Re: Anti Aliasing of Arbitrary Waveforms

jim wrote:

Jerry Avins wrote:

...

Scott asks for (a) general method(s) to avoid "aliasing" of
computer-generated signals: "What are the common method(s) for
preventing aliasing of arbitrary waveforms generated within a DSP
application?"

So when did he become an authority on the subject. Just because he asked for
general solution doesn't mean you have to pretend there is one.

Recall that I wrote "An impulse train is sufficient counterexample to demonstrate that there is no overall solution." There is, however, one that can be made to work in mist cases.

His sawtooth generated by incrementing a register with
overflow or other reset is a stalking horse, a cheap convenient test
signal to illustrate the properties of of the (apparently uncommon) method.

But he can't solve the problem of aliasing after the aliasing is done. So there
is no general solution because he hasn't come up with a good way to make a
signal that avoids unwanted low frequencies.

True, but he can avoid significant folding by generating the signal at a high enough rate and filtering before (or in conjunction with) decimating.

...

An impulse train is sufficient counterexample to demonstrate that there
is no overall solution. Nevertheless, most periodic waveforms' spectra
eventually roll off in amplitude as the frequency increases. For all of
those, generating the waveforms at sufficiently high sample rate solves
Scott's problem. Lowpass filtering the result to remove irrelevantly
high harmonics and most of the folded ones leaves a signal that can be
downsampled to the desired range without aliasing. (For efficiency, the
filtering and downsampling can be combined. There is no reason to
compute samples destined to be discarded.)

So you agree the current problem he is working has pretty much nothing to do
with the methods he will later use. I agree if he had chosen a different method
to start with it might work.

His choice of generating frequency won't work, but with proper attention to the numbers, his method can be made to work.

For most signals, including his arbitrarily chosen test signal, ir is a
very good solution.

No it won't work for this method of generating a signal at least not unless he
is very careful about what increments he chooses to use. In general for any
random increment that produces a high frequency (i.e. very large increments) he
will still get unwanted frequencies. And any filtering he does when down
sampling won't remove them because they are not high enough to be removed. The
low frequency noise generated is going to be directly proportional to the size
of his increment.

Not true at all. Assuming that the result is intended for the ear -- Scott seems to imply that -- the DAC is run ridiculously fast. Ridiculous or not, it can work. Let's do the numbers using the sawtooth as an example. Assume a 5 KHz sawtooth and a 20 KHz ultimate upper reproduction frequency. The harmonic amplitudes vary as 1/n, so if the fundamental amplitude has th relative amplitude 1, 10 KHz will be .5; 15 KHz, .333; 20 KHz, .25. These are in the final passband.

To determine the necessary generating rate, we must decide how much folding to allow below 20 KHz. Most applications would be satisfied with 30 dB attenuation, in line with what is acceptable stereo separation. Because more than one harmonic will fold into the passband, let's shoot for 40 dB. In voltage, that's 1%. So we want the hundredth harmonic to fold no lower than 20 KHz, which puts the 104th at DC. To meet this condition, the generating frequency needs to be at least 52*5. or 260 KHz. This needs to be proportionately higher to make higher-frequency sawtooths (sawteeth?). The generated waveforms need to be lowpassed to 20 KHz and decimated to the DAC rate. What don't you understand?

Jerry
--
Engineering is the art of making what you want from things you can get.
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** Posted from http://www.teranews.com **
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