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

Fred Marshall wrote:
"Jerry Avins" <jya@xxxxxxxx> wrote in message news:84-dnQsCK-uTGszZnZ2dnUVZ_uWdnZ2d@xxxxxxxxxx

Fred Marshall wrote:

...


I think this is an interesting question and I don't have an answer.

I think I do, at least if I understand what you're driving at. Every possible set of samples represents some bandlimited signal. To determine what that is, just feed them to the reconstruction process and examine the result. So a set of samples that originally represented a sinusoid but is modified by zeroing out all negative samples represents *some* bandlimited signal, but probably not the same signal that results from half-wave rectifying followed by band limiting.

Jerry


Jerry and Mark,

OK - I think we're (or I'm) getting somewhere.

Jerry. We had this discussion some time back. Not every possible set of samples represents some bandlimited signal. I had thought so too but was stopped cold with the sequence:
......1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1.......
****
Note the flip in the sign sequence in the middle at ****.
When this is reconstructed with sincs, it blows up. It demonstrates that not every sequence represents a bandlimited signal. It's the counter-example pathological case.

I do remember the discussion; it embarrasses me to have forgotten. I hereby make another unfounded claim which has a better chance to be true. Any set of actual samples, even if corrupted by aliasing, represents a bandlimited signal (that may not closely resemble the sampled signal).

But, we are practical guys and I'm going to "remember it while forgetting about it" .. that is, "remember it while ignoring it - for practical situations".

----end of digression-----

I emphasized and Jerry responded to the bandlimitedness question. But it seems I had missed the point because Mark is focused on aliasing and wants to reduce the resulting aliasing in selecting a limiting process. Gee, I didn't know that hard limiting *samples* would necessarily cause aliasing! So, I'm starting from scratch.
Clearly if one takes a sinusoid and samples it at a very high rate then there will be harmonics.
The question I had was "will be aliases from limiting as well"?
A similar question would be "can one define a spectral character at all in this situation"?

OK - so here's a way to look at it:

Hard limiting of samples cause aliasing because the negative frequency components contribute harmonics in the negative direction (moving to lower frequencies from fs). This puts aliased components below fs/2.
Simple but clear I hope. At least I learned something....

So, how to mitigate this without lowpass filtering the results of limiting?

Here's a flip answer:
The "nonlinearity" can include linear components, thus a lowpass filter....
But that isn't what Mark asks for.

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.

Try this:
Apply a sinusoid to a hard limiter such as a half-wave rectifier.
Compute the Fourier Series of the resulting waveform.
Eliminate all the higher order terms of the series.
Construct the resulting waveform.
[In general, there will be the Gibbs phenomenon due to truncation of the series.]

I don't like the stark example because most limiters for practical purposes are symmetrical.

This observation may lend some insight because you want to have the Gibbs phenomenon in order to meet your specification.

Now, ask the question: what process will cause this new waveform to be generated?
We know that a half-wave rectifier followed by a lowpass filter is one answer.
Are there others?
I think probably not because it would imply there is a multiplicity of processes that will result in the same waveform. But there aren't a multiplicity of lowpass filters of the exact same response - unless you get into implementation details and they aren't the point here.
I suppose it might be argued that there may be a multiplicity of nonlinearities combined with filters that might have identical outputs ... and that leads to making a soft limiter that requires less filtering I suppose.

Then there is the complication of input waveforms that aren't sinusoids...... I think this is the killer if the waveforms are complicated composites. Too much amplitude dependence, etc.

I would imagine that power supply designers have pondered this kind of question in an attempt to make things less expensive.

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? Most practical limiter polynomials will be symmetric, hence consist of only odd powers.

Jerry
--
Engineering is the art of making what you want from things you can get.
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