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FIR Window Filters

The FIR filters: Rectangular, Bartlett, Hanning, Hamming, Blackman, Kaiser, and Dolph-Chebyshev, are all “Window” FIR filters.  The use of the name "Window" arises from the fact that these filters are created by scaling a sinc (SIN(X)/X) pattern with a window to produce the desired frequency effect.  All FilterSolutions FIR Window filters create the same sinc pattern, which is the Fourier Transform of a square pulse.

The FIR selection then determines the correct window used to scale the sinc pattern; thereby creating the desired filter. While all of the Windows shown below appear similar, they are, nonetheless, unique. The Kaiser and Dolph-Chebyshev are adjustable. It is the uniqueness of each Window that creates the unique FIR Window filter:

 The first graph is a sinc pattern for the 29 tap low pass filter depicted in the subsequent table.  The pattern is used in the initial step of creating the FIR filter. The number of taps is the order of the filter, plus 1, to account for the 0 term. The value of the taps at the sinc curve is scaled by the Window and then is installed directly into the Z-Transform numerator.

The following table depicts the various Window functions supported by FilterSolutions. The sinc pattern, H(n), shown above, is scaled by one of the Windows shown below, W(n), to create the final Z-Transform numerator.  Each "n" on the X axis becomes a Z**n in the Z-Transform. The references to N in the equations refer to the total number of taps in the filter, 29 in this example: