0

What are $\cos(\omega_k), -\sin(\omega_k)$ in Chebyshev filter design in matrix form?

The Chebyshev filter design problem "via SOCP" (https://en.wikipedia.org/wiki/Second-order_cone_programming) is formulated:

$$\min_x \| A^{(k)}h -b^{(k)}\|, k=1,...,m$$

where

$A^{(k)}=\begin{matrix} 1 & \cos \omega_k &...& \cos(n-1)\omega_k \\ 0 & -\sin \omega_k & ... & -\sin(n-1)\omega_k \end{matrix}$

$b^{(k)}=\begin{matrix} Re\space H_{des}(\omega_k) \\ Img \space H_{des}(\omega_k) \end{matrix}$

$h=\begin{matrix} h_0 \\ ... \\ h_{n-1} \\ \end{matrix}$

mavavilj
  • 7,270
  • do you mean the same Chebyshev filter that is what electrical engineers mean by the term? – robert bristow-johnson Mar 06 '19 at 07:15
  • @robertbristow-johnson See I don't know. This appears in the context of "Chebyshev filter design using optimization", but I'm not sure whether it refers to that or something else. And what the $\sin,\cos$ are. It's some kind of basis on which the frequency response is written as a linear combination. – mavavilj Mar 06 '19 at 08:06
  • if it's "Chebyshev filter" and "frequency response", we're talking about the thing electrical engineers and DSPers know about. i have no idea what this "second-order code programming" is about, but i know Chebyshev filters pretty well. maybe you should ask at the DSP Stack Exchange. – robert bristow-johnson Mar 06 '19 at 09:24
  • @robertbristow-johnson SOCP is an optimization formulation for solving Chebyshev filter problems. I.e. it's about finding the coefficients for a desired filter by minimizing a "difference equation". So here $b$ is the desired response and $h$ is the coefficients. Should this then tell what the purpose of $A$ is since it's multiplied by the coefficients? – mavavilj Mar 06 '19 at 10:43
  • i don't get why there is an "optimization" problem. Chebyshev Type 1 and Type 2 filters are well-defined designs given particular specification for pass-band and stop-band. – robert bristow-johnson Mar 06 '19 at 20:53
  • @robertbristow-johnson Because the coefficients are found through minimizing? https://machinelearningmastery.com/solve-linear-regression-using-linear-algebra/ – mavavilj Mar 06 '19 at 20:57
  • Minimizing exactly what? Chebyshev filters offer a *tradeoff between minimizing passband ripple (for Type 1) or stopband ripple (for Type 2) and the width of the transition band. If you want to minimize both, you must increase the order of the filter. – robert bristow-johnson Mar 06 '19 at 21:27
  • @robertbristow-johnson As it says in the formulation in the question, minimizing the difference between desired response and chebyshev approximation. Or put otherwise, finding coefficients which minimize difference between desired response and the linear combination of those combinations in (Chebyshev?) basis. – mavavilj Mar 06 '19 at 21:28
  • well, this is curious. fitting Chebyshev polynomials to a target function is one thing (and a purely mathematical operation that conceptually precedes Chebyshev filters, which is another application of Chebyshev polynomials) but fitting a collection of Chebyshev filters to a target frequency response is another thing. i haven't groked how to relate the two problems. – robert bristow-johnson Mar 06 '19 at 23:07

0 Answers0