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White noise and the impulse function?

Started by Big Duck January 23, 2007
I am going to learn DSP!

I have been studying it for a few weeks (I have many years of RF under my 
belt).

I note that the impule function in a given window has a frequency response 
that is equal at every frequency in the DFT window.

I also know that white noise has a frequency response that has roughly equal 
magnitude across the sample frequencies in an FFT.

I am trying to make some sense of this.  Here are some thoughts on this. 
Please shoot down or verify:

1.  If you took an FFT of an impulse response but randomized the phase of 
each frequency in the FFT then you would get a time function that looked 
like white noise ?

2. The ear does not disriminate phase (so i have been told):
If you played white noise , or if you played a properly windowed (digital) 
impulse function at the right sample frequency  , then it would sound 
(roughly)the same to your ear.

3. It might be even better if you played an 1/128 impulse at a sample freq 
of 10 KHz and every period you jittered the location of the impulse signal?


Sorry if my DSP lingo is not quite right, but tell me if this equivelance of 
white noise/impulse function is true?



Big Duck wrote:
> I am going to learn DSP! > > I have been studying it for a few weeks (I have many years of RF under my > belt). > > I note that the impule function in a given window has a frequency response > that is equal at every frequency in the DFT window.
Equal in magnitude. Distinctive in phase.
> I also know that white noise has a frequency response that has roughly equal > magnitude across the sample frequencies in an FFT.
Equal in magnitude. Random in phase.
> I am trying to make some sense of this. Here are some thoughts on this. > Please shoot down or verify: > > 1. If you took an FFT of an impulse response but randomized the phase of > each frequency in the FFT then you would get a time function that looked > like white noise ?
Look and be like.
> 2. The ear does not disriminate phase (so i have been told): > If you played white noise , or if you played a properly windowed (digital) > impulse function at the right sample frequency , then it would sound > (roughly)the same to your ear.
They would sound very different. You have found a circumstance where phase matters. The impulse is localized in time: the noise is not.
> 3. It might be even better if you played an 1/128 impulse at a sample freq > of 10 KHz and every period you jittered the location of the impulse signal?
What is a 1/128 impulse?. The jitter would produce sidebands. Random noise would produce noisy sidebands.
> Sorry if my DSP lingo is not quite right, but tell me if this equivelance of > white noise/impulse function is true?
The amplitude spectra are the same. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Jan 23, 8:45 pm, Jerry Avins <j...@ieee.org> wrote:
> Big Duck wrote: > > > I note that the impule function in a given window has a frequency response > > that is equal at every frequency in the DFT window.
> Equal in magnitude. Distinctive in phase.
> > I also know that white noise has a frequency response that has roughly equal > > magnitude across the sample frequencies in an FFT.Equal in magnitude.
> Random in phase.
...
> The amplitude spectra are the same.
one other difference to think about is, while both white noise and the dirac impulse are nasty "functions" (having infinite power or infinite amplitude if we were ever really to make such stuff), they are, if they were decently behaved, in two different classes of signals. the white noise is a "power signal" and the dirac spike is an "energy signal". the way you compute power or energy spectrum is slightly different. with a power signal (like a sine wave that goes on forever), you conceptually have to window it (rect is fine), run that into your integrals, and divide by the width of the window and then let that width go to infinity (taking the limit). you don't do all of that normalization for an energy signal. so that's another reason they don't compare directly. r b-j
Big Duck wrote:

> I am going to learn DSP! > > I have been studying it for a few weeks (I have many years of RF under my > belt). > > I note that the impule function in a given window has a frequency response > that is equal at every frequency in the DFT window.
Correct
> > I also know that white noise has a frequency response that has roughly equal > magnitude across the sample frequencies in an FFT.
Here we have to get picky. White noise has an _expected_ frequency response that's equal in magnitude across the sample frequencies in an FFT. For any one measurement, though, each bin will have a considerable variation -- in general an FFT of a sample vector of white noise is _not_ at all flat.
> > I am trying to make some sense of this. Here are some thoughts on this. > Please shoot down or verify: > > 1. If you took an FFT of an impulse response but randomized the phase of > each frequency in the FFT then you would get a time function that looked > like white noise ?
Yes, but I doubt that it's distribution would be Gaussian, since Gaussian white noise doesn't give a constant-amplitude frequency response for a single real sample vector. Perhaps it would be Gaussian, which would make me ask if there is some other time-domain characteristic to it that's different from 'real' white noise. It would be an interesting exercise to see just what you _would_ get.
> > 2. The ear does not disriminate phase (so i have been told): > If you played white noise , or if you played a properly windowed (digital) > impulse function at the right sample frequency , then it would sound > (roughly)the same to your ear.
Yes and no. The ear doesn't discriminate phase to a point, but when the envelope variations get low enough (certainly 20Hz, probably higher, I don't know what the cutoff is) you start to notice.
> > 3. It might be even better if you played an 1/128 impulse at a sample freq > of 10 KHz and every period you jittered the location of the impulse signal? > > > Sorry if my DSP lingo is not quite right, but tell me if this equivelance of > white noise/impulse function is true?
You are confusing spectrum with distribution. Impulse noise that occurs at random times (i.e. a Poisson distribution) will be white, but without lots of filtering it _won't_ be Gaussian. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html