (41) yields

$${\mathbf{W}}_{MM} = {\mathbf{R}}_{{{\mathbf{H\hat{H}}}_{LS} }} {\mathbf{R}}_{{{\hat{\mathbf{H}}}_{LS} {\hat{\mathbf{H}}}_{LS} }}^{ - 1}$$

(42)

Using Eq. (39), it can be shown that the auto-correlation matrix ${\mathbf{R}}_{{{\hat{\mathbf{H}}}_{LS} {\hat{\mathbf{H}}}_{LS} }}$ is

$$\begin{aligned} {\mathbf{R}}_{{{\hat{\mathbf{H}}}_{LS} {\hat{\mathbf{H}}}_{LS} }} & = E\left[ {{\hat{\mathbf{H}}}_{LS} {\hat{\mathbf{H}}}^{H}_{LS} } \right] \\ & = E\left[ {\left( {{\mathbf{H}} + \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{H} {\mathbf W}} \right)\left( {{\mathbf{H}} + \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{H} {\mathbf W}} \right)^{H} } \right] \\ & = E\left[ {{\mathbf{HH}}^{H} } \right] + E\left[ {\left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{W}}^{H} {\mathbf{W}}} \right] \\ & = {\mathbf {R}}_{{{\mathbf{HH}}}} + \sigma_{w}^{2} \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} \\ \end{aligned}$$

(43)

However, the cross-correlation matrix ${\mathbf{R}}_{{{\mathbf{H\hat{H}}}_{LS} }}$ is found to be

$$\begin{aligned} {\mathbf{R}}_{{{\mathbf{H\hat{H}}}_{LS} }} & = E\left[ {{\mathbf{H\hat{H}}}^{H}_{LS} } \right] \\ & = E\left[ {{\mathbf{H}}\left( {{\mathbf{H}} + \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{H} {\mathbf{W}}} \right)^{H} } \right] \\ & = E\left[ {{\mathbf{HH}}^{H} } \right] + E\left[ {{\mathbf{HW}}^{H} {\mathbf{X}}\left( {{\mathbf{XX}}^{H} } \right)^{ - 1} } \right] \\ & \approx \;{\mathbf{R}}_{{{\mathbf{HH}}}} \\ \end{aligned}$$

(44)

Subsequently, the LMMSE channel estimates using Eqs. (38), (42), (43) and (44) are expressed as

$${\hat{\mathbf{H}}}_{MM} = {\mathbf{R}}_{{{\mathbf{HH}}}} \left\{ {{\mathbf{R}}_{{{\mathbf{HH}}}} + \sigma_{w}^{2} \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} } \right\}^{ - 1} {\hat{\mathbf{H}}}_{LS}$$

(45)

$$= {\mathbf{R}}_{{{\mathbf{HH}}}} \left\{ {{\mathbf{R}}_{{{\mathbf{HH}}}} + \sigma_{w}^{2} \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} } \right\}^{ - 1} \left( {{\mathbf{X}}^{H} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{H} {\mathbf{Y}}$$

(46)