biquad
A sweepable general purpose biquadratic digital filter.
Syntax
ares biquad asig, kb0, kb1, kb2, ka0, ka1, ka2 [, iskip]
Initialization
iskip (optional, default=0) -- if non-zero, intialization will be skipped. Default value 0. (New in Csound version 3.50)
asig -- input signal
biquad is a general purpose biquadratic digital filter of the form:
\[
a_0 y(n) + a_1 y(n-1) + a_2 y(n-2) = b_0 x(n) + b_1 x(n-1) + b_2 x(n-2)
\]
This filter has the following frequency response:
\[
H(z) = \frac{B(z)}{A(z)} = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{a_0 + a_1 z^{-1} + a_2 z^{-2}}
\]
This type of filter is often encountered in digital signal processing literature. It allows six user-defined k-rate coefficients.
Examples
Here is an example of the biquad opcode. It uses the file biquad.csd.
| Example of the biquad opcode. |
|---|
| <CsoundSynthesizer>
<CsOptions>
; Select audio/midi flags here according to platform
; Audio out Audio in No messages
-odac -iadc -d ;;;RT audio I/O
; For Non-realtime ouput leave only the line below:
; -o biquad.wav -W ;;; for file output any platform
</CsOptions>
<CsInstruments>
; Initialize the global variables.
sr = 44100
kr = 4410
ksmps = 10
nchnls = 2
; Instrument #1.
instr 1
; Get the values from the score.
idur = p3
iamp = p4
icps = cpspch(p5)
kfco = p6
krez = p7
; Calculate the biquadratic filter's coefficients
kfcon = 2*3.14159265*kfco/sr
kalpha = 1-2*krez*cos(kfcon)*cos(kfcon)+krez*krez*cos(2*kfcon)
kbeta = krez*krez*sin(2*kfcon)-2*krez*cos(kfcon)*sin(kfcon)
kgama = 1+cos(kfcon)
km1 = kalpha*kgama+kbeta*sin(kfcon)
km2 = kalpha*kgama-kbeta*sin(kfcon)
kden = sqrt(km1*km1+km2*km2)
kb0 = 1.5*(kalpha*kalpha+kbeta*kbeta)/kden
kb1 = kb0
kb2 = 0
ka0 = 1
ka1 = -2*krez*cos(kfcon)
ka2 = krez*krez
; Generate an input signal.
axn vco 1, icps, 1
; Filter the input signal.
ayn biquad axn, kb0, kb1, kb2, ka0, ka1, ka2
outs ayn*iamp/2, ayn*iamp/2
endin
</CsInstruments>
<CsScore>
; Table #1, a sine wave.
f 1 0 16384 10 1
; Sta Dur Amp Pitch Fco Rez
i 1 0.0 1.0 20000 6.00 1000 .8
i 1 1.0 1.0 20000 6.03 2000 .95
e
</CsScore>
</CsoundSynthesizer>
|
Here is another example of the biquad opcode used for modal synthesis. It uses the file biquad-2.csd. See the Modal Frequency Ratios appendix for other frequency ratios.
| Example of the biquad opcode for modal synthesis. |
|---|
| <CsoundSynthesizer>
<CsOptions>
; Select audio/midi flags here according to platform
; Audio out Audio in
-odac -iadc ;;;RT audio I/O
; For Non-realtime ouput leave only the line below:
; -o biquad-2.wav -W ;;; for file output any platform
</CsOptions>
<CsInstruments>
sr = 44100
kr = 4410
ksmps = 10
nchnls = 2
/* modal synthesis using biquad filters as oscillators
Example by Scott Lindroth 2007 */
instr 1
ipi = 3.1415926
idenom = sr*0.5
ipulseSpd = p4
icps = p5
ipan = p6
iamp = p7
iModes = p8
apulse mpulse iamp, 0
icps = cpspch( icps )
; filter gain
iamp1 = 600
iamp2 = 1000
iamp3 = 1000
iamp4 = 1000
iamp5 = 1000
iamp6 = 1000
; resonance
irpole1 = 0.99999
irpole2 = irpole1
irpole3 = irpole1
irpole4 = irpole1
irpole5 = irpole1
irpole6 = irpole1
; modal frequencies
if (iModes == 1) goto modes1
if (iModes == 2) goto modes2
modes1:
if1 = icps * 1 ;pot lid
if2 = icps * 6.27
if3 = icps * 3.2
if4 = icps * 9.92
if5 = icps * 14.15
if6 = icps * 6.23
goto nextPart
modes2:
if1 = icps * 1 ;uniform wood bar
if2 = icps * 2.572
if3 = icps * 4.644
if4 = icps * 6.984
if5 = icps * 9.723
if6 = icps * 12.0
goto nextPart
nextPart:
; convert frequency to radian frequency
itheta1 = (if1/idenom) * ipi
itheta2 = (if2/idenom) * ipi
itheta3 = (if3/idenom) * ipi
itheta4 = (if4/idenom) * ipi
itheta5 = (if5/idenom) * ipi
itheta6 = (if6/idenom) * ipi
; calculate coefficients
ib11 = -2 * irpole1 * cos(itheta1)
ib21 = irpole1 * irpole1
ib12 = -2 * irpole2 * cos(itheta2)
ib22 = irpole2 * irpole2
ib13 = -2 * irpole3 * cos(itheta3)
ib23 = irpole3 * irpole3
ib14 = -2 * irpole4 * cos(itheta4)
ib24 = irpole4 * irpole4
ib15 = -2 * irpole5 * cos(itheta5)
ib25 = irpole5 * irpole5
ib16 = -2 * irpole6 * cos(itheta6)
ib26 = irpole6 * irpole6
;printk 1, ib 11
;printk 1, ib 21
; also try setting the -1 coeff. to 0, but be sure to scale down the amplitude!
asin1 biquad apulse * iamp1, 1, 0, -1, 1, ib11, ib21
asin2 biquad apulse * iamp2, 1, 0, -1, 1, ib12, ib22
asin3 biquad apulse * iamp3, 1, 0, -1, 1, ib13, ib23
asin4 biquad apulse * iamp4, 1, 0, -1, 1, ib14, ib24
asin5 biquad apulse * iamp5, 1, 0, -1, 1, ib15, ib25
asin6 biquad apulse * iamp6, 1, 0, -1, 1, ib16, ib26
afin = (asin1 + asin2 + asin3 + asin4 + asin5 + asin6)
outs afin * sqrt(p6), afin*sqrt(1-p6)
endin
</CsInstruments>
<CsScore>
;ins st dur pulseSpd pch pan amp Modes
i1 0 12 0 7.089 0 0.7 2
i1 . . . 7.09 1 . .
i1 . . . 7.091 0.5 . .
i1 0 12 0 8.039 0 0.7 2
i1 0 12 0 8.04 1 0.7 2
i1 0 12 0 8.041 0.5 0.7 2
i1 9 . . 7.089 0 . 2
i1 . . . 7.09 1 . .
i1 . . . 7.091 0.5 . .
i1 9 12 0 8.019 0 0.7 2
i1 9 12 0 8.02 1 0.7 2
i1 9 12 0 8.021 0.5 0.7 2
e
</CsScore>
</CsoundSynthesizer>
|
See also
Standard filters: Biquad filters
Credits
Author: Hans Mikelson
October 1998
New in Csound version 3.49