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Homework Questions. ECE-430 Digital Signal Processing.
Instructor Name: Ioannis Kyriakides

1• a) Are these sinusoids periodic? If yes, compute their fundamental period.
i) cos(0.01πn)
ii) 3cos(5n + π/10)
b) A bandlimited signal has a frequency range of 800 MHz 900 MHz. What is
the minimum sampling rate with which we can sample this signal with no loss?

2• We would like to transmit the following message using bits:
x(t) = 3cos(200πt) + 4cos(800πt)
If the transmission channel carries a maximum of 8800 bits/s, find the resolution
of the digital signal at the receiver end.

3• Determine the properties
i) static or dynamic
ii) Linear or nonlinear
iii) Time invariant or time varying
iv) causal or noncausal
v) stable or unstable
y(n) = |x(n)|

4• a) What is the impulse response of the following system?
y(n) = ax(n − 3) + bx(n − 5) + cx(n − 6)
b) Determine the output of a system with impulse response
h(n) = {1, 3, 1}
and input:
x(n) = {2, 3, 1}
c) Determine the output of a system with impulse response:
h(n) = {1, 3, 1, 1}
and input:
x(n) = δ(n − 3)
d) Plot the block diagram representation of the following system
y(n) = by(n − 1) + ay(n − 2)x(n)nx(n − 1), where b is a constant

5• a) Prove that the following system is unstable
y(n) = nx(n)
b) What are the conditions on a and b so that the following system is stable?
y(n) = ax(n) + bx(n − 1)

6• Determine the output of a system with impulse response
1
h(n) = {3, 5, 1}
and input:
x(n) = {4, 1, 4}
a) Using convolution in the time domain
b) Using the z-transform
c) Using DTFT

7• a) Determine the impulse response (h(n)) of a causal LTI system described by
the following input-output relation.
y(n) = 2ay(n − 1) − ay(n − 2) + bax(n − 1)
b) Determine a above so that the system is stable

8• Determine the Fourier series coefficients of the following signal
a) cos(πn/2)

b) cos(πn3 2)

9• Determine the Fourier transform of the following signal using the Fourier trans-
form formula
x(n) = an u(n), −1 < a < 1

10• Determine the z-transform of the following signals
a) x(n) = {1, 4, 3, 5, 6, 1}
b) x(n) = 10(1/4)n u(n) + 10(1/4)n u(n − 3)
c) [an cos(0 n)u(n)] ∗ [an sin(0 n)u(n)]

11• Using Fourier domain analysis, determine the type of the following filter (Low-
pass, Band-pass or High-pass).
y(n) = (x(n − 1) + x(n) + x(n + 1))/3

12• Determine the output of a system with impulse response
h(n) = {3, 2, 1}
and input:
x(n) = {4, 1}
a) Using convolution in the time domain
b) Using the Fourier transform

13• a) Determine the condition on the real valued variable a so that the Fourier
transform of the following signal exists.
x(n) = an u(n)
b) Determine the Fourier transform of the following signal using the Fourier
transform formula (Do not use FT tables)

14• Determine the Discrete Fourier Transform (DFT) of the following signals
a) x(n) = {1, 4, 3, 5, 6, 1}

2
b) x(n) = 10(1/4)n u(n) + 10(1/4)n u(n − 3)
c) x(n) = 10(1/4)n u(n)(n − 5) + 10(1/4)n u(−n − 3)

15• Prove that
y(n) = x(n) ∗ h(n)
implies that
Y () = X()H()
where Y (), X(), and H() are the Fourier transforms of y(n), x(n), and h(n) re-
spectively.
(Hint: Take the Fourier transform of both sides of the signal)

16• A bandlimited signal has a frequency range of 800 MHz 900 MHz. What is the
minimum sampling rate with which we can sample this signal with no loss? What is
the sampling rate if we do not use demodulation techniques?

17
• We would like to transmit an analog message using bits. If the transmission
channel carries a maximum of 9900 bits/s, and the resolution ∆ of the digital signal
is 16/2047, choose a maximum allowable bandwidth and a maximum amplitude for
our analog message?

18•
a) What is the impulse response of the following system? What is the condition
on a, b, and c so that the system is stable?
y(n) = ax(n − 3) + bx(n − 5) + cx(n − 6)
b) Determine the output of a system with impulse response
h(n) = {1, 3, 1}
and input:
x(n) = {2, 3, 1}
c) Determine the output of a system with impulse response:
h(n) = {1, 3, 1, 1}
and input:
x(n) = δ(n)
d) Plot the block diagram representation of the following system
y(n) = ny(n − 1) + by(n − 3)nx(n)5x(n − 1), where b is a constant

19•
a) Show that the following system is stable.
y(n) = x(n)
b) Set conditions on a and b so that the following system is stable. Note that a
and b are not necessarily constants.
y(n) = ax(n) + bn x(n − 1)

20• Determine the output of a system with impulse response
h(n) = {1, 0, 0, 3, 2, 1}
and input:

