Solutions to Practice Problems
Practice Problem 21.1
Consider the signal from the oboe depicted below in time and frequency domain representations.
What is the maximum frequency present in the oboe signal?
1. Based upon this, what sampling rate must be exceeded in order to accurately reconstruct the signal from its
samples?
1
0.25
0.2
Voltage (V)
Voltage (V)
0.5
0
0.15
0.1
-0.5
0.05
-1
1
1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005
Time (sec)
0
0
1000
2000
3000
4000
5000
Frequency (Hz)
From the frequency plot, it looks like the highest frequency spike appears at 5.8 kHz. This means the sample rate
should be at least 2 (5.8 kHz)=11.6 kHz.
Practice Problem 21.2
Consider the following analog waveform. This waveform is sampled at a 500 Hz rate and quantized with a 2-bit quantizer
(i.e., A/D converter) The input range is -1.0 to +1.0 V.
a. Circle the sample points (first sample is at time t = 0 sec).
b. Indicate the quantization intervals and corresponding digital words.
c. Indicate the digital word assigned to each sample point.
d. What is the stream of binary bits generated after the A/D conversion is complete?.
e. What is the resulting bit rate from this A/D?
a.
b.
Since fs = 500 Hz, Ts = 1/fs = 1/500 Hz = 2 msec. A sample is taken every 2 msec.
Using a 2-bit quantizer, there are 22 = 4 quantization intervals. The resolution is
q
vmax  vmin 1  (1)

 0.5 V
2N
22
The intervals and digital words are as shown.
6000
c.
d.
e.
Digital words for each sample are as shown.
Binary bits: 10 11 11 10 00 01
Bit rate:
Rb 
2 bits 500 samples

 1000 bps
sample
sec