COURSE NOTES 6
Nonlinear Synthesis: Frequency Modulation & Waveshaping

Basic Frequency Modulation Instrument

• A - peak amplitude

• f_c - carrier frequency in hertz

• f_m - modulating frequency in hertz

• delta_f - frequency deviation in hertz

• I - index of modulation = delta_f/f_m

Spectrum of FM Sound

• The spectrum of a frequency modulated waveform consists of many sinusoidal components

• Carrier frequency at f_c with amplitude J₀(I)

• Sidebands at f_c + kf_m (k = ..., -3, -2, -1, 1, 2, 3) with amplitude J_k(I).

Bessel Functions

• Bessel functions of the first kind of order n are solutions to Bessel's differential equation:
  x²y'' + xy' + (x² - n²)y = 0

• Solution: J_n(x) = SUM_k=0..inf[((-1)^k(x/2)^n+2k)/(k!(n+k)!)

• Also: J_n+1(x) = (2n/x)J_n(x) - J_n-1(x)

More on Bessel Functions

• J_-k(I) = -J_k(I) for odd k

• J_-k(I) = J_k(I) for even k

• Aliasing can occur since f_c - kf_m can be negative and f_c + kf_m can be greater than R/2.

Example

• f_c = 440Hz, f_m = 440 Hz, delta_f = 880 Hz, A = 10000

• Thus, I=2.

• Amp at 440 Hz = J₀(2) * 10000
  Amp at 880 Hz = J₁(2) * 10000
  Amp at 0 Hz = J_-1(2) * 10000
  etc.

Simple FM Instrument

; p4 = amplitude
; p5 = carrier frequency
; p6 = modulating frequency
; p7 = index of modulation

instr 1
	k1 linseg 0, 0.2*p3, 1, 0.5*p3, 1, 0.3*p3, 0
	k2 = k1 * p4
	k3 = k1 * p6 * p7	// p6 * p7 = delta f
	a1 oscil k3, p6, 1
	a2 = a1 + p5
	a3 oscil k2, a2, 1
	out a3
endin

f1 0 2048 10 1
i1 0 2 20000 440 440 3
e

Harmonicity Ratio

• H = f_m/f_c

• Examples:
  H = 1/2 (frequency component appears between 0 Hz and carrier frequency: f_c-f_m)
  H = 2 (odd harmonics appear only)
  H = 1/sqrt(2) (inharmonic sounds like bells)

• Instruments:
  Horn (H=1, I = 5)
  Bassoon (H=1/5, I = 1.5)
  Clarinet (H=2/3, I = 4 to 6)
  Gong (H = sqrt(2), I = 10)

Waveshaping

• Waveshaping is a technique of distortion which creates complex spectra from simple tones by explicitly altering the shape of the original waveform

• Eavh waveshaping instrument is characterized by its transfer function

Transfer Function

• Let the transfer function of a waveshaping instrument be F(x), defined for x between -1 and 1.

• The new waveform is given by solving F(x) given x is an output from a unit-amplitude sinusoidal waveform.

• The process of waveshaping is equivalent to composition of functions.

Calculating the Output Spectrum

• If the input of a waveshaper is a sinusoid of unit amplitude, then the amplitudes of the harmonics of the output waveform can be determined based on Pascal's triangle. (SEE HANDOUT)

• Given a transfer function F(x) = c_nxⁿ + c_n-1x^n-1 + ... + c₁x + c₀ where x is the output of a unit-amplitude sinusoid, the output waveform from F(x) has harmonics h_j (j=0,1,...,n) whose amplitudes are given by the formulas:
  
  h₀ = SUM_k=0..n(c_kH[k,0]/2^k)
  h_j = SUM_k=0..n(ckH[k,0]/2^k-1) for j=1,...,n
  
  where H[k,j] is the table entry in Pascal's triangle (HANDOUT) at row k, column j (counting from 0)

Example

• F(x) = 4x⁵ + 6x⁴ - (11/3)x³ + (5/4)x - (7/4)

• h₀ = 1/2 (DC or average power), h₁ = 1, h₂ = 3, h₃ = 1/3, h₄ = 3/4, h₅ = 1/4

Spectral Matching

• In order to determine the actual transfer function needed to generate a desired output waveform, a technique called spectral matching is used.

• Chebyshev polynomials are used to reconstruct the original transfer function.

Chebyshev Polynomials

• The Chebyshev polynomial of the first kind of order k is defined by T_k(x) = cos(k arccos x)

• Also, T_n+1(x) = 2xT_n(x) - T_n-1(x)

• Thus, T₀(x) = 1, T₁(x) = x, T₂(x) = 2x² - 1, etc. (SEE HANDOUT)

Building the Transfer Function

• If hj is the amplitude of the jth harmonic of an audio waveform (with n harmonics) output by a waveshaper, then the transfer function required to generate this waveform is
  F(x) = h₀T₀(x) + h₁T₁(x) + ... + h_nT_n(x).

Example

• h₀ = 1/2, h₁ = 1, h₂ = 3, h₃ = 1/3, h₄ = 3/4, h₅ = 1/4

• F(x) = 4x⁵ + 6x⁴ - (11/3)x³ + (5/4)x - (7/4)

Properties of Waveshapers

• If a transfer function is an even function, then the output will consist of only even harmonics.

• An odd transfer function yields an output with odd harmonics only.

• The more extreme the change in slope is for a transfer function, the greater the number and intensity of harmonics that will be in the output waveform.

• The highest harmonic of the output waveform must be nf₀, for an input unit-amplitude sinusoid, where f₀ is the frequency of the sinusoid and n is the highest degree of F(x), the transfer function.

New Csound Commands and Function Generators

• ar table andx, ifn
  Extracts sample at index andx from function table number ifn.

• GEN09 - sum of sinusoids, specifying each sinusoid's partial (harmonic) number, amplitude (strength) and phase (in degrees)
  f # time size 9 pna stra phsa pnb strb phsb ...

• GEN13 - generate transfer function F(x) defined from -1 to 1 in wavetable given amplitude of desired audio spectrum
  f # time size 13 1 1 h0 h1 h2 h3 ... hn

• GEN03 - generate transfer function F(x) defined from -1 to 1 in wavetable given coefficients of transfer function equation
  f # time size 3 -1 1 c0 c1 c2 ... cn

Waveshaping in Csound (based on numerical example above)

; p4 = amplitude of note
; p5 = fundamental frequency

instr 1
	a1 oscil 255, p5, 1
	a2 = a1 + 256
	a3 table a2, 2
	a4 = a3 * p4
	out a4
endin

f1 0 256 9 1 1 90				; cosine wave (sine offset by 90 degrees)
f2 0 512 13 1 1 0.5 1 3 0.3333 0.75 0.25	; transfer function using harmonics
 OR
f2 0 512 3 -1 1 -1.75 1.25 0 -3.6667 6 4	; transfer function using coefficients

i1 0 5 20000 440
e

COURSE INFORMATION | HOMEWORK ASSIGNMENTS
COURSE PROJECT | CS240 HELP DESK