5.3 Design of linear-phase fir filters by general interpolation

Design of fir filters by general interpolation

If the desired interpolation points are not uniformly spaced between $0$ and $\pi$ then we can not use the DFT. We must take a different approach. Recall that for a Type I FIRfilter, $$A()=h(M)+2\sum_{n=0}^{M-1} h(n)\cos ((M-n))$$ For convenience, it is common to write this as $$A()=\sum_{n=0}^M a(n)\cos (n)$$ where $h(M)=a(0)$ and $\forall n, 1\le n\le N-1\colon h(n)=\frac{a(M-n)}{2}$ . Note that there are $M+1$ parameters. Suppose it is desired that $A()$ interpolates a set of specified values: $$\forall k, 0\le k\le M\colon A({}_k)=A_k$$ To obtain a Type I FIR filter satisfying these interpolation equations, one can set up a linear system of equations. $$\forall k, 0\le k\le M\colon \sum_{n=0}^M a(n)\cos (n{}_k)=A_k$$ In matrix form, we have $$\begin{pmatrix}1 & \cos {}_0 & \cos (2{}_0) & & \cos (M{}_0)\\ 1 & \cos {}_1 & \cos (2{}_1) & & \cos (M{}_1)\\ \\ 1 & \cos {}_M & \cos (2{}_M) & & \cos (M{}_M)\\ \end{pmatrix}\left(\begin{array}{c}a(0)\\ a(1)\\ \\ a(M)\end{array}\right)=\left(\begin{array}{c}A(0)\\ A(1)\\ \\ A(M)\end{array}\right)$$ Once $a(n)$ is found, the filter $h(n)$ is formed as $$\{h(n)\}=1/2\{a(M), a(M-1), , a(1), 2a(0), a(1), , a(M-1), a(M)\}$$

Example

In the following example, we design a length 19 Type I FIR. Then $M=9$ and we have 10 parameters. We can therefore have 10 interpolation equations. We choose:

The general interpolation problem is much more flexible than the uniform interpolation problem that the DFT solves. Forexample, by leaving a gap between the pass-band and stop-band as in this example, the ripple near the band edge is reduced(but the transition between the pass- and stop-bands is not as sharp). The general interpolation problem also arises as asubproblem in the design of optimal minimax (or Chebyshev) FIR filters.

Linear-phase fir filters: pros and cons

FIR digital filters have several desirable properties.

• They can have exactly linear phase.
• They can not be unstable.
• There are several very effective methods for designing linear-phase FIR digital filters.

• Linear-phase filters can have long delay between input and output.
• If the phase need not be linear, then IIR filters can be more efficient.