Finite impulse response (FIR) filters, also known as non-recursive filters and convolution filters, are digital filters that have a finite impulse response. FIR filters operate only on current and past input values and are the simplest filters to design. FIR filters perform a convolution of the filter coefficients with a sequence of input values and produce an equally numbered sequence of output values. The following equation defines the finite convolution an FIR filter performs.
where x is the input sequence to filter, y is the filtered sequence, and h is the FIR filter coefficients.
FIR filters have the following characteristics:
- FIR filters can achieve linear phase because of filter coefficient symmetry in the realization.
- FIR filters are always stable.
-
FIR filters allow you to filter signals using the convolution. Therefore, you generally can associate a delay with the output
sequence, as shown in the following equation:
delay = (n - 1)/2
where n is the number of FIR filter coefficients.
The following figure shows a typical magnitude and phase response of an FIR filter compared to normalized frequency.
In the previous figure, the discontinuities in the phase response result from the discontinuities introduced when you use the absolute value to compute the magnitude response. The discontinuities in phase are on the order of π. However, the phase is clearly linear.