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Hi all,

I have a stationary process X_k as an input to a Linear and Time-Invariant
system. Is the output process Y_k still a stationary process?
SSS(Strict-Sense Stationary) or at least WSS(Wide Sense Stationary)?

Thanks a lot!

kiki wrote:

Hi all,

I have a stationary process X_k as an input to a Linear and Time-Invariant
system. Is the output process Y_k still a stationary process?
SSS(Strict-Sense Stationary) or at least WSS(Wide Sense Stationary)?

Thanks a lot!



Well, think about it. A stationary process is one who's parameters are

independent of time, although the value of the signal resulting from the
process can be correlated to past or future values of itself.

A time-invariant system is one that modifies a signal in a
time-independent way, although the output signal at any given time may
depend on past (or future, if you don't care about causality) values of
the input signal.

So you tell me -- is it still stationary, or has it acquired a time
dependence?

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim wrote:

"
Well, think about it. A stationary process is one who's parameters are
independent of time, although the value of the signal resulting from
the
process can be correlated to past or future values of itself.

A time-invariant system is one that modifies a signal in a
time-independent way, although the output signal at any given time may
depend on past (or future, if you don't care about causality) values of

the input signal.

So you tell me -- is it still stationary, or has it acquired a time
dependence?
"

Your explanation seems to make sense.

However, if you have an LTI system with poles on the unit circle, one
can still say that it modifies the signal in a time-independent way,
but the output is not stationary. Can you refine your argument?
Regards,
Andor

Andor,

What about startup transients? I agree in the steady-state.

--RY

"Andor" <an2or@mailcircuit.com> writes:

Tim wrote:

"
Well, think about it. A stationary process is one who's parameters are
independent of time, although the value of the signal resulting from
the
process can be correlated to past or future values of itself.

A time-invariant system is one that modifies a signal in a
time-independent way, although the output signal at any given time may
depend on past (or future, if you don't care about causality) values of

the input signal.

So you tell me -- is it still stationary, or has it acquired a time
dependence?
"

Your explanation seems to make sense.

However, if you have an LTI system with poles on the unit circle, one
can still say that it modifies the signal in a time-independent way,
but the output is not stationary. Can you refine your argument?
Regards,
Andor

--
% Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC % She love the way Puccini lays down a tune, and
%%% 919-577-9882 % Verdi's always creepin' from her room."
%%%% <yates@ieee.org> % "Rockaria", *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Randy Yates wrote:

What about startup transients? I agree in the steady-state.

Well, then the system isn't, strictly speaking, LTI, is it? :-)
Ciao,

Peter K.

Randy Yates wrote:

Andor,

What about startup transients? I agree in the steady-state.

--RY

"Andor" <an2or@mailcircuit.com> writes:


Tim wrote:

"
Well, think about it. A stationary process is one who's parameters are
independent of time, although the value of the signal resulting from
the
process can be correlated to past or future values of itself.

A time-invariant system is one that modifies a signal in a
time-independent way, although the output signal at any given time may
depend on past (or future, if you don't care about causality) values of

the input signal.

So you tell me -- is it still stationary, or has it acquired a time
dependence?
"

Your explanation seems to make sense.

However, if you have an LTI system with poles on the unit circle, one
can still say that it modifies the signal in a time-independent way,
but the output is not stationary. Can you refine your argument?
Regards,
Andor



I missed Andor's post, so I'm answering it here.

You're probably thinking of the Wiener process, where you integrate
white noise starting at time t = zero. You are correct that this is not
stationary, but if you have a system that holds its output at zero until
time t = zero then the system is not time invariant. If your system
_is_ time invariant then you must have a signal that is zero for t < 0
and has energy for t > 0 -- and that's not a stationary signal.

Actually any "LTI" that is held at zero and only released at time t=0
will have an output that's non-stationary because you must be violating
one of the conditions above. To keep the overall system time invariant
you have to start things up at -infinity.

So I think I'm still right.

Now, granted, if you have an unstable or metastable system your output
signal in such a case will have infinite variance -- this is probably
why you want to hold things to zero until t = 0.

I suspect that trying to do the analysis with any sort of rigor while
allowing unstable or metastable systems would require many extra reams
of paper, but I think my assertions would hold true.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim wrote:

"If your system
_is_ time invariant then you must have a signal that is zero for t < 0
and has energy for t > 0 -- and that's not a stationary signal. "

Point taken.

"Now, granted, if you have an unstable or metastable system your output

signal in such a case will have infinite variance -- this is probably
why you want to hold things to zero until t = 0."

I guess that's what I had in mind. Best just forget about unstable
systems.

Regards,
Andor

Homework?
--
Stephan M. Bernsee
http://www.dspdimension.com