Signal Processing of Random Physiological Signals - Charles S. Lessard

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C H A P T E R 1 1

Digital Filters

This chapter will review the concept of filters before presenting an overview of digital filters. It is not the intent of the chapter to enable the reader to write digital filter programs, but rather to present the engineer with the ability to use application software with filters and to choose the proper filter for specific biomedical applications. Let us start with the basic question, “What are Filters?” By definition, “Filters are circuits that pass the signals of certain frequencies and attenuate those of other frequencies.”

11.1CLASSIFICATION OF FILTERS

The most common classification or types of filters are as follows:

1.Low-pass: pass low frequencies; block high frequencies

2.High-pass: pass high frequencies; block low frequencies

3.Band-pass: pass a band of frequencies

4.Band-reject: attenuate a band of frequencies

5.All-pass filter: used for phase shifting

In addition filters are also categorized as follows:

1.Butterworth

a.Low-pass filter

b.High-pass filter, etc.

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2.Chevbychev

a.Low-pass filter

b.High-pass filter, etc.

Other classifications that are also used in conjunction with the previous classification are

1.Voltage Controlled Voltage Source (VCVS),

2.Bi Quad,

3.Elliptical, and

4.Digital.

The most common digital filters fall into one of two classes:

1.Finite Impulse Response (FIR) and

2.Infinite Impulse Response (IIR)

Filters are generally described mathematically by their “Transfer Function” in the Time domain as differential equations, or in the Frequency domain as the ratio of two polynomials in “s-domain” or as j ω in radians. The order of a filter is defined by the degree of the denominator. The higher the order, the closer the filter approaches the ideal filter; however, the tradeoff an engineer must make is that higher order filters require more complex hardware, and consequently, are more expensive (costly).

11.1.1Low-Pass Butterworth Filter

In many applications, the most common low-pass filter used is referred to as an “All-Pole Filter,” which implies that the filter contained “All poles” and “No zeros” in the s-plane. In addition, the all pole, low-pass filter may be called a “Butterworth filter,” which means that the filter has an “over damped” response. Figure 11.1 shows the ideal response of a low-pass Butterworth filter with its cutoff frequency at ωc and the practical response for the same cutoff frequency. It should be noted that the low-pass Butterworth filter

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The “Transfer Function” or describing equation for the first-order Butterworth high-pass filter is given by (11.6) for the s-domain and rewritten in (11.7) in terms of frequency. It should be noted that the high-pass filter is not an “All Pole Filter,” since the equations show a zero or an s in the numerator of the equations. Hence, in the s-plane a zero exists at the origin, s = 0.

where Ahp = high-pass gain.

11.1.42nd-Order Butterworth High-Pass Filter

The transfer function for a second-order Butterworth high-pass filter is given by (11.8):

11.1.5Band-Pass Filters

Band-pass filters may be of the Butterworth or of the Chebyshev class. Band-pass filters are designed to pass a band of frequencies of bandwidth, B, with the center of the band around a center frequency (ω0 rad/sec) or f0 = ω0/2π Hz. Band-pass filter must designate two cutoff frequencies: a lower cutoff, ωL, and an upper cutoff, ωU.

The frequencies that are passed by the filter are those for which the transfer function, H (s), gain is greater than or equal to 0.707A0 as shown in Fig. 11.5.

11.1.5.1 Quality Factor

Quality Factor (Q) is a measure of narrowness of the band-pass filter. By definition, the quality factor, Q, is the ratio of the center frequency, ω0, to the bandwidth, B = ωU − ω L in radians per second; hence, the ratio may be expressed as Q = ω0/ B or Q = f0/ B. If the quality factor, Q, is greater than 10, the band-pass region is considered to be narrow. The gain of the band-pass filter is the magnitude at the center frequency. The transfer function for the band-pass filter is given by (11.9). Band-pass filters must be of second

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FIGURE 11.7: Chebyshev band-reject filter. Response of a fourth-order Chebyshev notch filter

as shown by on an oscilloscope

The transfer function for the band-reject filter is given by (11.10). Band-reject filters must be of second or higher order, since the filters require two slopes (high and low). Note that band-reject filters are not All-Pole filters, since the equation shows a pair of zeros in the numerator (s 2).

Figure 11.7 shows the response of a fourth-order Chebyshev band-reject (notch) filter. It should be noted that the maximum attenuation of the band-reject filter is the magnitude at the center frequency, ω0.

11.2DIGITAL FILTERS

11.2.1Classification of Digital Filters

The classification of digital filters is similar to the classification of analog filters

1.Chebyshev filters: low-pass or high-pass (Figs. 11.8 and 11.9)

a.Underdamped, can oscillate

b.Ripple factor

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11.2.2Infinite Impulse Response (IIR) Digital Filters

The Pole-zero transfer function of an Lth-order direct form Infinite Impulse Response Filter (IIR-filter) has the z-transform transfer function described by (11.11).

where the zs are the “z-transformation.”

General characteristics of the infinite impulse response filters are as follows:

a.Very sharp cutoff

b.Narrow transition band

c.Low-order structure

d.Low-processing time

There are several designs that can be used to implement (11.11), such as the direct, parallel, or cascade design. The “Direct” design is shown in Fig. 11.10, whereas a “Symmetric Lattice” of the IIR design is shown in Fig. 11.11.

FIGURE 11.10: Direct-form IIR digital filter. Note the numerator coefficients are the bL terms and the denominator coefficients are the aL terms