Block and Fast Block Sparse Adaptive Filtering for Outdoor Wireless Channel Estimation and Equalization

Abstract

Rayleigh’s distribution is mainly used when fading wireless medium does not have proper line of sight (LOS) path and is dominated by a large number of non-line of sight (NLOS) paths due to reflections of the received signal. Also because of relative motion of the base station and mobile station, a random frequency shift is generally introduced in the carrier, which can be realized in terms of Doppler spread. In case of Rayleigh’s fading channels, there are two critical problems for receiver design that is accurate estimation of channel coefficients followed by mitigation of channel impairments like inter symbol interference and fading in presence of user mobility. The accuracy of estimated channel state information is really crucial to design robust equalizer for reconstruction of bit sequence and the equalizer performance is affected by the fading rate and Doppler spread. The main research contributions of the paper is based on the exploitation of underlying sparseness of block adaptive filters through $l_{0}$-norm penalty for accurate estimation with stable convergence which helps to design computationally efficient adaptive models for estimation. The accuracy of the proposed sparse block and fast block models is tested using 16 QAM modulation format with Rayleigh’s fading wireless channel for outdoor environments. With the help of MATLAB simulations, the performance of the proposed sparse BLMS and FBLMS adaptive filtering are verified and the detail comparison results are presented.

Appendix

1.1 Base Band Channel Model for QAM

QAM modulated transmitted signal passing through the channel can be expressed as:

$$\begin{aligned} x_{bp} (t) & = \sum\limits_{k} {s_{c} } (k)p(t - kT)\cos \omega_{c} t \\ & - \sum\limits_{k} {s_{s} } (k)p(t - kT)\sin \omega_{c} t \\ \end{aligned}$$

$$= \text{Re} \sum\limits_{k} {(s_{c} (k) + j} s_{s} (k))p(t - kT)ej\omega_{c} t$$

(A.1)

where $s_{c} (k)$,$s_{s} (k)$ and p(t) are real. p(t) is the base band signal and $s_{c} (k) + js_{s} (k)$ represents the complex QAM symbol. The complex base band equivalent signal is

$$x_{lp} (t) = \sum\limits_{k} {(s_{c} (k) + js_{s} (k))} p(t - kT)$$

(A.2)

Assuming $x_{lp} (t)$ to be transmitted over a band pass channel $h_{bp} (t)$ with low pass equivalent $h_{lp} (t)$:

$$h_{bp} (t) = \text{Re} (h_{lp} (t)e^{{j\omega_{c} t}} )$$

(A.3)

So the channel out is:

$$\begin{aligned} y_{bp} & = (h_{bp} *x_{bp} )(t) = \text{Re} ((x_{lp} *h_{lp} )(t)ej\omega_{c} t) = \text{Re} \left[ {\left( {\sum\limits_{k} {(s_{c} (k) + js_{s} (k))} p(t - kT)*h_{lp} (t)} \right)ej\omega_{c} t} \right] \\ & = \text{Re} \left[ {\left( {\sum\limits_{k} {(s_{c} (k) + js_{s} (k))} h_{lp,e} (t - kT)} \right)ej\omega_{c} t} \right] = \text{Re} \left[ {y_{lp} (t)ej\omega_{c} t} \right] \\ \end{aligned}$$

(A.4)

where $h_{lp} ,e = (h_{lp} *p)(t)$ and $y_{lp} (t)$ denotes the complex signal

$$\begin{aligned} y_{lp} (t) & = \left[ {\left( {\sum\limits_{k} {(s_{c} (k) + js_{s} (k))} h_{lp,e} (t - kT)} \right)} \right] \\ & = y_{r} (t) + jy_{i} (t) \\ \end{aligned}$$

(A.5)

So

$$y_{bp} (t) = \text{Re} \left[ {y_{lp} (t)ej\omega_{c} t} \right] = y_{r} (t)\cos \omega_{c} t + jy_{i} (t)\sin \omega_{c} t$$

(A.6)

