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").

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Z-transform derivation

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

<math>y(n) = \sum_{k=0}^{P-1} h(k) x(n-k)<math>

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

<math>H(z) = \sum_{k=0}^{P-1} h(k) z^{-k} = h(0) + h(1) z^{-1} + \cdots + h({P-1})z^{-(P-1)}<math>

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

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)

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