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Additive White Gaussian Noise Question

Started by Necronomicon October 4, 2007
My understanding is that when you add AWGN to a carrier,
you make the amplitude a gaussian distribution, and then
add white noise to this (not really pure white noise, but as
high a BW as your simulation can handle).

However, if you amplitude modulate a carrier by any
AM % modulation, then your Fourier transform will
show side bands which will not be spectrally flat like
the added white noise.  (this indicates how we get AM-to-PM
effects, and also makes AM seem a bit of a misnomer,
because in actuality you still alter the frequency spectrum.
Whereas you can strip the AM off an FM signal, and still
not lose information).

So these added side bands from the Gaussian distribution
of the amplitude is what makes AWGN different from just pure
white noise on a carrier, right?

Thanks for any feedback.....

On Thu, 04 Oct 2007 16:33:55 -0700, Necronomicon wrote:

> My understanding is that when you add AWGN to a carrier, > you make the amplitude a gaussian distribution, and then > add white noise to this (not really pure white noise, but as > high a BW as your simulation can handle).
If you are doing simulation, and not analysis or real-world measurements, yes.
> > However, if you amplitude modulate a carrier by any AM % modulation, > then your Fourier transform will show side bands which will not be > spectrally flat like the added white noise.
That depends on how you're amplitude modulating the carrier -- "amplitude modulate" it with white noise & you'll get spectrally flat "side bands" (actually you'll spread the carrier energy all over every place). I use the quote marks because the amplitude of white noise is infinite, while regular old AM (A1x modulation) demands that the modulating signal be bounded. But if you generate a signal as (1 + x(t)) * sin(w * t), and x(t) is white noise, then you'll splatter your signal evenly from DC to light.
> (this indicates how we get > AM-to-PM effects, and also makes AM seem a bit of a misnomer, because in > actuality you still alter the frequency spectrum. Whereas you can strip > the AM off an FM signal, and still not lose information).
_Any_ modulation will alter the frequency spectrum of a carrier, so I don't see how AM is a misnomer.
> > So these added side bands from the Gaussian distribution of the > amplitude is what makes AWGN different from just pure white noise on a > carrier, right?
Huh? I'm not even sure what you're asking here. Carrier + noise gives you a spectral density with an impulse at your carrier frequency plus a smooth noise floor. Carrier * (noise + 1) gives you a spectral density with an impulse at your carrier frequency plus a smooth noise floor. In the frequency domain there's no difference, although there's certainly a difference in the time domain. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Necronomicon wrote:
> My understanding is that when you add AWGN to a carrier, > you make the amplitude a gaussian distribution, and then > add white noise to this (not really pure white noise, but as > high a BW as your simulation can handle). > > However, if you amplitude modulate a carrier by any > AM % modulation, then your Fourier transform will > show side bands which will not be spectrally flat like > the added white noise. (this indicates how we get AM-to-PM > effects, and also makes AM seem a bit of a misnomer, > because in actuality you still alter the frequency spectrum. > Whereas you can strip the AM off an FM signal, and still > not lose information). > > So these added side bands from the Gaussian distribution > of the amplitude is what makes AWGN different from just pure > white noise on a carrier, right?
I don't think so. AWGN stands for additive white Gaussian noise. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
On Oct 5, 12:33 pm, Necronomicon <radio...@aol.com> wrote:
