Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 77

Figure 3.35 Spatial wave on a ULA of L measurements y 1 ; n , K, y L ; n

sinusoidal oscillator. These coefficients are then modified and applied to a direction
finding (DF) system using a ULA.

3.12.3.1 Angular Frequency Estimation

Consider an AR process given by

x n þ 1 ¼ 2 cos ð 2 f Þ x n x n 1 ;

representing an oscillator with normalised angular frequency , 2 f.By

defining

2 cos 1

T

X n ðÞ , ½ x n þ 1 ; x n ; FðÞ , ;

1 0

this can be rewritten as X n ðÞ¼FðÞX n 1 ðÞ , where X n ðÞ is the state vector and

ðÞ is the state transition matrix. For the nth snapshot ð n ¼ 1 ; ...; N Þ let L

measurements y 1 ; n ; ...; y L ; n of x n ; ...; x n þ L 1 be written as

T

y k ; n ðÞ¼c X n ðÞþ k ; n ; k ¼ 1 ; ...; L ; ð 3 : 63 Þ

T , and

where c , ½ 0 ; 1 k ; n is additive noise. An L 1 measurement vector

T

Y n ðÞ , ½ y 1 ; n ; ...; y L ; n

can be expressed using (3.63) as

Y n ðÞ¼AðÞX n ðÞþG n ; ð 3 :

64 Þ