Finite impulse response
This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
FIR filters can be discrete-time or continuous-time, and digital or analog.
For a causal discrete-time FIR filter of order N, each value of the output sequence is a weighted sum of the most recent input values: where: This computation is also known as discrete convolution.
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.
The impulse response of the filter as defined is nonzero over a finite duration.
Including zeros, the impulse response is the infinite sequence: If an FIR filter is non-causal, the range of nonzero values in its impulse response can start before
FIR filters: The main disadvantage of FIR filters is that considerably more computation power in a general purpose processor is required compared to an IIR filter with similar sharpness or selectivity, especially when low frequency (relative to the sample rate) cutoffs are needed.
However, many digital signal processors provide specialized hardware features to make FIR filters approximately as efficient as IIR for many applications.
respectively denote the discrete-time Fourier transform (DTFT) and its inverse.
It is defined by a Fourier series: where the added subscript denotes
in units of cycles per sample, which is favored by many filter design applications.
Matched filters perform a cross-correlation between the input signal and a known pulse shape.
The FIR convolution is a cross-correlation between the input signal and a time-reversed copy of the impulse response.
[1] When a particular frequency response is desired, several different design methods are common: Software packages such as MATLAB, GNU Octave, Scilab, and SciPy provide convenient ways to apply these different methods.
In the window design method, one first designs an ideal IIR filter and then truncates the infinite impulse response by multiplying it with a finite length window function.
Multiplying the infinite impulse by the window function in the time domain results in the frequency response of the IIR being convolved with the Fourier transform (or DTFT) of the window function.
If the window's main lobe is narrow, the composite frequency response remains close to that of the ideal IIR filter.
The ideal response is often rectangular, and the corresponding IIR is a sinc function.
The result of the frequency domain convolution is that the edges of the rectangle are tapered, and ripples appear in the passband and stopband.
Working backward, one can specify the slope (or width) of the tapered region (transition band) and the height of the ripples, and thereby derive the frequency-domain parameters of an appropriate window function.
Another method is to restrict the solution set to the parametric family of Kaiser windows, which provides closed form relationships between the time-domain and frequency domain parameters.
The window design method is also advantageous for creating efficient half-band filters, because the corresponding sinc function is zero at every other sample point (except the center one).
An appropriate implementation of the FIR calculations can exploit that property to double the filter's efficiency.
Goal: Method: In addition, we can treat the importance of passband and stopband differently according to our needs by adding a weighted function,
The transfer function is: The next figure shows the corresponding pole–zero diagram.
But plots like these can also be generated by doing a discrete Fourier transform (DFT) of the impulse response.
[B] And because of symmetry, filter design or viewing software often displays only the [0, π] region.
The phase plot is linear except for discontinuities at the two frequencies where the magnitude goes to zero.
The size of the discontinuities is π, representing a sign reversal.
They do not affect the property of linear phase, as illustrated in the final figure.
