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Path: utzoo!linus!decvax!harpo!seismo!rochester!ee461
From: ee461@rochester.UUCP (VLSI class)
Newsgroups: net.audio
Subject: Re: Superstition revisited - let's play some math.
Message-ID: <2484@rochester.UUCP>
Date: Thu, 4-Aug-83 13:47:16 EDT
Article-I.D.: rocheste.2484
Posted: Thu Aug  4 13:47:16 1983
Date-Received: Thu, 4-Aug-83 17:10:38 EDT
References: <1750@rabbit.UUCP>
Organization: U. of Rochester, CS Dept.
Lines: 58

>From rabbit!jj:  "Furthermore, when the gentleman's tweeter was out of phase,
the speaker with the improperly phased tweeter (...) had a large zero in its
frequency response due to the improperly phased speaker."

Huh ?  Say what ?? Large zero ??? No way, the crossover will mess the phase so
that a total cancellation will not occur (if the initial design was good) OK,
here I come with some equations: Assume a 2-way speaker + 1st order crossover
+ some idealization (flat response and no phase shift introduced by drivers
itself). The transfer function for the woofer is: TW(f)=1/(1+jf/f0), for the
tweeter: TT(f)=(jf/f0)/(1+jf/f0), where f0 is the crossover frequency. Your ear
gets a sum of these signals, so when everything is correct, the total transfer
function is TW(f)+TT(f)= 1. IT IS FLAT, both phase and amplitude, which explains
why a 1st order crossover is nice to have in your speakers. Now switch the 
polarity of the tweeter; you get TW(f)-TT(f)= (1-jf/f0)/(1+jf/f0). THE 
AMPLITUDE RESPONSE is 1! No big zeros! Instead, there is a nice, smooth, 
continuos phase shift totaling 180 degrees over some 2 decades. Now  back to
CD players (the one with 180 degree phase shift): they utilize some quite steep
ultrasonic filters; 4-th order seems to be a reasonable guess. Associated slope
of the phase response is 180deg/decade, therefore it would appear that the 
particular CD player could have the sloping part of its phase response 
contained roughly between 2 and 20 kHz. Nice, smooth and continuous shift. 

THEREFORE: the effect encountered in the CD player with phase response screwed
up is roughly of the same nature as the effect obtained by switching the 
polarity of the tweeter in a speaker with the 1st order crossover. The 
difference is in the span of frequencies where the change occurs.
							QED

So there. Now a rough estimate of how much nouniformity can be expected
from a regular speaker, with a 4th order crossover. Idealization as before.
Assume transfer functions with 4th degree poles, e.g.: TW(f)=1/(1+jf/f0)**4,
TT(f)=(jf/f1)**4/(1+jf/f1)**4. Note, that f0 is NOT equal to f1, as otherwise
there would be a nice -9 dB dip in the amplitude response at f0=f1. Rather, f0 
and f1 are chosen in such a way, that -3dB points of each transfer function
coincide, which means that roughly 0.64*f0 = 1.71*f1. I.e., there is
about 1.5 octave difference between f0 and f1. The resulting misalignment
between phase characteristics causes a 90 deg peak, followed by a 90 deg dip
in the resulting phase response (all numbers approximate). These changes occur
on a span of some 7 octaves, with the distance between the +90 and -90 points
some 4 octaves, i.e. slightly more than a decade. This would be an answer to
the question about relative comparison between phase responses of speakers and 
CD's that originated this discussion.

As for the perceptible effects of the phase shift, I start to suspect that
this is a largely subjective issue. For example, Keith Ericson wrote that 
after experimenting with the polarity of his speakers he could not
definitely decide which sounded better. I think that phase coherent speakers
sound much more realistically. He apparently quotes somebody's opinion
that human ear is more sensitive to time delay than to the phase shift.
I've read opinions (in Abso!ute Sound) that were quite opposite and guys
argued there that it is phase that makes all the difference. It appears that 
everybody should pick his own conclusions accordingly to his preferences. I'm 
just wondering how much the general preferences (and reviewers of HiFi 
equipment) were affected by getting accustomed to the sound of the abundance 
of mediocre speakers all around.

				Krzysztof Kozminski
				(ee461@rochester)