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Statement of the Optimal Linear Phase FIR Filter Design Problem

Module by: John Treichler. E-mail the author

Equal-ripple Design

While other types of filters are often of interest, this note focuses on the lowpass linear phase filter. Even though it is not immediately obvious, virtually all of the analytical results developed in this note apply to the other types as well. This fact is amplified in the module Extension to Non-lowpass Filters.

It is known that the Parks-McClellan filter synthesis software package produces “optimal" filters in the sense that the best possible filter performance is attained for the number of “filter taps" allowed by the designer. “Optimal" can be defined various ways. The Parks-McClellan package uses the Remez exchange algorithm to optimize the filter design by selecting the impulse response of given length, termed here NN, which minimizes the peak ripple in the passband and stopband. It can be shown, though not here, that minimizing the peak, or maximum, ripple is equivalent to making all of the local peaks in the ripple equal to each other. This fact leads to three different names for essentially the same filter design. They are commonly called “equal-ripple" filters, because the local peaks are equal in deviation from the desired filter response. Because the maximum ripple deviation is minimized in this optimization procedure, they are also termed “minimax" filters. Finally, since the Russian Chebyshev is usually associated with minimax designs 1, these filters are often given his name.

The design template for an equal-ripple lowpass filter is shown in Figure 1.

Figure 1: Frequency Response of an Optimal Weighted Equal-ripple Linear Phase FIR Filter
Figure 1 (fig1.png)

The passband extends from 0 Hz to the cutoff frequency denoted fcfc. The gain in the passband is assumed to be unity. Any other gain is attained by scaling the whole impulse response appropriately. The stopband begins at the frequency denoted fstfst and ends at the so-called Nyquist or “folding" frequency, denoted by fs2fs2, where fsfs is the sampling frequency of the data entering the digital filter. In some references, [1] for example, the sampling rate fsfs is assumed to be normalized to unity just as the passband gain has here. The dependence on the sampling frequency is kept explicit in this note, however, so that its impact on design parameters can be kept visible.

The optimal synthesis algorithm is assumed here to produce an impulse response whose associated frequency response has ripples in both the passband and the stopband. The peak deviation in the passband is denoted δ1δ1 and the peak deviation in the stopband is denoted δ2δ2. It is commonly thought that an “equal-ripple" design forces δ1δ1 to equal δ2δ2. In fact this is not true. The local ripple peaks in the passband will all equal δ1δ1 and those in the stopband will all equal δ2δ2. For a given filter specification the two are linked together by a weight denoted WW, so that δ1=Wδ2δ1=Wδ2. In fact the Parks-McClellan routines insure the design of weighted equal-ripple filters. The choice of WW is discussed shortly.

An important design parameter is the transition band, denoted ΔfΔf, and defined as the difference between the stopband edge fstfst and the passband edge fcfc. Thus,

Δ f = f s t - f c . Δ f = f s t - f c .
(1)

In theory the required filter order NN is a function of all of the design parameters defined so far, that is, fs,fc,fst,δ1fs,fc,fst,δ1, and δ2δ2. The central point of this technical note is that under a large range of practical circumstances the required value of NN can be estimated using only fs,Δffs,Δf, and the smaller of δ1δ1 and δ2δ2.

Conversion of Specifications

While the parameters defined in the previous section relate directly to the theory of FIR filter design optimization, some of them differ from those usually employed to specify the performance of a filter. We discuss here the conversion of two of those, δ1δ1 and δ2δ2, into more traditional measures.

Passband Ripple: Figure 1 uses the parameter δ1δ1 to describe the peak difference between the template lowpass filter and the magnitude of the filter response actually attained. Traditionally this passband ripple has been specified in terms of the maximum difference in the power level transmitted through the filter in the passband. By this definition, the peak-to-peak passband ripple, abbreviated here as PBR, is given by

P B R = 10 l o g 10 ( 1 + δ 1 ) 2 ( 1 - δ 1 ) 2 . P B R = 10 l o g 10 ( 1 + δ 1 ) 2 ( 1 - δ 1 ) 2 .
(2)

Assuming that the nominal power transmission through the filter is unity, the numerator is the power gain at a ripple peak and the denominator is the gain at a trough. It is easily shown (see Appendix A) that when δ1δ1 is small compared to unity, or, equivalently, when the passband ripple is less than about 1.5 dB, then 2

P B R 17 . 36 δ 1 . P B R 17 . 36 δ 1 .
(3)

Stopband Ripple: The traditional specification for stopband ripple, abbreviated here as SBR, is the power difference between the nominal passband transmission level and the transmission level of the highest ripple in the stopband. For the equal ripple design shown in Figure 1, all stopband ripples have equal peak values and the nominal passband transmission is unity, that is, 0 dB. The stopband ripple, or more accurately, the minimum stopband power rejection, denoted SBR, is given by

S B R = 20 l o g 10 δ 2 . S B R = 20 l o g 10 δ 2 .
(4)

Example 1

Suppose a filter is specified to have a peak-to-peak passband ripple of 0.5 dB and a minimum stopband attenuation of 60 dB. Using the above equations we find that δ1=0.0288δ1=0.0288, δ2=.001δ2=.001, and the relative weighting, WW, therefore equals 28.8.

In discussing filter specifications it should be noted that the cutoff frequency fcfc shown in Figure 1 differs from the definition typically used in analog filter designs. The cutoff frequency is commonly defined as the 3 dB point, that is, that frequency at which the power transfer function falls to a value 3 dB below the nominal passband level. Instead the value of fcfc shown in Figure 1 is the highest frequency at which the specified passband ripple is still attained. In very few practical cases do the two definitions result in the same value.

Footnotes

  1. He developed the concept of minimax design and a set of polynomials which carry his name not from filter design, but from the optimal design of piston drive rods for steam locomotives. They are discussed more in "Filter Sizing" and Appendix B.
  2. Strictly speaking, the peak ripple excursions are equal in magnitude, not in decibels. This subtlety is completely negligible for small values of δ1δ1.

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