Processing of Signals - Digital Signal Processing: A Practitioner's Approach - page 14

1.5.2.1 Aliasing Error
A careful inspection of (Equation 1.33 and Figure 1.8) shows there is no real loss of
information except for the periodicity in the frequency domain. Hence the original
signal can be reconstructed by passing it through an appropriate lowpass filter.
However, there is a condition in which information loss occurs, which is

1 = T s 2f c , where f c is the cut-off frequency of the band-limited signal. If this

condition is not satisfied, a wraparound occurs and frequencies are not preserved.
But this aliasing is put to best use for downconverting the signals without using any
additional hardware, like mixers in digital receivers, where the signals are bandpass
in nature.

1.5.3 Quantiaation
In addition, the signal gets quantised due to finite precision analogue-to-digital
converters. The signal can be modelled as

y k ¼b y k cþ k ; ð 1 : 34 Þ

where k is a uniformly distributed (UD)

random number of 2 LSB. Sampled signal y k and b y k c are depicted in Figure 1.9(a)
and the quantisation error is shown in Figure 1.9(b). In this numerical example we
have used 10 ( 5) levels of quantisation, giving an error (

b y k c is a finite quantised number and

1

10 , which
can be seen in Figure 1.9(b). The process of moving 9 signal from one domain to the

k ) between

1

0.1

0.5 0.05

0

0

−0.05

− 0.5

− 0.1

−1

0 0.02 0.04 0.06 0.08 0.1

0 0.02 0.04 0.06 0.08 0.1

Seconds

(a) (b)

Figure 1.9 Quantised discrete signal j y k j

9

This movement is because we want to do digital signal processing.