Case Studies - Digital Signal Processing: A Practitioner's Approach - page 126

6.3.1 Problem Formulation
The two signals x ð t Þ and y ð t Þ are converted to the digital domain as

1

x k ¼ A x cos ð k t þ

k Þ;

2

y k ¼ A y cos ð k t þ

k Þ;

ð 6 : 6 Þ

where t is the sampling time.

6.3.2 Finding DoA
The prime objective here is to estimate the DoA ( ) and the quality of the DoA from

the digitised signals x k and y k of N samples. The principle of the method is to plot
x k , y k values in a 2D space and to find a best-fit straight line using linear regression.
The slope of this line gives tan(DoA), from which we compute DoA. The mean
square error is calculated and is used as a measure of the quality of the DoA
given by the system. This is shown in Figure 6.4(b) and the theory is elaborated
below.

6.3.3 Straight-Line Fit
The general equation for a straight line can be written as

y k ¼ mx k þ

¼ ð : Þ

c for k 1toN 6 7

¼ p 1 x k þ p 2 ; ð 6 : 8 Þ

where m ¼ p 1 and c ¼ p 2 are the slope and the abscissa of the straight line,

respectively. Let

t

x ¼½ x 1 ; x 2 ; ...; x N ; ð 6 : 9 Þ

t

y ¼½ y 1 ; y 2 ; ...; y N ; ð 6 : 10 Þ

t

u ¼½ 1 ; 1 ; ...; 1 ; ð 6 : 11 Þ

t

x

i ¼½

x i ; 1 : ð 6 : 12 Þ

A

We formulate an N 2 rectangular matrix given as

A ¼½xju; ð 6 : 13 Þ

t

p ¼½ p 1 ; p 2 : ð 6 : 14 Þ

Then we write the set of equations (6.8) in matrix form:

y ¼ A:p ð 6 : 15 Þ