## 8.1. Theoretical background

Unlike IIR filters, finite impulse response (FIR) filters are specific to the discrete-time domain and cannot be obtained by direct transposition of analog filters. The linear phase of the transfer function makes them useful for many applications.

### 8.1.1. *Transfer function and properties of FIR filters*

For this type of filter, also called non-recursive due to the implementation method, the transfer function is defined by:

where *h*[*n*] stands for the (finite) impulse response of the filter.

In order to avoid the phase discontinuities of the transfer function, *H*(ω) is expressed in the following form:

where the zero-phase transfer function *H*_{0} (ω) and the phase function θ(ω) are real and continuous.

In practice, the impulse response *h*[*n*] is required to be real, which means that the real part of *H* (ω) is even, while its imaginary part is odd. The linearity constraint for the phase function θ(ω) leads to the four FIR filter classes, whose properties are summarized in Table 8.1.

**Table 8.1**. *Properties of FIR filters*

Coefficients *a*_{i} and *b*_{i} in Table 8.1 can be obtained using the following equations:

The first two classes' filters are usually involved in pure filtering ...