biquad
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## biquad

biquad — A sweepable general purpose biquadratic digital filter.

## Description

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)

## Performance

asig -- input signal

biquad is a general purpose biquadratic digital filter of the form:

a0*y(n) + a1*y[n-1] + a2*y[n-2] = b0*x[n] + b1*x[n-1] + b2*x[n-2]

This filter has the following frequency response:

B(Z) b0 + b1*Z-1 + b2*Z-2
H(Z) = ---- = ------------------
A(Z) a0 + a1*Z-1 + a2*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 61. Example of the biquad opcode.

See the sections Real-time Audio and Command Line Flags for more information on using command line flags.

    <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 62. 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 = irpole12
        irpole3 = irpole13
        irpole4 = irpole14
        irpole5 = irpole15
        irpole6 = irpole16

        ; 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

biquada, moogvcf, rezzy

## Credits

Author: Hans Mikelson
October 1998

New in Csound version 3.49

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