Homework 3 of Principles of Communications Due date: 2013/12/05 1. (a) Plot the spectrum of a PAM wave produced by the modulating signal đ?‘š(t) = đ??´đ?‘š cos(2đ?œ‹đ?‘“đ?‘š đ?‘Ą) assuming a modulation frequency đ?‘“đ?‘š = 0.25 Hz, sampling period đ?‘‡đ?‘ = 1 s, and pulse duration đ?‘‡ = 0.45 s. (b) Using an ideal reconstruction filter, plot the spectrum of the filter output. Compare this result with the output that would be obtained if there were no aperture effect. 2.
A message signal đ?‘š(đ?‘Ą) is normalized to peak voltages of Âą1 V. The average message power equals 20 mW. To transmit the signal by binary PCM without compression, uniform quantization is adopted. If the signal-to-quantization-noise ratio (SQNR) is required to be at least 43 dB, determine the minimum number of bits required to code the uniform quantizer. Determine the SNR obtained with this quantizer.
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In a PAM scheme with đ?‘€ = 16, do the following. (a) Determine the minimum transmission bandwidth required to transmit data at a rate of 12000 bits/s with zero ISI. (b) Determine the transmission bandwidth if Nyquist criterion pulses with roll-o factor đ?‘&#x; = 0.2 are used to transmit data.
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The signal đ?‘š(đ?‘Ą) = 6 sin(2đ?œ‹đ?‘Ą) volts is transmitted using a 4-bit binary PCM system. The quantizer is of the midrise type, with a step-size of 1 volt. Sketch the resulting PCM wave for one complete cycle of the input. Assume a sampling rate of four samples per second, 1
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what samples taken at � = ¹ 8 , ¹ 8 , ¹ 8 , ‌, seconds. 1
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Multipath Some radio systems suffer from multipath distortion, which is caused by the existence of more than one propagation path between the transmitter and the receiver. Consider a channel, the output of which, in response to a signal đ?‘ (đ?‘Ą), is deďŹ ned as (in the absence of noise) đ?‘Ľ(đ?‘Ą) = đ?‘Ž1 đ?‘ (đ?‘Ą − đ?‘Ą01 ) + đ?‘Ž2 đ?‘ (đ?‘Ą − đ?‘Ą02 ) , where đ?‘Ž1 and đ?‘Ž2 are constant, and đ?‘Ą01 and đ?‘Ą02 represent transmission delays. It is proposed to use the three-tap delay-line-ďŹ lter of ďŹ gure 1 to equalize the multipath distortion produced by this channel. (a) Evaluate the transfer function of channel. (b) Evaluate the parameters of the tapped-delay-line ďŹ lter in terms of đ?‘Ž1 , đ?‘Ž2 , đ?‘Ą01 , and đ?‘Ą02 , assuming that đ?‘Ž2 ≪ đ?‘Ž1 and đ?‘Ą02 > đ?‘Ą01 .
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