Translations:Appendix G: Exercises/44/en

Because the autocorrelation of an all-pass operator is equal to the unit spike, it follows that its energy spectrum is flat and equal to unity. Its magnitude spectrum therefore is also flat and equal to unity. An all-pass operator, in passing a signal from input to output, does not alter the amplitude spectrum of the signal but does add to the phase spectrum of the signal. As a matter of terminology, any operator with a flat amplitude spectrum is called a phase-shift operator, because such an operator cannot change the shape of the magnitude spectrum of an input signal but can change only the phase spectrum of the input signal. An arbitrary phase-shift operator can be two-sided - that is, it can have both a causal component and an anticausal component. An all-pass operator can be described as a causal phase-shift operator. One further fact about phase-shift operators is of interest: The inverse of a phase-shift operator is equal to the reverse of the phase-shift operator. Moreover, this fact can be true only for an operator with a flat magnitude spectrum - that is, for a phase-shift operator. A phase-shift operator is one whose autocorrelation function is equal to the unit spike. However, the autocorrelation of an operator is the same as the convolution of the operator with its time reverse. Hence, the convolution of a phase-shift operator with its time reverse is equal to the unit spike. Because by definition the convolution of an operator with its inverse is equal to the unit spike, it follows that the inverse of a time-shift operator is the same as its time reverse. Write down the inverse of the all-pass operator given above. The causal component of the phase-shift operator becomes the anticausal component of its inverse. Both the all-pass operator and its inverse are one-sided; the all-pass operator is one-sided in the sense of having only a causal component. As we have said, an all-pass operator is a phase-shift operator that is causal in real time; by the same token, the inverse of an all-pass operator is not causal in real time. An all-pass operator can be described as a causal filter that adds a certain phase spectrum to the phase spectrum of an input signal without changing its magnitude spectrum. The inverse to this all-pass operator can be described as an anticausal filter that subtracts the phase spectrum from the phase spectrum of an input signal without changing its magnitude spectrum. It can be shown that any wavelet always can be represented as the convolution of an all-pass operator with the minimum-delay wavelet with the same autocorrelation as the given wavelet.