The Formula for Estimation of the FIR Filter Length For many lowpass filter designs the peak passband excursion δ1 exceeds the peak stopband excursion δ2 by a factor of ten or more. This ratio, earlier denoted as the weight W, was just evaluted in the previous section to have the value 28.8 for a typical set of specifications. In this example the stopband attenuation specification drives the required filter order. In this case, and with a few additional assumptions which will be enumerated later, the number of coefficients in the impulse response of a high-order FIR linear phase filter, denoted N, can be accurately estimated using the formula: N ≈ α f s Δ f , where the design parameter α is given by the equation: α = 0 . 22 + 0 . 0366 · S B R . As before, SBR is the minimum stopband attenuation compared to the nominal passband power transmission level, measured in decibels. Continuing from Example I "Statement of the Optimal Linear FIR Filter Design Problem" Suppose as before that the lowpass filter of interest is to have a peak-to-peak passband ripple (PBR) of 0.5 dB and a minimum stopband attenuation of 60 dB. Since W has been evaluated to be approximately 29 in this case, applies. Using , α is evaluated to be 2.42. Thus N is closely approximated by 2.42 times the reciprocal of the normalized transition bandwidth Δffs. To continue the example assume that the sampling rate is 8 kHz, that the cutoff frequency fc is 1530 Hz, and that the stopband edge fst is 2330 Hz. Thus Δf=800 Hz and Δffs=0.1, yielding an estimated filter order N of approximately 24. Executing the Parks-McClellan design program with these parameters happens to produce an impulse response which almost perfectly matches the desired result (e.g., peak stopband ripple of 60.07 dB as opposed to the stated objective of 60 dB).♠ Note that the required filter order N as estimated by and does not depend on the passband ripple PBR or on the exact values of the cutoff and stopband frequencies. Thus, when the conditions allowing the underlying assumptions to be met are true, estimating the required filter order N becomes very easy. provides the values of the design parameter α from for various degrees of stopband suppression. Given also is the range of the passband ripple for which the values of α apply. The column marked maximum passband ripple reflects the the assumption that the passband deviation δ1 is small compared to unity; specifically, the stated value of 1.74 dB corresponds to δ1=0.1. The rightmost column, denoted minimum passband ripple, is the limit imposed by the assumption that δ1>10·δ2. Of course FIR linear phase equal ripple filters can be designed with passband ripple extending beyond the stated range. However, as the PBR specification approaches either of these endpoints the validity of will degrade. The predicted filter length will err on the low side for small PBR values and be overly pessimistic for PBR > 1.74 dB. In such cases, an iteration on design might be necessary to obtain the desired filter characteristics. Stopband Attenuation (in dB) α Maximum Passband Ripple (in dB) Minimum Passband Ripple (in dB) 45 1.87 1.74 1.0 50 2.05 1.74 0.55 55 2.23 1.74 0.31 60 2.42 1.74 0.174 65 2.60 1.74 0.098 70 2.78 1.74 0.055

Derivation of the Formula This section describes the theoretical underpinnings of and . 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 N 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 N 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 and . 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 sinqq function, where q=πffs, by an N-point window function constructed from a Chebyshev polynomial. The sampled sinqq, 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 x between -1 and 1, and that for greater values of |x|, the magnitude grows as xM where M is the order of the polynomial. One such polynomial is shown in .

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 xM portion to correspond to the transition from the stopband to the passband. This is accomplished by invoking a change of variables relating x to the frequency f. The resulting equation is then evaluated at the several points to obtain an expression for the transition bandwidth Δf. The details of this manipulation are contained in Appendix C. They result in 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 δ1 is small compared to unity and N is large compared to unity, as already assumed, then Δf 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 Δf: 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", δ2 can be written as δ 2 = 10 - S B R 20 = e - 2 . 303 · S B R 20 . Substituting this into yields α = 0 . 22 + 0 . 0366 · S B R , which can be recognized as .

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 δ1≪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 and useful. An indication of how errors can enter the estimate of N under other conditions can be seen, however, by examining .

Comparison of the Transition Widths of Even and Odd Optimal Lowpass Filters (drawn from [1]) This figure shows the smallest value of Δf attainable with optimal equal-ripple linear phase filters of different lengths as a function of the cutoff frequency fc. and 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. shows that these generalities are not true. It can be seen that Δf varies somewhat as a function of fc and that there are particular choices of fc where a lower value of Δf 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 fc 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 shows that Δf is not truly independent of the cutoff frequency fc 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 and can be used with impugnity, even for highpass and bandpass filters.