Music 320 lab 2
October 12, 2000

lab 2: synthesis with sinusoids

discussion
  1. additive synthesis
     a. intuitions: sums of sinusoids and the DFT
     b. square wave as sum of sinusoids
	- visual demonstration

  2. ring modulation
     a. as an analog technique
	- Stockhausen example (Mantra for two pianos and electronics)
     b. multiplication of two signals produces sidebands
	- demonstrate mathematics for simplest case

	  (cos(omegac*t))*(cos(omegam*t))

	= e^(j*omegac*t)+e^(-j*omegam*t)   e^(j*omegac*t)+e^(-j*omegam*t)
	  ------------------------------ * ------------------------------
	                2                                2 

	= e^(j*(omegac+omegam)*t)+e^(-j*(omegac+omegam)*t)+e^(j*(omegac-omegaw)*t)+e^(-j*(omegac+omegam)*t)
	  -------------------------------------------------------------------------------------------------
						          4

	= cos((omegac+omegam)*t)   cos((omegac-omegam)*t
	  ---------------------- + ---------------------\
		     2                        2

     c. single-sideband modulation: use a complex sinusoidal modulator instead of a real sinusoid

  3. amplitude modulation
     a. as a radio technique
     b. bipolar carrier and unipolar modulator
	- produces sidebands without suppressing carrier
	- demonstrate mathematics for simplest case (dc offset)

	  cos(omegac*t) * (cos(omegam*t) + A)

	= A * cos(omegac*t) + ring modulation case above....

  4. frequency modulation
     a. as CCRMA history
	- Chowning and the DX7
	- computational economy relative to additive synthesis
	- perceptual economy: limited set of strong parameters controlling spectral content
	- Stria example
     b. modulator changes instantaneous frequency of carrier
	
	A * cos((omegac*t) + (I * cos(omegam*t)))

	- multiple sidebands "stealing" energy from carrier
	- sidebands at integer multiples of modulator frequency
	- sidebands "reflect" around zero: interesting
	  harmonic/inharmonic relationships
	- number of sidebands determined by modulation index (I)
	  - I = peak deviation / modulating frequency
	  - Carson's rule: I+1 significant sidebands
	- "tedious mathematical analysis" (Bessel functions)
     c. complex FM
	- driving the carrier with a more complicated signal
	- Schottstaedt suggests that three sinusoidal components 
	  are sufficient
	- or, cascade carrier/modulator pairs
	- or, modulate carrier with its own (lowpass filtered) output
     d. Bill Schottstaedt's FM page as a reference (including mathematics)