# Periodicity , shift invariance and stationarity

Started by November 25, 2004
```Continuing with Tim Wescott's article I would like to ask following
question

Periodic signals are defined as

signal which repeats after certain period i.e mathematically,

f(t)=f(t+p), where p is time period (say, time domain signal).

Can this signal be called  as "shift invariant"? Also some papers
write
"shift invariant" as "stationary" - meaning statistical property such
as
mean remain constant and autocorr does not change.

If I scale my signal every time it repeats i.e at every time period ;
also apply my scaling as dynamic, Can this signal be called as
periodic ? - I guesss shape of the signal does not change. Obviously
mean is not constant here.

What also confuses me is non-stationary periodic signals  - Is that
what I describe above or something else? I have heard plenty of
examples of non-stationary periodic signals  found in wavelet
analysis.

Regards,
Santosh
```
```santosh.nath@ntlworld.com (santosh nath) wrote in message news:<6afd943a.0411242346.5ba48050@posting.google.com>...
> Continuing with Tim Wescott's article I would like to ask following
> question
>
> Periodic signals are defined as
>
> signal which repeats after certain period i.e mathematically,
>
> f(t)=f(t+p), where p is time period (say, time domain signal).
>
> Can this signal be called  as "shift invariant"? Also some papers
> write
> "shift invariant" as "stationary" - meaning statistical property such
> as
> mean remain constant and autocorr does not change.

Sounds right to me.

> If I scale my signal every time it repeats i.e at every time period ;
> also apply my scaling as dynamic, Can this signal be called as
> periodic ? - I guesss shape of the signal does not change. Obviously
> mean is not constant here.

...which mean the mean changes, depending on the observation interval
which in turn means...?

> What also confuses me is non-stationary periodic signals  - Is that
> what I describe above or something else? I have heard plenty of
> examples of non-stationary periodic signals  found in wavelet
> analysis.
>

I prefer to avoid the philosophical aspects of such discussions.
The terms "periodic", "stationary", "ergodic" have very specific
meanings in terms of theoretical statistics.

Now, deciding whether any one signal belongs to this or that category,
is a completely different question. I prefer to base any particular
analysis on a *postulate* of the signal being linear, stationary,
periodic or whatever, and go on with the analysis based on such
assumptions. The important part is to be very aware of why you
impose the (perhaps unjustified!) assumption and what consequences
it has for the analysis.

It is very awkward to assume that a reflection series from a sonar
or seismic survey is a "stationary" time series, but one frequently
does so. Not because it's a very good description of the reality
(one could dispute such a claim until the cows come home, without
ever reacing an agreement), but because the mathemathical tools that
are designed to be used with stationary data are easy and convenient
to use. The price paid by using a more "accurate" mathematical
representation of the problem, is more often than not that the
mathemathical tools become too awkward to use, to be practical.

So a common way to do things is to postulate the signal to be, say,
stationary. If one needs to, one then points out what conclusions
from the analysis are shaky, and why. If one can tie some conclusion
to some property of the a priori postulate of the signal being
stationary, one have done the job. One have provided an analysis but
also pointed out the weak spot of the analysis. That is often the
best one can do.

I gave such a line of arguments in my thesis. The task was to analyze
a spatial signal that due to physics could be described on the form

s(x) = 1/sqrt(x) exp(jkx)

which is hardly stationary in x.

The signal processing tools worked for complex exponentials, so
what I did, was to say that "the method works when the 1/sqrt(x)
term is neglectable". The next task, then, was to investigate
the effects on non-neglectable sqrt(x) terms in order to recognize
them in the analysis, and also to find methods to decide pror to
analysis when the sqrt(x) terms were neglectable, and when they
are not.

So the task was to determine when the data were "sufficiently stationary"
in a certain sense. It worked quite well.

Rune
```
```
santosh nath wrote:
> Continuing with Tim Wescott's article I would like to ask following
> question
>
> Periodic signals are defined as
>
> signal which repeats after certain period i.e mathematically,
>
> f(t)=f(t+p), where p is time period (say, time domain signal).
>
> Can this signal be called  as "shift invariant"? Also some papers
> write
> "shift invariant" as "stationary" - meaning statistical property such
> as
> mean remain constant and autocorr does not change.

Shift invariant usually means that if you have some property of
your signal and then you discover that you should have relabeled the
time so that what was "t+1" is now "t" that you do not have to do
the same relabellings of the properties. When you stop to work this
through you will find that the above is slightly sloppy as it should
have been about the ensemble of signals of which you have a single
realization. Whether time is periodic or unbounded does not matter as
long as you use the corresponding shift operation. There is also a
periodic stationarity where as long as the relabellings is t+P for t,
like the weather which is stationary over years but not within years.
(Neglecting the 13000+ year orbital cycle, etc, etc, and even man

> If I scale my signal every time it repeats i.e at every time period ;
> also apply my scaling as dynamic, Can this signal be called as
> periodic ? - I guesss shape of the signal does not change. Obviously
> mean is not constant here.
>
> What also confuses me is non-stationary periodic signals  - Is that
> what I describe above or something else? I have heard plenty of
> examples of non-stationary periodic signals  found in wavelet
> analysis.

Are you thinking of something where each "repetition" is only almost
the same so it would not be "periodic"? In a periodic signal you would
have x(t), x(t+P), x(t+2P), ... identical but x(t+1), x(t+1+P), ...
could be different if there it was not shift invariant and so
non stationary in the periodic time.

>
> Regards,
> Santosh