> My understanding is that when you add AWGN to a carrier, > you make the amplitude a gaussian distribution, and then > add white noise to this (not really pure white noise, but as > high a BW as your simulation can handle). > > However, if you amplitude modulate a carrier by any > AM % modulation, then your Fourier transform will > show side bands which will not be spectrally flat like > the added white noise. (this indicates how we get AM-to-PM > effects, and also makes AM seem a bit of a misnomer, > because in actuality you still alter the frequency spectrum. > Whereas you can strip the AM off an FM signal, and still > not lose information). > > So these added side bands from the Gaussian distribution > of the amplitude is what makes AWGN different from just pure > white noise on a carrier, right? > > Thanks for any feedback.....
If you add white noise to FM and then remove the amplitude variations then there will be phase-noise introduced.
On Oct 4, 7:40 pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On Thu, 04 Oct 2007 16:33:55 -0700, Necronomicon wrote: > > My understanding is that when you add AWGN to a carrier, > > you make the amplitude a gaussian distribution, and then > > add white noise to this (not really pure white noise, but as > > high a BW as your simulation can handle). > > If you are doing simulation, and not analysis or real-world measurements, > yes. > > > > > However, if you amplitude modulate a carrier by any AM % modulation, > > then your Fourier transform will show side bands which will not be > > spectrally flat like the added white noise. > > That depends on how you're amplitude modulating the carrier -- "amplitude > modulate" it with white noise & you'll get spectrally flat "side bands" > (actually you'll spread the carrier energy all over every place). >
My point is that if you add the gaussian noise to the amplitude, the side bands will not be spectrally flat.
> I use the quote marks because the amplitude of white noise is infinite, > while regular old AM (A1x modulation) demands that the modulating signal > be bounded. But if you generate a signal as (1 + x(t)) * sin(w * t), and > x(t) is white noise, then you'll splatter your signal evenly from DC to > light. > > > (this indicates how we get > > AM-to-PM effects, and also makes AM seem a bit of a misnomer, because in > > actuality you still alter the frequency spectrum. Whereas you can strip > > the AM off an FM signal, and still not lose information). > > _Any_ modulation will alter the frequency spectrum of a carrier, so I > don't see how AM is a misnomer. >
My point is that pure FM does not alter the amplitude while pure AM still alters the frequency spectrum.
> > > > So these added side bands from the Gaussian distribution of the > > amplitude is what makes AWGN different from just pure white noise on a > > carrier, right? > > Huh? I'm not even sure what you're asking here. >
How would you differentiate AWGN from simple white noise + carrier?
HardySpicer wrote:
> On Oct 5, 12:33 pm, Necronomicon <radio...@aol.com> wrote: >> My understanding is that when you add AWGN to a carrier, >> you make the amplitude a gaussian distribution, and then >> add white noise to this (not really pure white noise, but as >> high a BW as your simulation can handle). >> >> However, if you amplitude modulate a carrier by any >> AM % modulation, then your Fourier transform will >> show side bands which will not be spectrally flat like >> the added white noise. (this indicates how we get AM-to-PM >> effects, and also makes AM seem a bit of a misnomer, >> because in actuality you still alter the frequency spectrum. >> Whereas you can strip the AM off an FM signal, and still >> not lose information). >> >> So these added side bands from the Gaussian distribution >> of the amplitude is what makes AWGN different from just pure >> white noise on a carrier, right? >> >> Thanks for any feedback..... > > If you add white noise to FM and then remove the amplitude variations > then there will be phase-noise introduced.
True. Also true for any other kind of noise or signal. So? Jerry -- Engineering is the art of making what you want from things you can get. &macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
On Oct 4, 7:33 pm, Necronomicon <radio...@aol.com> wrote:
> My understanding is that when you add AWGN to a carrier, > you make the amplitude a gaussian distribution,
You make the amplitude of the NOISE is a Gaussian distribution, not the carrier. You ADD this Gaussian white noise to the desired carrier signal, you do not change the desired carrier signal in any other way. Mark
Necronomicon wrote:

   ...

