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A digital filter is an operation that associates an input time series x[n] into an output one, y[n]. Methods developed in the LTPDA Toolbox deal with linear digital filters, i.e. those which fulfill that a linear combination of inputs results in a linear combination of outputs with the same coefficients (provided that these are not time dependent). In these conditions, the filter can be expressed as

described in these terms, the filter is completely described by the impulse response h[k], and can then be subdivided into the following classes:

- Causal: if there is no output before input is fed in.
- Stable: if finite input results in finite output.
- Shift invariant: if time shift in the input results in a time shift in the output by the same amount.

Digital filters can be described as difference equations. If we consider an input time series x and an output y, three specific cases can then be distinguished:

- Autoregressive (AR) process: the difference equation in this case is given by:

AR processes can be also classified as IIR Filters.

- Moving Averrage (MA) process:the difference equation in this case is given by:

MA processes can be also classified as FIR Filters.

- Autoregressive Moving Average (ARMA) process: the difference equation in this case contains both an AR and a MA process:

Signal Processing in LTPDA | IIR Filters |

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