What is an all-pass filter? Let us begin by reviewing a phase-shift filter. A phase-shift filter can be two sided. A phase-shift filter operates on the phase alone (that is, a phase-shift filter has a flat amplitude spectrum). A phase-shift filter usually is normalized so that its flat amplitude spectrum is equal to unity. Thus, in passing a signal from input to output, a normalized phase-shift filter does not alter the amplitude spectrum of the signal, but it does change the phase spectrum of the signal. If it is not normalized, a phase-shift filter multiplies the amplitude spectrum of the signal by a constant factor. For this discussion, let us assume that all phase-shift systems are normalized.
Another property of a phase-shift filter is important: The inverse of a phase-shift filter is equal to the reverse (with respect to time zero) of the phase-shift filter. Moreover, this fact can be true only for filters with a flat amplitude spectrum. Let us see why this property holds. A phase-shift filter has an autocorrelation equal to the unit spike. However, the autocorrelation is equal to the convolution of the filter with its time reverse. Hence, the convolution of the phase-shift filter with its time reverse is equal to the unit spike. It follows that the inverse of a phase-shift filter is equal to its time reverse.
Let us give a note on usage. In physics, the word amplitude can mean the extent of a vibration or oscillation, measured from the position of equilibrium. For example, we would say the amplitude of a wave and the amplitude of a seismic trace. In this sense, amplitude can be either positive or negative. The word magnitude represents a numerical quantity or value. In mathematics, magnitude often refers to a nonnegative quantity. For example, the magnitude of a real number is the absolute value of that real number. The magnitude of a complex number z, denoted , is defined to be the positive square root of the complex number times its complex conjugate. The magnitude of a complex number is always a nonnegative real number. The frequency spectrum is complex, and for that reason, we prefer to call the magnitude spectrum instead of the amplitude spectrum. However, the term amplitude spectrum is so prevalent that we often tend to favor it.
In Chapter 5, it is stated that any causal linear system can be described by its gain and its delay. Its gain is a measure of the increase or decrease of the magnitude of the output compared with the magnitude of the input. Delay is a measure of the time from the instant the input is activated to the instant that this input is felt significantly at the output. As we expect, both gain and delay depend on the frequency of the signal. The magnitude spectrum represents the gain, and the phase spectrum represents the delay.
The all-pass filter was introduced in Chapter 7. An all-pass filter is a causal phase-shift filter. In other words, an all-pass filter is a one-sided filter whose amplitude spectrum is flat and equal to unity. The logarithm of one is zero. The amplitude spectrum (or gain) of an all-pass system is equal to one for all frequencies. Thus, the logarithm of its gain is equal to zero for all frequencies. All of the information in an all-pass filter is contained in its phase spectrum. In passing a signal from input to output, an all-pass filter does not alter the amplitude spectrum of the signal, but the all-pass filter does add its phase to the phase spectrum of the signal. In passing a signal from input to output, an inverse all-pass filter does not alter the amplitude spectrum of the signal, but the inverse all-pass filter does subtract its phase from the phase spectrum of the signal. An all-pass filter converts a minimum-phase input wavelet into a nonminimum-phase wavelet with the same amplitude spectrum. The corresponding inverse all-pass filter converts the nonminimum-phase wavelet back into the original minimum-phase wavelet.
Also in this chapter
- The shaping filter
- Spiking filter
- White convolutional model
- Wavelet processing
- Nonminimum-delay wavelet
- Signature deconvolution
- Dual-sensor wavelet estimation
- Deconvolution: Einstein or predictive?
- Appendix I: Exercises