Communication Theory Homework 1 solution

Communication Theory Homework 1

1. A square wave has period T, amplitude A and duty cycle τ /T (the signal takes value A from
time 0 to time τ , then 0 from time τ to T). Find the Fourier series representation of this
signal.
2. Does the signal from the previous question have finite energy? Does it have finite power? If
the answer to either question is yes, find the value of the energy or power.
3. What is sinc(t) ∗ sinc(t)? Hint: use the convolution theorem.
4. Suppose a system acts on signal x and produces output y by the rule
y(t) = |x(t + 3)|.
Is the system linear? Is the system time invariant? Is the system causal?

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Communication Theory Homework 1

1. A square wave has period T, amplitude A and duty cycle τ /T (the signal takes value A from
time 0 to time τ , then 0 from time τ to T). Find the Fourier series representation of this
signal.
2. Does the signal from the previous question have finite energy? Does it have finite power? If
the answer to either question is yes, find the value of the energy or power.
3. What is sinc(t) ∗ sinc(t)? Hint: use the convolution theorem.
4. Suppose a system acts on signal x and produces output y by the rule
y(t) = |x(t + 3)|.
Is the system linear? Is the system time invariant? Is the system causal?
5. Let x(t) be the signal given by cos(2πf t) for 0 ≤ t ≤ T and 0 otherwise, where T = 1/f. Find
the Fourier transform, X(ω), of this signal. Demonstrate Parseval’s theorem by comparing
the norms of x and X.
6. Suppose we pass x(t) from the previous question through the system from question 4. Use
MATLAB to find the amplitude and phase of the output signal’s Fourier transform Y (ω).
Plot X(ω) as well – how do the signals compare?
7. Let z(t) = y(t) cos(64πt + θ). Write z as a sum of in-phase and quadrature components. Plot
the Fourier transform Z(ω) for θ = π/3, and comment on the effect of the modulation on the
amplitude and phase (comparing to Y (ω).
8. Write a function that takes as an input a time-domain signal and outputs the Hilbert transform
of that signal (also in the time-domain). Plot the Hilbert transforms of x, y and z in the
frequency and time domains.
9. Using MATLAB, demonstrate the orthagonality of a signal and its Hilbert transform for all
of x, y and z.
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