0.10 "notes on the design of optimal fir filters" appendix c

Using a chebyshev polynomial to estimate

We desire that the oscillatory portion of the polynomial shown in Figure 1 in the module titled "Filter Sizing" correspond to the stopband region of the filter response and the $x^M$ portion to correspond to the transition from the stopband to the passband. This is achieved byemploying a change of variables from frequency $f$ to the polynomial argument $x$ :

While many different types of variable changes could be employed, this one matches the boundary conditions (an obvious requirement) but happens toemploy the cosine function, a member of the same family used to define the Chebyshev polynomials.

With this change of variables we see that the transition band $\Delta f$ is defined by the difference between $x=1$ and $x=x_p$ . Using the closed, but nonintuitive form of the K-th order Chebyshev polynomial, valid for $\left|x\right|>1$ , we have that

To synthesize the desired impulse response using this windowing technique we multiply the resulting window function by the sampled sinc function. In thiscase, however, we desire that the cutoff frequency be as low as possible, limiting at zero Hz. The associated sinc function equals unity for allnon-zero coefficients of the impulse response. Since the final impulse response isthe point-by-point product of the window and the sampled sinc function, in this case the window itself is the resulting impulse response. It suffices thento examine the properties of the N-th order Chebyshev polynomial to see how the N-point optimal filter will behave.

To find the relationship between the required filter order $N$ and the attainable transition band $\Delta f$ , we first determine the proper value of $K$ and then evaluate [link] at the known combinations of $x$ and $P_K\left(x\right)$ . To select $K$ we note that all but one of the ripples in the polynomial's response are used in the stopband and these are splitevenly between the positive and negative frequencies. Thus a filter and window of order $N$ implies a Chebyshev polynomial of order

With this resolved we observe from Figure 1 in the module titled "Filter Sizing" that

These equations are manipulated to yield an expression for $x_p$ . [link] is then used to obtain values for $f_{st}$ , corresponding to $x=1$ , and $f_c$ , corresponding to $x=x_p$ . Their difference, defined earlier to be the transition band $\Delta f$ , is then given by

Under suitable conditions this equation can be simplified considerably. For example, in the limits of small ${\delta }_1$ and large $N$ , [link] reduces to Equation 4 in the module titled "Filter Sizing" .