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How to prevent aliasing caused by non-linear function implemented in the digital domain

Started by Mark April 27, 2006
"Jerry Avins" <jya@ieee.org> wrote in message 
news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net...

>> In some respects the specification contradicts itself. >> To limit means to add harmonics and, thus, aliases in the sampled signal. >> To lowpass filter means to remove some of those added frequencies and, >> thus, to *not* really limit. > > Things are muddled. Limiting produces harmonics of the limited signal, but > those harmonics needn't be aliases if the sample rate is high enough. > Consider what hams do (did?) to get more "punch" in their transmissions. > Crank the audio gain up to where overmodulation would be frequent, then > clip the audio to preclude overmodulation. The resulting waveform is rich > in harmonics that cause out-of-band sidebands ("splatter"), which are > removed by a low-pass filter. The final signal (which can still have a > mystery flaw: do you see it?) is obviously clipped -- a scope shows > that -- but bandlimited.
Well, might it be that the mystery flaw is that lower frequency fundamentals still have their harmonics in-band and will sound "not so good"? Fred
Fred Marshall wrote:

   ...

> Well, might it be that the mystery flaw is that lower frequency fundamentals > still have their harmonics in-band and will sound "not so good"?
Actually, being harmonic, they don't sound very bad. If the clipping level is set right at the overmodulation level. the low-pass filter output can exceed it. Just think of a square wave out of the clipper. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
"Jerry Avins" <jya@ieee.org> wrote in message 
news:pPGdnWqXycnoTM_ZnZ2dneKdnZydnZ2d@rcn.net...
> Fred Marshall wrote: > > ... > >> Well, might it be that the mystery flaw is that lower frequency >> fundamentals still have their harmonics in-band and will sound "not so >> good"? > > Actually, being harmonic, they don't sound very bad. If the clipping level > is set right at the overmodulation level. the low-pass filter output can > exceed it. Just think of a square wave out of the clipper.
Oh, yeah. Gibbs and all .....
Fred Marshall wrote:
> "Jerry Avins" <jya@ieee.org> wrote in message > news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net... > > > Aren't you impressed by the elegant simplicity of R.B-J.'s observation > > that the highest order harmonic generated by a soft limiter is the same as > > the order of the polynomial that represents it?
...
> > Yes, very.
oh c'mon, guys. it's just a degeneration of knowing when you multiply signals, you convolve their spectrums. the general case doesn't even have to be a sinusoid. bandlimited signals have spectrums of finite nonzero reach. squaring a signal doubles that reach. cubing it is squaring it and multiplying by another one more time so it triples the reach. we've had this before. another way to look at it is to consider Tchebyshev polynomials (and an arbitrary polynomial can be expressed as a sum of Tchebyshevs up to that order). so when a cosine wave guzzinta an Nth order Tchebshev: T_N( cos(w*t) ) = cos(N*arccos( cos(w*t) )) = cos(N*w*t) that's what i think is cool. a nice frequency multiplying non-linearity. the trick is to limit the nature of the non-linearity to these finite polynomials so you know how much to oversample. if it's has to be a high order, you're just screwed. another little trick is that you need not care about aliasing that folds down to the area that you'll LPF out. so a 5th order polynomial needs only to have an oversampling ratio of 3. those top 2 harmonics might alias, but won't get back into the baseband. when downsampling, you filter those aliased harmonics out. so i think the hard and fast rule is oversampling ratio = (polynomial order + 1)/2 what impresses me (because i haven't really figgered it out) is that ... 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 ... blows up, if that does, and if ... 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 ... reconstructs to a nice cosine at Nyquist (it does, doesn't it?), then that means ... 0 0 0 0 0 0 0 0 0 0 0 2 -2 2 -2 2 -2 2 -2 2 -2 2 ... is the component that blows up. but i would think it would just settle to a nice steady state 2*cosine. it must be the transient that is hell with all of those 1/n terms adding up to infinity (since the alternating signs of the data cancels the alternating signs of the wiggles in the sinc). r b-j
"robert bristow-johnson" <rbj@audioimagination.com> wrote in message 
news:1146280966.313879.11310@j33g2000cwa.googlegroups.com...
> Fred Marshall wrote: >> "Jerry Avins" <jya@ieee.org> wrote in message >> news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net... >> >> > Aren't you impressed by the elegant simplicity of R.B-J.'s observation >> > that the highest order harmonic generated by a soft limiter is the same >> > as >> > the order of the polynomial that represents it? > ... >> >> Yes, very. > > oh c'mon, guys.
hey, r b-j, I was serious. And, now that you've expounded on it, I really like the treatment you've given it. Ever' now and then I bump into something new (for me) that should have been obvious, etc. etc.... Well, it's obvious now. Thanks! ... Yeah, it's the signs of the wiggles in the sincs. So B strictly <fs/2 would seem to be the lesson one can teach from this simple example. Fred
robert bristow-johnson wrote:
> Fred Marshall wrote: > >>"Jerry Avins" <jya@ieee.org> wrote in message >>news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net... >> >> >>>Aren't you impressed by the elegant simplicity of R.B-J.'s observation >>>that the highest order harmonic generated by a soft limiter is the same as >>>the order of the polynomial that represents it? > > ... > >>Yes, very. > > > oh c'mon, guys. it's just ...
Hey, man: I meant it. It's one of those things I could have figured out but didn't. It's obvious .. once it's said out loud. I'm _always_ impressed by "Gee, I should have thought of that!" insights. Give yourself two pats on the back. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;