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covariance question

Started by Tom Derham January 1, 2005
If a zero-mean Gaussian random noise signal is at the input to a matched 
filter bank (so the input to each filter is identical), what is the nature 
of the outputs of each filter in the bank?
Assuming the filters are linear then each output should also be Gaussian, 
but are they correlated or made independent by the filters.. is there a way 
of calculating the covariance matrix from the transfer functions of the 
filters?

I'm a bit stuck on this conceptually (currently reading Papoulis, etc), so 
any help would be much appreciated

Thanks

Tom 


Tom Derham wrote:

> If a zero-mean Gaussian random noise signal is at the input to a matched > filter bank (so the input to each filter is identical), what is the nature > of the outputs of each filter in the bank?
If they are linear filters then it'll be zero-mean Gaussian.
> Assuming the filters are linear then each output should also be Gaussian, > but are they correlated or made independent by the filters.. is there a way > of calculating the covariance matrix from the transfer functions of the > filters? > > I'm a bit stuck on this conceptually (currently reading Papoulis, etc), so > any help would be much appreciated > > Thanks > > Tom >
If the filters' spectra do not overlap then the outputs of the filters will be uncorrelated. If the filters' spectra _do_ overlap then I have every faith that there will be cross-correlation, and that there's a way of calculating the covariance -- but I haven't faintest notion of what that is. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
On Sat, 01 Jan 2005 18:28:07 +0000, Tim Wescott wrote:

> Path: > newsbe2-gui.ntli.net!newspeer1-win.ntli.net!news-in.ntli.net!newsrout1-win. > ntli.net!ntli.net!sn-xit-02!sn-xit-01!sn-post-01!supernews.com!corp.superne > ws.com!not-for-mail > From: Tim Wescott <tim@wescottnospamdesign.com> > Newsgroups: comp.dsp > Subject: Re: covariance question > Date: Sat, 01 Jan 2005 18:28:07 -0800 > Organization: Posted via Supernews, http://www.supernews.com > Message-ID: <10ten1hlnb19j5a@corp.supernews.com> > User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.6) > Gecko/20040113 > X-Accept-Language: en-us, en > References: <SoGdnYZ4v6awrkrcRVnysw@pipex.net> > In-Reply-To: <SoGdnYZ4v6awrkrcRVnysw@pipex.net> > X-Complaints-To: abuse@supernews.com > Lines: 32 > Xref: newspeer1-win.ntli.net comp.dsp:168734 > X-Received-Date: Sun, 02 Jan 2005 02:29:09 GMT (newsbe2-gui.ntli.net) > MIME-Version: 1.0 > Content-Type: text/plain; charset=us-ascii; format=flowed > Content-Transfer-Encoding: 7bit > > > Tom Derham wrote: > >> If a zero-mean Gaussian random noise signal is at the input to a matched >> filter bank (so the input to each filter is identical), what is the nature >> of the outputs of each filter in the bank? > > If they are linear filters then it'll be zero-mean Gaussian.
If the filters' spectra _do_ overlap then I have
> every faith that there will be cross-correlation, and that there's a way > of calculating the covariance -- but I haven't faintest notion of what > that is.
Co variance is the ratio between the variance of two signals. I.e. take the variance of signal A and the variance of signal B. The co-variance is A/B. What it really tells you is how noisy (or reliable) A is compared to B. -- [=========] -==++""" . /. . . \ . """++==- -+"" \ .. . . | .. . | . . . / ""+- /\ +-"" `-----=====\ <O> /=====-----' ""-+ /\ / / ""="" \ \ / / \ \ // | \\ /") \ | / ("\ \o\ \*/ /o/ \ ) --**O**-- ( / /*\ / | \ |
"Robin Clark" <robinTEETH48gx@hotTEETHmail.com> wrote in message 
news:pan.2005.01.02.11.21.41.551837@hotTEETHmail.com...
> > Co variance is the ratio between the variance of two signals. > I.e. take the variance of signal A and the variance of signal B. > > The co-variance is A/B. > > What it really tells you is how noisy (or reliable) A is compared to B. > >
Hello Robin, I think you need to recheck your stats book. The above statements are wrong. Covarince is communitive so therefore cannot be written as a ratio of variances. Also recall the covariance of two indepenent random variables is zero, yet each comes from a distribution with nonzero variances. So clearly it cannot be what you stated. Loosely if A tends to be large when B is large, then the covariance is positive. Likewise if one of A, B tends to be small while the other is large, then the covariance is negative. And if one's values don't associated at all with the other's, then the covariance is zero. Clay
Tom Derham wrote:
> If a zero-mean Gaussian random noise signal is at the input to a
matched
> filter bank (so the input to each filter is identical), what is the
nature
> of the outputs of each filter in the bank? > Assuming the filters are linear then each output should also be
Gaussian,
> but are they correlated or made independent by the filters.. is there
a way
> of calculating the covariance matrix from the transfer functions of
the
> filters?
Yes it is. Define the filtering operation as y(t) = x(t) (*) h(t) where x(t) is the input signal, h(t) is the filter impulse response, y(t) is the filtered signal and (*) means convolution. There is a standard exercise in intro statistical DSP on how to show that the autocorrelation of filtered white noise is given by the filtering function, something like E[S_yy(f)] = E[S_xx(f)]|H(f)|^2 where S_yy(f) is the power spectrum density of y(t) and S_xx(f) is the power spectrum density of x(t). The power density is defined as the FT of the autocorrelation sequence, S_xx(f) = FT{r_xx(tau)}. Using the same kind of argument for the cross correlation sequences of the filtered outputs, it is possible to find the cross correlation sequences for the filtered sequences y_u(t) and y_v(t) defined as y_u(t) = x(t) (*) u(t) y_v(t) = x(t) (*) v(t), to be something like E[S_uv(f)] = S_xx(f) U(f)V(f). Here, U(f) and V(f) are defined as the FTs of u(t) and v(t) respectively, and one of them should be complex conjugated.
> I'm a bit stuck on this conceptually (currently reading Papoulis,
etc), so
> any help would be much appreciated
Papoulis is a good place to start. The main lines ought to be something like what I sketched above. Rune