/* IIR filter design using bilinear transform and prewarp. Transforms
2nd order s domain analog filter into a digital IIR biquad link. To
create a filter fill in a, b, Q and fs and make space for coef and k.
Example Butterworth design:
Below are Butterworth polynomials, arranged as a series of 2nd
order sections:
Note: n is filter order.
n Polynomials
-------------------------------------------------------------------
2 s^2 + 1.4142s + 1
4 (s^2 + 0.765367s + 1) * (s^2 + 1.847759s + 1)
6 (s^2 + 0.5176387s + 1) * (s^2 + 1.414214 + 1) * (s^2 + 1.931852s + 1)
For n=4 we have following equation for the filter transfer function:
1 1
T(s) = --------------------------- * ----------------------------
s^2 + (1/Q) * 0.765367s + 1 s^2 + (1/Q) * 1.847759s + 1
The filter consists of two 2nd order sections since highest s power
is 2. Now we can take the coefficients, or the numbers by which s
is multiplied and plug them into a standard formula to be used by
bilinear transform.
Our standard form for each 2nd order section is:
a2 * s^2 + a1 * s + a0
H(s) = ----------------------
b2 * s^2 + b1 * s + b0
Note that Butterworth numerator is 1 for all filter sections, which
means s^2 = 0 and s^1 = 0
Let's convert standard Butterworth polynomials into this form:
0 + 0 + 1 0 + 0 + 1
--------------------------- * --------------------------
1 + ((1/Q) * 0.765367) + 1 1 + ((1/Q) * 1.847759) + 1
Section 1:
a2 = 0; a1 = 0; a0 = 1;
b2 = 1; b1 = 0.765367; b0 = 1;
Section 2:
a2 = 0; a1 = 0; a0 = 1;
b2 = 1; b1 = 1.847759; b0 = 1;
Q is filter quality factor or resonance, in the range of 1 to
1000. The overall filter Q is a product of all 2nd order stages.
For example, the 6th order filter (3 stages, or biquads) with
individual Q of 2 will have filter Q = 2 * 2 * 2 = 8.
Arguments:
a - s-domain numerator coefficients, a[1] is always assumed to be 1.0
b - s-domain denominator coefficients
Q - Q value for the filter
k - filter gain factor. Initially set to 1 and modified by each
biquad section in such a way, as to make it the
coefficient by which to multiply the overall filter gain
in order to achieve a desired overall filter gain,
specified in initial value of k.
fs - sampling rate (Hz)
coef - array of z-domain coefficients to be filled in.
Note: Upon return from each call, the k argument will be set to a
value, by which to multiply our actual signal in order for the gain
to be one. On second call to szxform() we provide k that was
changed by the previous section. During actual audio filtering
k can be used for gain compensation.
return -1 if fail 0 if success.
*/
int af_filter_szxform(const FLOAT_TYPE* a, const FLOAT_TYPE* b, FLOAT_TYPE Q, FLOAT_TYPE fc,
FLOAT_TYPE fs, FLOAT_TYPE *k, FLOAT_TYPE *coef) {
FLOAT_TYPE at[3];
FLOAT_TYPE bt[3];
if(!a || !b || !k || !coef || (Q>1000.0 || Q< 1.0))
return -1;
memcpy(at,a,3*sizeof(FLOAT_TYPE));
memcpy(bt,b,3*sizeof(FLOAT_TYPE));
bt[1]/=Q;
/* Calculate a and b and overwrite the original values */
af_filter_prewarp(at, fc, fs);
af_filter_prewarp(bt, fc, fs);
/* Execute bilinear transform */
af_filter_bilinear(at, bt, k, fs, coef);
return 0;
}
/* IIR filter design using bilinear transform and prewarp. Transforms
2nd order s domain analog filter into a digital IIR biquad link. To
create a filter fill in a, b, Q and fs and make space for coef and k.
Example Butterworth design:
Below are Butterworth polynomials, arranged as a series of 2nd
order sections:
Note: n is filter order.
n Polynomials
-------------------------------------------------------------------
2 s^2 + 1.4142s + 1
4 (s^2 + 0.765367s + 1) * (s^2 + 1.847759s + 1)
6 (s^2 + 0.5176387s + 1) * (s^2 + 1.414214 + 1) * (s^2 + 1.931852s + 1)
For n=4 we have following equation for the filter transfer function:
1 1
T(s) = --------------------------- * ----------------------------
s^2 + (1/Q) * 0.765367s + 1 s^2 + (1/Q) * 1.847759s + 1
The filter consists of two 2nd order sections since highest s power
is 2. Now we can take the coefficients, or the numbers by which s
is multiplied and plug them into a standard formula to be used by
bilinear transform.
Our standard form for each 2nd order section is:
a2 * s^2 + a1 * s + a0
H(s) = ----------------------
b2 * s^2 + b1 * s + b0
Note that Butterworth numerator is 1 for all filter sections, which
means s^2 = 0 and s^1 = 0
Let's convert standard Butterworth polynomials into this form:
0 + 0 + 1 0 + 0 + 1
--------------------------- * --------------------------
1 + ((1/Q) * 0.765367) + 1 1 + ((1/Q) * 1.847759) + 1
Section 1:
a2 = 0; a1 = 0; a0 = 1;
b2 = 1; b1 = 0.765367; b0 = 1;
Section 2:
a2 = 0; a1 = 0; a0 = 1;
b2 = 1; b1 = 1.847759; b0 = 1;
Q is filter quality factor or resonance, in the range of 1 to
1000. The overall filter Q is a product of all 2nd order stages.
For example, the 6th order filter (3 stages, or biquads) with
individual Q of 2 will have filter Q = 2 * 2 * 2 = 8.
Arguments:
a - s-domain numerator coefficients, a[1] is always assumed to be 1.0
b - s-domain denominator coefficients
Q - Q value for the filter
k - filter gain factor. Initially set to 1 and modified by each
biquad section in such a way, as to make it the
coefficient by which to multiply the overall filter gain
in order to achieve a desired overall filter gain,
specified in initial value of k.
fs - sampling rate (Hz)
coef - array of z-domain coefficients to be filled in.
Note: Upon return from each call, the k argument will be set to a
value, by which to multiply our actual signal in order for the gain
to be one. On second call to szxform() we provide k that was
changed by the previous section. During actual audio filtering
k can be used for gain compensation.
return -1 if fail 0 if success.
*/
int af_filter_szxform(_ftype_t* a, _ftype_t* b, _ftype_t Q, _ftype_t fc, _ftype_t fs, _ftype_t *k, _ftype_t *coef)
{
_ftype_t at[3];
_ftype_t bt[3];
if(!a || !b || !k || !coef || (Q>1000.0 || Q< 1.0))
return -1;
memcpy(at,a,3*sizeof(_ftype_t));
memcpy(bt,b,3*sizeof(_ftype_t));
bt[1]/=Q;
/* Calculate a and b and overwrite the original values */
af_filter_prewarp(at, fc, fs);
af_filter_prewarp(bt, fc, fs);
/* Execute bilinear transform */
af_filter_bilinear(at, bt, k, fs, coef);
return 0;
}
