5.13 - Yeah, well what about square waves? I've seen square wavetests of digital systems that show a lot of ringing. Isn't that bad?
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This article is from the Audio Professional FAQ, by with numerous contributions by Gabe M. Wiener others.
5.13 - Yeah, well what about square waves? I've seen square wavetests of digital systems that show a lot of ringing. Isn't that bad?
Square waves are a mathematically precisely defined signal. One of the
ways to describe a perfect square wave is as the sum an infinite series
of sine waves in a precise phase, harmonic and amplitude relationship.
The relation is:
1 1 1 1
F(t) = sin(wt) + -sin(3wt) + -sin(5wt) + -sin(7wt) + -sin(9wt) ...
3 5 7 9
where t is time, w is "radian frequency", or 2 pi times frequency.
Remember, we require an infinite number of terms to describe a perfect
square wave. If we limit the number of terms to, say, 10 terms, (such as
the case with a 1 kHz square wave perfectly band limited to 20 kHz),
there simply aren't enough terms to describe a perfect square wave.
What will result is a square wave with the highest harmonic imposed on
top as "ringing." In fact, this appearance indicates that the phase
and frequency response is perfect out to 20 kHz, and the bandwidth
limiting is limiting the number of terms in the series.
Well, what would a perfect analog system do with square waves? As it
turns out, if you take a high quality 15 IPS tape recorder, bias and
adjust it for the flattest possible frequency response over the widest
possible bandwidth, the result looks remarkably like that of a good
digital system for exactly the same reasons.
On the other hand, adjust the analog tape recorder for a square wave
response that has no ringing, but the fastest possible rise time. Now
listen to it: it sounds remarkably dull and muffled compared to the
input. Why? Because in order to achieve that square wave response, it's
necessary to severely roll off the high end response in order to
suppress the high-frequency components needed to achieve fastest rise
time. [Dick]
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