Processing of Signals - Digital Signal Processing: A Practitioner's Approach - page 12

1.5.1 Windowing
The original signal could be very long and non-periodic, but due to physical
limitations we observe the signal only for a finite duration. This results in multi-
plication of the signal by a rectangular window function R ð t Þ , giving an observed
signal

~ y ð t Þ¼ y ð t Þ R ð t Þ; ð 1 : 29 Þ

where

1 ; 0 t < T w ;

R ð t Þ¼ ð 1 : 30 Þ

0 ; otherwise :

T w is the observation time interval. In general, R ð t Þ can take many forms and these
functions are known as windowing functions [5].

1.5.2 Sampling
In reality, a band-limited analogue signal y ð t Þ needs to be sampled, resulting in a
discrete-time signal f y k g , converting the function as a mapping from the integer line
to the real line I!R , where I denotes the set of integers. We express f y k g as

f y k g¼ 1
k ¼1

y ð t Þ R ð t Þð t kT s Þ

" # R ð t Þ:

1

¼

y ð t Þð t kT s Þ

ð 1 :

31 Þ

k ¼1

where ð t Þ is the unit impulse (Dirac delta function)

1 ; t ¼ 0 ;

ð t Þ¼ ð 1 : 32 Þ

0 ; t 6¼ 0 :

T s is the sampling time. The sampling process is defined via (1.31) and is shown
in Figure 1.7(a). This process generates a sampled or discrete signal. The
Fourier transform of the sampled signal y k

(Figure 1.7(b)) is computed and

shown in Figure 1.8.
The striking feature in Figure 1.8 is the periodic replication of the narrowband
spectrum. Revisiting Fourier series will help us to understand the effect of
sampling. The Fourier series is indeed a discrete spectrum or a line spectrum,
and results from the periodic nature of a signal in the time domain. Any
transformation from the time domain to the frequency domain maps periodicity