Filters Preserving Phase
In this chapter, linear phase and zero phase filters are defined and discussed.
(Symmetric Impulse Responses)
A linear-phase filter is typically used when a causal filter is needed to modify a signal's magnitude-spectrum while preserving the signal's time-domain waveform as much as possible. Linear-phase filters have a symmetric impulse response, e.g.,
We will show that every real symmetric impulse response corresponds to a real frequency response times a linear phase term , where is the slope of the linear phase. Linear phase is often ideal because a filter phase of the form corresponds to phase delay
(Even Impulse Responses)
A zero-phase filter is a special case of a linear-phase filter in which the phase slope is . The real impulse response of a zero-phase filter is even.11.1 That is, it satisfies
A zero-phase filter cannot be causal (except in the trivial case when the filter is a constant scale factor ). However, in many ``off-line'' applications, such as when filtering a sound file on a computer disk, causality is not a requirement, and zero-phase filters are often preferred.
It is a well known Fourier symmetry that real, even signals have real, even Fourier transforms . Therefore,
This follows immediately from writing the DTFT of in terms of a cosine and sine transform:
A real frequency response has phase zero when it is positive, and phase when it is negative. Therefore, we define a zero-phase filter as follows:
Recall from §7.5.2 that a passband is defined as a frequency band that is ``passed'' by the filter, i.e., the filter is not designed to minimize signal amplitude in the band. For example, in a lowpass filter with cut-off frequency rad/s, the passband is .
Under our definition, a zero-phase filter always has a real, even impulse response [ ], but not every real, even, impulse response is a zero-phase filter. For example, if is zero phase, then is not; however, we could call a ``-phase filter'' if we like (a zero-phase filter in series with a sign inversion).
Phase in the Stopband
Practical zero-phase filters are zero-phase in their passbands, but may switch between 0 and in their stopbands (as illustrated in the upcoming example of Fig.10.2). Thus, typical zero-phase filters are more precisely described as piecewise constant-phase filters, where the constant phase is 0 in all passbands, and over various intervals within stopbands. Similarly, practical ``linear phase'' filters are typically truly linear phase across their passbands, but typically exhibit discontinuities by radians in their stopband(s). As long as the stopbands are negligible, which is the goal by definition, the -phase regions can be neglected completely.
Example Zero-Phase Filter Design
Figure 10.1 shows the impulse response and frequency response of a length 11 zero-phase FIR lowpass filter designed using the Remez exchange algorithm.11.2 The matlab code for designing this filter is as follows:
N = 11; % filter length - must be odd b = [0 0.1 0.2 0.5]*2; % band edges M = [1 1 0 0 ]; % desired band values h = remez(N-1,b,M); % Remez multiple exchange designThe impulse response h is returned in linear-phase form, so it must be left-shifted samples to make it zero phase.
Figure 10.2 shows the amplitude and phase responses of the FIR filter designed by remez. The phase response is zero throughout the passband and transition band. However, each zero-crossing in the stopband results in a phase jump of radians, so that the phase alternates between zero and in the stopband. This is typical of practical zero-phase filters.
Elementary Zero-Phase Filter Examples
A practical zero-phase filter was illustrated in Figures 10.1 and 10.2. Some simple general cases are as follows:
- The trivial (non-)filter
has frequency response
, which is zero phase for all .
- Every second-order zero-phase FIR filter has an impulse
response of the form
- Extending the previous example, every order zero-phase real FIR
filter has an impulse response of the form
and frequency response
- There is no first-order (length 2) zero-phase filter, because,
to be even, its impulse response would have to be proportional to
. Since the bandlimited digital
impulse signal is ideally interpolated using bandlimited
interpolation [91,84], giving samples of
unit-amplitude sinc function having zero-crossings on the
integers, we see that sampling on the integers yields
an IIR filter:
- Similarly, there are no odd-order (even-length) zero-phase filters.
Odd Impulse Reponses
Note that odd impulse responses of the form are closely related to zero-phase filters (even impulse responses). This is because another Fourier symmetry relation is that the DTFT of an odd sequence is purely imaginary . In practice, Hilbert transform filters and differentiators are often implemented as odd FIR filters . A purely imaginary frequency response can be divided by to give a real frequency response. As a result, filter-design software for one case is easily adapted to the other .
Equivalently, an odd impulse response can be multiplied by in the time domain to yield a purely imaginary impulse response that is Hermitian. Hermitian signals have real Fourier transforms . Therefore, a Hermitian impulse response gives a filter having a phase response that is either zero or at each frequency.
Symmetric Linear-Phase Filters
As stated at the beginning of this chapter, the impulse response of every causal, linear-phase, FIR filter is symmetric:
Simple Linear-Phase Filter Examples
- The example of §10.2.1 was in fact a linear-phase FIR
filter design example. The resulting causal finite impulse response
was left-shifted (``advanced'' in time) to make it zero phase.
