75th or 150th harmonics? A bandlimited squarewave of 8 kHz @ 44.1 kHz samplerate is a sinewave. 3 * 8 kHz is already outside of the bandwidth. This means that the basic frequency must be pretty low to get a square wave shape.
- Uli 2018-06-13 20:40 GMT+02:00 robert bristow-johnson <[email protected]> : > > > ---------------------------- Original Message ---------------------------- > Subject: Re: [music-dsp] Playing a Square Wave > From: "Neil Goldman" <[email protected]> > Date: Wed, June 13, 2018 11:16 am > To: [email protected] > [email protected] > -------------------------------------------------------------------------- > > >> such a simple wave like the square wave, just two signal levels with a > >> near instantaneous jump between them > > > > I think I disagree with this definition of a square wave. This is what a > > perfect, ideal one would look like, but even in reality I don't think any > > system (digital or analog) can exactly produce a perfect square and the > > infinite bandwidth it takes for the infinite number of harmonics. > > > > Even assuming a magically perfect and noiseless analog square wave > > generator, at the very least your speaker cones can't teleport between > two > > positions. And I'm no electrical engineer but the circuitry must also > have > > some kind of damping effect on the extreme ranges? I mean even the > > infinitesimally faint harmonics that will exist in the Mhz range, Ghz > > range, etc up to infinity. > > > > If you remove or dampen any of those higher harmonics, even ones well > > beyond the range of human hearing, this perfect square shape takes on a > bit > > of a ripple shape. At what point is it no longer a "true" square wave? > > > > So I would argue that a perfect square wave doesn't exist anywhere except > > in theory, and it's more useful to define it by its harmonic series. And > at > > that point it doesn't look easier or more complex than any other common > > waveshape. > > > Neil here hits it pretty much spot on. Theo's equipment and Theo's > hearing does not have infinite bandwidth. i sorta doubt that he would be > able to discern the difference between a "perfect" square wave and the same > waveform passed through a nearly perfect brickwall filter with the upper > edge at 20 kHz. they'll sound the same. > > so then the task to generate this square wave digitally is to generate a > bandlimited representation keeping all of the harmonics up to our limit of > hearing. for a square wave at middle C, you need to break it down to > harmonics (say, using Fourier Series), keep all of the harmonics up to the > 75th harmonic (about 20 kHz), ditch the rest, and then digitally reproduce > that waveform. > > there is BLIT (or integrating a BLIT to get a square wave), but those > BLITs are stored in some kinda wavetable, and i don't see why not just > represent the bandlimited waveform itself in a wavetable. > > so using an FFT of a large size. let's say N=64K points. in that 64K > buffer, draw out a single cycle of the analog waveform you seek; square, > saw, triangle, PWM, sync-saw, sync-square, or even a single cycle of a > recorded waveform. > > FFT the bastard. bin 0 will have DC. bin 1 and bin N-1 will have the > amplitude and phase of the 1st harmonic. bins 2 and N-2 have the 2nd > harmonic (the evens should be zero for a square wave), etc. > > then for middle C, go up to the 76th bin and zero all of them up to the > (N-76)th bin. that will bandlimit your waveform. > > then inverse FFT the thing. > > the waveform you have left in 64K points is a bandlimited square wave that > will sound just fine for around middle C. you can reduce the size of that > waveform by decimating (or down-sampling) down to about a 256-point > waveform for storage. but for wavetable synth playback, you really want > that waveform represented with more points than 256. i would recommend > 2048 or 4096 points. > > then you have to do this again for square waves at other pitches. say one > octave lower, then you need to keep 150 harmonics. or an octave higher > than middle C, you need to bandlimit it to 37 harmonics. > > then, in keeping with wavetable synthesis, you keep all of these > bandlimited square waves (say 2048 points per waveform) in memory and you > mix them proportionately as the note goes up and down the keyboard. i > think that two waveforms per octave is generally good enough. but maybe > you want 3 or 4. > > so BLIT, wavetable, or some other mathematical algorithm to generate a > bandlimited square on the fly (i did something like that for Kurzweil, but > i ain't saying what it was, and i don't recommend it anyway). that's how > you do these things. > > > > -- > > > r b-j [email protected] > > "Imagination is more important than knowledge." > > > > > > > > > _______________________________________________ > dupswapdrop: music-dsp mailing list > [email protected] > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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