(1) A sinusoidal signal cos(ω0t) + j sin(ω0t) where ω = 2πf0t sampled at a rate fS such that
|f0| ≤ fS /2. The resulting sampled sinusoid is reconstructed by an arbitrary reconstructor
H(f). Determine the analog signal at the output of the reconstructor when H(f) is
i. an ideal reconstructor
ii. a staircase reconstructor equalized by digital filter defined as
for the interval – fS /2 ≤ f ≤ fS /2
(2) The signal x(t) = 4sin(3πt)sin(2πt) + cos(5πt), where t is in milliseconds, is sampled at a
rate of 3 kHz. Determine the signal xa(t) aliased with x(t). Determine two other signals
x1(t) and x2(t) that are different from each other and from x(t), yet they are aliased with
the same xa(t) that you found.
(3) A signal that is sampled at sampling frequency 20 kHz is filtered by a digital filter
designed to act as an ideal low pass filter with cutoff frequency 4 kHz. The filter signal is
then fed into a staircase D/A, and then into a low pass anti-image postfilter. The overall
reconstructor is required to suppress the spectral images cause by sampling by more than
A = 70 dB. What should be the least stringent specifications for the analog postfilter that
will satisfy this requirement?
(4) Consider a pure sinusoidal signal of frequency f0 such that x(t) = cos(2πf0t) without the
complex portion. Prove that the spectrum of the sampled sinusoid x(nT) is given by
(5) For the following input signals and impulse responses, determine the output signal y(t) by
i. x(t) = e^-3t and h(t) = e^6t
ii. x(t) = 2t2 and h(t) = 1 / (3
iii. x(t) = 4t and h(t) = 6t – 1
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