Appendix F. Digital Filter Terminology

The first step in becoming familiar with digital filters is to learn to speak the language used in the filter business. Fortunately, the vocabulary of digital filters corresponds very well to the mother tongue used for continuous (analog) filters—so we don’t have to unlearn anything that we already know. This appendix is an introduction to the terminology of digital filters.

Allpass filter—an IIR filter whose magnitude response is unity over its entire frequency range, but whose phase response is variable. Allpass filters are typically appended in a cascade arrangement following a standard IIR filter, H₁(z), as shown in Figure F–1.

Figure F–1. Typical use of an allpass filter.

An allpass filter, H_ap(z), can be designed so that its phase response compensates for, or equalizes, the nonlinear phase response of an original IIR filter[1–3]. Thus, the phase response of the combined filter, H_combined (z), is more linear than the original H₁(z), and this is particularly desirable in communications systems. In this context, an allpass filter is sometimes called a phase equalizer.

Allpass filters have the property that the numerator polynomial coefficients in the filter’s H(z) transfer function are a reverse-order version of the denominator polynomial coefficients. For example, the following transfer function describes a 2nd-order allpass filter:

Equation F–1

where the numerator polynomial coefficients are [B, A, 1] and the denominator polynomial coefficients are [1, A, B].

Attenuation—an amplitude loss, usually measured in dB, incurred by a signal after passing through a digital filter. Filter attenuation is the ratio, at a given frequency, of the signal amplitude at the output of the filter divided by the signal amplitude at the input of the filter, defined as

Equation F–2

For a given frequency, if the output amplitude of the filter is smaller than the input amplitude, the ratio in Eq. (F–2) is less than one, and the attenuation is a negative number.

Band reject filter—a filter that rejects (attenuates) one frequency band and passes both a lower- and a higher-frequency band. Figure F–2(a) depicts the frequency response of an ideal band reject filter. This filter type is sometimes called a notch filter.

Figure F–2. Filter symbols and frequency responses: (a) band reject filter; (b) bandpass filter.

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Bandpass filter—a filter, as shown in Figure F–2(b), that passes one frequency band and attenuates frequencies above and below that band.

Bandwidth—the frequency width of the passband of a filter. For a lowpass filter, the bandwidth is equal to the cutoff frequency. For a bandpass filter, the bandwidth is typically defined as the frequency difference between the upper and lower 3 dB points.

Bessel function—a mathematical function used to produce the most linear phase response of all IIR filters with no consideration of the frequency magnitude response. Specifically, filter designs based on Bessel functions have maximally constant group delay.

Butterworth function—a mathematical function used to produce maximally flat filter magnitude responses with no consideration of phase linearity or group delay variations. Filter designs based on a Butterworth function have no amplitude ripple in either the passband or the stopband. Unfortunately, for a given filter order, Butterworth designs have the widest transition region of the most popular filter design functions.

Cascaded filters—a filtering system where multiple individual filters are connected in series; that is, the output of one filter drives the input of the following filter as illustrated in Figures F–1 and 6–37(a).

Center frequency (f₀)—the frequency lying at the midpoint of a bandpass filter. Figure F–2(b) shows the f_o center frequency of a bandpass filter.

Chebyshev function—a mathematical function used to produce passband or stopband ripples constrained within fixed bounds. There are families of Chebyshev functions based on the amount of ripple, such as 1 dB, 2 dB, and 3 dB of ripple. Chebyshev filters can be designed to have a frequency response with ripples in the passband and a flat stopband (Chebyshev Type I), or flat passbands and ripples in the stopband (Chebyshev Type II). Chebyshev filters cannot have ripples in both the passband and the stopband. Digital filters based upon Chebyshev functions have steeper transition region roll-off but more nonlinear-phase response characteristics than, say, Butterworth filters.

CIC filter—cascaded integrator-comb filter. CIC filters are computationally efficient, linear-phase, recursive, FIR, lowpass filters used in sample rate change applications. Those filters are discussed in Chapter 10.

Coefficients—see filter coefficients.

Cutoff frequency—the highest passband frequency for lowpass filters (and the lower passband frequency for highpass filters) where the magnitude response is within the peak-peak passband ripple region. Figure F–3 illustrates the f_c cutoff frequency of a lowpass filter.

Figure F–3. A lowpass digital filter frequency response. The stopband relative amplitude is –20 dB.

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Decibels (dB)—a unit of attenuation, or gain, used to express the relative voltage or power between two signals. For filters, we use decibels to indicate cutoff frequencies (–3 dB) and stopband signal levels (–20 dB) as illustrated in Figure F–3. Appendix E discusses decibels in more detail.

Decimation filter—a lowpass digital FIR filter where the output sample rate is less than the filter’s input sample rate. As discussed in Section 10.1, to avoid aliasing problems, the output sample rate must not violate the Nyquist criterion.

Digital filter—computational process, or algorithm, transforming a discrete sequence of numbers (the input) into another discrete sequence of numbers (the output) having a modified frequency-domain spectrum. Digital filtering can be in the form of a software routine operating on data stored in computer memory or can be implemented with dedicated hardware.

Elliptic function—a mathematical function used to produce the sharpest roll-off for a given number of filter taps. However, filters designed by using elliptic functions, also called Cauer filters, have the poorest phase linearity of the most common IIR filter design functions. The ripples in the passband and stopband are equal with elliptic filters.

Envelope delay—see group delay.

Filter coefficients—the set of constants, also called tap weights, used to multiply against delayed signal sample values within a digital filter structure. Digital filter design is an exercise in determining the filter coefficients that will yield the desired filter frequency response. For an FIR filter, by definition, the filter coefficients are the impulse response of the filter.

Filter order—a number describing the highest exponent in either the numerator or denominator of the z domain transfer function of a digital filter. For tapped-delay line FIR filters, there is no denominator in the transfer function and the filter order is merely the number of delay elements used in the filter structure. Generally, the larger the filter order, the better the frequency-domain performance, and the higher the computational workload, of the filter.

Finite impulse response (FIR) filter—defines a class of digital filters that have only zeros on the z plane. The key implications of this are: (1) FIR filter impulse responses have finite time durations, (2) FIR filters are always stable, and (3) FIR filters can have exactly linear phase responses (so long as the filters’ impulse response samples are symmetrical, or antisymmetrical). For a given filter order, digital FIR filters have a much more gradual transition region roll-off (poorer performance) than digital IIR filters. FIR filters can be implemented with both nonrecursive (tapped-delay line) and recursive (CIC filters, for example) structures.

Frequency magnitude response—a frequency-domain description of how a filter interacts with input signals. The frequency magnitude response in Figure F–3 is a curve of filter attenuation (in dB) versus frequency. Associated with a filter’s magnitude response is a phase response.

Group delay—the negative of the derivative of a filter’s frequency-domain phase response with respect to frequency, G(ω) = –d(H_ø(ω))/d(ω). If a filter’s complex frequency response is represented in polar form as

Equation F–3

where digital frequency ω is continuous and ranges from –π to π radians/sample, corresponding to a cyclic frequency range of –f_s/2 to f_s/2 Hz, then the filter’s group delay is defined as

Equation F–4