Definition of a Stationary Process

rge11x wrote:

> "[...] Ooh, it is done. Specifically I have a copy of a PhD thesis ("An
> 
> Atmospheric Noise Model with Application to Low Frequency Navigation
> Systems", D.A. Feldman, MIT, 1972), where the atmospheric noise in the
> LF and MF (300-ish kHz) radio bands is called "non-stationary". "
> 
> Feldman is right: Atmospheric noise is very non-stationary. It depende
> on the time of day, seasons, sun spots activity, cosmic rays, human
> activity, and such. You may reasonably assume stationarity only over a
> short time scale, such as seconds. It clearly has strong periodic
> components over days, months and years.
> 
So if I look at the clock and see that it's 11:58:32 I know just exactly 
how the parameters of the noise are different from where they were at 
11:58:18?  If so, then by the definition that I was giving it's 
stationary over a period of seconds.  If not, then please explain your 
reasoning, and provide me with a sufficient definition of stationarity, 
or a reference thereto.

BTW:  I'm _not_ arguing that it's stationary over the course of a day or 
a year; you certainly _can_ look at the hour on the clock and the day on 
the calender and make predictions about atmospheric noise.

-- 

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

Herman Rubin wrote:

> In article <10q9v0jjsiie267@corp.supernews.com>,
> Tim Wescott  <tim@wescottnospamdesign.com> wrote:
> 
>>According to Athanasios Papoulis in "Probability, random variables, and 
>>stochastic processes", McGraw-Hill 1984, a stationary stochastic process 
>>is one whose statistical properties are invariant to a shift in the 
>>origin.  I've had problems with this definition ever since I heard it, 
>>and I'm hoping that someone will be able to clarify (hopefully with 
>>appropriate references).
> 
> 
>>It seems to me that the term "non-stationary" is overused, i.e. that 
>>processes that are stationary but are not well-behavied Gaussian 
>>processes are classified as non-stationary.
> 
> 
>>Let's assume that you have two random processes, x(t) and y(t).  x(t) is 
>>white and stationary, and y(t) is stationary and bandlimited.
> 
> 
> There is a little difficulty with proceeding from this.
> A "white" continuous time process does not exist; its
> integral does.  But we can ignore this.
> 
Please do.  I suspect that assertion could spawn another philosophical 
debate -- assume that when an engineer says "white" he really means "has 
a spectrum that is flat well beyond any limits that are going to affect 
the problem at hand".
> 
>>Now make three processes out of them:
> 
> 
>>w(t) = x(t) + y(t),
>>s(t) = x(t) * cos(42 * t),
>>z(t) = x(t) + y(t).
> 
> 
>>Now obviously w(t) is stationary,
> 
> 
> Assuming the processes are independent, or something else
> sufficiently strong.
> 
> 		and s(t) isn't (s(t) is 
> 
>>cyclostationary, but let's not touch that with a ten-foot pole).
> 
> 
>>But what about z(t)?
> 
> 
> From what I see on my screen, there is no difference between
> the w and z processes.  Do you mean * instead of +?  Some of
> your later material seems to indicate this.

Yes, it should be a product.  I "tidied" up the post for readability and 
clobbered the meaning of my central question in the process.
> 
>  To my simple mind it would seem that Papoulis's 
> 
>>definition results in the conclunsion that z(t) is stationary, because 
>>(a) it is the result of a memoryless operation on two stationary 
>>processes, and (b) you can't look at the clock and gain any a-priori 
>>knowledge about z(t).  On the other hand one could argue that z(t) is a 
>>white Gaussian process whose variance changes with time, according to 
>>the value of y(t).
> 
> 
> This argument is false in more than one way.  For one, z is
> no longer Gaussian.  For the other, you are being quite 
> ambiguous in your use of "white"; as I remarked above, white
> noise does not exist, and how it is "used" needs to be made
> precise.  
> 
The first or the second?  I believe that the first one matches the 
definition given, i.e. that the noise in question is stationary.  Yet I 
see the second argument apparently used in practice, I think because you 
could measure x(t) over a small enough interval and see white Gaussian 
noise, then measure it again to see that it's parameters have changed 
and conclude that it's not stationary.

> In fact, 
> 
> 	q(t) = x(t) * cos(42 * (t + U)),
> 
> where U is uniform between 0 and 1/42, is a strictly stationary
> Gaussian process, except for the problem of white noise, as
> long as U is independent of x.
> 

I know.  I considered using another function in my example, but cos was 
the one that dribbled off my finger tips as I was asking the question.

-- 

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

Maybe this example will help. Take a resistor R connected to a bandpass
filter (input/output matched to R) centered at 1MHz whose bandwidth is
1Hz. Connect the output to a high impedance voltmeter. The measured
voltage will be randomly fluctuating with average 0 and mean square
4kT. The wave form will look like a 1MHz sinusoid for a few hundred
milliseconds, then it will change its phase and amplitude randomly. The
amplitudes, in fact, will have normal distribution at any given time.
This means that if you take a replica of this hypothetical setup, and
replicate it, say, in 100 resistor/filter/voltmeter combinations, and
make a histogram of the measured voltage, then it will have a nice bell
curve shape centered at 0V and its variance 4kT when measured at any
given instant. Moreover, if you keep them at the same temperature,
pressure, humidity, whatever, the bell shaped histogram over the 100
experimental setups stays roughly the same, and also their correlations
to any order. This is stationarity in practice.

Now imagine that you slowly change the temperature T of the resistor,
then the variance 4kT of the voltage at the output of the filter will
also drift with the temperature. If T varies periodically, then the
noise variance will also have the same periodicty. It is still normally
distributed at any given instant but it is not a stationary process any
more. If you replicate the same experiment and measure the histogram of
the voltage you do get the bell shaped curve but it will change its
width as the temperature changes.

I know of no process that is truly stationary: being stationary is a
mathematical idealization, just like any other concept applied in
physics. After all, there is no such a thing as a resistor, either. At
high enough frequency, it is a complicated resonating and radiating
thing. If you wait long enough everything changes to some extent. But
in the example above most resistors will be stationary over 1sec so a
1Hz filter is adequate to characterize the spectral distribution of the
thermal noise generated by it. If you, instead, take a 1microHz filter
you may have difficulty verifying the Nyquist distribution because 1
million seconds is very long time in electronics.

Another example that might give you more comfort is the effect of phase
of an oscillator on signal detection. Do you really care how the
oscillator drifts over a year or over a microsendsecond when you detect
a signal whose bandwidth is about 1MHz and lasts for 1second? What you
care about is obviously what happens over a second sampleda million
times (once a microsecond) with respect to phase detection but then you
stabilize your oscillator so with its slow frequency drift it will not
leave your receiver. The phase of the crystal oscillator is
nonstationary over a year's time but over the detection time scale it
is a reasonable assumption, and all radios are built assuming that
because so far nobody is really worried about 1microbit/second
transmission rates.