User:Kapsberger/filters

A linear filter modifies its source's frequency spectrum by multiplying its own frequency response (the spectrum of its impulse response) with it.

In order to use the filter, we want to be able to describe its effect on the input (also called its response) independently of the form of that input. The trick is to use a unit impulse as the input: once we know the response of the filter to an impulse, we can describe its response to a longer signal simply by using the linearity principle of linear systems, which we will describe below.

The unit impulse is described as $$\delta[n] = \begin{cases} 1 & n=0\\0 & n\ne 0 \end{cases}\,$$.

The integral on the unit impulse is 1. The corresponding magnitude spectrum is flat with a magnitude of 1.

A longer signal can be filtered through the two stages of scaling and superposition of the linearity principle.

Linearity Principle
The Linearity Principle of linear systems states that a signal can be scaled when passing through the system and that two signals passing through the system together can be added, or superposed.

Scaling: $$\alpha x[n] \to \alpha y[n]\,$$

Superposition: $$x_1[n] + x_2[n] \to y_1[n] + y_2[n]\,$$

Linearity Principle: $$\alpha x_1[n] + \beta x_2[n] \to \alpha y_1[n] + \beta y_2[n]\,$$

Filters in Speech Recognition
Both the unit impulse and white noise have flat magnitude spectra. They therefore have the same amplitude at each frequency of their spectrum, leading to the fact that in the source-filter model, the multiplication of the source by the vocal tract response simply results in the shape of the filter being applied to the source. The speech frequency spectrum obtained is the magnitude spectrum resulting from Fourier Analysis or Fourier Transform.