NAME ____________________________________________
ES 442 Homework # 6
(Spring 2016 – Due April 11, 2016 )
Print out homework and do work on the printed pages.
Textbook: B. P. Lathi & Zhi Ding, Modern Digital and Analog Communication
Systems, 4th edition, Oxford University Press, New york, 2009.
Problem 1 Sampling a Signal Consisting of Two Sinusoids (20 points)
We are given a signal m(t) = cos(2 100 t) + cos(2 220 t) which is ideally sampled at a
sampling frequency of fs = 300 Hz. The sampled output y(t) is passed through a lowpass filter with bandwidth 150 Hz. What frequency components pass through the lowpass filter? Hint: Sketch the two-sided spectrum of the sampled output look for what
passses through the LPF.
Problem 2 Sampling (20 points)
Homework 6
The signal m(t), whose spectrum is shown below, is ideally sampled at fs = 20 Hz.
(a) Sketch the spectrum of the sampled output y(t) for | f | < 40 Hz.
(b) Can the original signal be recovered with a sampling rate of 20 Hz?
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Problem 3 Time Domain Multiplexing (20 points)
Three signals (m1(t), m2(t) and m3(t)) are to be multiplexed. Signals m1(t) and m2(t) both
have a 5 kHz bandwidth, however, signal m3(t) has a 10 kHz bandwidth. Design a
commutating (rotating) switching system so that each signal is sampled at its Nyquist
rate.
Problem 4 S0/Nq for PCM (15 points)
A low-pass signal m(t) with a 3 kHz bandwidth and an amplitude range from -5 volts to
+5 volts is sampled at the Nyquist rate. It is then converted to an eight-bit code using
uniform quantization. The mean-squared value of the message signal S0 is 2 voltssquared.
Homework 6
(a) Calculate the normalized power for quantization noise Nq.
(b) Calculate the signal-to-quantization noise ratio (S0/Nq) in decibels (dB).
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Problem 5 PCm for CDs (25 points)
A compact disc recording system samples each of the two stereo signals with a 16-bit
analog-to-digital converter with a sampling rate of 44.1 kHz.
(a) Determine the output signal-to-quantized noise ratio (S0/Nq) for a full amplitude
sinusoidal signal.
(b) The bit stream of the digitized data is augmented by the addition of error correcting
bits, clock extraction bits, etc. These additional bits represent an overhead of 100% (in
other words the number of bits per second is doubled). Determine the output bit rate
with the overhead bits included for stereo operation.
(c) The CD disc can record approximately 1 hour of music. Determine the number of bits
Homework 6
stored on a typical CD from the above information.
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