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Computer Sound Synthesis 2

Computer Sound Synthesis 2. MUS_TECH 335 Selected Topics. Filters continued. Complex Numbers Complex Plane. imaginary. x + yi. y. real. x. Cartesian coordinates. Im. polar form A q. A. q. Re. A = x 2 + y 2. y x. q = arctan( ). Im. A. y = A sin q. q. Re.

vincent-cabrera
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Computer Sound Synthesis 2

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  1. Computer Sound Synthesis 2 MUS_TECH 335 Selected Topics

  2. Filters continued

  3. Complex NumbersComplex Plane imaginary x + yi y real x Cartesian coordinates

  4. Im polar form A q A q Re A= x2 + y2 y x q= arctan( )

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

  6. Complex Numbers Addition (a+bi) + (c+di) = (a+c) + (b+d)i Multiplication A1q1A2q2 = A1A2(q1+ q2) (a+bi)(c+di) = ac + adi + bci + bdi2 = (ac - bd) + (ad + bc)i . -1

  7. Meaning of i? i represents a 90 degree rotational shift i i = i2 = -1 represents a 180 degree rotational shift .

  8. imaginary z-plane unit circle frequency real Nyquist -3/2 SR -1/2 SR 0 1/2 SR 3/2 SR

  9. Geometric Interpretation of Magnitude Response img z-plane = pole |P1| = zero |Z1 | real |Z1| | Z2| |P1| |P2| |H| proportional to

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

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

  12. Take a look at Movie Demonstrations http://www.ece.msstate.edu/~hagler/Aug1996/011/cd/Demos/Z2freq/index.htm

  13. 2nd-order Filters 2-zero filter a1 k a2 z-1 z-1 non-recursive k 2-pole filter z-1 z-1 b1 b2 recursive

  14. 2-zero band-reject a f 0 SR/2 2-pole band-pass (x+yi) a f 0 (x-yi) SR/2 -1 (x + yi)(x - yi) = x2 - xyi + xyi - y2i2 = x2 + y2

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

  16. Visit Pole-Zero Filter Design Applet http://www.earlevel.com/Digital%20Audio/PoleZero.html

  17. two-pole filter y(n) k x(n) z-1 z-1 -b1 -b2 difference equation: y(n) = k x(n) - b1 y(n-1) - b2 y(n-2) b1 = 2r cos q b2 = -r2 The coefficients depend on the position of the poles, here expressed in polar form, qand r.

  18. Programming Implementation output input old1 old2 z-1 z-1 k feedback loop -b1 -b2 Program Flow output = k input - b1 old1 - b2 old2; old2 = old1; old1 = output; * * *

  19. Biquad Filter 1 a1 a2 out k z-1 z-1 b1 b2 combined IIR/FIR 2nd-order

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