3
x(n) = {5, 1}
a) Using convolution in the time domain
b) Using the z-transform

21• Using the z-transform determine the output of a system y(n) (expressed in the
time-domain) with impulse response:
h(n) = {1, 0, 0, 3, 2, 1}
and input:
x(n) = {5, 1}

22• a) Determine the impulse response (h(n)) of a causal LTI system described by
the following input-output relation.
y(n) = 2ay(n − 1) − a2 y(n − 2) + bax(n − 1)
b) Determine a above so that the system is stable

23• Determine the Fourier series coefficients of the following signal
a) cos(3πn/2)

b) cos(πn4/ 2)

24• Determine the Fourier transform of the following signal using the Fourier trans-
form formula
x(n) = an u(n − 5) − 1 < a < 1

25• Determine the z-transform of the following signals
a) x(n) = {1, 4, 3, 5, 6, 1}
b) x(n) = 10(1/4)n u(n) + 10(1/4)n u(n − 3)
c) [an cos(0 n)u(n)] ∗ [an sin(0 n)u(n)]

25
• a) Determine the condition on the real valued variable a so that the Fourier
transform of the following signal exists.
x(n) = an u(n − 5)
b) Determine the Fourier transform of the following signal using the Fourier
transform formula (Do not use FT tables)

26• Determine the Discrete Fourier Transform (DFT) of the following signals
a) x(n) = {1, 4, 3, 5, 6, 1}
b) x(n) = 10(1/4)nu(−n − 3) + 10(1/4)nu(n − 3)
c) x(n) = 10(1/4)n u(n)δ(n − 5) + 10(1/4)n u(n − 3)
Convolve the following signals (graphically)
{0, 1, 2}
{1, 2}
b) Convolve the following signal with the unit impulse (graphically)
{1, 2}
c) Convolve the following signals (both algebraically and graphically)
δ(n − 2), u(n − 3)
4
27• For the following systems, determine whether they are i) linear or non linear,
ii) Time varying or invariant, iii) stable or unstable. (fully justify your answers).
y(n) = 13 [x(n − 2) + x(n − 1) + x(n)]
y(n) = x(n2 )

28• A DSP student (A) who hasn’t studied the sampling theorem properly, sampled
the following signal at 200 Hz. The student then provided the discrete time signal
to another student (B - who knows the sampling theorem and signal reconstruction
very well) telling them that the signal was sampled at 200 Hz.
x(t) = 3cos(600πt) + 2cos(100πt)
a) Derive the discrete time signal that student A created in its most simplified
form.
b) Explain how student (B) can try to recover a continuous signal from the dis-
crete time signal that student A provided.
c) Will student B manage to recover the original signal? If not then explain how
the recovered signal will look like. (What are the frequencies that student B will see
in the recovered signal and how do those frequencies compare to the original signal).
Note: just describe what happens to the frequencies - do not derive the recovered
signal.

29• a) Determine the 8-point DFT, X(k), of the following signal:
x(n) = an u(n), where a is a constant
b) Determine and plot the inverse DFT, y(n), of X(k)
c) Determine the error e(n) = |y(n) − x(n)|. What are the factors affecting this
error?

30• a) Determine the 4-point DFT matrix (Show all steps in detail)
b) Explain the structure of the DFT matrix by first showing that
k+N/2 k
WN = −WN
c) Show that the DFT matrix you derived above is unitary by multiplying the
matrix with its complex conjugate

31• The 4-point DFTs of w(n), x(n), y(n), z(n) are respectively:
W(k) = {1, 0, 1, 1}, X(k) = {1, 0, 0, 1}, Y(k) = {4, 0, 2, 1}, Z(k) = {1, 1, 1, 1, }
Using the inverse DFT matrix find w(n), x(n), y(n), and z(n)

32• Find the 8-point DFT of the signal shown below using
a) The DFT summation formula b) The DFT matrix
x(n) = {0, 0, 0, 1, 0, 0, 1, 0}
Hint: you can save a lot of time by taking into account the sparse nature of x(n).

33• A continuous-time non-periodic signal x(t) with duration 5 sec is sampled at
twice its maximum frequency with sampling period T s = 0.05 sec. The signal is
digitized resulting to x(n) and the 64-point DFT of the signal, X(k) is taken and
stored on digital media. The original x(t) and x(n) were not stored.

5
a) Determine the frequency spacing (in rad/s) of the DFT samples.
b) Will someone be able to retrieve the original x(n)? Fully justify your answer.
c) What is the maximum frequency found in the original signal x(n), and what is
the maximum frequency found in X(k)?
d) How should the DFT of x(n) be taken so that one can retrieve x(t)?

34• Verify that this impulse response of an LTI (hint start from sys eq and go to
impulse)
h(n) = 2( 21 )n u(n)
is associated with the system equation
y(n) = 12 y(n − 1) + 2x(n)
a) Identify the z-transform of y(n), and the z-transform of the impulse response
h(n) denoted as H(z). Plot the ROC of H(z) and indicate the poles and zeros on the
plot. Is the system stable and ...

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