Filter design by convex iteration
From Wikimization
(Difference between revisions)
Line 55: | Line 55: | ||
- | Define <math>I_n\,</math>... | + | Define <math>I_n\triangleq\,</math>... |
- | + | By spectral factorization, <math>R(\omega)=|H(\omega)|^2\,</math> , | |
- | + | an equivalent problem is: | |
- | + | ||
<math>\begin{array}{lll} | <math>\begin{array}{lll} | ||
\textrm{minimize} &\|\textrm{diag}(G)\|_\infty\\ | \textrm{minimize} &\|\textrm{diag}(G)\|_\infty\\ | ||
- | \textrm{subject~to} &\frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2 | + | \textrm{subject~to} &\frac{1}{\delta_1^2}\leq R(\omega)\leq\delta_1^2 &\omega\in[0,\omega_p]\\ |
- | & R(\omega)\leq\delta_2^2 | + | & R(\omega)\leq\delta_2^2 & \omega\in[\omega_s\,,\pi]\\ |
- | & R(\omega)\geq0 | + | & R(\omega)\geq0 & \omega\in[0,\pi]\\ |
& R(\omega) = r(0)+\!\sum\limits_{t=1}^{N-1}2r(t)\cos(\omega t)\\ | & R(\omega) = r(0)+\!\sum\limits_{t=1}^{N-1}2r(t)\cos(\omega t)\\ | ||
- | & r(n)=\frac{1}{n}\textrm{trace} | + | & r(n)=\frac{1}{n}\textrm{trace}(I_{n-N}G) &n=1\ldots N\\ |
- | & r(n)=\frac{1}{n-N}\textrm{trace} | + | & r(n)=\frac{1}{n-N}\textrm{trace}(I_{n-N}G) &n=N+1\ldots 2N-1\\ |
& G\succeq0\\ | & G\succeq0\\ | ||
& \textrm{rank}(G) = 1 | & \textrm{rank}(G) = 1 | ||
\end{array}</math> | \end{array}</math> |
Revision as of 21:32, 23 August 2010
where
For a low pass filter, frequency domain specifications are:
To minimize maximum magnitude of , the problem becomes
A new vector
is defined by concatenation of time-shifted versions of .
Then
is a positive semidefinite rank 1 matrix.
Summing along each of subdiagonals gives entries of the autocorrelation function of .
In particular, the main diagonal of holds squared entries of .
Minimizing is therefore equivalent to minimizing .
Define ...
By spectral factorization, , an equivalent problem is: