Two basic questions regarding random processes and systems

hI all,

I have the following two questions regarding random processes and systems.

1. I read from Proakis' book that stationary random processes are infinite 
energy signal, so the Foureir transform of stationary random processes do 
not exist...

My questions are: a) For what kind of signal does Foureir transform exist? 
b) What about WSS random processes?

2. The absolute integral appears a lot: Integrate(|h(t)|, t from -inf to 
+inf)
My question is:

Integrate(|h(t)|, t from -inf to +inf) < +inf

vs.

Integrate(|h(t)|^2, t from -inf to +inf) < + inf

which one is a subset of the other one? i.e. which one implies the other 
one?

Thanks a lot 

In article <d53ctl$q58$1@news.Stanford.EDU>,
 "kiki" <lunaliu3@yahoo.com> wrote:

> 2. The absolute integral appears a lot: Integrate(|h(t)|, t from -inf to 
> +inf)
> My question is:
> 
> Integrate(|h(t)|, t from -inf to +inf) < +inf
> 
> vs.
> 
> Integrate(|h(t)|^2, t from -inf to +inf) < + inf
> 
> which one is a subset of the other one? i.e. which one implies the other 
> one?

Neither, unless you add some hypotheses. For example, if h is 
bounded, then the first condition implies the second.

"kiki" <lunaliu3@yahoo.com> writes:

> hI all,

Hi kiki,

> I have the following two questions regarding random processes and systems.
>
> 1. I read from Proakis' book that stationary random processes are infinite 
> energy signal, so the Foureir transform of stationary random processes do 
> not exist...
>
> My questions are: a) For what kind of signal does Foureir transform exist? 

The constraints on a signal that are required in order for the Fourier transform
of that signal to exist are specified in the so-called Dirichlet conditions. 

  http://cnx.rice.edu/content/m10089/latest?format=pdf

One of those conditions is that the signal be a finite-energy signal (inequality
5 in the paper cited above).

> b) What about WSS random processes?

Same problem. Wide-sense stationary processes still have infinite energy, and
thus no FT.

> 2. The absolute integral appears a lot: Integrate(|h(t)|, t from -inf to 
> +inf)
> My question is:
>
> Integrate(|h(t)|, t from -inf to +inf) < +inf
>
> vs.
>
> Integrate(|h(t)|^2, t from -inf to +inf) < + inf
>
> which one is a subset of the other one? i.e. which one implies the other 
> one?

These integrals yield the total energy and the total power,
respectively, in the signal.  A finite-energy signal is also a finite
power signal, i.e.,

  finite energy --> finite power.

That is the only implication one can make.

Note that it is a common trick to use Dirac delta functions with
periodic signals (which are infinite-energy signals and thus shouldn't
have FTs) so that we can play around with them as if they had Fourier
transforms, but I don't think this is rigorously acceptable.
-- 
%  Randy Yates                  % "With time with what you've learned, 
%% Fuquay-Varina, NC            %  they'll kiss the ground you walk 
%%% 919-577-9882                %  upon."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO
http://home.earthlink.net/~yatescr

In article <64y2d6kz.fsf@ieee.org>, Randy Yates <yates@ieee.org> 
wrote:

> > 2. The absolute integral appears a lot: Integrate(|h(t)|, t from -inf to 
> > +inf)
> > My question is:
> >
> > Integrate(|h(t)|, t from -inf to +inf) < +inf
> >
> > vs.
> >
> > Integrate(|h(t)|^2, t from -inf to +inf) < + inf
> >
> > which one is a subset of the other one? i.e. which one implies the other 
> > one?
> 
> These integrals yield the total energy and the total power,
> respectively, in the signal.  A finite-energy signal is also a finite
> power signal, i.e.,
> 
>   finite energy --> finite power.
> 
> That is the only implication one can make.

That's false for certain h: for example, h(t) = 1/sqrt(t) for 0 < 
t < 1, h(t) = 0 elsewhere.

in article waderameyxiii-A24B0C.19534701052005@comcast.dca.giganews.com, The
World Wide Wade at waderameyxiii@comcast.remove13.net wrote on 05/01/2005
22:53:

> In article <64y2d6kz.fsf@ieee.org>, Randy Yates <yates@ieee.org>
> wrote:
> 
>>> 2. The absolute integral appears a lot: Integrate(|h(t)|, t from -inf to
>>> +inf)
>>> My question is:
>>> 
>>> Integrate(|h(t)|, t from -inf to +inf) < +inf
>>> 
>>> vs.
>>> 
>>> Integrate(|h(t)|^2, t from -inf to +inf) < + inf
>>> 
>>> which one is a subset of the other one? i.e. which one implies the other
>>> one?
>> 
>> These integrals yield the total energy and the total power,
>> respectively, in the signal.  A finite-energy signal is also a finite
>> power signal, i.e.,
>> 
>> finite energy --> finite power.
>> 
>> That is the only implication one can make.
> 
> That's false for certain h: for example, h(t) = 1/sqrt(t) for 0 <
> t < 1, h(t) = 0 elsewhere.

a finite energy signal has zero mean power and a finite power signal has
infinite energy.

