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.21000t/: (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.

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