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Recursive Digital Filters: A Concise Guide

The book provides a quick introduction to recursive digital
filters. The goal is to get the reader to the point where he can
understand and use these filters as quickly as possible. The amount of
mathematical background material is kept to a minimum and many examples are included.

More information ...

All the software on this page is free and distributed under the terms of the GNU General Public License. It is written in ANSI C and should compile with any C compiler.

Digital signal generation software can be found here.

The following are command line filter programs. They read data from stdin and write to stdout. Both input and output streams are in ascii format.

- bwlpf.c - 2, 4, 6,... order lowpass filter.
Usage: bwlpf n s f Butterworth lowpass filter. n = filter order 2,4,6,... s = sampling frequency f = half power frequency

- cheblpf.c - 2, 4, 6,... order lowpass filter.
Usage: cheblpf n e s f Chebyshev lowpass filter. n = filter order 2,4,6,... e = epsilon [0,1] s = sampling frequency f = cutoff frequency

- bwhpf.c - 2, 4, 6,... order highpass filter.
Usage: bwhpf n s f Butterworth highpass filter. n = filter order 2,4,6,... s = sampling frequency f = half power frequency

- chebhpf.c - 2, 4, 6,... order highpass filter.
Usage: chebhpf n e s f Chebyshev highpass filter. n = filter order 2,4,6,... e = epsilon [0,1] s = sampling frequency f = cutoff frequency

- bwbpf.c - 4, 8, 12,... order bandpass filter.
Usage: bwbpf n s f1 f2 Butterworth bandpass filter. n = filter order 4,8,12,... s = sampling frequency f1 = upper half power frequency f2 = lower half power frequency

- chebbpf.c - 4, 8, 12,... order bandpass filter.
Usage: chebbpf n e s f1 f2 Chebyshev bandpass filter. n = filter order 4,8,12,... e = epsilon [0,1] s = sampling frequency f1 = upper half power frequency f2 = lower half power frequency

- bwbsf.c - 4, 8, 12,... order bandstop filter.
Usage: bwbsf n s f1 f2 Butterworth bandstop filter. n = filter order 4,8,12,... s = sampling frequency f1 = upper half power frequency f2 = lower half power frequency

- chebbsf.c - 4, 8, 12,... order bandstop filter.
Usage: chebbsf n e s f1 f2 Chebyshev bandstop filter. n = filter order 4,8,12,... e = epsilon [0,1] s = sampling frequency f1 = upper half power frequency f2 = lower half power frequency

The following files are for a library of functions to calculate Butterworth filter coefficients. There are functions for lowpass, bandpass, highpass, and bandstop filters. If you just want an efficient implementation of these filters then see the programs listed above.

The following are example programs that use liir.c.

- bwlp.c - lowpass filter coefficient calculator.
- bwhp.c - highpass filter coefficient calculator.
- bwbp.c - bandpass filter coefficient calculator.
- bwbs.c - bandstop filter coefficient calculator.
- Makefile - creates libiir.a and compiles example programs.

The example programs take the order of the filter and the cutoff
frequencies as command line input parameters. Run the programs with no
parameters to get a usage message showing how to list the parameters. The
output is a file that lists the `c`

and `d`

coefficients
of the filter (see the documentation in iir.txt for an explaination of what this
means)

**Calculating cutoff frequencies:** The cutoff frequencies are
specified as a fraction of π. If `f`

is the frequency
at which the data is sampled (samples per second) and _{s}`f`

is the
cutoff frequency you want, then the parameter you would input to the program
is `2*f/f`

. For example if your data is sampled at 5000
samples/sec and you want a bandpass filter from 500 to 1000 Hz the lower
cutoff would be 2*500/5000=0.2 and the upper cutoff would be
2*1000/5000=0.4. Here's a calculator that will calculate this for you._{s}

**Other recursive filter programs:**

rffr.c - Calculates the frequency response of a recursive
filter using the coefficient file created by one of the above programs.

rdf.c - Filters data from an input file, using a
coefficient file generated by one of the above programs.

A general nonrecursive filter is implemented as follows:

`y[n] = c`

_{0}*x[n] + c_{1}*x[n-1] + ... + c_{M}*x[n-M]

The n^{th} output is a linear function of the n^{th} input
and the previous M inputs. The c_{k} coefficients are calculated to
give the filter a specific frequency response.

The three programs listed below calculate the c_{k} coefficients for
implementing a lowpass, bandpass, and highpass filter. The filters are simple
(no smoothing) approximations to ideal filters, and have linear phase. The
programs take command line input parameters. The parameters are the number of
coefficients and the cutoff frequencies. The more coefficients, the sharper
the cutoff. The cutoff frequencies are specified as a fraction of Pi. See the
discussion under "Recursive Digital Filters" for how to calculate the cutoff
frequencies. An additional parameter is the number of zeros to pad on to the
end of the coefficient list. This is useful if you want to take the FFT of the
coefficients in order to apply the filter in the frequency domain, or if you
just want to see what the frequency response of the filter is. You can set
this parameter to zero. Running the programs with no parameters will print out
a usage message showing the correct order in which the parameters should be
given.

tdlpf.c - Calculates lowpass filter coefficients.

tdbpf.c - Calculates bandpass filter coefficients.

tdhpf.c - Calculates highpass filter coefficients.

goertzel.c - A program for doing spectrum calculations using the Goertzel algorithm. It allows spectrum calculation at arbitrary points, whereas the FFT calculates the spectrum at evenly spaced points.

fft.c - FFT program. Takes the FFT of a data (time domain) file, and outputs to file the complex FFT data. The input file needs to be a text (ascii) file. All lines at the top of the input file that start with # are ignored. One data point per line is assumed.

extract.c - Extracts the real or imaginary parts or the magnitude or the phase of an FFT file that was created by the program fft.

A Simple Recursive Digital Filter (pdf, 7 pages, 132 KB, Oct 3 2006)

Calculating the Frequency Spectrum of a Signal (pdf, 10 pages, 157 KB)

Fourier Transform of a Sampled Signal (pdf, 5 pages, 93 KB)

Spectrum Magnification: When An FFT Is Not Enough (pdf, 9 pages, 147 KB)

Fourier Series Expansion of Functions in Two or More Dimensions (pdf, 3 pages, 79 KB)

Chebyshev Polynomials (pdf, 6 pages, 154 KB)