High Pass Filter Design
High pass filters are used to separate the frequency components of a signal and pass the higher frequencies while filtering out the lower frequencies. This divides the frequency spectrum into a low-frequency stop band, a transition band and a high-frequency pass band. A high pass filter is characterized by the limits and characteristics of each band and by a few overall characteristics.
The stop band extends from DC (0 Hz) up to the beginning of the transition band, and is characterized by a rejection level which represents the minimum acceptable attenuation. The transition band covers the portion of the spectrum between the stop band and the pass band; it represents an unspecified region within which no constraints are placed on the filter. A narrow transition band indicates a sharp transition between the stop and pass bands, and will, in general, force either a poorer fit or a longer filter. The pass band extends from the transition band upwards, and is characterized by a desired gain value (usually +0 dB or x1) and a ripple value which measures the maximum desired deviation from that gain value.
In addition to the limits and characteristics of each band, an FIR filter design depends on the sampling frequency of the input data. In fact, the usable frequency range is between DC (0 Hz) and half the sampling frequency, which is called the folding frequency. A filter designed for data sampled at one frequency will not behave the same with data sampled at a different rate: the band characteristics remain the same, but the band limits change proportionally with the change in sampling frequency.
Two additional characteristics of an FIR filter are its length and symmetry. The length refers to the number of “taps” or coefficients; increasing the length generally improves the fit at the expense of design speed, execution speed and sample delay. Even length and odd length filters display significantly different behavior. Filter symmetry (even, odd or none) refers to the symmetry of the coefficients and is independent of the evenness or oddness of filter length, although there is some interplay between the two characteristics.
High pass filters are typically designed to have even symmetry and odd length. Even symmetry, even length filters are inappropriate, as are odd symmetry, odd length filters; both of these cases force the folding frequency response to be zero. Odd symmetry, even length designs are sometimes used, especially to provide phase shifting effects or non-integer sample delays.
As an example, consider the following high pass audio filter designed with even symmetry and a sampling frequency of 10 KHz:
|Stop||0 KHz||2 KHz||20 dB|
|Pass||2.5 KHz||5 KHz||0.2||3|
When this filter is generated with 25 taps, the filter coefficients and frequency response graph (in decibels) are as follows:
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