Algorithms and Derivations

Part 1: Time Delay Estimate Problem:We consider the use of LMS adaptive filter for the estimation of time delay between two measured signals. This is a problem which occurs in a diverse range of applications including radar, sonar, and biomedical signal analysis among them. In the simplest arrangement measurements x(i) and d(i), i0, are made at sensors S1, S2 say separated by a distance r.

Fig.1

The form of measurement sensor varies accordingly to the application. In sonar, for example, the measurements are generated at the output of two hydrophones. The delay D is related to the angle of arrival of the signal through simple geometry. Reffering to above figure we have,

D = (r sin)/

Where is the propagation velocity of the signal through the medium. Thus, the estimation of bearing angle is reduced to the estimation of delay D.

The structure of adaptive filter is shown in below figure,

Fig.2

In above figure, Z-1 is the unit delay line, i.e.,

Z-1 x(i) = x(i-1), i0

The order of the FIR filter is L. we choose LD.

Least Mean Square Algoritm:From above figure we have

y(i) = (0+1z-1+..+Dz-D+Lz-L)x(i)

= (0x(i)+1x(i-1)+..+Dx(i-D)+Lx(i-L)

= [(0+1+..+D+L] [x(i) x(i-1) . X(i-D).x(i-L)] = 0T(i) (1)

Parameter vector is an estimate of optimal parameters f0, obtained based on the measurements (observations) upto time i-1. Thus we may write

y(i)=(i-1)T(i)

where

(i-1)T = [0(i-1),1x(i-1),..,Dx(i-1),Lx(i-1)]

(2)

Note that the error e(i) is given by

e(i)=d(i)-y(i)=d(I)-(i-1)T(i)

It is not difficult to see that the vector of optimal parameters is

oT=[0,0,0,1,0,.0]

When f(i) f0 , then e(i) 0

I I

The LMS algorithm is given as follows

(i)=(i-1) + (i) e(i)

With (i), and e(i) defined by (1) and (3), respectively.

Instead of (5) we can use the following normalized gradient algorithm.

Normalized Gradient Algorithm:(i)=(i-1) + [ (i) e(i)]/r(i) , 0