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

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

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

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

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

• do you mean the same Chebyshev filter that is what electrical engineers mean by the term? – robert bristow-johnson Mar 6 at 7:15
• @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 6 at 8:06
• 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 6 at 9:24
• @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 6 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 6 at 20:53