hw 8 matlab problem
DESCRIPTION
matlab problem for ee518TRANSCRIPT
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EE518 PMP Digital Signal Processing Winter 2014 Homework 8, due March 11
Prof. Les Atlas, [email protected] http://www.ee.washington.edu/class/pmp518/2012wtr/
TA: David Perlmutter, [email protected] Page 1 of 2
Homework #8, due Tuesday 3/11, at 9:00 (beginning of discussion):
A. Oppenheim & Schafer: problems 7.61, 7.62 a-d and f (skip part e)
The problem below is derived from McClellan et. al., MATLAB Exercises for Signal
Processing, Prentice-Hall, 1998.
B. MATLAB Project Background
As Discussed in Text and Lecture:
Definition of the Discrete Fourier Transform (DFT)
Analysis:
1
0
[ ] , 0,1,2,..., 1[ ]
0,
N
kn
N
n
x n W k NX k
otherwise
Synthesis:
1
0
1[ ] , 0,1,2,..., 1
[ ]
0,
N
kn
N
k
X k W n Nx n N
otherwise
where the zeros outside the calculated N-point range are directly implied.
The following DFTs are rather difficult to compute by hand, yet the results are not hard to
visualize with MATLAB. To this end, the important duality principal of Fourier analysis will be
demonstrated by the following exercises.
Exercise 1
Aliased Sinc Sequence
Once mastered, the duality principal if quite powerful. As a starting example of its use in
computing, consider the DFT of an aliased sinc signal, called asinc. According to duality, the rectangular pulse and the asinc function are, in time to frequency, or in its dual, in frequency to
time, transform pairs. We will see both possibilities below.
a. The DTFT of a rectangular pulse is an aliased sinc function in ; the N-point DFT just
samples the asinc uniformly in frequency , namely at 2 / , N k 0,1,2,..., 1 k N . For an even-symmetric1 L-point pulse, the resulting DTFT is
2 /
sin /[ ] , , 0,1,2,..., 1
sin /
k N
kL NR k asinc L k N
k N
Generate a 7-point pulsewidth boxcar(7), and compute its N=16 point DFT. Verify that the correct transform values were obtained. Repeat for a N=21 point DFT. Explain why
the N=21 point DFT has so many values equal to zero.
1 DFT symmetries are closely related, and essentially identical to one period of the discrete Fourier series (DFS)
symmetries seen in Table 8.1 on page 634 of Oppenheim and Schafer text.
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EE518 PMP Digital Signal Processing Winter 2014 Homework 8, due March 11
Prof. Les Atlas, [email protected] http://www.ee.washington.edu/class/pmp518/2012wtr/
TA: David Perlmutter, [email protected] Page 2 of 2
b. For the dual of the above part (a), the time sequence is taken to be an asinc function, now in n. Use MATLAB to calculate and plot the N-point DFT of the following sampled asinc
sequence:
0
sin 9 /[ ] for 0,1,..., 1
sin /
n Na n n N
n N
Assume that N is greater than 9, say 16 or 21N N .
c. Find the N-point DFT of the following shifted asinc sequence:
1
sin 9 ( 1) /[ ] for 0,1,..., 1
sin ( 1) /
n Na n n N
n N
Since the asinc function is periodic, this shift must be a circular shift.
d. Note that the factor in the numerator of 0 1[ ] or [ ]a n a n must be odd! Try replacing the 9
with 10 in either of the previous parts, and compute the DFT. Explain why there is so
much difference between the even and odd cases.