5.3 Design of linear-phase fir filters by general interpolation

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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:

$\forall k, {}_{k}=\{0, 0.1\pi , 0.2\pi , 0.3\pi \}0\le k\le 3\colon A({}_{k})=1$
$\forall k, {}_{k}=\{0.5\pi , 0.6\pi , 0.7\pi , 0.8\pi , 0.8\pi , 1.0\pi \}4\le k\le 9\colon A({}_{k})=0$
To solve this interpolation problem in Matlab, note that the matrix can be generated by a single multiplication of a columnvector and a row vector. This is done with the command C = cos(wk*[0:M]); where wk is a column vector containing the frequency points. To solve the linear system ofequations, we can use the Matlab backslash command.

N = 19; M = (N-1)/2;wk = [0 .1 .2 .3 .5 .6 .7 .8 .9 1]'*pi;Ak = [1 1 1 1 0 0 0 0 0 0]';C = cos(wk*[0:M]);a = C/Ak; h = (1/2)*[a([M:-1:1]+1); 2*a([0]+1); a(1:M]+1)];[A,w] = firamp(h,1);plot(w/pi,A,wk/pi,Ak,'o') title('A(\omega)')xlabel('\omega/\pi')

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.
On the other hand,
• 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.

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