>> That depends on how you're amplitude modulating the carrier -- "amplitude >> modulate" it with white noise & you'll get spectrally flat "side bands" >> (actually you'll spread the carrier energy all over every place). >> > > My point is that if you add the gaussian noise to the > amplitude, the side bands will not be spectrally flat.
Modulating and adding are very different. Think about what happens when addding 60 Hz to an AM radio signal. ...
>> _Any_ modulation will alter the frequency spectrum of a carrier, so I >> don't see how AM is a misnomer. >> > > My point is that pure FM does not alter the amplitude > while pure AM still alters the frequency spectrum.
Regardless of the modulation scheme, sending information requires bandwidth. When information is modulated onto an AM carrier, the result occupies a band of frequencies around the carrier. If a very narrow filter is used to leave only the carrier, it will no longer have amplitude variations. ...
> How would you differentiate AWGN from > simple white noise + carrier?
You evidently recognize that noise can be white only within some interesting frequency band and not from DC to gamma rays and beyond. As for your question, the difference is that your AWGN has no carrier. AWGN is a particular kind of noise, and the concept assumes something about its introduction to the system. It is *white* and *Gaussian*, and of course it is *noise*. But before all, it is *additive*, which is to say, that the system behaves linearly at the point where the addition takes place. Does that clear anything up? Jerry -- Engineering is the art of making what you want from things you can get. &macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
Mark <makolber@yahoo.com> writes:

> On Oct 4, 7:33 pm, Necronomicon <radio...@aol.com> wrote: >> My understanding is that when you add AWGN to a carrier, >> you make the amplitude a gaussian distribution, > > You make the amplitude of the NOISE is a Gaussian distribution, not > the carrier. > > You ADD this Gaussian white noise to the desired carrier signal, you > do not change the desired carrier signal in any other way.
I'm getting confused by the OP's haziness myself! Sheesh! You're right, Mark - what is Gaussian is the noise. That is, the distribution of the noise amplitude - the values of the individual samples of noise - is Gaussian. (Gauss - neat name, eh?) And if the noise is "white," it's power spectrum is flat. If you add a sine wave (i.e., a "carrier") to Gaussian noise, you get a sine wave in noise. (Duh.) It's power spectrum is not flat but has a spike at the carrier frequency. However, and I think this is where the OP may be confused, if the input to an AM modulator is bandlimited "white" noise, then since the input spectrum is flat, the sidebands will also be flat. There. Did that help? -- % Randy Yates % "The dreamer, the unwoken fool - %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Eldorado Overture', *Eldorado*, ELO http://www.digitalsignallabs.com
Randy Yates wrote:
> Mark <makolber@yahoo.com> writes: > >> On Oct 4, 7:33 pm, Necronomicon <radio...@aol.com> wrote: >>> My understanding is that when you add AWGN to a carrier, >>> you make the amplitude a gaussian distribution, >> You make the amplitude of the NOISE is a Gaussian distribution, not >> the carrier. >> >> You ADD this Gaussian white noise to the desired carrier signal, you >> do not change the desired carrier signal in any other way. > > I'm getting confused by the OP's haziness myself! Sheesh! > > You're right, Mark - what is Gaussian is the noise. That is, the > distribution of the noise amplitude - the values of the individual > samples of noise - is Gaussian. (Gauss - neat name, eh?) And if > the noise is "white," it's power spectrum is flat. > > If you add a sine wave (i.e., a "carrier") to Gaussian noise, you > get a sine wave in noise. (Duh.) It's power spectrum is not flat > but has a spike at the carrier frequency. > > However, and I think this is where the OP may be confused, if the input to > an AM modulator is bandlimited "white" noise, then since the input > spectrum is flat, the sidebands will also be flat. > > There. Did that help?
Probably, but with this OP, you need to emphasize that adding is not modulating. Necronomicron (a tiny bit dead?): Noise is confusing. Let's stay with sine waves for now. Suppose we have a signal at 1 MHz and add to it some 60 Hz. Displayed on a scope, the MHz signal will have fuzzy tops and bottom, as if it were modulated. It isn't modulated. Speed up the sweep so individual MHz cycles show, and you will see that the tops and bottoms move in the same direction. If the MHz signal were modulated, the actual amplitude would grow and shrink, that is, when the tops get taller, the bottoms go deeper. Added noise will have an effect only if it ends up in the receiver's passband, and it will modulate the carrier only if it is introduced into the transmitter's modulator (or if there are nonlinearities in the receiver). Jerry -- Engineering is the art of making what you want from things you can get. &macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;