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Numerical problems on different topics, Exams of Digital Communication Systems

Visvesvaraya Regional College of EngineeringDigital Communication Systems

Digital Communication Systems

Typology: Exams

2017/2018

Uploaded on 04/05/2018

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EEE482F: Problem Set 1
1. A digital source emits −1.0 and 0.0V levels with a probability of 0.2 each,
and +3.0 and +4.0V levels with a probability of 0.3 each. Evaluate the
average information of the source.
2. Consider a source that produces 6 messages with probabilities 1
2,1
4,1
8,1
16,
1
32, and 1
32 . Determine the average information content of a message.
3. A given source alphabet consists of 300 words, of which 15 occur with
probability 0.06 each and the remaining 285 words occur with probability
0.00035 each. If 1000 words are transmitted each second, what is the
average rate of information transmission.
4. A numeric keypad has the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Assume that
the probability of sending any one digit is the same as that for sending any
of the other digits. Calculate how often the buttons must be pressed in
order to send out information at a rate of 2 bits/second.
5. Consider a voice-grade telephone circuit with a bandwidth of 3kHz.
Assume that the circuit can be modelled as an additive white Gaussian
noise (AWGN) channel.
(a) What is the capacity of such a circuit if the SNR is 30dB.
(b) What is the minimum SNR required for a data rate of 4800 bits/s on
such a voice grade circuit?
(c) Repeat part (b) for a data rate of 19200 bits/s.
6. A 100 kbit/s data stream is to be transmitted on a voice-grade telephone
circuit with a bandwidth of 3kHz. Is it possible to achieve error-free
transmission with a SNR of 10dB?
7. Answer the following:
(a) Find the average capacity in bits per second that would be required to
transmit a high-resolution black-and-white TV signal at the rate of 32
1
pf3
pf4
pf5
pf8

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EEE482F: Problem Set 1

  1. A digital source emits − 1 .0 and 0.0V levels with a probability of 0.2 each, and + 3 .0 and + 4 .0V levels with a probability of 0.3 each. Evaluate the average information of the source.
  2. Consider a source that produces 6 messages with probabilities 12 , 14 , 18 , 161 , 1 32 , and^

1

  1. Determine the average information content of a message.

  2. A given source alphabet consists of 300 words, of which 15 occur with probability 0.06 each and the remaining 285 words occur with probability 0 .00035 each. If 1000 words are transmitted each second, what is the average rate of information transmission.

  3. A numeric keypad has the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Assume that the probability of sending any one digit is the same as that for sending any of the other digits. Calculate how often the buttons must be pressed in order to send out information at a rate of 2 bits/second.

  4. Consider a voice-grade telephone circuit with a bandwidth of 3kHz. Assume that the circuit can be modelled as an additive white Gaussian noise (AWGN) channel. (a) What is the capacity of such a circuit if the SNR is 30dB. (b) What is the minimum SNR required for a data rate of 4800 bits/s on such a voice grade circuit? (c) Repeat part (b) for a data rate of 19200 bits/s.

  5. A 100 kbit/s data stream is to be transmitted on a voice-grade telephone circuit with a bandwidth of 3kHz. Is it possible to achieve error-free transmission with a SNR of 10dB?

  6. Answer the following: (a) Find the average capacity in bits per second that would be required to transmit a high-resolution black-and-white TV signal at the rate of 32

pictures per second if each picture is made up of 2 × 106 picture elements (pixels) and 16 different brightness levels. All pixels are assumed to be independent and all levels have equal likelihood of occurrence. (b) For colour TV, this system additionally provides for 64 different shades of colour. How much more system capacity is required for a colour system compared to the black-and-white system? (c) Find the required capacity if 100 of the possible brightness-colour combinations occur with a probability of 0.003 each, 300 of the combinations occur with a probability of 0.001, and 624 of the combinations occur with a probability of 0.00064.

  1. Assume that a computer terminal has 110 characters on its keyboard and that each character is sent using binary words. (a) What are the number of bits required to represent each character? (b) How fast can characters be sent (characters/sec) over a telephone line channel having a bandwidth of 3.2kHz and a signal-to-noise ratio of 20dB? (c) What is the entropy of each character if each is equally likely to be sent?
  2. In a binary PCM system, if the quantising noise is not to exceed ± P percent of the peak-to-peak analogue level, show that the number of bits in each PCM word needs to be

n ≥ (log 2 10 ) log 10

P

= 3 .32 log 10

P

  1. The information in an analogue voltage waveform is to be transmitted over a PCM system with a ± 0 .1% accuracy (full scale). The analogue waveform has an absolute bandwidth of 100Hz and an amplitude range of −10 to +10V. (a) Determine the minimum sampling rate needed.

