265 lines
12 KiB
Plaintext
265 lines
12 KiB
Plaintext
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IIR Digital Filter Functions
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============================
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An IIR filter is also known as a recursive digital filter because its output
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is a function of previous outputs as well as the input. If x[n] represents the
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nth input to the filter and y[n] is the nth output of the filter then a
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general iir filter is implemented as follows:
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y[n] = c0*x[n] + c1*x[n-1] + ... + cM*x[n-M] - ( d1*y[n-1] + d2*y[n-2] + ... + dN*y[n-N])
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This means that the nth output is a linear function of the nth input, the
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previous M inputs, and the previous N outputs. The c and d coefficients are
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calculated to give the filter a specific frequency response. The number of
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coefficients, M and N, will vary depending on the type of filter. There are
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many different kinds of iir filters and many different ways to calculate the
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coefficients. Listed below are filter types (currently only Butterworth
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filters) and the functions that can be used to calculate the c and d
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coefficients for lowpass, highpass, bandpass, and bandstop implementations of
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the filter.
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I. Butterworth Filters
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-------------------
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A Butterworth filter is also known as a maximally flat filter because its
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frequency response is characterized by no ripple in the pass band and stop
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band.
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A. Lowpass functions
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The example program that shows how to use all the lowpass functions is
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bwlp.
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double *dcof_bwlp( int N, double fcf );
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This fuction calculates the d coefficients for a Butterworth lowpass
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filter. The coefficients are returned as an array of doubles.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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fcf = filter cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A pointer to an array of doubles is returned. The size of the
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array is equal to N+1, one more than the filter order. The first
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element of the array is d0, the coefficient of y[n], which will
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always be equal to 1. The second element of the array is d1, the
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coefficient of y[n-1], and so on. The calling program must free
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the array when finished with it.
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int *ccof_bwlp( int n );
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This fuction calculates the c coefficients for a Butterworth lowpass
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filter. The coefficients are returned as an array of integers.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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Return value:
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A pointer to an array of integers is returned. The size of the
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array is equal to N+1, one more than the filter order. The first
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element of the array is c0, the coefficient of x[n], which is the
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current input to the filter. The second element of the array is
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c1, the coefficient of x[n-1], and so on. The calling program
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must free the array when finished with it.
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double sf_bwlp( int n, double fcf );
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This fuction calculates the scaling factor for a Butterworth lowpass
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filter. The scaling factor is what the c coefficients must be
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multiplied by so that the frequency response of the filter has a
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maximum magnitude of 1.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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fcf = filter cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A double that is scaling factor.
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B. Highpass functions
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The example program that shows how to use all the highpass functions is
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bwhp.
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double *dcof_bwhp( int N, double fcf );
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This fuction calculates the d coefficients for a Butterworth highpass
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filter. The coefficients are returned as an array of doubles.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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fcf = filter cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A pointer to an array of doubles is returned. The size of the
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array is equal to N+1, one more than the filter order. The first
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element of the array is d0, the coefficient of y[n], which will
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always be equal to 1. The second element of the array is d1, the
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coefficient of y[n-1], and so on. The calling program must free
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the array when finished with it.
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int *ccof_bwhp( int n );
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This fuction calculates the c coefficients for a Butterworth highpass
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filter. The coefficients are returned as an array of integers.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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Return value:
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A pointer to an array of integers is returned. The size of the
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array is equal to N+1, one more than the filter order. The first
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element of the array is c0, the coefficient of x[n], which is the
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current input to the filter. The second element of the array is
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c1, the coefficient of x[n-1], and so on. The calling program
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must free the array when finished with it.
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double sf_bwhp( int n, double fcf );
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This fuction calculates the scaling factor for a Butterworth highpass
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filter. The scaling factor is what the c coefficients must be
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multiplied by so that the frequency response of the filter has a
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maximum magnitude of 1.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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fcf = filter cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A double that is scaling factor.
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C. Bandpass functions
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The example program that shows how to use all the bandpass functions is
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bwbp.
