Digital Signal Processing Reference
In-Depth Information
2.4
MATLAB assignment. Let us have similar array configuration as above. Now
two BPSK modulated signals are impinging the array. Our
N ¼
200 obser-
vations are contaminated by complex white second order circular Gaussian
noise such that the signal to noise ratio (SNR) is 10 dB. Plot the MUSIC
pseudo spectrum. Study the traces of array covariance and pseudo-covariance
matrices. What can you say about the circularity of the observations?
2.5
MATLAB assignment. Consider again a uniform linear array of eight elements.
Two QPSKmodulated signals are impinging on the array from directions of arri-
val 72
8
and 66
8
. Now we have an
1
-contaminated noise model obeying mixture
of two complex second order circular Gaussian distributions
f
(
n
)
¼
(1
2
1
)
f
(0,
s
2
I
)
þ 1 f
(0, 50
s
2
I
). With
1 ¼
0.1 we have 10 percent outliers present in the
sample. The signal to noise ratio would be 10 dB in the absence of outliers. Plot
the MUSIC pseudo-spectrum using sample covariance matrix based estimator
and sign covariance matrix based estimator. What can you say about the robust-
ness of the estimators?
2.6
MATLAB assignment. Write a function called
glrtcirc(Z)
for the GLRT
test statistic
2
n
ln
l
n
of circularity, where the argument
Z
of the function is a
k n
snapshot data matrix. After doing so,
a)
generate
k n
data matrix
Z
consisting of
n
indenpendent random samples
from circular
T
k
,
v
using result (2.14). Transform the data by
Z!S ¼ GZ
,
where
G
can be any nonsingular
k k
matrix. Then verify that computed
values of test statistics
glrtcirc(Z)
and
glrtcirc(S)
coincide,
meaning that GLRT test statistic is invariant under invertible linear data
transformations.
b)
Write a function for the adjusted GLRT test statistic
2
l
n
/
(1
þ
ˆ
/
2) of circu-
larity and reproduce the chi-square plots of Figure 2.2.
2.7
Show that the kurtosis matrix
C
kur
(
.
) and the pseudo-kurtosis matrix
P
kur
(
.
)
defined in (2.21) possess IC-property. If sources are symmetric, that is,
s
i
¼
d
2
s
i
, for
i ¼
1,
...
,
d
, then show that any scatter matrix
C
(
.
) or pseudo-scatter
matrix
P
(
.
) possess IC-property.
REFERENCES
1. H. Abeida and J.-P. Delmas, MUSIC-like estimation of direction of arrival for noncircular
sources.
IEEE Trans. Signal Processing
, 54(7), (2006).
2. S. I. Amari, A. Cichocki, and H. H. Yang, “A new learning algorithm for blind source sep-
aration,” in D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, Eds.,
Advances in Neural
Information Processing Systems 8
, pages 757-763. MIT Press, Cambridge, MA, (1996).
3. J. Anem¨ ller, T. J. Sejnowski, and S. Makeig, Complex independent component analysis of
frequency-domain electroencephalographic data.
Neural Networks
, 16: 1311-1323,
(2003).
4. P. Billingsley,
Probability and Measure
(3rd ed). Wiley, New York, 1995.
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