Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 20

spectrum 1

1 1

T Xðf þ f s Þ , . , are overlapped; that is, there are many
overlapping portions in the sampled signal spectrum.
From Figure 2.6 , it is clear that the sampled signal spectrum consists of the scaled baseband
spectrum centered at the origin, and its replicas centered at the frequencies of nf s

T Xðf Þ and its replicas T Xðf f s Þ ,

(multiples of the

sampling rate) for each of n ¼ 1 ; 2 ; 3 ; . .
If applying a lowpass reconstruction filter to obtain exact reconstruction of the original signal
spectrum, the following condition must be satisfied:

f s f max f max (2.4)

Solving Equation (2.4) gives

f s 2 f max (2.5)

In terms of frequency in radians per second, Equation (2.5) is equivalent to

u s 2 u max (2.6)

This fundamental conclusion is well known as the Shannon sampling theorem, which is formally
described below:

For a uniformly sampled DSP system, an analog signal can be perfectly recovered as long as the sampling rate is at
least twice as large as the highest-frequency component of the analog signal to be sampled.

We summarize two key points here.

1. The sampling theorem establishes a minimum sampling rate for a given band-limited analog signal
with highest-frequency component f max . If the sampling rate satisfies Equation (2.5) , then the
analog signal can be recovered via its sampled values using the lowpass filter, as described in
Figure 2.6 ( b).
2. Half of the sampling frequency f s = 2 is usually called the Nyquist frequency (Nyquist limit) or
folding frequency. The sampling theorem indicates that a DSP system with a sampling rate of f s
can ideally sample an analog signal with a maximum frequency that is up to half of the
sampling rate without introducing spectral overlap (aliasing). Hence, the analog signal can be
perfectly recovered from its sampled version.

Let us study the following example.

EXAMPLE 2.1
Suppose that an analog signal is given as
xðtÞ¼5cosð2p$1; 000tÞ; for t 0

and is sampled at the rate 8,000 Hz.
a. Sketch the spectrum for the original signal.

b. Sketch the spectrum for the sampled signal from 0 to 20 kHz.