Block NLMS/F-based equalizer design and channel capacity analysis for indoor IEEE 802.11 fading wireless channels

Abstract

In this paper, adaptive decision feedback-based model is used to design an efficient equalizer to mitigate the detrimental effects during the signal transmission through wireless channels having non-ideal frequency response. The proposed adaptive equalizer is designed using normalized block least mean square/fourth algorithm which yields low bit error rate and mean square error in spite of high noisy and fading channel conditions. The constellations as well as eye diagrams are also taken as performance measuring criteria for indoor wireless channel which is designed by using IEEE 802.11 model with exponential power delay profile. The channel capacity using the proposed model is also analyzed to verify the system performance. Combination of block processing approach and the elegant structure of the equalizer achieves higher symbol estimation accuracy and stable convergence which is quite suitable for indoor fading channel characteristics. The performance comparison of the proposed model is presented with detail simulation results.

Appendix

1.1 Exponential PDP

The exponential PDP for an indoor channel is expressed as the average power of the channel that decreases exponentially with channel delay:

$$ p(\tau ) = 1/\tau_{d} {\text{e}}^{{ - \tau /\tau_{d} }} $$

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$\tau_{d}$ is the parameter to determine the PDP and the maximum excess delay is given by:

$\tau_{m} = - \tau_{d} \ln V$,V is the ratio of non-negligible path power to the first path power, i.e.,

$$ V = P(\tau_{m} )/P(0) = \exp ( - \tau_{d} /\tau_{m} ) $$

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The discrete time model of Eq. (23) with sampling period $T_{s}$ then can be written as:

$$ P(p) = 1/\sigma_{\tau } {\text{e}}^{{ - pT_{s} /\sigma_{\tau } }} ,\;p = 0,1, \ldots p_{\rm{max} } $$

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p is the discrete index and $p_{\rm{max} }$ is the index of the last path and $p_{\rm{max} } = [\tau_{m} /T_{s} ]$.

The total power can be calculated as:

$$ P_{total} = \sum\nolimits_{p = 0}^{{p_{\rm{max} } }} {P(p) = } 1/\sigma_{\tau } \left( {1 - {\text{e}}^{{ - (p_{\rm{max} } + 1)T_{s} /\sigma_{\tau } }} /1 - {\text{e}}^{{ - T_{s} /\sigma_{\tau } }} } \right) $$

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To normalize the total power as given in Eq. (4), Eq. (25) is again modified as:

$$ P(p) = P(0){\text{e}}^{{ - pT_{s} /\sigma_{\tau } }} ,\;p = 0,1, \ldots p_{\rm{max} } $$

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$$ {\text{where}}\;P(0)\;{\text{is}}\;{\text{the}}\;{\text{first}}\;{\text{path}}\;{\text{power}}\;{\text{and}}\;P(0) = 1/\left( {p_{\text{total}} .\sigma_{\tau } } \right) $$

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The probability density function (PDF) of Rayleigh’s distribution of a time varying signal statistics with flat fading channel of different multi-path component may be given as

$$ p(u) = \left\{ {\begin{array}{*{20}c} {u/\sigma^{2} \;\exp \left( { - u^{2} /2\sigma^{2} } \right) \ldots \left( {0u \le \infty } \right)} \\ {0 \ldots (u < 0)} \\ \end{array} } \right. $$

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where $\sigma$  rms. value of is received signal and $\sigma^{2}$ is time-averaged signal before envelop detection.

Then, the probability of not exceeding a specified value U for the received signal is given as:

$$ p(u) = {\text{pr}}(u \le R) = \int_{0}^{U} {p(u)} {\text{d}}u = 1 - \exp \left( { - U^{2} /2\sigma^{2} } \right) $$

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$$ u_{\text{mean}} = E\left[ u \right] = \int\limits_{0}^{\infty } {up(u){\text{d}}u = } \sigma \sqrt {\pi /2} = 1.2533\sigma $$

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and the variance of the Rayleigh distribution is given by $\sigma_{u}^{2}$:

$$ \begin{aligned} \sigma_{u}^{2} = & E\left[ {u^{2} } \right] - E^{2} \left[ u \right] = \int\limits_{0}^{\infty } {u^{2} } p(u){\text{d}}u - \sigma^{2} \pi /2 \\ = & \sigma^{2} \left( {2 - \pi /2} \right) = 0.4292\sigma^{2} \\ \end{aligned} $$

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1.2 Doppler spectrum

The slow moving user terminal of the model associated with a bell-shaped Doppler spectrum as expressed in Eq. (33) as:

$$ S_{D} (f) = \frac{1}{{1 + 6\left( {\frac{f}{{f_{d} }}} \right)^{2} }} $$

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$S_{D} (f)$ in linear scale and $f_{d}$, being the Doppler spectrum is expressed as $v/\lambda$ where ‘v’ is the velocity of the environment and $\lambda$ is the wavelength. The coherence time of the given Doppler spectrum is mathematically presented by:

$$ T_{c} = \frac{3}{{2\pi f_{d} }}{\text{In}}(2) $$

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1.3 Property of a signum function

A real number defined by a function $\text{sgn} \;x$ is defined by the following property:

$$ \text{sgn} \;x = \left\{ \begin{aligned} 1,\quad {\text{if}}\;0 < x \hfill \\ - 1,\quad {\text{if}}\;x < 0 \hfill \\ 0,\quad {\text{otherwise}} \hfill \\ \end{aligned} \right. $$

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