Digital Signal Processing Reference
In-Depth Information
The first term on the right-hand side is a convolution of the Gaussian pdf with
itself and is therefore a Gaussian. The second term is the convolution of
h
(
x
0
)
with itself and is in general nonzero. By choosing
h
(
.
) subject to the mild
constraints mentioned in Ex. E.1 we can get infinitely many nonzero shapes for
the second term. In particular, the pdf of
z
is not Gaussian even though it is
a sum of two Gaussian random variables! While this appears to violate what
we mentioned around Eq. (E.58), it happens because
x
0
and
x
1
are not
jointly
Gaussian, that is, the vector defined in Eq. (E.59) is not a Gaussian vector.
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