| 1. |
If X(ƒ), the Fourier
Transform of the signal x(t), is band limited such that X(ƒ)
=0 for |ƒ| 
W. |
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(a) |
Find Xs(ƒ), the
Fourier Transform of  which
is the output of the ideal sampler and the sampling frequency ƒs=1/Ts.(15%) |
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(b) |
Plot Xs(ƒ)
and explain what is the sampling theorem.(5%) |
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| 2. |
Consider the signals of the following analog and digital
communication systems. |
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(a) |
Describe the AM,PM and FM signals s(t) with
the message signal m(t) and the carrier frequency ƒc.(8%) |
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(b) |
Describe the M-ary PSK,M-ary FSK and M-ary ASK signals si(t)
with the symbol energy Es.(8%) |
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(c) |
Find the dimension of each modulation scheme in (b).(4%) |
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| 4. |
An obsolete 2400 bps modem uses QAM to transmit one of 16 symbols, S0,...,S15,
illustrated as below, every 1/600 seconds. The receiver output is  ,
as signal Si is transmitted. N1
& N2 are i.i.d.~N (0, 2).
(Gaussian pdf with mean=0 and variance=2)
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(a) |
Under the ML (maximum likelihood) decision rule, sketch the
acceptance regions based upon the receiver outouts X1
& X2.(10%) |
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(b) |
Let Hi be the event that symbol Si
was transmitted and let C be the event that the correct symbol is
decoded. Find the probability Pr[C|H1].(10%) |
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| 4. |
What is the intersymbol interference?(5%)What is its effect
on a digital communication system?(5%)Describe the principles of the techniques
to reduce this effect.(5%) |
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| 5. |
Given the transmitted waveforms, s1(t)=Acos(2 ƒct)
and s2(t)=-Acos(2ƒct)
with A =50 mV, under the coherent detection and AWGN channel.
Assume that the single-sided noise power spectral density is N0=10-11
Watts/Hz and the signal power is normalized relative to a 1  load. |
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(a) |
Please find the error probability for a BPSK system with a
bit rate of 10 M bps.(15%) |
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(b) |
If there is a fixed phase error 
(=60°) between the transmitter and the receiver, what is the effect
of this phase error on the average probability of error of the system?(10%) |
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