A finite impulse response (FIR) filter is a type of a digital filter, that is normally implemented through digital electronic computation. The Z-transform of an FIR filter has only zeros and no poles. The number of coefficients in an FIR filter is its order (sometimes referred to as "taps").

Z-transform derivation

Given a time-invariant input signal x(n) and a P^th-order FIR filter h(n), the convolution of x with h is defined as follows:

The z-transform of h(n), denoted H(z) is defined as follows:

The z-transform of y(n) is then Y(z) = H(z)X(z).

Properties

A FIR filter has a number of useful properties which sometimes make it preferable to an infinite impulse response filter:

• FIR filters are inherently stable
• Require no feedback
• Can have linear phase

An FIR filter has linear phase if and only if its coefficients are symmetric about the center coefficient.

See also

• Digital filter
• Electronic filter
• Filter (signal processing)
• Infinite impulse response
• Z-transform (specifically Z-transform#LCCD equation)