Filter design by convex iteration
From Wikimization
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For low pass filter, the frequency domain specifications are: | For low pass filter, the frequency domain specifications are: | ||
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<math> | <math> | ||
\begin{array}{ll} | \begin{array}{ll} | ||
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A new vector <math>g \in \texttt{C}^\texttt{N*N} </math> is defined as concatenation of time-shifted versions of <math>h </math>, ''i.e.'' | A new vector <math>g \in \texttt{C}^\texttt{N*N} </math> is defined as concatenation of time-shifted versions of <math>h </math>, ''i.e.'' | ||
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<math> | <math> | ||
g = \left[ | g = \left[ | ||
Revision as of 16:19, 23 August 2010
where
For low pass filter, the frequency domain specifications are:
To minimize the maximum magnitude of , the problem becomes
A new vector is defined as concatenation of time-shifted versions of
, i.e.
Then is a positive semidefinite matrix of size
with rank 1. Summing along each 2N-1 subdiagonals gives entries of the autocorrelation function of
. In particular, the main diagonal holds squared entries
of
. Minimizing
is equivalent to minimizing
.
Using spectral factorization, an equivalent problem is