1.   Consider the periodic sawtooth signal x (t) shown in Figure Q1 below .

Figure Q1

a)    Find the Fourier Series expansion of x(t).

b)    Sketch the magnitude of the first six spectral components |cn | = *an(2) + bn(2) .

c)    Find the average power of x(t) .

d)    How many harmonics would be required to retain at least 90% of the signal’s power .

e)    The signal is passed through a low-pass filter with a cut-off frequency of 4.5 kHz (i.e. allowing only frequencies below 4.5 kHz to pass) and through a clamping circuit that adds 1 V d.c.  Given A  = π and T  =  1 ms , find the Fourier Series expansion of the resulting signal.

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2.   a)    Sketch the signal  x1 (t) = cos10t  in the time and frequency domains .

b)   Given that  e一atu(t)  and   a +1j幼   (a 米 0) represent a Fourier pair, show that the Fourier Transform of  x2 (t) = e一a ltl   is  X2 (幼) = a22  .

c)    Sketch x2 (t) and its Fourier Transform X2 (幼).

d)   The above energy signals x1 (t) and x2 (t) are inputs into a system with output y(t) = x1 (t) . x2 (t).  Sketch y(t) and Y(幼).

e)    State Parseval’s Theorem and verify it for the signal f(t) = e一atu(t).

Hint:    =  tan一1 

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4.   a)

A discrete-time system has the following difference equation:

y [n] = x [n] + 0.5y[n − 1] + 0.25y[n − 2]

Figure Q4

Find the transfer function H(z).

b)

Find the z-transform for the discrete time input signal:

x [n] = {1, 0, 1, 0, 0, 0, ⋯}

c)

Use the transfer function you found in part (a) and the input signal from part

(b) to find Y(z), the z-transform of the corresponding output signal.

d)

Find the discrete time approximation for the second derivative of an input signal, and its corresponding block diagram.

e)

Find the output Y(z) of a system with impulse response  ℎ[n] = nE  ∙ u [n] subject to an input signal x [n] = n ∙ u [n].

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