From: Les Cargill on
dvsarwate wrote:
> On Mar 8, 4:19 pm, Les Cargill <lcargil...(a)comcast.net> wrote:
>> Jerry Avins wrote:
>>> Les Cargill wrote:
>>>> robert bristow-johnson wrote:
>>> ...
>>>>> are you trying to introduce a specified amount of specific harmonics
>>>>> to a pure sine wave?
>>>> No, for any random PCM stream.
>>> As I see it, the answer is a non-sequitur.
>>> ...
>>> Jerry
>> How about "for any *arbitrary* PCM stream?"
>>
>> --
>> Les Cargill
>
> Perhaps the OP should be more clear as to what exactly he
> wants. For a pure sinusoid, say s(t) = 3cos(wt), whether as an
> analog signal or as a PCM stream, we can add in second
> harmonic distortion by adding (say) 0.1s^2(t) to s(t) and
> subtracting off the DC term thus introduced. But using
> a cubic adds in a third harmonic and also changes the
> amplitude of the fundamental. Is this OK? Or does
> harmonic distortion in this instance mean 0.1cos(3wt)
> gets added to s(t) with no change in the fundamental?
>

My understanding is that there's no change to the fundamental.
You just calculate the interesting harmonics ( second,
third, nth) and mix them in. But the original signal may
be relatively harmonically complex to begin with.

And by "harmonics", those will be vector Fourier
quantities, not single sine tones.

> Turning to more arbitrary signals, if s(t) = 3 sin(w1t) + 4 sin(w2t),
> then adding in (say) 0.1s^2(t) to s(t) adds in second harmonics
> of w1 and w2 but also intermodulation products at frequencies
> w1+w2 and w1-w2. Is this OK? Or are looking to change
> s(t) = 3 sin(w1t) + 4 sin(w2t) by adding in (say)
> 0.1sin(2w1t) + 0.2sin(2w2t)? Pure harmonic distortion and
> *no* intermodulation distortion?

I would expect the difference tone signals to be part of this -
just as if you opened the appropriate stop on an organ. But
realize - I am not 100% sure what is possible - if you can
quantify the difference tones, I'd think you could subtract
those out as well.

If it's really just a matter of banging the DFT at multiples
of t, then that's what I'm looking for.

> And what exactly does the
> OP mean by harmonic distortion of a random or arbitrary
> data stream which is not necessarily periodic at all.
> Absent a fundamental, what is *harmonic* distortion?


I suppose harmonics of any element of the Fourier series
of the signal. Obviously, if it's white noise, the result
will be white noise.

>
> --Dilip Sarwate

--
Les Cargill
From: Les Cargill on
Jerry Avins wrote:
> Les Cargill wrote:
>> Jerry Avins wrote:
>>> Les Cargill wrote:
>>>> robert bristow-johnson wrote:
>>>
>>> ...
>>>
>>>>> are you trying to introduce a specified amount of specific harmonics
>>>>> to a pure sine wave?
>>>>
>>>> No, for any random PCM stream.
>>>
>>> As I see it, the answer is a non-sequitur.
>>>
>>> ...
>>>
>>> Jerry
>>
>>
>> How about "for any *arbitrary* PCM stream?"
>
> That doesn't fix it. I wasn't picking nits. The question was, given any
> arbitrary waveform, what harmonics do you want to add to it.

The usual suspects - second, third, fifth.

> An answer
> pins down a process that applies to any waveform, whether it be random
> or have a familiar name.
>

Ah! Thanks!

> Jerry

--
Les Cargill
From: Mark on
On Mar 8, 8:20 pm, Les Cargill <lcargil...(a)comcast.net> wrote:
> Jerry Avins wrote:
> > Les Cargill wrote:
> >> Jerry Avins wrote:
> >>> Les Cargill wrote:
> >>>> robert bristow-johnson wrote:
>
> >>>   ...
>
> >>>>> are you trying to introduce a specified amount of specific harmonics
> >>>>> to a pure sine wave?
>
> >>>> No, for any random PCM stream.
>
> >>> As I see it, the answer is a non-sequitur.
>
> >>>   ...
>
> >>> Jerry
>
> >> How about "for any *arbitrary* PCM stream?"
>
> > That doesn't fix it. I wasn't picking nits. The question was, given any
> > arbitrary waveform, what harmonics do you want to add to it.
>
> The usual suspects - second, third, fifth.
>
> > An answer
> > pins down a process that applies to any waveform, whether it be random
> > or have a familiar name.
>
> Ah! Thanks!
>
> > Jerry
>
> --
> Les Cargill- Hide quoted text -
>
> - Show quoted text -

I may be missing something in the discussion but I think the simple
answer to your question is:

the second harmonic component is created by the A*x^2 term and the the
third harmonic component is created by the B*x^3 term etc of the
transfer function. The (complex) magnitude of the coefficents A and
B etc determine the magnitude (and phase) of the varous harmonics.

