Filter Design using Lp Norms

In terms of $L_p$ norm minimization, the filter design problem becomes:

$$\min_h \left\Vert W(\omega_k)\left[H(\omega_k) - D(\omega_k)\right]\right\Vert _p$$

Where

• $\left\Vert \cdot \right\Vert _p$ denotes the Lp norm (defined below)
• $\omega_k =$ discrete frequency (possibly non-uniform)
• $H(\omega_k) =$ frequency response of our filter
• $D(\omega_k) =$ desired (complex) frequency response
• $W(\omega_k) =$ (optional) weighting function

This formulation applies to both FIR and IIR filters replacing $h$ by the direct-form coefficients $[B,A]$ .
(With $A=1$ , $B=h$ ).

We'll look at some specific cases below