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double *dcof_bwbp( int n, double f1f, double f2f );
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This fuction calculates the d coefficients for a Butterworth bandpass
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filter. The coefficients are returned as an array of doubles. Note
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that, although there is no upper limit on the filter order, if the
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bandwidth, f2f - f1f, is very small, the coefficients returned may
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not give the desired response due to numerical instability in the
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calculation. This problem should not occure if the filter order is
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kept less that or equal to 10. For very small bandwidths you should
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always verify the frequency response using a program such as rffr.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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f1f = lower cutoff frequency as a fraction of pi. Range = [0,1].
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f2f = upper cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A pointer to an array of doubles is returned. The size of the
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array is equal to 2N+1, one more than twice the filter order. The
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first element of the array is d0, the coefficient of y[n], which
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will always be equal to 1. The second element of the array is d1,
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the coefficient of y[n-1], and so on. The calling program must
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free the array when finished with it.
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int *ccof_bwbp( int n );
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This fuction calculates the c coefficients for a Butterworth bandpass
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filter. The coefficients are returned as an array of integers.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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Return value:
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A pointer to an array of integers is returned. The size of the
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array is equal to 2N+1, one more than twice the filter order. The
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first element of the array is c0, the coefficient of x[n], which
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is the current input to the filter. The second element of the
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array is c1, the coefficient of x[n-1], and so on. The calling
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program must free the array when finished with it. Note that ck
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for all odd k, c1, c3, c5, and so on, will be equal to zero for
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this filter.
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double sf_bwbp( int n, double f1f, double f2f );
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This fuction calculates the scaling factor for a Butterworth bandpass
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filter. The scaling factor is what the c coefficients must be
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multiplied by so that the frequency response of the filter has a
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maximum magnitude of 1.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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f1f = lower cutoff frequency as a fraction of pi. Range = [0,1].
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f2f = upper cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A double that is scaling factor.
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D. Bandstop functions
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The example program that shows how to use all the bandstop functions is
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bwbs.
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double *dcof_bwbs( int n, double f1f, double f2f );
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This fuction calculates the d coefficients for a Butterworth bandstop
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filter. The coefficients are returned as an array of doubles. Note
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that, although there is no upper limit on the filter order, if the
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bandwidth, f2f - f1f, is very small, the coefficients returned may
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not give the desired response due to numerical instability in the
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calculation. This problem should not occure if the filter order is
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kept less that or equal to 10. For very small bandwidths you should
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always verify the frequency response using a program such as rffr.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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f1f = lower cutoff frequency as a fraction of pi. Range = [0,1].
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f2f = upper cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A pointer to an array of doubles is returned. The size of the
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array is equal to 2N+1, one more than twice the filter order. The
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first element of the array is d0, the coefficient of y[n], which
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will always be equal to 1. The second element of the array is d1,
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the coefficient of y[n-1], and so on. The calling program must
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free the array when finished with it.
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double *ccof_bwbs( int n, double f1f, double f2f );
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This fuction calculates the c coefficients for a Butterworth bandstop
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filter. The coefficients are returned as an array of doubles.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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f1f = lower cutoff frequency as a fraction of pi. Range = [0,1].
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f2f = upper cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A pointer to an array of doubles is returned. The size of the
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array is equal to 2N+1, one more than twice the filter order. The
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first element of the array is c0, the coefficient of x[n], which
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is the current input to the filter. The second element of the
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array is c1, the coefficient of x[n-1], and so on. The calling
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program must free the array when finished with it. Note that ck
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for all odd k, c1, c3, c5, and so on, will be equal to zero for
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this filter.
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double sf_bwbs( int n, double f1f, double f2f );
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This fuction calculates the scaling factor for a Butterworth bandstop
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filter. The scaling factor is what the c coefficients must be
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multiplied by so that the frequency response of the filter has a
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maximum magnitude of 1.
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Parameters:
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N = filter order. Range = [1, 20 or more] no fixed upper limit.
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f1f = lower cutoff frequency as a fraction of pi. Range = [0,1].
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f2f = upper cutoff frequency as a fraction of pi. Range = [0,1].
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Return value:
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A double that is scaling factor.
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