COURSE NOTES
Filters and Subtractive Synthesis

Subtractive Synthesis

• Opposite of additive synthesis

• Start with an audio signal with an equal amount of energy at all frequencies (white noise)

• Use filters to remove frequency components from original audio signal to create desired signal

Filtering in the Frequency Domain

• Let  X[Z] be a signal expressed in the complex frequency domain

• The process of filtering is equivalent to multiplying the signal X[Z] by some filtering signal H[Z] to generate a new signal Y[Z]: Y[Z]=X[Z]H[Z].

Filtering in the Time Domain

• In the time domain, the filtering process is expressed as convolution.

• A time signal x[n] filtered by a filter defined by impulse response h[n] yields a new signal y[n]: y[n] = x[n] * y[n]. (* represents convolution here, not multiplication)

• h[n] (the impulse response) is the output of the filter in the time domain when fed a signal of {1,0,0,0,...}, that is, a single impulse and then no other input.

Convolution

• Given: x[] = original waveform, h[] = impulse response of filter

• Output signal after filtering is y[j] = x[j] * h[j] = SUM_m=0...j(x[m]h[j-m]) for j=0,1,...

• Example: x[n] = {2, 2, 3, 3, 4}, h[n] = {1, 1, 2}. Thus, y[n] = {2, 4, 9, 10, 13, 10, 8}.

Digital Filter Properties

• Linearity: If filter output for input signal x₁[n] is y₁[n] and output for input signal x₂[n] is y₂[n], then the output for input Ax₁[n] + Bx₂[n] is Ay₁[n] + By₂[n].

• Causal: A filter's output is only dependent on past and present inputs.

• Time invariance: The response of a filter does not change as time goes by.

• Recursive: Some output of the filter is fed back into the filter to determine future outputs.

Digital Filter Equation

• If a filter is linear, causal and time-invariant, it can be described by the equation:
  
  y[n] = SUM_i=0..M(a_ix[n-i]) - SUM_i=1..N(b_iy[n-i])

• The response of any digital filter can be characterized by its complex transfer function H[Z].

• If we know the transfer function H[Z] of a filter, we can calculate the Z-transform of its response to any signal by multiplying H[Z] by the Z-transform of the input signal.

The Z Transform

• Given a discrete waveform x[n] defined for all values of n, its Z-transform is defined as X[Z] = SUM_n=-inf...inf(x[n]Z^-n) where Z is a complex variable and inf is infinity.

• Z^-1 is the unit sample operator (represents one unit of delay)

• H[Z] = Y[Z]/X[Z]

• Z Transform is a more general case of the Fourier transform (replace Z with e^iw and what do you get?

• x[n-k] <==> Z^-kX[Z]

Poles and Zeroes of a Filter

• Take the Z transform of the digital filter equation and solve for H[Z].

• Y[Z] = SUM_i=0..M(a_iz^-iX[Z]) - SUM_i=1..N(b_iz^-iY[Z])

• H[Z] = Y[Z]/X[Z] = (a₀ + a₁Z^-1 + ... + a_MZ^-M)/(1 + b₁Z^-1 + ... + b_NZ^-N)
  = a0(PROD_i=1..M (1 - z_iZ^-1)/PROD_i=1..N(1 - p_iZ^-1)

• z_i represent zeroes, p_i represent poles

• Zeroes cause H[Z] to become zero - these represent frequencies that are attenuated (weakened) as they pass through the filter

• Poles cause H[Z] to become infinite - these represent frequencies that are amplified (strengthened) as they pass through the filter

Filter Example

• Filter equation: y[n] = x[n] + 0.8y[n-1]

• Take Z transform: Y[Z] = X[Z] + 0.8Z^-1Y[Z]

• Solve for H[Z] = Y[Z]/X[Z] = 1/(1 - 0.8Z^-1)

• Plot H[e^iw] for w=0..pi for frequency response of filter from f=0 to f=R/2.

Program filter.cpp

• Set zeroes and poles of the filter (amplitude and phase), number of zeroes and poles (M,N) and number of calculation points (K)

• Plot resulting list of numbers to yield frequency response curve of filter.

• Program also displays H[Z] equation: From this, you can derive original filter equation for y[n].

Csound Noise Sources

• ar rand xamp
  Generates a uniform random sequence of data values between -xamp and xamp

• ar buzz xamp, xcps, knh, ifn
  Generates a set of knh equal-strength harmonic partials, beginning with the fundamental frequency (xcps), whose combined amplitude is xamp. ifn is the function table number for a sine wave.

Types of Filters

• Lowpass - weakens high frequencies

• Highpass - weakens low frequencies

• Bandpass - weakens frequencies outside of a specified bandwidth of frequencies

• Band-reject - opposite of bandpass filter

Filters In Csound

• ar tone asig, khp
  First order recursive lowpass filter

• ar atone asig, khp
  First order recursive highpass filter

• asig = original audio signal, khp = half-power point (frequency where filter causes power of signal to be reduced by a factor of 2)

• ar reson asig, kcf, kbw, kscl
  Second order bandpass filter

• ar areson asig, kcf, kbw, kscl
  Second order band-reject filter

• asig = original audio signal
  kcp = center frequency of filter
  kbw = bandwidth (between half-power points)
  kscl = scale factor (1 means peak output = 1)

Csound example (subtractive synthesis using white noise)

; p4 = noise amp, p5 = peak bw, p6 = rate of bandwidth open
and close
instr 1
  a1 rand p4
  k1 oscil p5, p6, 1
  a2 tone a1, kl
     out a2
endin

f1 0 1024 8 0 256 0.5 256 1 256 0.5 256 0   ; in score

(GEN08 - generates a cubic spline point through endpoints; smooth curve, no discontinuities)

Csound example (subtractive synthesis using pink noise)

; p4 = note amp, p5 = note fundamental frequency
instr 1
  a1 buzz p4, p5, 6, 1
  a2 reson a1, p5, p5/2
  a3 reson a1, 3*p5, p5/2
  a4 reson a1, 5*p5, p5/2
  a5 balance a2+a3+a4, a1     ; equalize power with original signal
     out a5
endin

f1 0 8192 10 1		; in score

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