What is a perfect impulse signal made of?

[domtw]

By perfect impulse signal, I mean the pure mathematical object.

How is it constructed?

I am not sure if I will be able to express clearly enough my idea but here I try:

My intuition is that a perfect impulse signal is the sum of the sine wave of all the frequencies considered, each of the sine plot lasting for only 1 wavelength and all of them having their maximum (positive) amplitude at t=0

Is that right?
Or, is it one possibility among others?
Or my brain is cooking too much?

In the case it's right, the sine plot for 20Hz will last for 0.05ms, it will be constructed in the way that its maximum amplitude is centered at t=0 while the sine plot for 20KHz will last for 0.00005ms it will be constructed in the way that its maximum amplitude is centered at t=0 .
Once all the other reduced sine plot (one for each frequency) lasting for a short tlme:their respective wave length have been created we can sum them all to built up the impulse signal...

[Jmmlc]

Hello,

I am sorry to tell you that, unfortunately, a perfect pulse from the mathematical point of view is the result of the summation of an infinity of sine waves of the same alplitude (infinity of frequencies) each of one lasting an infinite number of periods (= wavelength).

Best regards from Paris, France

Jean-Michel Le Cléac'h

Originally Posted by domtw

My intuition is that a perfect impulse signal is the sum of the sine wave of all the frequencies considered, each of the sine plot lasting for only 1 wavelength and all of them having their maximum (positive) amplitude at t=0

[PLB]

Hi domtw,

I think the construction of an impulse is best illustrated by summing an infinite number of cosine waveforms, all of different frequency and all of which start at t=0. In the construction of the impulse, the cosine waveform (rather than a sine waveform) is used because at t=0 the cosine has its maximum value. So when t=0, the magnitude of an infinite number of cosine waveforms summed together, must be infinitely high.
At any time greater than t=0, we can assume that there are as many positive as there are negative half-wave cycles and that their sum must result in a magnitude of zero.
So an ideal impulse will have an infinitely high magnitude at t=0 and zero magnitude at any time greater than t=0.
Since all of the cosine waveforms start at t=0, the Fourier transform of the impulse contains phase information as well as magnitude. Hope this helps.

Regards.

Peter

[domtw]

Hello Jean Michel,
Don't be sorry, I am here to learn...

Thanks PLB,
You are right, a cosine is better than a sine...

I'll probably come back with more questions later as I need to assimilate and play with what I learned today...

Thanks to you two!

[JohnPM]

Sounds like you have things a little back to front. A perfect impulse is a delta function in time, i.e. it is zero except at t=0. In a sampled data world a perfect impulse is one sample non-zero, all others zero. Decomposing that impulse into sine & cosines, complex exponentials or any other basis functions are just attempts at representing the function in forms that may be more amenable for other processing - Fourier transform is just one way of performing such a decomposition.

[wavey1]

A signal step or impulse does not correspond to an infinite number of periods of an infinite sum of periodic signals. As JohnPM correctly points out - the leading edge of a square wave which is by definition - an impulse, consists of an infinite number of periodic signals of various frequency with infinitesmal duration.

[domtw]

I guess JohnPM is right if we are talking about a perfect impulse.
That is my fault, I have used the word "perfect" when I should have used "good approximation of a perfect impulse".
Please forgive my errors and continue helping me understanding this better.
Also forgive the way I explain mathematical thing as I studied math a long time ago and that was in French...

The following is just an essai to try to understand better, it has no value whatsoever, I only would like to know if it is right in principle.

I now suppose that a good approximation of a perfect impulse is by summing a great number of cosine wave form.

For this approximation of a perfect impulse there are no phase variation, for all frequencies the phase is 0.
In this case, the amplitude function of the frequency f and time t would be:
A(f,t)=cos(2.pi.f.t)

Example of a wave form for 30Hz



Then I consider that CPI(t), sum of the amplitude wave form from 20Hz to 20KHz in steps of 20Hz, is a good approximation of a perfect impulse.




Then I introduce an artificial phase variation that looks kind of similar to what we often see in speaker measurement: Phase(t)


In this case, the amplitude of a waveform would be:
Ar(f,t)=cos(2.pi.f.t+Phase(t))

Example for 30Hz


Finally the impulse response including the phase variation would be:


Is that correct in principle ?
Thanks for your attention and clarification...

[wavey1]

Rather than obsess over the "perfect" impulse for the purposes of evaluating a system's audio band performance - it makes sense to merely consider that which is adequate to the task. In that case, if you are going to adopt this discretized approach, it makes sense to apply the Nyquist-Shannon sampling theorem:


Nyquist?Shannon sampling theorem - Wikipedia, the free encyclopedia



for audible frequencies. This would mean that your "perfect impulse" would have a time duration of no more than .000025 seconds, corresponding to a frequency of 40Khz. Extending this pulse any further in time would change the definition of what you are trying to do from impulse response to steady state driven response. Other than this, I have no issue with what you are proposing.

[domtw]

I will read and understand this theorem later, thanks...

Just noticed that I made a mistake in my previous post, please read Phase(f) instead of Phase(t)
The graphs from Mathcad are correct...

Otherwise, I would like to know if you think that the sum of the amplitude wave form from 20Hz to 20KHz in steps of 20Hz, is a good approximation of a perfect impulse?
with the amplitude being A(f,t)=cos(2.pi.f.t)

[Pano]

I agree with John. In the digital domain the perfect impulse is 1 sample on, all others off.
At least that is what I see when looking at the waveform at the sample level.