Filter design by convex iteration
From Wikimization
(Difference between revisions)
Line 26: | Line 26: | ||
g = \left[ | g = \left[ | ||
\begin{array}{c} | \begin{array}{c} | ||
- | h( | + | h(n) \\ |
- | h( | + | h(n-1) \\ |
\vdots \\ | \vdots \\ | ||
- | h( | + | h(n-N) \\ |
\end{array} | \end{array} | ||
\right] | \right] | ||
Line 46: | Line 46: | ||
& R(\omega)\leq\delta_2^2, & \omega\in[\omega_s,\pi]\\ | & R(\omega)\leq\delta_2^2, & \omega\in[\omega_s,\pi]\\ | ||
& 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}\textrm{trace(\texttt{I})_{n-N}\,g & for n=1,\hdots,N\\ | ||
+ | & r(n) = frac{1}{n-N}\textrm{trace(\texttt{I})_{n-N}\,g & for n=N+1,\hdots,2N-1\\ | ||
\end{array} | \end{array} | ||
</math> | </math> |
Revision as of 17:02, 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