This section describes the theoretical underpinnings of Equation 1 and Equation 2. A clear understanding of this section is not required to use the Parks-McClellan software routines or to enjoy the remainder of this technical note.

As discussed in Section 2, the Parks-McClellan synthesis algorithm uses the Remez exchange algorithm to optimally select the values of the NN impulse response coefficients in such a way as to minimize the weighted peak difference between the desired magnitude frequency response and the actual one. Since the solution to this optimization problem does not have a closed form, it is not easy to generalize its properties. To learn about its properties and to develop appropriate design rules, McClellan, Rabiner, and others synthesized thousands of filters and measured their properties. Curves with this sort of information are presented in [1], along with a complicated empirical formula for the filter order NN in terms of all of the parameters specifying the filter. While this work is not immediately useful for design work, a limiting case uncovered by those workers does provide some insight into the optimal filter solutions and leads to the simple rules compressed into Equation 1 and Equation 2.

Suppose we desire to design a high-order, FIR, linear phase filter for which the passband is as narrow as possible. Looking again at Figure 1 from the module titled "Statement of the Optimal Linear Phase FIR Filter Design Problem" with this in mind reveals that all of the ripple behavior for such a filter will occur in the stopband. Such a filter, or a very close approximation to it, can be synthesized using another FIR filter design method, that of multiplying a sampled sinqqsinqq function, where q=πffsq=πffs, by an NN-point window function constructed from a Chebyshev polynomial. The sampled sinqqsinqq, or sinc, function is the inverse z-transform of a perfect lowpass filter. It cannot be used directly since it extends infinitely far into both forward and backward time. A finite duration impulse response is obtained by multiplying the “perfect" response by a finite-duration window function. The one discussed here uses Chebyshev polynomials as their basis. These polynomials are discussed in Appendix B They all have the property that the polynomials' peak magnitude is unity for values of xx between -1 and 1, and that for greater values of |x||x|, the magnitude grows as xMxM where MM is the order of the polynomial. One such polynomial is shown in Figure 1.

Figure 1: A Chebyshev Polynomial (drawn from [1])

We desire that the oscillatory portion of the polynomial correspond to the stopband region of the filter response and the xMxM portion to correspond to the transition from the stopband to the passband. This is accomplished by invoking a change of variables relating xx to the frequency ff. The resulting equation is then evaluated at the several points to obtain an expression for the transition bandwidth ΔfΔf. The details of this manipulation are contained in Appendix C. They result in the following equation:

If δ1δ1 is small compared to unity and NN is large compared to unity, as already assumed, then ΔfΔf is closely approximated by

When the argument of the hyperbolic cosine is large, the function can be approximated as

With suitable manipulation we find that

Substituting this expression for the inverse hyperbolic cosine yields a simple formula for ΔfΔf:

Rewriting this equation shows that NN must equal or exceed:

where αα is given by

Rewriting equation 4 from the module titled "Statement of the Optimal Linear Phase FIR Filter Design Problem", δ2δ2 can be written as

Substituting this into Equation 9 yields

which can be recognized as Equation 2.

The derivation just presented assumes that the filter of interest is a lowpass design, the filter order is high (>20>20 or so), that the passband ripple is small (that δ1≪1δ1≪1), and that the filter uses all degrees of freedom except one in the stopband, that is, that the filter has the lowest possible cutoff frequency. In fact not all of these conditions have to be met to make the design Equation 1 and Equation 2 useful. An indication of how errors can enter the estimate of NN under other conditions can be seen, however, by examining Figure 2.

Figure 2: Comparison of the Transition Widths of Even and Odd Optimal Lowpass Filters (drawn from [1])

This figure shows the smallest value of ΔfΔf attainable with optimal equal-ripple linear phase filters of different lengths as a function of the cutoff frequency fcfc. Equation 1 and Equation 2 predict that the transition bandwidth is constant as a function of cutoff frequency and that it always gets smaller as the filter order NN increases. Figure 2 shows that these generalities are not true. It can be seen that ΔfΔf varies somewhat as a function of fcfc and that there are particular choices of fcfc where a lower value of ΔfΔf is actually attainable with a lower filter order rather than a higher one. It would appear that, for a given filter order NN, some values of fcfc are “hard" to attain a small transition bandwidth and others are “easy". This is in fact true and the reason for it will be discussed in "Why does alpha Depend on the Cutoff Frequency fc?".

While Figure 2 shows that ΔfΔf is not truly independent of the cutoff frequency fcfc and monotonic in the filter order NN, the significant variations appear only for low filter orders. If NN is greater than 20 or so, and the other conditions listed above hold true, as they usually do, then Equation 1 and Equation 2 can be used with impugnity, even for highpass and bandpass filters.