logo (11/11/82)
(Note: this is a duplicate posing of an article that went out minutes ago with a horribly wrong name. please forgive the duplicate posting.) At the recent AES show, I spoke with an engineer from Sounstream and a product manager (ex-engineer?) from Sony. Both of them claimed that the low pass filters that they used in the input and output stages of their digital recorders were minimum phase (in what freq range? hmmmm) and were associated with appropriate all pass filters to make the phase shift (nearly?) zero for all frequencies. As discussed in a recent article in JAES, a constant phase shift is probably not acceptable for audio reproduction (especially if the shift is 180 degrees). For those who are not familiar with the terminology, minimum phase means that the circuit has a linear phase shift vs frequency curve. David (Reisner) uucp : ...!ucbvax!sdcsvax!logo arpanet : sdcsvax!logo@nosc
vax2:toms (11/13/82)
I feel that I must reply (at length it turns out) to the following statement: "For those who are not familiar with the terminology, minimum phase means that the circuit has a linear phase shift vs amplitude curve." Wait a minute! The term 'minimum phase' does not mean that a system has a linear phase characteristic. In fact, it implies the opposite. Given the desired amplitude response of a filter, there are a number of ways to realize that filter (assuming that at least one zero exists in the transfer function of the filter). For a filter whose transfer function has N zeros, there are 2^N ways to implement the filter. Each of these possible implementations has an identical amplitude response, but differs in the amount of phase shift applied to the signal. The minimum phase realization of the filter is that filter which exhibits the desired amplitude response while producing the least amount of phase shift. There is only one such filter for the desired amplitude response. The phase response of a minimum phase filter is the Hilbert transform of its amplitude response. All minimum phase filters exhibit the characteristic that the zeros of their transfer function lie in the left-half of the s-plane. (All poles must also lie in the left-half of the s-plane for the filter to be stable.) If all the zeros of such a minimum phase filter are mirrored about the imaginary axis, so that they all lie in the right-half of the s-plane, the maximum phase realization of the filter results. This filter, naturally enough, produces the maximum amount of phase shift while realizing the desired amplitude response. Each of these two filters exhibit the desired amplitude response; they differ substantially in the amount of phase shift applied to the signal. All other possible realizations of the filter produce a phase shift somewhere between these two extremes. A linear phase filter is an entirely different beast. The term "linear phase" means that a graph of the phase response, when plotted on a log-log scale, is a straight line. A linear phase filter which also has a constant amplitude response has a perfectly undistorted output. (Since the signal is not changed at all in this case, it is debatable whether this theoretical circuit could properly be called a filter.) A filter with constant amplitude response but nonlinear phase response produces a distorted output signal. This is because it takes different amount of time for signal components of different frequencies to pass through the filter. The resulting distortion is called delay or phase distortion. Note that it is a *linear* phase response that ensures that no phase response is added to the signal; a *constant* phase response does add phase distortion. A filter with a phase response of exactly zero meets both these criteria. However, such a filter would have to propagate the input signal with no delay whatsoever. Since the speed of light is finite, a zero phase response filter is clearly impossible to realize. All linear phase filters have at least one zero in the right-half of the s-plane. Thus linear phase filters cannot be minimum phase filters. Likewise minimum phase filters cannot exhibit linear phase response. These two sets of filters are mutually exclusive. The study of minimum linear phase systems is a fascinating subject which has applications in a number of areas. Linear phase response is important whenever it is desirable to minimize distortion. Audio reproduction is an obvious application. Minimum phase systems are studied in such diverse fields as control theory, communication theory and system identification. There are a number of sources of information available on these subjects. While writing this article I have been referring to "Digital Signal Processing" by Oppenheim and Schafer and "Theory and Applications of Digital Signal Processing" by Rabiner and Gold. Both books have good discussions of linear phase and minimal phase systems and of the relationships between the two. Tom Sager John Fluke Mfg. Co. Seattle, WA