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Filter Sizing

Module by: John Treichler. E-mail the author

The Formula for Estimation of the FIR Filter Length

For many lowpass filter designs the peak passband excursion δ1δ1 exceeds the peak stopband excursion δ2δ2 by a factor of ten or more. This ratio, earlier denoted as the weight WW, 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 NN, can be accurately estimated using the formula:

N α f s Δ f , N α f s Δ f ,
(1)

where the design parameter αα is given by the equation:

α = 0 . 22 + 0 . 0366 · S B R . α = 0 . 22 + 0 . 0366 · S B R .
(2)

As before, SBR is the minimum stopband attenuation compared to the nominal passband power transmission level, measured in decibels.

Example 1: 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 WW has been evaluated to be approximately 29 in this case, Equation 1 applies. Using Equation 2, αα is evaluated to be 2.42. Thus NN is closely approximated by 2.42 times the reciprocal of the normalized transition bandwidth ΔffsΔffs. To continue the example assume that the sampling rate is 8 kHz, that the cutoff frequency fcfc is 1530 Hz, and that the stopband edge fstfst is 2330 Hz. Thus Δf=800Δf=800 Hz and Δffs=0.1Δffs=0.1, yielding an estimated filter order NN 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 NN as estimated by Equation 1 and Equation 2 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 NN becomes very easy.

Table 1 provides the values of the design parameter αα from Equation 2 for various degrees of stopband suppression. Given also is the range of the passband ripple for which the values of αα apply. The column marked maximum passband ripple reflects the the assumption that the passband deviation δ1δ1 is small compared to unity; specifically, the stated value of 1.74 dB corresponds to δ1=0.1δ1=0.1. The rightmost column, denoted minimum passband ripple, is the limit imposed by the assumption that δ1>10·δ2δ1>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 Equation 2 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.

Table 1: Table 1: Values of the Design Parameter α as a Function of the Minimum Stopband Attenuation
Stopband Attenuation (in dB) α 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 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])
Figure 1 (fig2.png)

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:

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 ] . 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 ] .
(3)

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

Δ f = f s π N ( c o s h - 1 ( 1 δ 2 ) ) . Δ f = f s π N ( c o s h - 1 ( 1 δ 2 ) ) .
(4)

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 1 δ 2 = c o s h y e y 2
(5)

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 . y l o g e 2 δ 2 = l o g e 2 - l o g e δ 2 .
(6)

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

Δ f = f s π N ( l o g e 2 - l o g e δ 2 ) . Δ f = f s π N ( l o g e 2 - l o g e δ 2 ) .
(7)

Rewriting this equation shows that NN must equal or exceed:

N α f s Δ f N α f s Δ f
(8)

where αα is given by

α = l o g e 2 - l o g e δ 2 π . α = l o g e 2 - l o g e δ 2 π .
(9)

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

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

Substituting this into Equation 9 yields

α = 0 . 22 + 0 . 0366 · S B R , α = 0 . 22 + 0 . 0366 · S B R ,
(11)

which can be recognized as Equation 2.

Caveats

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 δ11δ11), 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])
Figure 2 (fig3.png)

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.

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