### Bilinear Transform Substitutions

Source: https://github.com/git/freeverb3/blob/master/doc/Audio-EQ-Cookbook.txt

Explanation of the substitutions used in the bilinear transform for converting analog filter designs to digital, including trigonometric identities.

```mathematica
FYI:   The bilinear transform (with compensation for frequency warping)
substitutes:

                                  1         1 - z^-1
      (normalized)   s  <--  ----------- * ----------
                              tan(w0/2)     1 + z^-1

   and makes use of these trig identities:

                     sin(w0)                               1 - cos(w0)
      tan(w0/2) = -------------           (tan(w0/2))^2 = ------------- 
                   1 + cos(w0)                             1 + cos(w0) 

```

```mathematica

   resulting in these substitutions:


                 1 + cos(w0)     1 + 2*z^-1 + z^-2
      1    <--  ------------- * -------------------
                 1 + cos(w0)     1 + 2*z^-1 + z^-2


                 1 + cos(w0)     1 - z^-1
      s    <--  ------------- * ----------
                   sin(w0)       1 + z^-1

                                      1 + cos(w0)     1         -  z^-2
                                  =  ------------- * -------------------
                                        sin(w0)       1 + 2*z^-1 + z^-2


                 1 + cos(w0)     1 - 2*z^-1 + z^-2
      s^2  <--  ------------- * -------------------
                 1 - cos(w0)     1 + 2*z^-1 + z^-2
```

```mathematica

   The factor:

                    1 + cos(w0)
                 -------------------
                  1 + 2*z^-1 + z^-2

   is common to all terms in both numerator and denominator, can be factored
   out, and thus be left out in the substitutions above resulting in:


                 1 + 2*z^-1 + z^-2
      1    <--  -------------------
                    1 + cos(w0)


                 1         -  z^-2
      s    <--  -------------------
                      sin(w0)


                 1 - 2*z^-1 + z^-2
      s^2  <--  -------------------
                    1 - cos(w0)


   In addition, all terms, numerator and denominator, can be multiplied by a
   common (sin(w0))^2 factor, finally resulting in these substitutions:


      1         <--   (1 + 2*z^-1 + z^-2) * (1 - cos(w0))

      s         <--   (1         -  z^-2) * sin(w0)

      s^2       <--   (1 - 2*z^-1 + z^-2) * (1 + cos(w0))

      1 + s^2   <--   2 * (1 - 2*cos(w0)*z^-1 + z^-2)


   The biquad coefficient formulae above come out after a little
   simplification.
```

--------------------------------

### Low-Shelf Filter Biquad Coefficients

Source: https://github.com/git/freeverb3/blob/master/doc/Audio-EQ-Cookbook.txt

Formulas for calculating the biquad coefficients for a digital low-shelf filter. These are derived using the bilinear transform.

```mathematica
lowShelf: H(s) = A * (s^2 + (sqrt(A)/Q)*s + A)/(A*s^2 + (sqrt(A)/Q)*s + 1)

            b0 =    A*( (A+1) - (A-1)*cos(w0) + 2*sqrt(A)*alpha )
            b1 =  2*A*( (A-1) - (A+1)*cos(w0)                   )
            b2 =    A*( (A+1) - (A-1)*cos(w0) - 2*sqrt(A)*alpha )
            a0 =        (A+1) + (A-1)*cos(w0) + 2*sqrt(A)*alpha
            a1 =   -2*( (A-1) + (A+1)*cos(w0)                   )
            a2 =        (A+1) + (A-1)*cos(w0) - 2*sqrt(A)*alpha
```

--------------------------------

### Peaking Equalizer Biquad Coefficients

Source: https://github.com/git/freeverb3/blob/master/doc/Audio-EQ-Cookbook.txt

Formulas for calculating the biquad coefficients for a digital peaking equalizer filter. These are derived using the bilinear transform.

```mathematica
peakingEQ:  H(s) = (s^2 + s*(A/Q) + 1) / (s^2 + s/(A*Q) + 1)

            b0 =   1 + alpha*A
            b1 =  -2*cos(w0)
            b2 =   1 - alpha*A
            a0 =   1 + alpha/A
            a1 =  -2*cos(w0)
            a2 =   1 - alpha/A
```

--------------------------------

### Notch Filter Biquad Coefficients

Source: https://github.com/git/freeverb3/blob/master/doc/Audio-EQ-Cookbook.txt

Formulas for calculating the biquad coefficients (b0, b1, b2, a0, a1, a2) for a digital notch filter. These are derived using the bilinear transform.

```mathematica
notch:      H(s) = (s^2 + 1) / (s^2 + s/Q + 1)

            b0 =   1
            b1 =  -2*cos(w0)
            b2 =   1
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha
```

--------------------------------

### All-Pass Filter (APF) Biquad Coefficients

Source: https://github.com/git/freeverb3/blob/master/doc/Audio-EQ-Cookbook.txt

Formulas for calculating the biquad coefficients for a digital all-pass filter. These are derived using the bilinear transform.

```mathematica
APF:        H(s) = (s^2 - s/Q + 1) / (s^2 + s/Q + 1)

            b0 =   1 - alpha
            b1 =  -2*cos(w0)
            b2 =   1 + alpha
            a0 =   1 + alpha
            a1 =  -2*cos(w0)
            a2 =   1 - alpha
```

--------------------------------

### High-Shelf Filter Biquad Coefficients

Source: https://github.com/git/freeverb3/blob/master/doc/Audio-EQ-Cookbook.txt

Formulas for calculating the biquad coefficients for a digital high-shelf filter. These are derived using the bilinear transform.

```mathematica
highShelf: H(s) = A * (A*s^2 + (sqrt(A)/Q)*s + 1)/(s^2 + (sqrt(A)/Q)*s + A)

            b0 =    A*( (A+1) + (A-1)*cos(w0) + 2*sqrt(A)*alpha )
            b1 = -2*A*( (A-1) + (A+1)*cos(w0)                   )
            b2 =    A*( (A+1) + (A-1)*cos(w0) - 2*sqrt(A)*alpha )
            a0 =        (A+1) - (A-1)*cos(w0) + 2*sqrt(A)*alpha
            a1 =    2*( (A-1) - (A+1)*cos(w0)                   )
            a2 =        (A+1) - (A-1)*cos(w0) - 2*sqrt(A)*alpha
```

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