DSP II ASSIGNMENT

Problem definition

Estimation of random channel impulse response h(n) using LMS, VSS-LMS, NLMS and RLS algorithms and compare their speed of convergence

A random channel needs to be identified in respect of its impulse response. Adaptive algorithm allows to identify a channel without any prior knowledge about the channel. Algorithms need to be compared by their convergence rate. Obviously with higher convergence rate, computational cost increases. But with modern high processing power, computational power is trivially important.

Significance

Objective

The main objective is to determine the impulse response of any random channel, representing the channel as an FIR filter.

Next task is to compare the convergence rate of LMS, NLMS, VSS-LMS, RLS

Channel

System model

Adaptive Filter, w

Adaptive algorithm

Input, xChannel Output, d

Filter output, y

Error, e

noise

Mathematical formulation

LMS algorithm:

here, n is increasing iterations is the step size

NLMS algorithm:

Mathematical formulation

Vss-LMS algorithm:

the new step size is given by:μ0, if it is

between MinStep and MaxStepMinStep, if μ0 < MinStep

MaxStep, if μ0 > MaxStep

Mathematical formulation

RLS algorithm:Initializing the algorithm by setting:

for each instant of time, n =1,2,…

,

h(n) from different algorithmsActual LMS NLMS Vss-LMS RLS

0.09760.04290.01460.04210.93

0.0969 0.0417 0.0140 0.0415 0.9290 -0.0006 -0.0005 -0.0005 -0.0007 -0.0003

0.0967 0.0414 0.0140 0.0417 0.9289 -0.0009 -0.0006 -0.0005 -0.0010 -0.0004

0.0980 0.0432 0.0153 0.0425 0.9305 0.0005 -0.0000 -0.0001 0.0010 0.0005

0.0971 0.0419 0.0140 0.0416 0.9291 -0.0006 -0.0004 -0.0004 -0.0006 -0.0003

Comparison of convergence rate

100 200 300 400 500 600

0.5

1

1.5

2

2.5

3

3.5

Iterations

MS

E

LMS

NLMS

Vss-LMS

RLS

Discussion

From the table, we can see that even after adding some noise, the algorithms fit the data quite well. RLS is the best fitting algorithms.

From the learning curves, we can find that RLS has the best learning feature. Vss-LMS, NLMS, LMS are gradually degrading in learning the filter.

Conclusion

Here, an unknown channel was identified with the help of adaptive filtering and it was identified well, even after adding some noise. The learning curves of the algorithms were plotted and they performed as expected.