1. (Samples and sequences) Consider the signal
x(t) = ?(t)+ ? m2(t – m)
m?Z\0
where ? is the rectangular pulse and Z\0 is the set of all integers other than zero. Plot x and show
that it is absolutely integrable and square integrable, but not periodic. Now consider the sequence of
samples cn = x(n) of the signal x. Plot the sequence c and show that it is periodic, but neither absolutely
summable, nor square summable. Hint:
8 1 p2
m2 = 6 .
m=1
2. (Raised cosine) Plot the signal
? 1 1 ??? 4 1 < t =
3
4
3
< t = 1
4
-4
?1 1 p
22 2 + cos 2pt – x(t) = ???1 1 p 1
4 22 2 4 + cos 2pt + – < t = –
?0 otherwise
and find its Fourier transform ˆx = Fx. Plot the Fourier transform. Is the Fourier transform square
integrable? Is it absolutely integrable?
3. (Finite impulse response filter) Design a low pass finite impulse response filter with cuttoff frequency
c = 2400 Hz and sample period P = 1
F where F = 8000 Hz. Ensure the filter satisfies the following
properties:
• has no more that 81 taps,
• affects the amplitude of frequencies in the interval [0, 2300 Hz] by no more than 10%,
• attenuates the amplitude of frequencies in the interval [2500 Hz,F/2] by more than 90%.
Plot the discrete impulse response and magnitude spectrum of this digital filter.
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