energy at every multiple

[Ross C]

I don’t understand what it means for there to be energy at every multiple of F0. Please see attached image.

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[Simon]

You are confusing the time domain and frequency domain. A signal can have energy at some frequency F without there being an impulse occurring every 1/F seconds.

Develop your intuitions with this tool (tip: you can use your mouse to draw any waveform you like; also, use the ‘Mag/Phase View’ rather than sines and cosines) and let me know if that helps.

[Ross C]

I was starting from https://speech.zone/courses/speech-processing/module-2-basics-speech-production/videos/impulse-train/ at 1m00s (i.e. it is in time domain)

After some Slack debate, I think we got to this answer:

The impulse train is a composite wave, which means it’s made up of a bunch of different sine/cosine waves. The way this wave would occur is if you have a cosine wave at every multiple of the fundamental frequency, at the same amplitude. When you add them together, you’ll get this graph on your image. If you do a Fourier transform (decomposing the complex wave back to a series of simple cosine waves) of this graph, you’ll get the frequency domain graph where indeed it’ll show that there is a frequency spike at every integer multiple of F0, which is what made up the impulse train in the first place.

Thanks for the tool!

[Matt M]

I have kind of a related question:

What determines if a wave has energy at every harmonic, and how much energy there is at each? E.g. Why does the impulse train have those properties whilst the sine wave doesn’t?

Is this factor(s) shown by the time-domain waveform anywhere?

I can’t get any kind of intuitive sense for it when just looking at a wave, is my problem.

[Simon]

It’s really not easy to look at a waveform and guess how much energy it will have at each harmonic (except in some special cases like the above). That’s why we prefer to inspect signals in the frequency domain.

Sounds like you are spending too much time looking at waveforms and not enough time with the spectrum?

Periodic signals

All periodic signals have energy only at multiples of the fundamental frequency (which are called the harmonics). We can see the periodicity in the waveform, but not much else.

How much energy at each harmonic is what differentiates one signal from another.

Special cases (where inspecting the waveform makes sense)

The sine wave is the simplest case: it has energy at the fundamental frequency only and no energy at all the other multiples.

The impulse train has an equal amount of energy at every multiple of the fundamental.

A square wave has energy at all the odd multiples of the fundamental and no energy at the even multiples.

The general case (where inspecting the waveform is of limited use)

Voiced speech has energy at all multiples of the fundamental (in common with the impulse train) but the amount of energy varies with frequency (why?) and so voiced speech does not sound like an impulse train.

[Matt M]

So, combining discussion on Slack and your explanation, is the following correct?

—

The sine wave is a pure tone because it’s made up of only one wave, itself.

The impulse train is the composite of a bunch of basis functions, and thus involves many (co)sine waves (each at a harmonic of its fundamental frequency and each with its own weighting in the Fourier transform).

In the impulse train specifically, we have a basis function at each of its harmonics (whereas in other signals, we might skip some) which causes it to have some amount of energy at each harmonic, and these basis functions all have equal weighting (unlike most signals) which causes each of the harmonics to have an equal amount of energy?

Whereas if for instance I took the series of basis functions that comprises the impulse train and doubled the weighting on the 400Hz one (but no others), I’d get a spectrogram identical to the impulse train except that the 400Hz point would have twice as much energy.

And because the sine wave only has one wave making it up, there’s only one frequency (its own) that has energy.

[Simon]

Yes, that’s all correct.

[Author]

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