2.2 Least squared error design of fir filters  (Page 10/13)

with the elements of the $M+1$ by $M+1$ matrix ${\mathbf{C}}_{\mathbf{w}}$ as

and the $M+1$ by 1 vector of intermediate values ${\mathbf{a}}_{\mathbf{w}}$ is given by an inverse DTFT of the weighted ideal amplitude response in

and

Solving [link] for the optimal $\mathbf{a}$ which minimizes the integral weighted squared error [link] is formally done by

and more accurately done by special numerical algorithms. The case for even $N$ is easily derived by using Equation 40 from FIR Digital Filters to derive [link] . The actual length- $N$ filter coefficients $h\left(n\right)$ are then found from $a\left(n\right)$ using Equation 41 from FIR Digital Filters . Note that if the weighting is unity across the pass, transition, and stop bands, ${\mathbf{C}}_{\mathbf{w}}$ is the identity matrix. ${\mathbf{C}}_{\mathbf{w}}$ gives the effects of the weighting.

A similar formula was derived by Fleischer [link] , Tufts, Rorabacher and Mosier [link] , by Schüssler [link] , by Oetken, Parks, and Schüssler [link] , and by Burrus, Soewito, and Gopinath [link] , [link] in addressing similar problems.

If the integrals in [link] and [link] can be analytically evaluated, the solution of the weighted squared error approximation is obtained bysolving $M+1$ equations. Fortunately these equations can be analytically evaluated for several interesting cases as was done for the unweightedcase. The even length- $N$ case is derived in a similar way using Equation 40 from FIR Digital Filters and Equation 41 from FIR Digital Filters . An alternate formulation could modify the first column.

In most practical situations where specifications are set in the frequency domain, these filters are described in terms of frequency bands.We have already seen the idea of single pass, stop, and transition bands. We now allow multiple pass and stopbands separated by multiple transitionbands. In order to obtain analytical solutions of [link] and [link] and to be consistent with usual practice, we restrict ourselves to constant weights over each separately defined frequency band. The errorin [link] now becomes

where the weights are constant over each band and are given by $W\left(\omega \right)=W_k$ in the $k^{th}$ band defined by ${\omega }_k<\omega <{\omega }_{k+1}$ . The desired amplitude $A_{d_k}\left(\omega \right)$ is likewise defined in the $k^{th}$ band and is hopefully simple enough to allow analytical evaluation of the formula [link] for the ideal impulse response.

The form of [link] causes [link] to become

which has an analytical solution as given by

which for $F$ band edges has terms that are indeterminate for $n=m\ne 0$ with values

and for $n=m=0$ as

Since the matrix elements are functions of $(n-m)$ and $(n+m)$ , $\mathbf{C}$ is the sum of a Toeplitz and a Hankel matrix. This matrix can always becalculated and it simply depends on the set of band edges ${\omega }_k$ and the band weights $W_k$ but not on the ideal amplitude response $A_d\left(\omega \right)$ . The case for even $N$ is similar but uses Equation 8 from Constrained Approximation and Mixed Criteria rather than Equation 7 from Constrained Approximation and Mixed Criteria with [link] to derive an appropriate form of [link] .