Useful Information: Filters
---------------------------


BUTTERWORTH FILTER

Lowpass

                         1
  atennuation = -------------------
                SQR((f/fc)^(2*N)+1)

Highpass

                         1
  atennuation = -------------------
                SQR((fc/f)^(2*N)+1)

  where N = order

--------------------------------------------------------------------------------
1st ORDER FILTER

Lowpass

           1
  Re = ----------
       (f/fc)^2+1

           1
  Im = ---------
       f/fc+fc/f

                       1
  atennuation = ---------------
                SQR((f/fc)^2+1)

  phase = arctan(R/Xc)

Highpass

                      1
  atennuation = ---------------
                SQR((fc/f)^2+1)

  phase = arctan(Xc/R)

--------------------------------------------------------------------------------
2nd ORDER FILTER

Lowpass

              G*w0^2
  H(s) = -----------------
         s^2+2*d*w0*s+w0^2

Highpass

              G*s^2
  H(s) = -----------------
         s^2+2*d*w0*s+w0^2

  where G  = gain in passband
        w0 = cutoff (rad/sec)

--------------------------------------------------------------------------------
NOISE BANDWIDTH OF BUTTERWORTH FILTER

Order     Equivalent Noise Bandwidth

 1           3dB_BW * PI/2

 2           3dB_BW * PI/(2*SQR(2))

 3           3dB_BW * PI/3

 high        3dB_BW (approx)

--------------------------------------------------------------------------------
CASCADED 1st ORDER FILTERS

rel. bandwidth = SQR(2^(1/N)-1)

  where N = number of stages

--------------------------------------------------------------------------------
GAUSSIAN FILTER

Gaussian impulse response

              1
  F(t) = ----------- * exp(-t^2/(4*v))
         2*SQR(PI*v)          

Gaussian frequency response

  F(w) = exp(-v*w^2)

--------------------------------------------------------------------------------
GAUSSIAN FILTER APPROXIMATION TO RAISED COSINE

for impulse response . . x 1 x . . (ideal would be 0 0 0 1 0 0 0)

  if x = 0.01

  F(t) = exp(-0.46*(t/T)^2)
  F(f) = exp(-0.54*(f*T)^2)

--------------------------------------------------------------------------------
GAUSSIAN APPROXIMATION FILTER

|H(jw)|^2 = 1 + 2*(w/wc)^2 + (2^2/2!)*(w/wc)^4 + (2^3/3!)*(w/wc)^6 + ... 

ref. Handbook of Filter Synthesis
Anatol Zverev, 1967

--------------------------------------------------------------------------------
'RIAA' EQUALISATION FILTER

       C1       C2
   ,---||---,---||---,
---|        |        |------
   '--/\/\--'--/\/\--'   |
       R1       R2       /
                         \ R
                         /
                         |

               R1+R2+R
low freq gain  -------
                  R

             1
1st break  -----
           C1*R1

              1     R1+R2
lead break  ----- * -----
            C1+C2   R1*R2

             1
2nd break  -----
           C2*R2

--------------------------------------------------------------------------------
TWIN TEE FILTER

       R         R
  ,--/\/\----,-/\/\--,
--|    C     |   C   |--
  '---||--,--|--||---'
          |  |
          /  -
      R/2 \  - 2C
          /  |
          |  |
         -'--'-

                    1
notch frequency  --------
                 2*PI*C*R

--------------------------------------------------------------------------------
FILTER DESIGN USING ZVEREV TABLE DATA

PI low pass prototype

   --,--L -,--
     |     |
     C     C
     |     |

w = 2*PI*fc

L = R/w * Ln

C = 1/w * Cn

Table of normalised element values (e.g. Zverev)

N   C1  L2  C3  L4  .
2   .   .   .   .
3   .   .   .
4   .   .
.   .
.


for T sections C1 -> L1, L2 -> C2 etc.
 normalised values remain the same

Highpass:
 transform C' -> L and L' -> C by  L = 1/C'  C = 1/L'

Bandpass:
 transform  C -> L1&C1 in parallel
 and        L -> C2&L2 in series

 by  C1 = Q*C  L1 = 1/C1
     L2 = Q*L  C2 = 1/L2

  where f0 = geometrical centre frequency
  where B  = bandwidth
  where Q  = f0/B

--------------------------------------------------------------------------------
SALLEN & KEY FILTER

Optimum sensitivity to component changes with regard to stability of Q at

Gain = 4/3