```
```hello,
why people are making very simple concepts this complicated.
Stability,Shift invarienc", one should go through the good books like
1. Ditigal Signal Processing by Allen Openheam.

2. Digital Signal Processing by Rabiner and Gold.

3. Digital Signal Processing, A computer based appoach by SK Mithra.

4. Probability, random variables, and stochastic processes by Papouls.

Sorry for saying like this.

Best Regards,

Athreya Sathish

```
```Athreya Sathish wrote:

>  why people are making very simple concepts this complicated.

Well, I thought a newsgroup was a place to ask questions :-).

> Stability,Shift invarienc", one should go through the good books like

Now that's a good suggestion, but please don't stop people from asking
questions! You might think you know all the answers, but sometimes you
find that when you ask an expert a dumb question, both of you end up
learning a lot.

```
```"Athreya Sathish" <saikumar.mangapuram@de.bosch.com> wrote in message news:<co4sgl\$64q\$1@ns2.fe.internet.bosch.com>...
> hello,
>    why people are making very simple concepts this complicated.
> Stability,Shift invarienc", one should go through the good books like
>     1. Ditigal Signal Processing by Allen Openheam.
>
>     2. Digital Signal Processing by Rabiner and Gold.
>
>     3. Digital Signal Processing, A computer based appoach by SK Mithra.
>
>     4. Probability, random variables, and stochastic processes by Papouls.
>
> Sorry for saying like this.
>
>    Best Regards,
>
> Athreya Sathish

Hi Sathish,
You look to be new in this group! I guess Rune is ,no way, less
intelligent than you. He is a thoughtful contributor - check with his
articles.I am not sure how serious you are reading those books -
spelling mistakes "Openheam",
"Mithra" and "Papouls" are not expected from a person who treat these
books as "simple".  In my opinion, Alan V.  Oppenheim talks mainly
about LTI system throughout the book(1975 EDITION) with less focus on
other topics. Was that my article describing?

I would be more happy if you help me getting these answers  than
ignoring.

Santosh
```
```On 25 Nov 2004 18:43:33 -0800, "ks" <karthik301176@gmail.com> wrote:

>Athreya Sathish wrote:
>
>>  why people are making very simple concepts this complicated.
>
>Well, I thought a newsgroup was a place to ask questions :-).
>
>> Stability,Shift invarienc", one should go through the good books like
>
>Now that's a good suggestion, but please don't stop people from asking
>questions! You might think you know all the answers, but sometimes you
>find that when you ask an expert a dumb question, both of you end up
>learning a lot.

Hi ks,