−1 −0.5 0 0.5 1 −

−0.

0

1

ei

eo

μ= μ=

(a) Sketch the complete μ = 10 characteristic that will handle input voltages over the range −5V to +5V. (b) Plot the corresponding expander characteristic. (c) Draw a 16-level nonuniform quantiser characteristic that corresponds to the μ = 10 compression characteristic.

  1. For a 4-bit PCM system, calculate and sketch a plot of the output S / N (in decibels) as a function of the relative input level, 20 log( x rms/ V ) for (a) A PCM system that uses μ = 10 law companding. (b) A PCM system that uses uniform quantisation (no companding). Assume that a triangular-type waveform is present at the input so that < | x | >=

3 x rms/2. Which of these systems is better to use in practice? Why?

  1. Suppose 100 voltage levels are employed to transmit 100 equally likely messages. Assume λ = 3 .5 and the system bandwidth B = 104 Hz. (a) Calculate S /η. (b) If an integrate-and-dump filter is employed to determine which level is sent, calculate the probability of an error when sending the k th level. Assume that the only errors possible are in choosing the k − 1 or the k + 1 levels.
  1. Consider a baseband unipolar communication system with equally likely signalling:

s ( t ) =

  • A , 0 < t ≤ T (binary 1) 0 , 0 < t ≤ T (binary 0) Assume that the receiver uses a simple RC LPF with a time constant of RC = τ where τ = T and 1/ T is the bit rate. (By “simple” it is meant that the initial conditions of the LPF are not reset to zero at the beginning of each bit interval.) (a) For signal alone at the receiver input, evaluate the approximate worst-case signal to ISI ratio (in decibels) out of the LPF at the sampling time t = t 0 = nT , where n is an integer. (b) Evaluate the signal to ISI ratio (in decibels) as a function of the parameter K , where t = t 0 = ( n + K ) T and 0 < K ≤ 1. (c) What is the optimum sampling time to use to maximise the signal-to-ISI power ratio out of the LPF?
  1. For unipolar baseband signalling with pulses

s ( t ) =

  • A , 0 < t ≤ T (binary 1) 0 , 0 < t ≤ T (binary 0), (a) Find the matched-filter frequency response and show how the filtering operation can be implemented by using an integrate-and-dump filter. (b) Show that the equivalent bandwidth of the matched filter is B eq = 1 /( 2 ∗ T ) = R /2.
  1. Equally likely polar signalling is used in an baseband communication system. Gaussian noise having a PSD of N 0 /2 W/Hz plus a polar signal with a peak level of A volts is present at the receiver input. The receiver uses a matched-filter circuit having a voltage gain of 1000. (a) Find the expression for P  as a function of A , N 0 , T , and VT , where

equivalent impulse response is

he ( t ) =

e − t^ , t ≥ 0 e − t 2 , t < 0.

(a) Plot the impulse response. (b) Design a transversal filter to force four points (at the sampling times) to zero. (c) Plot the impulse response that includes the zero-forcing equalising filter.

  1. A DM system is tested with a 10kHz sinusoidal signal, 1V peak-to-peak, at the input. It is sampled at 10 times the Nyquist rate. (a) What is the step size required to prevent slope overload and to minimise granular noise? (b) What is the power of the granular noise? (c) If the receiver input is bandlimited to 200kHz, what is the average-signal/quantising-noise power ratio?
  2. Assume that the input to a DM is 0. 1 t^8 − 5 t + 2. The step size of the DM is 1V, and the sampler operates at 10 samples/sec. Over a time interval of 0 to 2 sec, sketch the input waveform, the delta modulator output, and the integrator output. Denote the granular noise and the slope overload regions.
  3. A delta modulator is to be designed to transmit the information of an analogue waveform that has a peak-to-peak level of 1V and a bandwidth of 3.4kHz. Assume that the waveform is to be transmitted over a channel where the frequency response is extremely poor above 1MHz. (a) Select the appropriate step size and sampling rate for a sine-wave test signal and discuss the performance of the system using the parameter values you have selected.

(b) If the DM system is to be used to transmit the information of a voice (analogue) signal, select the appropriate step size when the sampling rate is 25kHz. Discuss the performance of the system under these conditions.