In the name of GOD.
Sharif University of Technology
Digital Signal Processing Dr. M.T. Manzuri
CE 40-763 Fall 2012
Homework 2
Problem 1
An LTI has an impulse response h[n] for which the -transform is
( )
∑
[ ]
| |
(a) Plot the pole-zero pattern for ( ).
(b) Using the fact that that signals of the form
are eigenfunctions of LTI systems,
determine the system output for all if the input [ ] is
[ ]
( )
( )
Problem 2
Shown in figure below is the pole zero plot for the z-transform ( ) of a sequence [ ]
1
Determine what can be inferred about the associated region of convergence from each of
the following statements.
(a) [ ] is right-sided.
(b) The Fourier transform of
[ ] converges.
Problem 3
(a) For the following -transform determine the inverse -transform
( )
| |
| |
(b) Determine the -transform (including the ROC) of the following sequences. Also
sketch the pole-zero plots and indicate the ROC on your sketch.
( )
[ ]
[
]
Problem 4
Calculate the N-point DFT of aperiodic sequence [ ] of length
follows:
[ ]
, which is defined as
{
Problem 5
Suppose we have two four-point sequence [ ] and [ ] as follows
[ ]
(
)
[ ]
(a) Calculate the four-point DFT [ ].
(b) Calculate the four-point DFT [ ]
[ ]( ) [ ] by doing the circular convolution directly.
(c) Calculate [ ]
(d) Calculate [ ]of part (c) by multiplying the DFTs of [ ] and [ ] and performing
the inverse DFT.
2
Problem 6
Oppenheim Problem 9.14.
Problem 7
Oppenheim Problem 9.16.
3