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Derivation of the formula

This section describes the theoretical underpinnings of [link] and [link] . A clear understanding of this section is not required to use the Parks-McClellan software routines orto 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 N impulse response coefficients in such a way as to minimize the weighted peak differencebetween the desired magnitude frequency response and the actual one. Since the solution to this optimization problem does not have a closed form, it isnot easy to generalize its properties. To learn about its properties and to develop appropriate design rules, McClellan, Rabiner, and others synthesizedthousands 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 N in terms of all of the parameters specifying the filter. While this work is not immediately useful for design work, a limiting caseuncovered by those workers does provide some insight into the optimal filter solutions and leads to the simple rules compressed into [link] and [link] .

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. Sucha filter, or a very close approximation to it, can be synthesized using another FIR filter design method, that of multiplying asampled s i n q q function, where q = π f f s , by an N -point window function constructed from a Chebyshev polynomial.The sampled s i n q q , or sinc, function is the inverse z-transform of a perfect lowpass filter. It cannot be used directly since it extendsinfinitely far into both forward and backward time. A finite duration impulse response is obtained by multiplying the “perfect" response bya 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 x between -1 and 1, and that for greater values of | x | , the magnitude grows as x M where M is the order of the polynomial. One such polynomial is shown in [link] .

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 x M portion to correspond to the transition from the stopband to the passband. This is accomplished byinvoking a change of variables relating x to the frequency f . The resulting equation is then evaluated at the several points to obtain anexpression for the transition bandwidth Δ f . The details of this manipulation are contained in Appendix C . They resultin the following equation:

Δ f = f s π ( N - 1 ) [ c o s h - 1 ( 1 + δ 1 δ 2 ) - { ( c o s h - 1 ( 1 + δ 1 δ 2 ) ) 2 - ( c o s h - 1 ( 1 - δ 1 δ 2 ) ) 2 } 1 2 ] .

If δ 1 is small compared to unity and N is large compared to unity, as already assumed, then Δ f is closely approximated by

Δ f = f s π N ( c o s h - 1 ( 1 δ 2 ) ) .

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

1 δ 2 = c o s h y e y 2

With suitable manipulation we find that

y l o g e 2 δ 2 = l o g e 2 - l o g e δ 2 .

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

Δ f = f s π N ( l o g e 2 - l o g e δ 2 ) .

Rewriting this equation shows that N must equal or exceed:

N α f s Δ f

where α is given by

α = l o g e 2 - l o g e δ 2 π .

Rewriting equation 4 from the module titled "Statement of the Optimal Linear Phase FIR Filter Design Problem" , δ 2 can be written as

δ 2 = 10 - S B R 20 = e - 2 . 303 · S B R 20 .

Substituting this into [link] yields

α = 0 . 22 + 0 . 0366 · S B R ,

which can be recognized as [link] .

Caveats

The derivation just presented assumes that the filter of interest is a lowpass design, the filter order is high ( > 20 or so), that the passband ripple is small (that δ 1 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 [link] and [link] useful. An indication of how errors can enter the estimate of N under other conditions can be seen, however, by examining [link] .

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

This figure shows the smallest value of Δ f attainable with optimal equal-ripple linear phase filters of different lengths as a function of thecutoff frequency f c . [link] and [link] predict that the transition bandwidth is constant as a function of cutoff frequency and that it always gets smaller as the filter order N increases. [link] shows that these generalities are not true. It can be seen that Δ f varies somewhat as a function of f c and that there are particular choices of f c where a lower value of Δ f is actually attainable with a lower filter order rather than a higher one.It would appear that, for a given filter order N , some values of f c 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 [link] shows that Δ f is not truly independent of the cutoff frequency f c and monotonic in the filter order N , the significant variations appear only for low filter orders. If N is greater than 20 or so, and the other conditions listed above hold true, as they usually do, then [link] and [link] can be used with impugnity, even for highpass and bandpass filters.

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Source:  OpenStax, Notes on the design of optimal fir filters. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10553/1.3
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