The QAM demodulator separates $y_{r} (t)$ and $y_{i} (t)$. But as $h_{lp,e} (t)$ is complex,$y_{r} (t)$ is a linear combination of $s_{c} (k)$ and the interference component $s_{s} (k)$. Similarly $y_{i} (t)$ is the linear combination of $s_{s} (k)$ and $s_{c} (k)$. As $y_{r} (t)$ and $y_{i} (t)$ are sampled at the output of QAM demodulator,$y_{lp} (nT)$ is the discrete version of $y_{lp} (t)$. The equivalent complex discrete-time channel is given as $h_{d} (n) = h_{lp,e} (nT)$ and the corresponding sampled signal is $y_{d} (n) = y_{lp} (nT)$. Thus the expression of output signal is

$$y_{d} (n) = \sum\limits_{k} {(s_{c} (k) + js_{s} (k))} h_{d} (n - k) + q(n)$$

(A.7)

where $q(n)$ represents the channel noise.

1.2 Sparsity Introduced Through Norm Penalty

The size of a vector $\vec{w}$ is called the norm of $\vec{w}$, which can be represented as $L_{1}$-norm, $L_{2}$-norm, $L_{p}$-norm, … $L_{\infty }$-norm etc.

$\vec{w} = (w_{1} ,w_{2} ,w_{3} \ldots w_{n} )$, ($w_{i} \in C$ for $i = 1,2, \ldots n$)

In general the $L_{\infty }$-norm is defined as:

$$\left\| {\vec{w}} \right\|_{\infty } = \sqrt[\infty ]{{\sum\nolimits_{i} {w_{i}^{\infty } } }}$$

(A.8)

Let $w_{j}$ is the highest entry in the vector $w_{i}$, then by the property of the infinity,

$$w_{j}^{\infty } = w_{i}^{\infty } ,\quad \forall_{i} \ne j$$

$${\text{which}}\;{\text{implies}}\quad \sum\limits_{i} {w_{i}^{\infty } } = w_{j}^{\infty }$$

(A.9)

and $\left\| {\vec{w}} \right\|_{\infty } = \sqrt[\infty ]{{\sum\nolimits_{i} {w_{i}^{\infty } } }} = \sqrt[\infty ]{{w_{j}^{\infty } }} = \left| {w_{i} } \right|$

So the $L_{\infty }$-norm is defined as:

$$\left\| {\vec{w}} \right\|_{\infty } = \hbox{max} (\left| {w_{i} } \right|)$$

(A.10)

In that context the $L_{p}$-norm or $l_{p}$-norm for $p \in N,\;1 \le p < \infty$ can be defined as:

$$\left\| {\vec{w}} \right\|_{p} = \sqrt[p]{{\sum\nolimits_{i} {w_{i}^{p} } }}$$

(A.11)

The cost function for LMS family using $l_{p}$-norm penalty can be represented as:

$$E_{{l_{p} }} (k) = 1/2\left| {er(k)} \right|^{2} + \gamma_{p} \left\| {w(k)} \right\|_{{l_{p} }}$$

(A.12)

Using gradient descent method the p-norm based update equation can be derived as:

$$w(k + 1) = w(k) + \mu er(k)x(k) - {{\left( {\rho_{p} \left( {\left\| {w(k)_{{l_{p} }} } \right\|} \right)^{1 - p} \text{sgn} (w(k))} \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{p} \left( {\left\| {w(k)_{{l_{p} }} } \right\|} \right)^{1 - p} \text{sgn} (w(k))} \right)} {\left| {w(k)} \right|^{1 - p} }}} \right. \kern-0pt} {\left| {w(k)} \right|^{1 - p} }}$$

(A.13)

where $\rho_{p} = \mu \gamma_{p}$

when the entry of $w(k)$ approaches to zero then the above equation is modified as:

$$w(k + 1) = w(k) + \mu er(k)x(k) - {{\left( {\rho_{p} \left( {\left\| {w(k)_{{l_{p} }} } \right\|} \right)^{1 - p} \text{sgn} (w(k))} \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{p} \left( {\left\| {w(k)_{{l_{p} }} } \right\|} \right)^{1 - p} \text{sgn} (w(k))} \right)} { \in_{p} + \left| {w(k)} \right|^{1 - p} }}} \right. \kern-0pt} { \in_{p} + \left| {w(k)} \right|^{1 - p} }}$$

(A.14)

here $\in_{p}$ indicates the bounds.