Case Studies - Digital Signal Processing: A Practitioner's Approach - page 125

where

d i is the projected element distance and is the angular rotation of the

elements.

x i ð t Þ¼ sin 2 ½ ft þ ' i : ð 6 : 2 Þ

The phase detector output m 1 ð t Þ is obtained from r ð t Þ¼ sin ð 2 ft Þ and x i ð t Þ :

h

ð

i

cos m 1 ð

Þ¼ ð' i Þ¼ Þ ð Þ : ð : Þ

t cos D cos i cos t 6 3

Signal m 1 ð t Þ is passed through an FM discriminator and another phase detector,

giving

x ~ ð t Þ¼ A x cos ð t þ 1 ð t ÞÞ; ð 6 : 4 Þ

~

y ð t Þ¼ A y cos ð t þ 2 ð t ÞÞ; ð 6 : 5 Þ

where A x ¼ K y is zero in a least-squares

sense for a given system. Traditional analogue systems present DoA by inputting
~

K x cos ðÞ , A y ¼ K y sin ðÞ and e ¼ K x

x ð t Þ and y ~ ð t Þ to an X-Y scope, shown in Figure 6.4(a). The phase distortions 1 ð t Þ

and are the undesired phase modulations due to multiple reflections,

instrumentation errors, etc., and the plot between

2 ð t Þ

x ~ ð t Þ and y ~ ð t Þ results in a

distorted figure of eight. The ratio between the amplitudes of Þ

has the DoA information in it and the sinusoids are in phase with each other.
However, finding a direct ratio is catastrophic and here we formulate it as a
DSP problem.

x ~ ð t Þ and y ~ ð t

1

1

0.5

0.5

0 0

−0.5 −0.5

− 1 −1

− 0.5 0 0.5

−0.5 0 0.5

Digital

Analogue

(b)

(a)

Figure 6.4 DOA using DSP in a rotary system