Matrix Formulation: Optimal $L_2$ Design, Cont'd

In matrix notation, our filter design problem can be stated

$$\min_x \left\Vert Ax-b \right\Vert _2^2$$

where, for zero-phase filters,

$$A \mathrel{\stackrel{\Delta}{=}}\left[ \begin{array}{ccccc} 1 & 2\cos(\omega_0) & \dots & 2\cos\left[\omega_0(L/2)\right] \\ 1 & 2\cos(\omega_1) & \dots & 2\cos\left[\omega_1(L/2)\right] \\ \vdots & & & \\ 1 & 2\cos(\omega_{N-1}) & \dots & 2\cos\left[\omega_{N-1}(L/2)\right] \\ \end{array} \right]$$

$$x \mathrel{\stackrel{\Delta}{=}}h$$

and $b=[D(\omega_k)]$ is the desired frequency response at the specified frequencies.

• The chosen grid frequencies $\omega_k$ provide
  • effective weighting (denser points $\Rightarrow$ more weight)
  • transition bands (gaps in the grid)
• Usually there is an explicit weighting allowed, so that $\left\Vert W(\omega_k)\left[H(\omega_k) - D(\omega_k)\right]\right\Vert _2^2$ is minimized

• The function firls in Octave and the Matlab Signal Processing Tool Box implements frequency-domain weighted-least-squares FIR filter design