Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 50

2.6. Assuming a continuous signal is given as

xðtÞ¼ 10cos ð 2 p$ 5 ; 500 tÞþ 5sin ð 2 p$ 7 ; 500 tÞ; t 0

for

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 filter the sampled signal in order to recover the original signal;
c. determine the frequency/frequencies of aliasing noise.
2.7. Assuming a continuous signal is given as

xðtÞ¼ 8cos ð 2 p$ 5 ; 000 tÞþ 5sin ð 2 p$ 7 ; 000 tÞ; for t 0

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 filter the sampled signal in order to recover the original signal;
c. determine the frequency/frequencies of aliasing noise.
2.8. Assuming a continuous signal is given as

xðtÞ¼ 10cos ð 2 p$ 5 ; 000 tÞþ 5sin ð 2 p$ 7 ; 500 tÞ; for t 0

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 filter the sampled signal in order to recover the original signal;
c. determine the frequency/frequencies of aliasing noise.
2.9. Given a Butterworth type second-order anti-aliasing lowpass filter ( Figure 2.35 ) , determine
the values of circuit elements if we want the filter to have a cutoff frequency of 1,000 Hz.

C 2

V in

R 1 R 2

V o

+

−

Choose

C 2

14142
(2

.

RR C f c ) ==

C 1

1 2

π

2

1
(2

C =

1

2

RRC π f c )

122

FIGURE 2.35

Filter circuit in Problem 2.9.