Digital Signal Processing Reference
In-Depth Information
We would like to generate samples from a circular generalized Gaussian dis-
tribution (GGD)—also called the exponential power distribution. We can use the
procedure given in [57] to generate GGD samples with shape parameter
c
and
scaling
s
using the expression [
gamrnd
(1
/
2c,
s
)]
1
/
2
c
where the MATLAB
(www.mathworks.com) function
gamrnd
generates samples from a gamma
distribution with shape parameter 1
/
2
c
and scale parameter
s
.
Explain why using this procedure directly to generate samples for the magni-
tude,
r
, will not produce samples with the same shape parameter as the bivariate
case. How can you modify the expression [
gamrnd
(1
/
2
c
,
s
)]
1
/
2
c
so that the
resulting samples will be circular-distributed GGDs with the shape parameter
c
when the expression given in (1.74) is used.
Hint:
A simple way to check for the form of the resulting probability
density function is to consider the case
c ¼
1, that is, to consider the Gaussian
special case.
1.4
Using the two mappings given in Proposition 1, Eqs. (1.25) and (1.26), and real-
valued conjugate gradient algorithm given in Section 1.3.1, derive the complex
conjugate gradient algorithm which is stated in Section 1.3.4.
N
notation by
1.5
Write the widely linear estimate given in (1.37) using the
C
defining
v
1
v
2
v ¼
and show that the optimum widely linear vector estimates can be written as
v
1,opt
¼
[
CPC
P
]
1
[
pPC
q
]
and
v
2,opt
¼
[
C
P
C
1
P
]
1
[
q
P
C
1
p
]
N
, where (
.
)
2
is the complex conjugate of the inverse.
Use the forms given above for
v
1, opt
and
v
2, opt
to show that the mean-square
error between a widely linear and linear filter
J
diff
is given by the expression
in (1.38).
in
C
1.6
Given a finite impulse response system with the impulse response vector
w
opt
with coefficients
w
opt,
n
for
n ¼
1,
...
,
N
.
Show that, if the desired response is written as
d
(
n
)
¼ w
opt
x
(
n
)
þ
v
(
n
)
where
x
(
n
)
¼
[
x
(
n
)
x
(
n
1)
x
(
N
1)]
T
and both the input
x
(
n
) and the noise
term
v
(
n
) are zero mean and
x
(
n
) is uncorrelated with both
v
(
n
) and
v
(
n
), then
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