1. (Samples and sequences) Consider the signal

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. 1

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