Randy, i'm not sure what you mean by "total" power.

a finite power signal is:

                      tau/2
  lim      1/tau    integral{ |x(t)|^2 dt}    < inf
 tau->inf            -tau/2


and a finite energy signal is:

                      tau/2
  lim               integral{ |x(t)|^2 dt}    < inf
 tau->inf            -tau/2


the finite magnitude condition:

                      tau/2
  lim               integral{ |x(t)|   dt}    < inf
 tau->inf            -tau/2

is stronger than having a finite energy, but is sometimes needed to get
certain other integrals (like the Fourier Transform) to converge.  however,
even if the integral magnitude does not converge, the energy spectrum will
be defined as long as it's a finite energy signal.  and the power spectrum
will be defined if it's a finite power signal.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

robert bristow-johnson <rbj@audioimagination.com> writes:

> in article waderameyxiii-A24B0C.19534701052005@comcast.dca.giganews.com, The
> World Wide Wade at waderameyxiii@comcast.remove13.net wrote on 05/01/2005
> 22:53:
> 
> > In article <64y2d6kz.fsf@ieee.org>, Randy Yates <yates@ieee.org>
> > wrote:
> > 
> >>> 2. The absolute integral appears a lot: Integrate(|h(t)|, t from -inf to
> >>> +inf)
> >>> My question is:
> >>> 
> >>> Integrate(|h(t)|, t from -inf to +inf) < +inf
> >>> 
> >>> vs.
> >>> 
> >>> Integrate(|h(t)|^2, t from -inf to +inf) < + inf
> >>> 
> >>> which one is a subset of the other one? i.e. which one implies the other
> >>> one?
> >> 
> >> These integrals yield the total energy and the total power,
> >> respectively, in the signal.  A finite-energy signal is also a finite
> >> power signal, i.e.,
> >> 
> >> finite energy --> finite power.
> >> 
> >> That is the only implication one can make.
> > 
> > That's false for certain h: for example, h(t) = 1/sqrt(t) for 0 <
> > t < 1, h(t) = 0 elsewhere.
> 
> a finite energy signal has zero mean power and a finite power signal has
> infinite energy.
> 
> Randy, i'm not sure what you mean by "total" power.
> 
> a finite power signal is:
> 
>                       tau/2
>   lim      1/tau    integral{ |x(t)|^2 dt}    < inf
>  tau->inf            -tau/2
> 
> 
> and a finite energy signal is:
> 
>                       tau/2
>   lim               integral{ |x(t)|^2 dt}    < inf
>  tau->inf            -tau/2

I concur. The information in my post was wrong - my apologies to the OP. 

> the finite magnitude condition:
> 
>                       tau/2
>   lim               integral{ |x(t)|   dt}    < inf
>  tau->inf            -tau/2
> 
> is stronger than having a finite energy, but is sometimes needed to get
> certain other integrals (like the Fourier Transform) to converge.  however,
> even if the integral magnitude does not converge, the energy spectrum will
> be defined as long as it's a finite energy signal.  and the power spectrum
> will be defined if it's a finite power signal.

If by "power spectrum" you mean |X(w)|^2, I agree. If you mean "power
spectral density, I disagree. As a counterexample, take the case in
which Rxx(t) = \delta(t). Then the power spectrum is defined even though
the signal has infinite energy,

  Sxx(w) = \int_{-\infty}^{+\infty} Rxx(t) e^{-j*w*t} dt
         = 1.

I would think there's a similar situation for the energy spectral density.
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

robert bristow-johnson <rbj@audioimagination.com> writes:
> [...]
> Randy, i'm not sure what you mean by "total" power.

I mean the "average power" in the signal over all time:

   \lim_{\tau \rightarrow \infty} 1/\tau \int_{-\tau/2}^{+\tau/2} |x(t)|^2 dt

Contrast this to the power in the signal at some specific point in time t_0
and over some averaging time T,

  1/T \int_{t_0-T/2}^{t_0+T/2} |x(t)|^2 dt.

