Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 24

and is sampled at the rate of 8,000 Hz.
a. Sketch the spectrum of the sampled signal up to 20 kHz.
b. Sketch the recovered analog signal spectrum if an ideal lowpass filter with a cutoff frequency of 4 kHz is used to
filter the sampled signal (yðnÞ¼xðnÞ in this case) to recover the original signal.
Solution:
a. Using Euler's identity, we get
xðtÞ¼ 3
2

þ 5
2

þ 5
2

þ 3
2

e j2p$3;000t e j2p$2;000t e j2p$2;000t e j2p$3;000t

The two-sided amplitude spectrum for the sinusoid is displayed in Figure 2.12 :

Xf

s ()

2.5/T

f kHz

5 6 8 10 11 13 14 16 18 19

−11 −10 −6 −5 −3 −2 2 3

FIGURE 2.12
Spectrum of the sampled signal in Example 2.2.

Y ()

f

kHz

−3 −2 2 3

FIGURE 2.13

Spectrum of the recovered signal in Example 2.2.

b. Based on the spectrum in (a), the sampling theorem condition is satisfied; hence, we can recover the original
spectrum using a reconstruction lowpass filter. The recovered spectrum is shown in Figure 2.13 .

EXAMPLE 2.3
Assume an analog signal is given by
xðtÞ¼5cosð2p 2; 000tÞþ1cosð2p 5; 000tÞ; for t 0

and is sampled at a rate of 8,000 Hz.
a. Sketch the spectrum of the sampled signal up to 20 kHz.

b. Sketch the recovered analog signal spectrum if an ideal lowpass filter with a cutoff frequency of 4 kHz is used to

recover the original signal (yðnÞ¼xðnÞ in this case).