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Low pass filtration of white noise

Started by Unknown March 18, 2007
runech@gmail.com wrote:

   ...

> I can do a lot of practical things but I would also like to know the > theory. I won't settle with observing a result. I would like to be > able to explain it. > > This is not homework, but even if it was does it really matter for > you?
Because doing student's homework for them doesn't help them learn. Helping them to get past a difficulty so they can do the homework themselves is the way most of us try to handle that.
> R.A Alters: Wide Band Systems and Gaussianity, IEEE Transactions on > Information Theory, IT-21, November 1975, pp 679-82 states that if the > "duration" of the impulse response of the filter is greater than the > "duration" of the autocorrelation of the input then the pdf will tend > towards the gaussian distribution independent of the pdf of the input > signal. > > So is that the way to do it or can it be expressed in another way?
"The result of the convolution of the two above signals *is a Gaussian* distributed signal" and "if the "duration" of the impulse response of the filter is greater than the "duration" of the autocorrelation of the input then the pdf will *tend towards the Gaussian* [emphases added] distribution independent of the pdf of the input signal" are very different statements. Adding (almost) any two random signals will produce a result that tends toward Gaussian. Revisit the Central Limit Theorem. http://mathworld.wolfram.com/CentralLimitTheorem.html Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Rune Allnor wrote:
> On 18 Mar, 11:17, not...@gmail.com wrote: >> Hello, >> >> The scenario: >> >> A white noise signal. (Uniformly distributed between 0 and 1) > > Does this qualify as white noise? I always thought that > the noise had to be Gaussian to be white...
Gaussian is a distribution that extends to infinity in both directions. The distribution of most random-number generators is uniform. Have you not come across a Rayleigh distribution? Those too can be white. Whiteness (or other color) has to do with spectral content. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
On Mar 18, 6:46 pm, Jerry Avins <j...@ieee.org> wrote:
> run...@gmail.com wrote: > > ... > > > I can do a lot of practical things but I would also like to know the > > theory. I won't settle with observing a result. I would like to be > > able to explain it. > > > This is not homework, but even if it was does it really matter for > > you? > > Because doing student's homework for them doesn't help them learn. > Helping them to get past a difficulty so they can do the homework > themselves is the way most of us try to handle that.
That's also want I wanted.A few hints for a specific problem. Not a finished paper.
> > > R.A Alters: Wide Band Systems and Gaussianity, IEEE Transactions on > > Information Theory, IT-21, November 1975, pp 679-82 states that if the > > "duration" of the impulse response of the filter is greater than the > > "duration" of the autocorrelation of the input then the pdf will tend > > towards the gaussian distribution independent of the pdf of the input > > signal. > > > So is that the way to do it or can it be expressed in another way? > > "The result of the convolution of the two above signals *is a Gaussian* > distributed signal" and "if the "duration" of the impulse response of > the filter is greater than the "duration" of the autocorrelation of the > input then the pdf will *tend towards the Gaussian* [emphases added] > distribution independent of the pdf of the input signal" are very > different statements. Adding (almost) any two random signals will > produce a result that tends toward Gaussian. Revisit the Central Limit > Theorem.http://mathworld.wolfram.com/CentralLimitTheorem.html > > Jerry > -- > Engineering is the art of making what you want from things you can get. > =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF I was focused on the way the filter was designed in the frequency domain rather than the fact that it is a convolution of two random variables. So for all practical distributions a convolution of two independent random signals will tends toward Gaussian. Thank you all
runech@gmail.com writes:
> [...] > So for all practical distributions a convolution of two independent > random signals will tends toward Gaussian.
I think you're misinterpreting the situation. In your problem, you are processing a random input signal (the uniformly-distributed signal) with a linear system that has a sinc() response in the frequency domain. The impulse response (the square wave) is not random. -- % 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://home.earthlink.net/~yatescr
On Mar 18, 8:22 pm, Randy Yates <y...@ieee.org> wrote:
> run...@gmail.com writes: > > [...] > > So for all practical distributions a convolution of two independent > > random signals will tends toward Gaussian. > > I think you're misinterpreting the situation. > > In your problem, you are processing a random input signal (the > uniformly-distributed signal) with a linear system that has a sinc() > response in the frequency domain. The impulse response (the square > wave) is not random. > -- > % Randy Yates % "The dreamer, the unwoken fool - > %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." > %%% 919-577-9882 % > %%%% <y...@ieee.org> % 'Eldorado Overture', *Eldorado*, ELOhttp://home.earthlink.net/~yatescr
I know the problem. But I don't understand the theory in this specific situation. I can easily show in Matlab that the filtration of a random uniformly distributed signal with a square pulse results in a Gaussian signal. The course notes states thats it is rather impossible to say anything about the distribution of a filtrated signal. That's way I turned to you guys. I know the sum of a N independent random variables tends toward Gaussianity. But how can I relate the central limit theorem to this situation when convolving a RV with the impulse response. I thought I had it.. /Rune
runech@gmail.com wrote:
> On Mar 18, 6:46 pm, Jerry Avins <j...@ieee.org> wrote: >
-- snip --
> > I was focused on the way the filter was designed in the frequency > domain rather than the fact that it is a convolution of two random > variables. > > So for all practical distributions a convolution of two independent > random signals will tends toward Gaussian. > > Thank you all >
There are practical distributions (atmospheric noise in LF and MF is one) that have essentially infinite variance, and so don't work with the central limit theorem. But aside from that, yes. -- 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
runech@gmail.com wrote:

-- snip --

> > > I know the problem. But I don't understand the theory in this specific > situation. > > I can easily show in Matlab that the filtration of a random uniformly > distributed signal with a square pulse results in a Gaussian signal. > The course notes states thats it is rather impossible to say anything > about the distribution of a filtrated signal. That's way I turned to > you guys. > > I know the sum of a N independent random variables tends toward > Gaussianity. But how can I relate the central limit theorem to this > situation when convolving a RV with the impulse response. > > I thought I had it.. >
You're not convolving a single random variable with an impulse response. When you get a sampled sequence of outputs from a random process, each sample is a random variable. If the process is white then each random variable is independent. When you convolve the sequence of random variables, you're weighting each one by some amount, then summing it to all of the other (weighted) samples. At that point you have one random variable that is the weighted sum of a whole bunch of independent random variables. You can go straight from that observation into the central limit theorem. -- 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
> I know the sum of a N independent random variables tends toward > Gaussianity. But how can I relate the central limit theorem to this > situation when convolving a RV with the impulse response. >
You said the noise is white, so each sample is independent of each other... Therefore if your signal is N samples long, you have N indenpendent RV being summed up during the convolution with the impulse response of your fiter... if the noise wasn't white....than thats would be a different story... note that the output of your filter is not white (samples are now correlated)
Just noticed that Tim beat me to the post...my answer is the same idea
of what he said...:)

On Mar 18, 4:19 pm, "Ikaro" <ikarosi...@hotmail.com> wrote:
> > I know the sum of a N independent random variables tends toward > > Gaussianity. But how can I relate the central limit theorem to this > > situation when convolving a RV with the impulse response. > > You said the noise is white, so each sample is independent of each > other... > Therefore if your signal is N samples long, you have N indenpendent RV > being summed up during the convolution with the > impulse response of your fiter... > > if the noise wasn't white....than thats would be a different story... > > note that the output of your filter is not white (samples are now > correlated)