Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 21

Solution:
a. Since the analog signal is sinusoid with a peak value of 5 and frequency of 1,000 Hz, we can write the sine wave
using Euler's identity:

e j2p1;000t þ e j2p1;000t

2

Þ¼2:5e j2p1;000t þ 2:5e j2p1;000t

5cosð2p 1; 000tÞ¼5$ð

which is a Fourier series expansion for a continuous periodic signal in terms of the exponential form (see Appendix B). We
can identify the Fourier series coefficients as

c 1 ¼ 2:5 and c 1 ¼ 2:5

Using the magnitudes of the coefficients, we then plot the two-side spectrum as shown in Figure 2.7A.
b. After the analog signal is sampled at the rate of 8,000 Hz, the sampled signal spectrum and its replicas centered
at the frequencies nf s , each with a scaled amplitude of 2:5=T , are as shown in Figure 2.7B :

X ()
2.

f kHz

−1 1

FIGURE 2.7A

Spectrum of the analog signal in Example 2.1.

Xf
s

()

2.5 / T

f kHz

−1 1 7 8 9 15 16 17

−9 −8 −7

FIGURE 2.7B

Spectrum of the sampled signal in Example 2.1.

Notice that the spectrum of the sampled signal shown in Figure 2.7B contains the images of the original
spectrum shown in Figure 2.7A ; that the images repeat at multiples of the sampling frequency f s (for our example,
8 kHz, 16kHz, 24kHz, . ); and that all images must be removed, since they convey no additional information.

2.2 SIGNAL RECONSTRUCTION
In this section, we investigate the recovery of analog signal from its sampled signal version. Two
simplified steps are involved, as described in Figure 2.8 . First, the digitally processed data yðnÞ are
converted to the ideal impulse train y s ðtÞ , in which each impulse has amplitude proportional to digital
output yðnÞ , and two consecutive impulses are separated by a sampling period of T ; second, the analog