Digital Signal Processing Reference
In-Depth Information
5
0
1/(1 + (−1.18)z
−1
+ (0.84)z
−2
−1
−0.5
0
0.5
1
(a) Normalized Freq
1
1
−
0
0.5
−1
−0.5
0
0.5
1
(b) Normalized Freq
1
2
0
0
−1
0
10
20
30
−0.5
(c) Sample, Impulse Response
1
0
Unit Circle
−1
−1
0
10
20
30
−1
−0.5
0
0.5
1
(d) Sample, Impulse Response
(e) Real
Figure 4.40:
(a) Magnitude of frequency response of an LTI system constructed using the poles shown
in plot (e); (b) Phase response of same; (c) Real part of impulse response of same; (d) Imaginary part of
impulse response of same.
4.14
EXERCISES
1. Compute the correlation at the zeroth lag (CZL) of the sequences [0.5,-1,0,4] and [1,1.5,0,0.25].
2. Devise a procedure to produce a sequence (other than a sequence of all zeros) which yields a
CZL of 0 with a given sequence
x
[
n
]
. Note that there is no one particular answer; many such procedures
can be devised.
3. Compute the CZL of the sequences 0.9
n
and (-1)
n
0.9
n
for
n
= 0:1:10.
4. Write a simple script to compute the CZL of two sine waves having length
N
, and respective
frequencies of
k
1
and
k
2
; (a) verify the relationships found in (4.6); (b) compute the CZL of two sine
waves of length 67 samples, using the following frequency pairs as the respective frequencies (in cycles)
of the sine waves: .[1,3],[2,3],[2,5], [1.5,4.5], [1.55,4.55].
5. Use Eqs. (4.7) and (4.8) to derive the correlation coefficients for the following sequences, and
then use Eqs. (4.9) and (4.10) to reconstruct the original sequences from the coefficients. The appropriate
range for
k
is shown is parentheses.
a) [1,-1] (k = 0:1:1)