Is there something wrong with this language?
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

in article xxp7jih3n02.fsf@usrts005.corpusers.net, Randy Yates at
randy.yates@sonyericsson.com wrote on 05/02/2005 07:06:

>> the finite magnitude condition:
>> 
>>                      tau/2
>>  lim               integral{ |x(t)|   dt}    < inf
>> tau->inf            -tau/2
>> 
>> is stronger than having a finite energy, but is sometimes needed to get
>> certain other integrals (like the Fourier Transform) to converge.  however,
>> even if the integral magnitude does not converge, the energy spectrum will
>> be defined as long as it's a finite energy signal.  and the power spectrum
>> will be defined if it's a finite power signal.
> 
> If by "power spectrum" you mean |X(w)|^2, I agree.

i haven't been differentiating between "power spectrum" and "power spectral
density".

i would thing that |X(w)|^2 would be the "energy spectral density" and for
the finite energy signal, x(t).

 
                                   tau/2
   Sxx(w)  =   lim              |integral{ x(t) exp(-j*w*t) dt}|^2
              tau->inf            -tau/2
 
 
 
                                   tau/2
   Rxx(T)  =   lim               integral{ x(t-T/2) * conj(x(t+T/2))  dt}
              tau->inf            -tau/2
 
 
for a finite power signal, i would define "power spectrum" or "power
spectral density" to be

 
                                   tau/2
   Sxx(w)  =   lim      1/tau   |integral{ x(t) exp(-j*w*t) dt}|^2
              tau->inf            -tau/2
 

 
                                   tau/2
   Rxx(T)  =   lim      1/tau    integral{ x(t-T/2) * conj(x(t+T/2))  dt}
              tau->inf            -tau/2
 

 
and you can show that in both cases Sxx(w) = FT{ Rxx(t) } .  i know these
are the definitions exactly in the books, but i think they are equivalent,
and i like keeping the symmetry.
    

> If you mean "power
> spectral density, I disagree. As a counterexample, take the case in
> which Rxx(t) = \delta(t). Then the power spectrum is defined even though
> the signal has infinite energy,
> 
> Sxx(w) = \int_{-\infty}^{+\infty} Rxx(t) e^{-j*w*t} dt
>        = 1.

it also has infinite power.  i am still sorta loathe to actually do
anything, either theoretically or practically with white noise because of
that.

> I would think there's a similar situation for the energy spectral density.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Randy Yates wrote:
> robert bristow-johnson <rbj@audioimagination.com> writes:
> > [...]
> > Randy, i'm not sure what you mean by "total" power.
>
> I mean the "average power" in the signal over all time:
>
>    \lim_{\tau \rightarrow \infty} 1/\tau \int_{-\tau/2}^{+\tau/2}
|x(t)|^2 dt
>
> Contrast this to the power in the signal at some specific point in
time t_0
> and over some averaging time T,
>
>   1/T \int_{t_0-T/2}^{t_0+T/2} |x(t)|^2 dt.
>
> Is there something wrong with this language?

not too much.  "total" power means that i am summing up a bunch of
different powers somewhere and there is a total sum.  "average" or
"mean" is exactly correct.  "total energy" has more meaning to me in
the present context than "total power".

r b-j

"robert bristow-johnson" <rbj@audioimagination.com> writes:

> Randy Yates wrote:
> > robert bristow-johnson <rbj@audioimagination.com> writes:
> > > [...]
> > > Randy, i'm not sure what you mean by "total" power.
> >
> > I mean the "average power" in the signal over all time:
> >
> >    \lim_{\tau \rightarrow \infty} 1/\tau \int_{-\tau/2}^{+\tau/2}
> |x(t)|^2 dt
> >
> > Contrast this to the power in the signal at some specific point in
> time t_0
> > and over some averaging time T,
> >
> >   1/T \int_{t_0-T/2}^{t_0+T/2} |x(t)|^2 dt.
> >
> > Is there something wrong with this language?
> 
> not too much.  "total" power means that i am summing up a bunch of
> different powers somewhere and there is a total sum.  

Yes, I see how the phrase could be taken that way. However, it doesn't
necessarily mean that. Webster's (http://www.webster.com/) defines
"total" in definition 2 as

  not lacking any part or member that properly belongs to it <gave us a
  total rundown of the events> 

It is this sense that I mean. 

> "average" or
> "mean" is exactly correct.  

True, but one can have an "average" or "mean" over a finite extent at
a specific place in the signal as well. We need a term that does not
impart the wrong connotation but also distinguishes the
infinite-extent case from this case.

> "total energy" has more meaning to me in
> the present context than "total power".

I see your point, but you don't have to choose that connotation of
total, as I stated above. 
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124