Source Filter Model Query

[Isobel W]

In the equation y[t]=0.1x[t-4]+0.3x[t-3]+0.5x[t-2]+0.3x[t-1]+0.1x[t]
I understand that the part in bold is the trivial filter – 1x the input at that time – the impulse train.
However, I still wonder where the coefficients come from? Are these coefficients the Fourier coefficients? To recreate a speech signal would you do a DFT transform on a speech sound such as [a] and use the coefficients to reproduce this artificially?
Further, I am probably missing something obvious here but why do four coefficients produce only two resonant peaks?
Thanks

[Simon]

The form of filter you give only has terms involving x[.] on the right hand side (“RHS” in maths jargon). This is a Finite Impulse Response (FIR) filter, and you can explore that in one of the Module 2 Jupyter notebooks.

The equation operates in the time domain, and the co-efficients are simply weights applied to input (x) samples: the output is nothing more than a weighted sum of input samples.

Importantly, there is no term involving the output (y) on the RHS. This means there is no “feedback”. For any given input, the filter’s output will only continue for a finite time (the “F” in FIR) after the input stops.

In contrast, if we start putting some y[.] terms on the RHS, then there will be some feedback, and the filter can potentially produce output for an infinite duration after the input has ceased. This form of filter can exhibit resonance, and so is the form we will use to model the vocal tract filter.

You can explore Infinite Impulse Response (IIR) filters in one of the Module 2 Jupyter notebooks.

Try the notebooks, then post follow-up questions.

[Isobel W]

Hello, I have had a look at the notebooks now and I understand that there are two main types of filters, the finite and the infinite impulse response and that the latter is more useful for simulating the vocal tract .
What I am unclear where we get the coefficients from in the operation to create the filter. How does this relate to the Fourier transform? Do we do the fourier transform first and then filter it? In the end I am struggling with how fourier is or isn’t related to filtering.
e.g – we want to recreate a sound [a] and so we use and IIR filter x by the impulse train – what does fourier then give us on top of this – why not use the fourier series when this series tells us the amount of each frequency in the signal?

[Simon]

The co-efficients in the filter equations are weights applied to speech samples. This representation of the filter is actually very hard to interpret. The relationship between the coefficients and the frequency response is complicated and not something we are going to cover in Speech Processing.

For this course, the way we will understand the filter’s behaviour is empirical rather than theoretical. We will excite the filter with an impulse and inspect the output impulse response. We can take the DFT of that to obtain the filter’s frequency response.

A key point is that there are multiple domains in which we can define and characterise the filter. One is the filter coefficients in the difference equation. Another is the filter’s impulse response – a pitch period of speech waveform. Yet another is its frequency response.

So, where do the filter co-efficients come from? Conceptually, this is straightforward: take a pitch period of speech and find a set coefficients that gives that output, when the input is an impulse. You might imagine doing that by trial and error (and can do exactly that in the notebooks).

The formal method for finding a set of filter coefficients that give the desired output speech signal involves solving a set of simultaneous equations; there are various methods available for this. They are out-of-scope for this course: we just want you to understand at a conceptual level.

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