Finite Impulse Response (FIR) filters are digital filters which have - as the name suggests - a finite impulse response. These filters do not have an internal feedback and can be applied simply by convolving an input sequence $x[n]$ with the filter's impulse response $h[n]$ to get the output sequence $y[n]$ :

$y[n] = x[n] * h[n]$

For FIR filters, the impulse response $h[n]$ is a one-dimensional array containing the $N$ filter coefficients. The following example shows the coefficients, respectively the impulse response, for a filter of the order $N=5$ . A filter of the order $N$ has $M=N+1$ coefficients, since $b_0$ - the gain of the undelayed signal - is not considered in this case. The filter order equals the maximum delay of the filter.

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The Difference Equation¶

The difference equation can be used for calculating the output $y[n]$ of an FIR filter. It is a sum over the multiplication of all filter coefficients with the respective delayed sample of the input signal $x[n]$ . For the fifth order FIR this results in:

$y[n] = b_0 x[n] + b_1 x[n-1]+ b_2 x[n-2]+ b_3 x[n-3]+ b_4 x[n-4] + b_5 x[n-5]$

The general form of the difference equation for FIR filters is the following sum equation:

$y[n] = \sum\limits_{i=0}^{i=N} b_i x[n-i]$

Implementation Structure¶

Another important representation of a filter is the implementation structure. It can be directly derived from the difference equation and vice versa. The signal flow diagram below shows the direct form of a fifth order FIR, which is canonical. A canonical implementation of a filter uses the minimum number of delay elements.

Each $z^{-1}$ block represents a delay by one sample.

Exercise¶

The plot below shows an impulse response $h[n]$ of an FIR filter.

1: Determine the order and the coefficients of the filter.
2: Write the difference equation.
3: Draw the signal flow graph with coefficients.
4: Calculate $y[n]$ for $x[n] = [1,1,1,1,1]$