Filter
[see page 14, Modifies] the properties of a signal.
A filter can be [see page 1, defined] as \[ y[k] = x[k] \times h[k] \] where \(x[k]\) is the value of the input signal at time \(k\), and \(h[k]\) is the filter transformation. \(y[k]\) is the signal outputted by the filter.
Typically characterised by it's [see page 14, frequency-response] in the frequency-domain:
| FR | Meaning |
|---|---|
| Low Pass | Allow only low frequencies through. |
| High Pass | Allow only high frequencies through. |
| Band Pass | Lets through a certain range of frequencies. |
| Band Stop | Cuts away certain frequency ranges. |
When a signal is sent through one of these filters only the parts of the signal the filter lets through are [see page 14, included] in the output.
The output of a linear-filter in time-domain is a discrete Convolution of the filter and the input signal.
Definition and Complexity
A filters behaviour is completely characterised by its poles and zeros. The [see page 3, complexity] of a filter is characterise by the order of these P&Z. We define order as the number of past input values involved in the calculation (eg. a filter using two values is labelled a \nth{2}-order filter).
Applications in speech-processing
- [see page 10, Notch Filtering] - Removing interference and noise.
- [see page 11, Pre-emphasis] - Lip radiation offsets our speech spectrum, filters can offset it back.
- [see page 12, Modelling speech production] - Source Filter Model (eg. [see page 13, gammatone filter]).
- Modelling the auditory system