Linear Predictive Coding (LPC) – what is the residual?

[Waylon L]

After watching the video, I’m still quite confused about exactly what a residual is. Why does inverse filtering create the residual and how can the residual help smoothing.

[Simon]

The residual is a special waveform. It is what you need to input to the filter in order to exactly reconstruct the speech signal.

The filter is not a perfect simulation of the vocal tract. The vocal folds also do not generate a perfect impulse train. Therefore, putting a impulse train through an LPC filter will not produce perfectly natural speech.

We really like the simple form of the filter because it is easy to solve equations to find its coefficients, given a frame of natural speech waveform. So, we account for the imperfections in the model by replacing the pulse train with the residual, which contains all the information “left over” from the original speech waveform that the filter was not able to model (that’s why it’s called the “residual”).

Join smoothing:

• we can manipulate the filter coefficients of a few frames around each join, if we wish to remove discontinuities in the spectral envelope.
• we can use PSOLA on the residual (it’s just a waveform) to manipulate the fundamental frequency, if we wish to remove pitch discontinuities across joins

So, in residual-excited LPC (RELP) we still need to use PSOLA to manipulate F0 and duration of the residual! Why not do that processing on the speech waveform (i.e., TD-PSOLA)? It’s because PSOLA works better on residual waveforms than on speech waveforms. This is because the residual is closer to an impulse train and so overlap-add creates fewer artefacts.

[Yichao L]

In the video for inverse filtering, I found it hard to perceive why the spectrum of the residual is flat, like the spectrum of a white noise?

Would you mind elaborate a bit more on this? (I attached the slide below where the “flat spectrum”, circled in red dash line, appeared).

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[Simon]

The spectral envelope of the speech signal (the output of the source-filter model) is equal to the frequency response of the filter. That is, the filter is solely responsible for shaping the speech spectral envelope.

In the frequency domain, we multiply frequency responses together, so the residual spectrum has to be flat so that when it is multiplied by the filter frequency response we get the speech spectral envelope.

In the idealised (simplified) version of the model, the residual is replaced with a train of impulses. This has a flat spectral envelope, as you will have discovered in the lab when analysing that signal. It has harmonic structure, but all harmonics are of the same amplitude.

[Author]

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