Computer Sound Synthesis 2 MUS_TECH 335 Selected Topics

Filters continued

Complex Numbers Complex Plane y x imaginary real x + yi Cartesian coordinates

Im Re  A = x 2 + y 2   = arctan( ) yxyx A polar form A 

Re  x = A cos  y = A sin  A Im Relationship of polar form to Cartesian form.

Complex Numbers Addition (a+bi) + (c+di) = (a+c) + (b+d)i Multiplication A 1  1 A 2  2 = A 1 A 2  (  1 +  2 ) (a+bi)(c+di) = ac + adi + bci + bdi 2 = (ac - bd) + (ad + bc)i.

Meaning of i? i represents a 90 degree rotational shift i i = i 2 = -1 represents a 180 degree rotational shift.

imaginary real z-plane unit circle 0-1/2 SR-3/2 SR3/2 SR1/2 SR Nyquist frequency

Geometric Interpretation of Magnitude Response img real |Z 1 | |P 1 | |H| proportional to |Z 1 | | Z 2 | |P 1 | |P 2 | = zero = pole z-plane

|Z | f a SR/2 = zero high-pass low-pass 0 a = 1 a = -1 z-plane f a SR/2 0 First-order Non-recursive Filter

|P | f a SR/2 = pole high-pass low-pass 0 b =.9 Z-plane f a SR/2 0 b = -.9 First-order Recursive Filter

Take a look at Movie Demonstrations

2 nd -order Filters z -1 k b1b1 ka2a2 a1a1 b2b2 non-recursive recursive 2-zero filter 2-pole filter

f a SR/2 2-pole band-pass 2-zero band-reject 0 f a SR/2 0 (x+yi) (x-yi) (x + yi)(x - yi) = x 2 - xyi + xyi - y 2 i 2 = x 2 + y 2

f low Q Q = freq BW * * * r  Q freq RQ = 1 = Used in SuperCollider: high Q Q increases as pole approaches the unit circle

Visit Pole-Zero Filter Design Applet

two-pole filter z -1 k -b 1 z -1 -b 2 difference equation: x(n) y(n) y(n) = k x(n) - b 1 y(n-1) - b 2 y(n-2) b 1 = 2r cos  b 2 = -r 2 The coefficients depend on the position of the poles, here expressed in polar form,  and r.

Programming Implementation z -1 -b 1 z -1 -b 2 input output k old1old2 feedback loop Program Flow output = k input - b 1 old1 - b 2 old2; old2 = old1; old1 = output; ***

Biquad Filter z -1 k b1b1 b2b2 1 a1a1 a2a2 out combined IIR/FIR 2 nd -order