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Slide 2 Sound Synthesis Part II: Oscillators, Additive Synthesis & Modulation Slide 3 Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE Slide 4 Simple Oscillator Oscillator 3 strategies Oscillator 3 strategies Mathematical equation based oscillator Wavetable oscillator IIR-Based oscillator Solve math function for each sample Ex: y = sin(x) + Accurate -Inefficient Non real-time applications Pre-computed and stored in memory + Fast (Look-up table) - Memory Unstable filter that generates waveform of desired amplitude and frequency. + Fast + Memory efficient Sound synthesis Slide 5 Wavetable Oscillator Example of a wavetable (N = 16) Store N values sampled over one cycle Phase increment: SI=N f0/fs Slide 6 Wavetable Oscillator (example) Parameters N = 16 F0 = 220 Fs = 1kHz SI = 16 * 220/1000 SI = 3.52 Increase quality: Increase sampling rate interpolate Slide 7 Wavetable Oscillator Distortions Quantization: Eg, pure tone F0=440Hz, Fs=8,192Hz Truncate N=16 Truncate N=32 Truncate N=512 Interpolation: truncate, mean, linear Aliasing Slide 8 Wavetable Oscillator Interpolation Truncation (0 th level interpolation) Slide 9 Wavetable Oscillator Interpolation (2) Rounding (slightly better 0 th order) Slide 10 Wavetable Oscillator Interpolation (3) Linear (First order interpolation) Slide 11 Wavetable Oscillator Interpolation (4) Quadratic (Second order interpolation) Slide 12 Wavetable Oscillator Interpolation (5) Cubic (Third order interpolation) Slide 13 Wavetable Oscillator Interpolation (6) Signal to (interpolation) Noise Ratio (SNR) (eg, pure tone F0=220Hz, Fs=8,192Hz) Truncation: SNR = 6 k 11 dB Rounding: SNR = 6 k 5 dB Linear:SNR = 12 (k 1) dB (Moore, 1977; Hartman, 1987) (k = log2(N) and N is the table length) Conclusion: For increasing quality, increase number of samples, and use interpolation. Slide 14 Wavetable Oscillator Interpolation (7) Pure tone F0=440Hz, Fs=8,192Hz Truncate N=16 Truncate N=32 Truncate N=512 Slide 15 Wavetable Oscillator Aliasing Aliasing: One of the biggest problem for modern digital sound synthesisers (sampling freq fs=48kHz, Nyquist freq fn=fs/2=24kHz). How to avoid aliasing? Storing a band-limited version of the waveform in the table (in memory) Or, generate an aliasing-free signal from frequency-limited Fourier series representation. Slide 16 Aliasing (2) Several sinusoids can fit a set of samples. Aliasing when sampling rate is low! Example: Signal: f0 = 0.9Hz (red) Sampling at: fs = 1Hz, Nyquist freq fn = 0.5Hz perceived fa=|n*fs-f0|=0.1Hz (blue) (n such that fa < fn) Slide 17 Aliasing (3) Square wave, 563 Hz fundamental, 48kHz sampling rate. Generated using perfect square waveform Generated using a limited Fourier series. Slide 18 Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE Slide 19 Time Envelope (1) ADSR Envelope Attack Decay Sustain Release Important is: Duration Shape Linear Exponential Other (functional, table) Slide 20 Linear vs. Exponential Envelope Recall: amplitude perception is (nearly) logarithmic linear decay logarithmic (perceived) fading Exponential decay linear (perceived) fading Note: Exponential decay never reaches zero set min value A) LinearB) Exponential Slide 21 Oscillator as an Envelope Generator Advantages: wavetable interpolated shape. Easy encoding of several repetitions. Drawback: attack and decay times are affected by overall duration! Alternative: interpolated function generator fc A fm Slide 22 Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE Slide 23 Simple Instrument Helmholtz model Waveform Constant frequency Envelope Envelope feeds varying amplitude to the oscillator. ASD Envelope AMP FREQ PHASE AMP ATTACK DURATION DECAY Slide 24 Simple Instrument (2) Envelope generator used as a signal processor. Oscillator feeds varying amplitude to the envelope generator. Allows to process the amplitude of a natural (recorded) sound through an envelope. AMPFREQ PHASE ASD Envelope AMP ATTACK DURATION DECAY Slide 25 Limitations of the Simple Instrument Helmholtz model Waveform Constant frequency Envelope Limitations: Amplitudes of all spectral components vary simultaneously. All spectral components are perfect (integer) harmonics.... unlike real sounds! ASD Envelope AMP FREQ PHASE AMP ATTACK DURATION DECAY Slide 26 Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE Slide 27 Types of synthesis Sound Synthesis Additive synthesis Distortion techniques Subtractive synthesis Granular synthesis Analysis based Physical modelling Slide 28 Additive Synthesis FREQ + Slide 29 Additive Synthesis (2) Analysis: Frequency and amplitude envelopes can be obtained from analysis (spectrogram) Flexibility: Virtually any sound can be synthesised. Allows for the generation of new, natural sounding functions. Quality: Can realize sounds that are indistinguishable from real tones by skilled musicians (Risset, Computer Study of Trumpet Tones, 1966) Slide 30 Additive Synthesis (3) But... Require large amount of data to describe a sound Each oscillator requires two functions Functions are only valid for limited range of pitch and loudness! Analysis for a given pitch and loudness will not give the same timbre when extrapolated for different pitch and loudness. Requires very large library of function sets! Just too much control? Slide 31 Plan Simple Oscillator (wavetable) Envelope control Simple Instrument (Helmholtz) Additive Synthesis Modulation Summary WF AMPFREQ PHASE Slide 32 Modulation Modulation: Alteration of amplitude, phase or frequency of an oscillator, in accordance to another signal (Dodge & Jerse, 1997) Vocabulary: Carrier oscillator: modulated oscillator Carrier wave: modulated signal (prior to modulation) Spectral components of modulated signal : Carrier components: come only from carrier Sidebands: come from both carrier & modularion Slide 33 Amplitude Modulation Carrier: Frequency: fc Modulating Frequency: fm Amplitude m*AMP Modulation index: m m=0 no modulation m>0 modulation m=1 full modulation AMP fc m*AMP fm AMP + Slide 34 Amplitude Modulation (2) Carrier frequency fc Unaffected by modulation index Sidebands fc+/-fm Amplitude m/2*AMP Energy split equally between lower/higher When m=1, sidebands 6dB below carrier Perception If fm>10Hz -> two tones, additional loudness. If fm tremolo m/2*AMP AMP fc-fmfc+fmfc Amplitude Frequency Pure tone fc=220Hz Tremolo fc=220Hz, fm=6Hz, m=1 Slide 35 Amplitude Modulation (3) Slide 36 Ring Modulation Modulation is applied directly to carriers amplitude. A=0 no signal! Alters frequency! If both sinusoidals: Only sidebands: fc-fm and fc+fm! Amplitude A/2 Eq. to signal multiplication fc A fm A/2 fc-fmfc+fmfc Amplitude Frequency fc A fm A * Slide 37 Vibrato Modulation Modulating signal applied to the carriers frequency. Slight wavering of pitch Pitch varying between fc-v