Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 156

^
s o ( t )

^

k

≈ θ ( t )

Input PM Signal,

θ ( k ) ≈ Σ i=0 x ( i )

x ( k )

Sampler &
ADC

DAC &
LPF

x ( t )

N −bits

G 1

x ( t ) = A sin[

ω o t +

( t )]

θ

z −1

[ θ ( t ) = α m ( t ) ]

DCO

v o ( k )

Fig. 3.21 Circuit diagram for PM demodulation using SDPLL1

Figure 3.21 shows the circuit diagram for PM demodulation using 1st-order
SDPLL.

3.5 Linear Estimation and Adaptive Filtering

A transmitted signal x(n) is normally corrupted by random noise during the
transmission process so that the received signal y(n) may be relatively difficult to
detect. To enhance detectability, it is advisable to use the best possible means to
reduce the effects of the noise. If one knows the sent signal exactly, one can use
matched filtering. In many situations, however, only partial (as opposed to exact)
information about the signal is available. The general process of forming an
approximation to the noise free signal based on partial information is called
estimation. This section deals with the issue of estimation, and in particular esti-
mation based on the use of linear filters. Throughout the section the estimated
signal will be denoted by x ð n Þ:
There are various different forms of partial information which may be avail-
able. One may, for example, know the spectral region occupied by the sent signal,
and the spectral region occupied by the undesirable noise. Often the noise will
occupy a wider bandwidth than the signal and so one can simply use a BPF to
remove all noise outside the band of the sent signal—the filtered signal could then
be considered to be the estimated signal. This is a very simplistic approach to
estimation, and one that may well not be optimal. During the 1940s Norbert
Wiener studied the problem of how to optimally estimate a sent signal, given a
knowledge of the sent signal's spectrum and the received signal's spectrum. He
found that the optimal estimate is given by the output of the filter whose transfer
function is:

H o ð s Þ¼ S yx ð s Þ
S xx ð s Þ

e as ð 3 : 44 Þ

where a is the delay of the Wiener filter H o (s), S xx (s) is the PSD of the original
signal, S yx (s) is the cross spectral density (CSD) between the original and observed
signals.