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Assignment 2

The document is a practice problem set for the course EE527L: Advanced Signal Analysis and Processing, covering discrete time signals and systems. It includes various problems related to periodicity, signal periods, continuous time Fourier transform (CTFT), energy calculations, signal sketching, convolution, and stability of linear time-invariant (LTI) systems. Students are encouraged to reach out to the TA for any doubts regarding the problems.

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10 views2 pages

Assignment 2

The document is a practice problem set for the course EE527L: Advanced Signal Analysis and Processing, covering discrete time signals and systems. It includes various problems related to periodicity, signal periods, continuous time Fourier transform (CTFT), energy calculations, signal sketching, convolution, and stability of linear time-invariant (LTI) systems. Students are encouraged to reach out to the TA for any doubts regarding the problems.

Uploaded by

princenani634
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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EE527L : Advanced Signal Analysis and Processing (Aug-Dec

2023)

Practice Problem Set 2

Note: This practice sheet is based on discrete time signals and systems properties. If
you have any doubts, please contact the TA.
1. State the reason(s) if the following signals are periodic. If they are periodic, deter-
mine their period.
√
(a) sin[ 2πn + 1.2]
π
(b) ej 2 n
2. Find the period of: x1 (t) = cos(πt/4), y1 [n] = cos(πn/4), x2 (t) = cos(3πt/8) and
y2 [n] = cos(3πn/8).
(a) Find the frequency of these signals.
(b) What do you observe about the fundamental periods of x1 , x2 compared to y1 , y2
and their frequencies?
3. The CTFT of a continuous time signal x(t) is shown in the figure below. Draw the
CTFT of the sampled signal with aliasing and without aliasing. What should be the
sampling frequency so that there is no aliasing in the sampled signal?

Figure 1: CTFT of a signal x(t)

4. Find the period of x(t) = sin(4πt/7) and y[n] = sin(4πn/7).


5. Find the energies of the signals depicted in the Figure 1.
6. For the signals shown in Figure 1, sketch the following signals without the interpo-
lation of the missing samples:
(a) f (−k)
(b) f (k + 6)
(c) f (3k)
(d) f (k/3)

1
7. Sketch the following signals for varying values of k:
(a) u[k − 2] − u[k − 6]
(b) (k − 2)(u[k − 2] − u[k − 6]) + (−k + 8)(u[k − 6] − u[k − 9])
8. By direct evaluation of the convolution sum, determine the unit step response (x[n] =
u[n]) of an LTI system whose impulse response is
h[n] = a−n u[−n], 0 < a < 1,
Pn=∞
9. Prove that a P
LTI system is stable if and only if n=−∞ | h[n] |< ∞. Hint: First
n=∞
prove that if n=−∞ | h[n] |< ∞ then for any bounded input you get a bounded
Pn=∞
output using convolution equation. Then prove that if n=−∞ | h[n] |= ∞ then for
h∗ [−n]
a bounded input x1 [n] we get an unbounded output. Here x1 [n] = |h[−n]| whenever
1
h[n] ̸= 0.
10. Find the convolution of x[n] = [1, 2, 3, 4] and h[n] = [1, 2, 1, − 1] using Method
↑ ↑
1 and Method 2 discussed in the class. Note: Arrow points the sample at n = 0 in
above sequences with numbers on the right representing n > 0.

1 See Chapter 2 of “Discrete-Time Signal Processing”, Oppenheim A V, Schafer R W and Buck J R, Pearson Education

(2010).

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