Parseval theorem - Fourier series, Fourier transform [Complex math question]

[nForce]

When analyzing electrical signals, we use Fourier transform for aperiodic signals and Fourier series for periodic ones.
But why do we use only power for periodic ones, and only energy for aperiodic in terms of Parseval's theorem?

Thank you.

[rhb]

Definition of the discrete transform vs the continuous transform.  The discrete signal has infinite energy, so by convention one uses the power over a single period.

In practice it is assumed that you know which version to use.  It is the statement of equivalence of the time and frequency domain results that matter.

There are a lot of minor distinctions like this. For the most part it doesn't matter.  But sometimes it does and woe if you overlook it.  I have an entire book which is just theorems related to the Fourier transform.  I rarely ever reference it, but it's a real time saver if I find myself dealing with edge cases.

[coppice]

When analyzing electrical signals, we use Fourier transform for aperiodic signals and Fourier series for periodic ones.
But why do we use only power for periodic ones, and only energy for aperiodic in terms of Parseval's theorem?

Thank you.

Parseval's theorem is basically just a statement of the conservation of energy, so it always works in terms of energies over the long term. However, the repetitive nature of periodic signals means its also true that the average power of the whole signal over a period and the sum of the average powers of the components over a period are equal.

I use the words "long term" because the energy over the infinite is problematic. Unless the signal is exactly zero for most of history, the energy over the whole of history tends to infinite.

If you look at a lot of DSP code, people can be really sloppy talking about power when they mean energy and vice versa. Take care that when they say power or energy they really mean what they say.

[Wimberleytech]

When analyzing electrical signals, we use Fourier transform for aperiodic signals and Fourier series for periodic ones.
But why do we use only power for periodic ones, and only energy for aperiodic in terms of Parseval's theorem?

Thank you.

Not sure I understand what you are getting at.  Parseval's theorem just informs me that I can calculate energy in the time domain and frequency domain and they will be the same. 

[nForce]

When analyzing electrical signals, we use Fourier transform for aperiodic signals and Fourier series for periodic ones.
But why do we use only power for periodic ones, and only energy for aperiodic in terms of Parseval's theorem?

Thank you.

Not sure I understand what you are getting at.  Parseval's theorem just informs me that I can calculate energy in the time domain and frequency domain and they will be the same.

That is correct, I thought that it was connected with Parseval's theorem. So because we can't calculate power for aperiodic and we can't calculate energy for periodic? So periodic signals are power signals, and aperiodic signals are energy signals?

[coppice]

So periodic signals are power signals, and aperiodic signals are energy signals?

Any signal has a power level for every instant of its history. Any signal also has an energy content over any defined period, which is just the integral of the power over that period. If the period is infinite, as is the case with a signal being passed through a true Fourier transform (as opposed to the ones we use in the real world, which are truncated in time) the energy will integrate to infinity for any signal which is not exactly zero for most of history. This makes Parseval's Theorem a little hard to deal with, as infinity is problematic.

For a periodic signal you can truncate the Fourier transform to exactly one period of the signal, with no side effects, and get finite values for the total energy and the component energies over that period. That makes it easy to apply Parseval's theorem.

[TheUnnamedNewbie]

When analyzing electrical signals, we use Fourier transform for aperiodic signals and Fourier series for periodic ones.
But why do we use only power for periodic ones, and only energy for aperiodic in terms of Parseval's theorem?

Thank you.

So periodic signals are power signals, and aperiodic signals are energy signals?

If the signals are periodic, it means it has a finite period and will thus repeat infinity many times. In other words, it's energy must either integrate to infninity or be zero. In other words, speaking about the energy of the signal is meaningless. However, we can consider the average power of the signal. (These are hence called power signals).

If the signal is non-periodic, the signal can in fact have a finite amount of energy (But doesn't have to. For example, the signals x(t) = t or y(t) = exp(t) have no period yet have infinity energy) . Examples include impulses, datastreams, etc. Yes, they are zero for most of the time, but you can also think of signals with a finite amount of instances t_zero where x(t) = 0, without having infinite energy. If you want I can go find some examples. If it has a finite amount of energy, it will have a zero average power (which is obvious from the definition of average power). Hence. there is no point talking about the power of this signal, and we talk about the energy instead. These are energy signals. Again, note that not all non-periodic signals are energy signals!


Because of the above, it makes no sense to try and work with the power of a aperiodic signal because it would just end up as zero. Likewise, we can't use energy for a periodic signal, because we can't make any meaningfull statement about the energy.

[rstofer]

I have an entire book which is just theorems related to the Fourier transform.  I rarely ever reference it, but it's a real time saver if I find myself dealing with edge cases.

Could you post the title and author?  I might want to buy a copy.
Thanks!

[rhb]

A Handbook of Fourier Theorems
D.C. Champeney
Cambridge University Press 1987
ISBN 0-521-36688-7

It's slightly under 200 pages.  I've only had to use it a few times, but the author tried to collect all the fine print related to the Fourier transform and series in one place; written as compactly as easy understanding would permit.  I've been glad to have it, as the alternative was searching though a much larger set of books.  It is intended for readers who are very familiar with applying Fourier theory and find themselves wandering down a dark alley in a dubious neighborhood.  The first 3/4ths of the book is on the continuous case.  The treatment of the discrete case is a bit thin. 

I got beat up pretty severely once resampling by FFT for sample rates which did not have a common divisor.  It took 2 weeks to figure out the cause of the phase shifts at the end of the series.  The DFT is defined on the semi-closed interval [-Pi:Pi).  Champeney states it incorrectly as being on the closed interval. To resample properly requires than m*dT1 equal n*dT2.  so both series need to be padded with zeros to enforce that condition.  This is the one instance in my career where I violated my rule of never writing code until I had run out of excuses for not writing.  It was a *very* painful two weeks.  But I'd been sampling by Fourier transform for years.  Just not for series with long stretches of missing data.