Filter design by convex iteration
From Wikimization
(Difference between revisions)
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& R(\omega)\geq0, & \omega\in[0,\pi]\\ | & R(\omega)\geq0, & \omega\in[0,\pi]\\ | ||
& \textrm{trace}(gg^\texttt{H}) = 1 &\\ | & \textrm{trace}(gg^\texttt{H}) = 1 &\\ | ||
| - | & r(n) = \frac{1}{n} trace(\texttt{I}_{n-N})\,g & \\ | + | & r(n) = \frac{1}{n} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for} n=1,\hdots,N\\ |
| - | & r(n) = \frac{1}{n-N} trace(\texttt{I}_{n-N})\,g & \\ | + | & r(n) = \frac{1}{n-N} \textrm{trace}(\texttt{I}_{n-N})\,g & \textrm{for} n=N+1,\hdots,2N-1\\ |
\end{array}</math> | \end{array}</math> | ||
Revision as of 17:16, 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