Impulse Train as Input Tutorial

[Isobel W]

Am really struggling with these questions in the tutorial about the DFT of a single impulse:
What does the magnitude spectrum show?
What does the phase spectrum show?
How might this be useful for modelling the vocal source?
I can see that the magnitude remains the same but other than that I am really stuck, I read further on and saw that this means a single impulse can be linked with any frequency but I don’t understand why.
I know that an impulse train contains an equal amount of energy at every harmonic – is it simply that by using the DFT we can change the fundamental frequency of the impulse train to suit our needs? Am probably way off base

[Catherine Lai]

If you look at the magnitude spectrum of a single impulse you’ll see that all the DFT output frequencies have non-zero magnitudes. This means if you send in a single impulse you potentially excite every possible frequency.

If we look at the DFT equation, we see that this happens because the impulse input sequence has exactly one non-zero value, e.g. [0,1,0,0,0,0,0]. This means that when we multiply it with a DFT phasor, we basicaly select one complex number sitting on the unit circle (magnitude 1) as the DFT output, no matter which DFT output frequency we’re analysing. That’s how we end up with non-zero magnitude for every DFT output when we apply the DFT to a single impulse.

If the mechanics of the DFT equation are too much right now, don’t worry!
Another way to think of an impulse is just as a burst of energy – like an infinitely short burst of air through your vocal folds (i.e. the source of the source-filter model). On it’s own it doesn’t tell you much. But if you put energy at different frequencies into a filter, you’ll get an idea of what that filter’s properties are by seeing which frequencies are boosted by the filter and which are attenuated. It’s a bit like blowing across the top of a bottle to make a flute-like sound come out.

The basic idea is that if you send an impulse into a filter (in mathematical terms, we’d do a convolution) and perform the DFT on the filter’s output, you can then see how that filter shapes the frequency spectrum. For example, does the filter boost low frequencies but dampen high frequencies (i.e. a low pass filter)? Or the opposite (i.e. a high pass filter)?

The physical filter we’re most interested in modelling for Speech Processing is, of course, the human vocal tract, but we could also think about other sorts of tube like objects a trumpet. In this case, if you put in impulses (i.e. air flow with glottal pulses) at regular intervals (i.e. an impulse train) then you’ll produce a wave with a fundamental period (T0) matching the time between impulses, and so you get a fundamental frequency of F0=1/T0. We also know that the DFT of an impulse train has a non-zero magnitude at every integer multiple of F0 (the harmonics, as you mentioned above).

In this way, we can model the pitch of the human voice (more frequent impulse, makes for a higher F0). But alongside this, each impulse also potentially excites the resonant properties of the filter. For our voices, the properties of the filter depend on how we shape/constrict the vocal track using articulators like our tongues. The frequencies that get boosted are the resonances of the vocal tract, but if you’ve already done some phonetics you might also know those resonances as formants: changing the vocal tract filter changes which vowels and consonants we hear!

Side note: If we go back to the discussion of DFT outputs including a magnitude and phase angle, we can also interpret the magnitude spectrum of a single impulse as saying that if we want to create a single impulse from a bunch of cosines, we basically need to add up cosines of all the frequencies we can get our hands on and each needs to be slightly shifted in phase. Similarly, if we want to make an impulse train from cosines, we need to add together versions of all cosines matching the frequencies of all the harmonics. The main takeaway from that is that it’s actually pretty hard work to make an impulse (sharp, spikey) from sinusoids (curvy)!

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