7.1. DIGITAL SIGNALS 231
erefore, to avoid any aliasing or distortion of the discrete signal frequency content and
to be able to recover or reconstruct the frequency content of the original analog signal, one must
have f
s
2f
max
. is is known as the Nyquist rate. at is, the sampling frequency should be at
least twice the highest frequency in the analog signal. Normally, before any digital manipulation,
a front-end anti-aliasing lowpass analog filter is used to limit the highest frequency of the analog
signal.
Let us further examine the aliasing problem by considering an under-sampled sinusoid as
depicted in Figure 7.4. In this figure, a 1 kHz sinusoid is sampled at f
s
D 0:8 kHz, which is less
than the Nyquist rate of 2 kHz. e dashed-line signal is a 200 Hz sinusoid passing through
the same sample points. us, at the sampling frequency of 0.8 kHz, the output of an A/D
converter is the same if one uses the 1 kHz or 200 Hz sinusoid as the input signal. On the other
hand, over-sampling a signal provides a richer description than that of the signal sampled at the
Nyquist rate.
7.1.2 QUANTIZATION
An A/D converter has a finite number of bits (or resolution). As a result, continuous amplitude
values get represented or approximated by discrete amplitude levels. e process of converting
continuous into discrete amplitude levels is called quantization. is approximation leads to
errors called quantization noise. e input/output characteristic of a 3-bit A/D converter is
shown in Figure 7.5 illustrating how analog voltage values are approximated by discrete voltage
levels.
Quantization interval depends on the number of quantization or resolution levels, as illus-
trated in Figure 7.6. Clearly, the amount of quantization noise generated by an A/D converter
depends on the size of the quantization interval. More quantization bits translate into a narrower
quantization interval and, hence, into a lower amount of quantization noise.
In Figure 7.7, the spacing between two consecutive quantization levels corresponds
to one least significant bit (LSB). Usually, it is assumed that quantization noise is signal-
independent and is uniformly distributed over 0:5 LSB and 0:5 LSB. Figure 7.7 also shows the
quantization noise of an analog signal quantized by a 3-bit A/D converter and the corresponding
bit stream.
7.1.3 A/D AND D/A CONVERSIONS
Because it is not possible to process an actual analog signal by a computer program, an analog
sinusoidal signal is often simulated by sampling it at a very high sampling frequency. Consider
the following analog sine wave:
x.t/ D cos.21000t/: (7.4)
Let us sample this sine wave at 40 kHz to generate 0.125 s of x.t/. Note that the sampling
interval T
s
D 2:5 10
5
s is very short, and thus x.t / appears as an analog signal. Sampling
involves taking samples from an analog signal every T
s
seconds. e above example generates
232 7. DIGITAL SIGNALS AND THEIR TRANSFORMS
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0 0.5 1 1.5
2 2.5 3 3.5 4
x 10
-3
Time
1 kHz 200 Hz
Amplitude
Figure 7.4: Ambiguity caused by aliasing.
111
010
101
100
011
010
001
000
76543210
Analog
Input
Digital Output
-½ LSB
½ LSB
Quantization Error
Analog
Input
76543210
(a) (b)
Figure 7.5: Characteristic of a 3-bit A/D converter: (a) input/output transfer function; (b) ad-
ditive quantization noise.

Get Anywhere-Anytime Signals and Systems Laboratory, 2nd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.