Finite Impulse Response (FIR) Filters

Finite Impulse Response (FIR) filters are defined by scaled and time-delayed versions of the filter input signal only, as given by the following difference equation:
$\displaystyle y[n] = b_{0} x[n] + b_{1} x[n - 1] + b_{2} x[n - 2] \ldots, \hspace{0.2in} n = 0, 1, 2, \ldots,$
where the input $x[n] \ensuremath{\stackrel{\Delta}{=}}0$ and output $y[n] \ensuremath{\stackrel{\Delta}{=}}0$ for .
An FIR filter can be represented by a block diagram as shown in Fig. 3 below.

Figure 3: An FIR filter block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.5,3.0) \put(0.27,0.0){\epsfig{f... ...1.17){$b_{2}$} \put (2.4,2.72){$y[n]$} \end{picture} \end{center} \end{figure}$
The $z^{-1}$ terms represent unit delays. Note that while this representation for a delay element is common and widely accepted in the signal processing community, the specification of delay in terms of powers of is a -domain characterization (to be described below) while the block diagram itself is a time-domain representation.
With respect to the filter block diagram, FIR filters make use of feed-forward terms only.
The impulse response of an FIR filter is only as long as the maximum delayed input term in its difference equation.
The summation of feedforward input terms can result in destructive signal interference, or cancellations, at certain frequency values.
The FIR filter is said to have an order equivalent to the number of unit delays in its difference equation.