Problems – Chapter 2
2.1
You want to transmit the world “HOW” using an 8-ary system.
(a) Encode the world “HOW” into a sequence of bits, using 7-bit ASCII coding,
followed by an eighth bit for error detection, per character. The 8th bit is chosen so
that the number of ones in the 8 bits is an even number. How many total bits are there
in the message?
ASCII (H): 0001001, ASCII (O): 1111001, ASCII (W): 1110101
(b) Partition the bit stream into k = 3 bit segments. Represent each of the 3-bit
segments as an octal number (symbol). How many octal symbols are there in the
message?
(c) If the system were designed with 16-ary modulation, how many symbols would be
used to represent the word “HOW”
(d) If the system were designed with 256-ary modulation, how many symbols would
be used to represent the word “HOW”
2.2
We want to transmit 800 characters/s, where each character is represented by its 7-bit
ASCII codeword, followed by an 8th bit for error detection per character, as in
problem 2.1. A multilevel PAM waveform with M = 16 levels is used.
(a) What is the effective transmitted bit rate?
(b) What is the symbol rate?
2.3
An analog signal is sampled at its Nyquist rate 1/Ts, and quantized using L
quantization levels. The derived signal is then transmitted on some channel.
(a) Show that the time duration, T, of one bit of the transmitted binary encoded signal
must satisfy T ≤ Ts /(log 2 L)
(b) When is the equality sign valid?
2.4
Consider an audio signal with spectral components limited to the frequency band 300
to 3300 Hz. Assume that a sampling rate of 8000 samples/s will be used to generate a
PCM signal. Assume that the ratio of peak signal power to average quantization noise
power at the output needs to be 30 dB.
(a) What is the minimum number of uniform quantization levels needed, and what is
the minimum number of bits per sample needed?
(b) Calculate the system bandwidth (as specified by the main spectral lobe of the
signal) required for the detection of such a PCM signal.
2.5
A waveform, x ( t ) = 10 cos(1000 t + π / 3) + 20 cos( 2000 t + π / 6) is to be
uniformly sampled for digital transmission
(a) What is the maximum allowable time interval between sample values that will
ensure perfect signal reproduction?
(b) If we want to reproduce 1 hour of this waveform, how many sample values need
to be stored?
2.6
Use the method of convolution to illustrate the effect of undersampling the waveform
3
x ( t ) = cos 2πf 0 t for a sampling rate of f s = f 0
2
2.7
The information in an analog waveform, whose maximum frequency fm = 4000 Hz, is
to be transmitted using a 16-level PAM system. The quantization distortion must not
exceed ± 1% of the peak-to-peak analog signal.
(a) What is the minimum number of bits per sample of bit bits per PCM word that
should be used in this PAM transmission system?
(b) What is the minimum required sampling rate, and what is the resulting bit rate?
(c) What is the 16-ary PAM symbol transmission rate?
2.8
In the compact disc (CD) digital audio system, an analog signal is digitized so that the
ratio of the peak-signal power to the peak-quantization noise power is at least 96 dB.
The sampling rate is 44.1 kilosamples/s.
(a) How many quantization levels of the analog signal are needed for (S/N)peak = 96
dB?
(b) How many bits per sample are needed for the number of levels found in part (a)?
(c) What is the data rate in bits/s?
2.9
(a) Consider that you desire a digital transmission system, such that the quantization
distortion of any audio source does not exceed ± 2% of the peak-to-peak analog
signal voltage. If the audio signal bandwidth and the allowable transmission
bandwidth are each 4000 Hz, and sampling takes place at the Nyquist rate, what value
of bandwidth efficiency (bits/s/Hz) is required?
(b) Repeat part (a) except that the audio signal bandwidth is 20 kHz (hi-fi), yet the
available transmission bandwidth is still 4000 Hz.
2.10
An analog voltage between -5V and 5V must be quantized using 3 bits. Quantize each
of the following samples, and record the quanization error for each:
(a) -3.4 V
(b) 0.0 V
(c) 0.625 V
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