Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 56

Output of A/D

0

Same instant

−0.2

y k

x k

−0.4

Phase correction

O

−0.6

O

Interpolated y k

o

−0.8

− 1

364 366 368 370 372 374 376

k

Figure 3.14 Correcting the phase shift

The signals x ð t Þ or 4 y ð t Þ are passed through an analogue multiplexer, with the
help of a control signal. The digitised signals x k and y k are interleaved to yield
z k ¼ x k [ y k and passed to the processor for further processing in the digital domain.
Ideally, the requirement is to sample the signals at the same time. However, to
reduce the amount of hardware, we decide to use only a single A/D converter. Refer
to Figure 3.14; what we need is the interpolated value of y k ¼ ^ y k corresponding to

x k . It implies that we should not take the two sequences f x k g and f y k g for further
processing, but the sequences f x k g and f^ y k g . And this implies three things:

1. The interleaved sequence z k needs to be separated as x k and y k .
2. We reconstruct the sequence y k from y k by passing through an APF.
3. We use this pair of sequences f x k g and f^ y k g as if they are sampled simulta-

neously.

Obtaining the sequence f y k g is known as phase correction and is shown in
Figure 3.14. This should be done for the entire sequence with linear phase
characteristics. It is typically done using an APF.

3.5.2 Delay Estimation Using an APF
We demonstrate an adaptive APF for estimating the delay between two sinusoidal
signals. An APF has a flat frequency response and a variable phase response. We
best use this principle by varying the pole and zero positions. Consider two
omnidirectional antenna elements spaced at a distance d = 2 to avoid the grating

lobe phenomenon (similar to the aliasing problem in the time domain). Estimating
the value of

is important to find out the direction of an emitter producing
narrowband signals.

'

4

Mutually exclusive or.