- While the trivial ``bypass filter''
(§10.2.2), the ``bypass filter with a unit delay,''
is linear phase. It is (trivially) symmetric
about time , and the frequency response is
is a pure linear phase term
having a slope
of samples (radians per radians-per-sample), or seconds
(radians per radians-per-second). The phase- and group-delays are
each 1 sample at every frequency.
- The impulse response of the simplest lowpass filter studied in
Chapter 1 was
Since this impulse response is symmetric about time samples,
it is linear phase, and
, as derived
in Chapter 1. The phase delay and group delay are both sample at
each frequency. Note that even-length linear-phase filters cannot be
time-shifted (without interpolation) to create a corresponding
zero-phase filter. However, they can be shifted to make a
near-zero-phase filter that has a phase delay and group delay equal to
half a sample at all passband frequencies.
Software for Linear-Phase Filter Design
The Matlab Signal Processing Toolbox covers many applications with the following functions:
Methods for FIR filter design are discussed in the fourth book of the music signal processing series , and classic references include [64,68]. There is also quite a large research literature on this subject.
Antisymmetric Linear-Phase Filters
In the same way that odd impulse responses are related to even impulse responses, linear-phase filters are closely related to antisymmetric impulse responses of the form , . An antisymmetric impulse response is simply a delayed odd impulse response (usually delayed enough to make it causal). The corresponding frequency response is not strictly linear phase, but the phase is instead linear with a constant offset (by ). Since an affine function is any function of the form , where and are constants, an antisymmetric impulse response can be called an affine-phase filter. These same remarks apply to any linear-phase filter that can be expressed as a time-shift of a -phase filter (i.e., it is inverting in some passband). However, in practice, all such filters may be loosely called ``linear-phase'' filters, because they are designed and implemented in essentially the same way .
Note that truly linear-phase filters have both a constant phase delay and a constant group delay. Affine-phase filters, on the other hand, have a constant group delay, but not a constant phase delay.
There are no linear-phase recursive filters because a recursive filter cannot generate a symmetric impulse response. However, it is possible to implement a zero-phase filter offline using a recursive filter twice. That is, if the entire input signal is stored in a computer memory or hard disk, for example, then we can apply a recursive filter both forward and backward in time. Doing this squares the amplitude response of the filter and zeros the phase response.
To show this analytically, let denote the output of the first filtering operation (which we'll take to be ``forward'' in time in the normal way), and let be the impulse response of the recursive filter. Then we have
Using this result and applying the convolution theorem (§6.3) twice gives the z transform
If the filter were complex, then we would need to conjugate its coefficients when running it backwards.
In summary, we have thus shown that forward-backward filtering squares the amplitude response and zeros the phase response. Note also that the phase response is truly zero, never alternating between zero and . No matter what nonlinear phase response a filter may have, this phase is completely canceled out by forward and backward filtering. The amplitude response, on the other hand, is squared. For simple bandpass filters (including lowpass, highpass, etc.), for which the desired gain is 1 in the passband and 0 in the stopband, squaring the amplitude response usually improves the response, because the ``stopband ripple'' (deviation from 0) is squared, thereby doubling the stopband attenuation in dB. On the other hand, passband ripple (deviation from 1) is only doubled by the squaring (because ).
A Matlab example of forward-backward filtering is presented in §11.6 (in Fig.11.1).
Phase Distortion at Passband Edges
For many applications (such as lowpass, bandpass, or highpass filtering), the most phase dispersion occurs at the extreme edge of the passband (i.e., in the vicinity of cut-off frequencies). This phenomenon was clearly visible in the example of Fig.7.6.4. Only filters without feedback can have exactly linear phase (unless forward-backward filtering is feasible), and such filters generally need many more multiplies for a given specification on the amplitude response . One should keep in mind that phase dispersion near a cut-off frequency (or any steep transition in the amplitude response) usually appears as ringing near that frequency in the time domain. (This can be heard in the upcoming matlab example of §11.6, Fig.11.1.)
For musical purposes, , or the effect that a filter has on the magnitude spectrum of the input signal, is usually of primary interest. This is true for all ``instantaneous'' filtering operations such as tone controls, graphical equalizers, parametric equalizers, formant filter banks, shelving filters, and the like. (Elementary examples in this category are discussed in Appendix B.) Notable exceptions are echo and reverberation , in which delay characteristics are as important as magnitude characteristics.
When designing an ``instantaneous'' filtering operation, i.e., when not designing a ``delay effect'' such as an echo unit or reverberator, the amplitude response should be as smooth as possible as a function of frequency . Smoother amplitude responses correspond to shorter impulse responses (when the phase is zero, linear, or ``minimum phase'' as discussed in the next chapter). By keeping impulse-responses as short as possible, phase dispersion is minimized, and ideally inaudible. Linearizing the phase response with a delay equalizer (a type of allpass filter) does not eliminate ringing, but merely shifts it in time. A general rule of thumb is to keep the total impulse-response duration below the time-discrimination threshold of hearing in the context of the intended application.
Implementation Structures for Recursive Digital Filters