Is that what you were asking?

Mark

From: Les Cargill on
Mark wrote:
> On Mar 8, 8:20 pm, Les Cargill <lcargil...(a)comcast.net> wrote:
>> Jerry Avins wrote:
>>> Les Cargill wrote:
>>>> Jerry Avins wrote:
>>>>> Les Cargill wrote:
>>>>>> robert bristow-johnson wrote:
>>>>> ...
>>>>>>> are you trying to introduce a specified amount of specific harmonics
>>>>>>> to a pure sine wave?
>>>>>> No, for any random PCM stream.
>>>>> As I see it, the answer is a non-sequitur.
>>>>> ...
>>>>> Jerry
>>>> How about "for any *arbitrary* PCM stream?"
>>> That doesn't fix it. I wasn't picking nits. The question was, given any
>>> arbitrary waveform, what harmonics do you want to add to it.
>> The usual suspects - second, third, fifth.
>>
>>> An answer
>>> pins down a process that applies to any waveform, whether it be random
>>> or have a familiar name.
>> Ah! Thanks!
>>
>>> Jerry
>> --
>> Les Cargill- Hide quoted text -
>>
>> - Show quoted text -
>
> I may be missing something in the discussion but I think the simple
> answer to your question is:
>
> the second harmonic component is created by the A*x^2 term and the the
> third harmonic component is created by the B*x^3 term etc of the
> transfer function. The (complex) magnitude of the coefficents A and
> B etc determine the magnitude (and phase) of the varous harmonics.
>
> Is that what you were asking?
>
> Mark
>

I believe it is.

--
Les Cargill
From: robert bristow-johnson on
On Mar 9, 11:30 pm, Jerry Avins <j...(a)ieee.org> wrote:
> Les Cargill wrote:
> > Mark wrote:
> >> On Mar 8, 8:20 pm, Les Cargill <lcargil...(a)comcast.net> wrote:
> >>> Jerry Avins wrote:
> >>>> Les Cargill wrote:
> >>>>> Jerry Avins wrote:
> >>>>>> Les Cargill wrote:
> >>>>>>> robert bristow-johnson wrote:
> >>>>>>   ...
> >>>>>>>> are you trying to introduce a specified amount of specific
> >>>>>>>> harmonics
> >>>>>>>> to a pure sine wave?
> >>>>>>> No, for any random PCM stream.
> >>>>>> As I see it, the answer is a non-sequitur.
> >>>>>>   ...
> >>>>>> Jerry
> >>>>> How about "for any *arbitrary* PCM stream?"
> >>>> That doesn't fix it. I wasn't picking nits. The question was, given any
> >>>> arbitrary waveform, what harmonics do you want to add to it.
> >>> The usual suspects - second, third, fifth.
>
> >>>> An answer
> >>>> pins down a process that applies to any waveform, whether it be random
> >>>> or have a familiar name.
> >>> Ah! Thanks!
>
> >>>> Jerry
> >>> --
> >>> Les Cargill- Hide quoted text -
>
> >>> - Show quoted text -
>
> >> I may be missing something in the discussion but I think the simple
> >> answer to your question is:
>
> >> the second harmonic component is created by the A*x^2 term and the the
> >> third harmonic component is created by the B*x^3 term etc of the
> >> transfer function.   The (complex) magnitude of the coefficents A and
> >> B etc determine the magnitude (and phase) of the varous harmonics.
>
> >> Is that what you were asking?
>
> >> Mark
>
> > I believe it is.
>
> But be aware that the x^3 term can affect the amplitude of the
> fundamental/ I need to think more about the phase.

i don't think that memoryless mapping functions can do much about
phase except maybe flip the polarity.

now, for Les and Mark. yes, in general, an Nth-order polynomial will
create, out of a single sinusoid, new frequencies that are harmonic up
to the Nth harmonic. But the generation of harmonics is level
dependent (as it is for any non-linear distortion).

now, if you *know* or can assume the level of your sinusoid, you can
create an Nth-order polynomial that will create a specified level of
each harmonic up to the Nth harmonic. you define your Nth-order
polynomial as a sum of Tchebyshev polynomials up to the Nth-order
Tchebyshev. if a normalized sinusoid goes into a 5th-order
Tchebyshev, what comes out is only the 5th harmonic and no others.
now, does that help you, Les, regarding the "generate harmonic
distortion analytically"?

otherwise, i think you need to be more specific about the nature of
your input signal.

r b-j