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.
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 $x[n] = [1,1,1,1,1]$
