Editing Finite impulse response (section)

==Definition==

[[File:FIR Filter.svg|thumb|300px|right|A direct form discrete-time FIR filter of order ''N''.  The top part is an ''N''-stage delay line with ''N'' + 1 taps.  Each unit delay is a ''z''<sup>−1</sup> operator in [[Z-transform]] notation.]]

[[File:FIR Lattice Filter.png|thumb|300px|right|alt=A depiction of a lattice type F I R filter|A lattice-form discrete-time FIR filter of order ''N''.  Each unit delay is a ''z''<sup>−1</sup> operator in [[Z-transform]] notation.]]

For a [[causal filter|causal]] [[discrete-time]] FIR filter of order ''N'', each value of the output sequence is a weighted sum of the most recent input values''':'''

:<math>\begin{align}
 y[n] &= b_0 x[n] + b_1 x[n-1] + \cdots + b_N x[n-N] \\
      &= \sum_{i=0}^N b_i\cdot x[n-i],
\end{align}</math>

where''':'''
*<math display="inline"> x[n]</math> is the input signal,
*<math display="inline"> y[n]</math> is the output signal,
*<math display="inline"> N</math> is the filter order; an <math display="inline"> N</math><sup>th</sup>-order filter has <math display=inline> N + 1 </math> terms on the right-hand side
*<math display="inline"> b_i </math> is the value of the impulse response at the ''i'''th instant for <math display=inline> 0 \le i \le N </math> of an <math display="inline"> N^\text{th}</math>-order FIR filter.  If the filter is a direct form FIR filter then <math display="inline"> b_i </math> is also a coefficient of the filter.

This computation is also known as [[discrete convolution]].

The <math display=inline> x[n-i]</math> in these terms are commonly referred to as ''{{visible anchor|tap}}s'', based on the structure of a [[Digital delay line|tapped delay line]] that in many implementations or block diagrams provides the delayed inputs to the multiplication operations.  One may speak of a ''5th order/6-tap filter'', for instance.

The impulse response of the filter as defined is nonzero over a finite duration. Including zeros, the impulse response is the infinite sequence''':'''
:<math>
  h[n] = \sum_{i=0}^N b_i\cdot \delta[n-i] = 
  \begin{cases}
    b_n & 0 \le n \le N\\
      0 & \text{otherwise}. 
  \end{cases}
</math>

If an FIR filter is non-causal, the range of nonzero values in its impulse response can start before <math>n=0</math>, with the defining formula appropriately generalized.