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102 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
1. T s =0.1 ms
2. T s =1ms
3. T s =0.01 sec
P3.20 We implement the following analog filter using a discrete filter.
x a (t) −→ A/D x(n)
−→ h(n) y(n)
−→ D/A −→ y a (t)
The sampling rate in the A/D and D/A is 8000 sam/sec, and the impulse response is
h(n) =(−0.9) n u(n).
1. What is the digital frequency in x(n) ifx a (t) =10cos (10, 000πt)?
2. Determine the steady-state output y a (t) ifx a (t) =10cos (10, 000πt).
3. Determine the steady-state output y a (t) ifx a (t) =5sin(8, 000πt).
4. Find two other analog signals x a (t), with different analog frequencies, that will give
the same steady-state output y a(t) when x a(t) =10cos(10, 000πt) isapplied.
5. To prevent aliasing, a prefilter would be required to process x a (t) before it passes to
the A/D converter. What type of filter should be used, and what should be the largest
cutoff frequency that would work for the given configuration?
P3.21 Consider an analog signal x a (t) =cos(20πt), 0 ≤ t ≤ 1. It is sampled at T s =0.01, 0.05,
and 0.1 sec intervals to obtain x(n).
1. For each T s plot x(n).
2. Reconstruct the analog signal y a (t) from the samples x(n) using the sinc interpolation
(use ∆t =0.001) and determine the frequency in y a (t) from your plot. (Ignore the end
effects.)
3. Reconstruct the analog signal y a (t) from the samples x(n) using the cubic spline
interpolation, and determine the frequency in y a (t) from your plot. (Again, ignore the
end effects.)
4. Comment on your results.
P3.22 Consider the analog signal x a (t) =cos (20πt + θ) , 0 ≤ t ≤ 1. It is sampled at T s =0.05
sec intervals to obtain x(n). Let θ =0,π/6, π/4, π/3, π/2. For each of these θ values,
perform the following.
1. Plot x a (t) and superimpose x(n) onitusing the plot(n,x,’o’) function.
2. Reconstruct the analog signal y a (t) from the samples x(n) using the sinc interpolation
(Use ∆t =0.001) and superimpose x(n) onit.
3. Reconstruct the analog signal y a (t) from the samples x(n) using the cubic spline
interpolation and superimpose x(n) onit.
4. You should observe that the resultant reconstruction in each case has the correct
frequency but a different amplitude. Explain this observation. Comment on the role of
phase of x a (t) onthe sampling and reconstruction of signals.
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