Finite Impulse Response - Definition

Definition

The output y of a linear time invariant system is determined by convolving its input signal x with its impulse response b.

For a discrete-time FIR filter, the output is a weighted sum of the current and a finite number of previous values of the input. The operation is described by the following equation, which defines the output sequence y in terms of its input sequence x:

\begin{align} y &= b_0 x + b_1 x + \cdots + b_N x \\ &= \sum_{i=0}^{N} b_i x \end{align}

where:

  • is the input signal,
  • is the output signal,
  • are the filter coefficients, also known as tap weights, that make up the impulse response,
  • is the filter order; an th-order filter has terms on the right-hand side. The in these terms are commonly referred to as taps, based on the structure of a 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.

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