Low Pass Filter Design

Low pass filters are used to separate the frequency components of a signal and pass the lower frequencies while filtering out the higher frequencies. This divides the frequency spectrum into a low-frequency pass band, a transition band and a high-frequency stop band. A low pass filter is characterized by the limits and characteristics of each band and by a few overall characteristics.

The pass band extends from DC (0 Hz) up to the beginning of the transition band, 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. The transition band covers the portion of the spectrum between the pass band and the stop 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 pass and stop bands, and will, in general, force either a poorer fit or a longer filter. The stop band extends from the transition band upwards, and is characterized by a rejection level which represents the minimum acceptable attenuation.

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 filterare 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.

Low pass filters are typically designed to have even symmetry and odd length. Even symmetry is necessary, as odd symmetry filters force the DC response to be zero. Even symmetry, even length designs are also possible, and may be particularly useful when it is absolutely necessary to guarantee zero response at the folding frequency or to provide non-integer sample delays.

As an example, consider the following low pass audio filter designed with even symmetry and a sampling frequency of 10 KHz:

When the filter is generated with 7 taps, the coefficients and frequency response graph are as follows:

Taps: -0.0870309 0.0468421 0.352007 0.529105 0.352007 0.0468421 -0.0870309

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