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nonstationary autocorrelation but stationary PSD

Started by fisico32 May 17, 2010
Hello Forum

the nonstationary autocorrelation function R(t1, t2) depends on two
variables t1,t2. If t2-t1=tau, then R(t1, t1+tau)

It turns out that the power spectral density S(w), for a nonstationary
autocorrelation function is equal to:

S(w)= FT {<R(t,t+tau)>}, the Fourier transform of the "long-time average of
R(t, t+tau)" is the power spectral density.

Why is S(w) not a function of time as well? I can see that mathematically,
but conceptually it seems that if the autocorrelation changes with time,
also the PSD would change with time....
But maybe the "instantaneous S(w)" would, correct?
So S(w)= FT {<R(t,t+tau)>} is actually the average power spectral
density...

Only if the autocorrelation is indepedent of time, <R(t, t+tau)>=R(tau) 
and the S(w) is direcly and purely the FT{R(tau)}.

Thanks
fisico32
fisico32 wrote:
> Hello Forum > > the nonstationary autocorrelation function R(t1, t2) depends on two > variables t1,t2. If t2-t1=tau, then R(t1, t1+tau) > > It turns out that the power spectral density S(w), for a nonstationary > autocorrelation function is equal to: > > S(w)= FT {<R(t,t+tau)>}, the Fourier transform of the "long-time average of > R(t, t+tau)" is the power spectral density.
By whose definition? Cite your sources.
> Why is S(w) not a function of time as well? I can see that mathematically, > but conceptually it seems that if the autocorrelation changes with time, > also the PSD would change with time.... > But maybe the "instantaneous S(w)" would, correct? > So S(w)= FT {<R(t,t+tau)>} is actually the average power spectral > density... > > Only if the autocorrelation is indepedent of time, <R(t, t+tau)>=R(tau) > and the S(w) is direcly and purely the FT{R(tau)}.
Are you a student? You seem to be in the student trap of believing that the knowledge you are learning is the same regardless of who you ask or who writes your text book. There are a lot of different ways to present this material, and part of the job of the textbook writer is to find the right way to do so. S(w) is not a function of time as well because whoever wrote the document (textbook, paper, monograph, whatever) you are currently studying it defined it that way. It may, indeed, be the popular definition. But there's no reason that you couldn't define S(w, t) as the expected value of the amplitude of the Fourier transform of the signal at that time -- in which case it'd just be the Fourier transform of the autocorrelation function with all due scaling or square-rooting or whatever your particular framework demands. You are correct to be puzzled -- either definition is equally useful in its own way, and is an equally bad fit to the reality of a signal whose parameters change with time. If this doesn't help, I hope it at least make you feel better... -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
Hello,
I am a "trapped" student in fact.

The book is  "Probabilistic Methods of Signal and Systems Analysis" by
Cooper and McGillem page 252.

It all starts with truncating an infinite extent realization, from -T to T,
to get X_T(t) which admits the ordinary Fourier transform F(w).

The spectral density of a random process (stationary or nonstationary) is
defined as limit for T->infinity of the E[|F(w)|^2]/T , the expectation
value of the modulus square of the F(w), which is FT of the truncated
realization. 

|F(w)|^2=F(w)F(-w), is split into two iterated integrals to end up with
E[X(t1)X(t2)] inside as integrand for one of them....




>fisico32 wrote: >> Hello Forum >> >> the nonstationary autocorrelation function R(t1, t2) depends on two >> variables t1,t2. If t2-t1=tau, then R(t1, t1+tau) >> >> It turns out that the power spectral density S(w), for a nonstationary >> autocorrelation function is equal to: >> >> S(w)= FT {<R(t,t+tau)>}, the Fourier transform of the "long-time average
of
>> R(t, t+tau)" is the power spectral density. > >By whose definition? Cite your sources. > >> Why is S(w) not a function of time as well? I can see that
mathematically,
>> but conceptually it seems that if the autocorrelation changes with
time,
>> also the PSD would change with time.... >> But maybe the "instantaneous S(w)" would, correct? >> So S(w)= FT {<R(t,t+tau)>} is actually the average power spectral >> density... >> >> Only if the autocorrelation is indepedent of time, <R(t, t+tau)>=R(tau)
>> and the S(w) is direcly and purely the FT{R(tau)}. > >Are you a student? You seem to be in the student trap of believing that >the knowledge you are learning is the same regardless of who you ask or >who writes your text book. There are a lot of different ways to present >this material, and part of the job of the textbook writer is to find the >right way to do so. > >S(w) is not a function of time as well because whoever wrote the >document (textbook, paper, monograph, whatever) you are currently >studying it defined it that way. It may, indeed, be the popular
definition.
> >But there's no reason that you couldn't define S(w, t) as the expected >value of the amplitude of the Fourier transform of the signal at that >time -- in which case it'd just be the Fourier transform of the >autocorrelation function with all due scaling or square-rooting or >whatever your particular framework demands. > >You are correct to be puzzled -- either definition is equally useful in >its own way, and is an equally bad fit to the reality of a signal whose >parameters change with time. > >If this doesn't help, I hope it at least make you feel better... > >-- >Tim Wescott >Control system and signal processing consulting >www.wescottdesign.com >
On May 18, 10:18=A0am, "fisico32" <marcoscipioni1@n_o_s_p_a_m.gmail.com>
wrote:
> Hello Forum > > the nonstationary autocorrelation function R(t1, t2) depends on two > variables t1,t2. If t2-t1=3Dtau, then R(t1, t1+tau) > > It turns out that the power spectral density S(w), for a nonstationary > autocorrelation function is equal to: > > S(w)=3D FT {<R(t,t+tau)>}, the Fourier transform of the "long-time averag=
e of
> R(t, t+tau)" is the power spectral density. > > Why is S(w) not a function of time as well? I can see that mathematically=
,
> but conceptually it seems that if the autocorrelation changes with time, > also the PSD would change with time.... > But maybe the "instantaneous S(w)" would, correct? > So S(w)=3D FT {<R(t,t+tau)>} is actually the average power spectral > density... > > Only if the autocorrelation is indepedent of time, <R(t, t+tau)>=3DR(tau) > and the S(w) is direcly and purely the FT{R(tau)}. > > Thanks > fisico32
Yes, if the process is non-stationary then in theory we should have a time-varying spectrum. However, you can still get a spectrum that combines any changes into a normal spectrum. What information you would get of course is another matter. anyway, that's why we use sliding windows and short-term FFT etc Often people who write books only concern themselves with the stationary problem but try and be fancy by defining things to be non-stationary and considering the stationary case to be a special case! Hardy Hardy