Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10 5/3/83; site rochester.UUCP 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)