Wavefolding

Wavefolding¶

Wavefolding is a special case of waveshaping, working with periodic transfer functions. Depending on the pre-gain, the source signal gets folded back, once a maximum of the transfer function is reached. Compared to the previously introduced soft clipping or other methods of waveshaping, this adds many strong harmonics.

Periodic Shaping Function¶

A simple basic transfer function is a sine with the appropriate scaling factor. The pre-gain $g$ is the parameter for controling the intensity of the folding effect:

$$y[n] = sin( g \frac{\pi}{2} x[n])$$

For an input signal $x$, limited to values between $-1$ and $1$, respectively for gains $g\leq1$, this results in a sinusoidal waveshaping function with saturation:

Text(0,0.5,'y')

When the input signal exceeds the boundaries $-1$ and $1$, the signal does not clip but is folded back. This can be achieved by amplifying the input with an additional gain:

For a gain of $g=3$, the time-domain output signal looks as follows:

Spectrum for a Sinusoidal Input¶

The spectrum of wavefolding can be calculated by expressing the folding term as a Fourier series: The Jacobi–Anger expansion can be used for this purpose, with the pre-gain $g$:

$$\sin(g \sin(x)) = 2 \sum\limits_{m=1}^{\infty} J_{2m-1}(g) \sin((2m-1)x)$$

At this point it is already apparent that the resulting signal contains harmonics at odd integer multiples of the fundamental frequency $f_m = 100 \mathrm{Hz}\ (2 m -1)$. Their gain is determined by first kind Bessel functions $J_{2m-1}(g)$:

For the DFT this leads to:

$$\begin{eqnarray} X[k] &=& 2 \sum\limits_{m=1}^{\infty} J_{2m-1}(g) \sin((2m-1)x) \sum\limits_{n=0}^{N-1} e^{-j 2 \pi k \frac{n}{N}} \\ % % % X[k] &=& 2 \sum\limits_{n=0}^{N-1} \sum\limits_{m=1}^{\infty} J_{2m-1}(g) \sin((2m-1)x)\ e^{-j 2 \pi k \frac{n}{N}} \\ % % % X[k] &=& 2 \sum\limits_{n=0}^{N-1} \sum\limits_{m=1}^{\infty} J_{2m-1}(g) \frac{1}{2} \left( e^{j (2m-1)x} - e^{-j(2m-1)x} \right) % \sin((2n-1)x)\ e^{-j 2 \pi k \frac{n}{N}} \\ % % % X[k] &=& \sum\limits_{n=0}^{N-1} \sum\limits_{m=1}^{\infty} J_{2m-1}(g) \left( e^{-j 2 \pi k \frac{n}{N} + j (2m-1)x } - e^{-j 2 \pi k \frac{n}{N} -j(2m-1)x} \right) % \sin((2n-1)x)\ \end{eqnarray}$$

With $x = 2 \pi \frac{f_0}{f_s} n$

$$X[k] = \sum\limits_{n=0}^{N-1} \sum\limits_{m=1}^{\infty} J_{2m-1}(g) \left( e^{-j 2 \pi k \frac{n}{N} + j (2m-1) 2 \pi \frac{f_0}{f_s} } - e^{-j 2 \pi k \frac{n}{N} -j(2m-1) 2 \pi \frac{f_0}{f_s}} \right) % \sin((2n-1)x)\$$

1 Hints on this by Peyam Tabrizian can be found here: https://youtu.be/C641y-z3aI0

DFT Plots¶

The below plots show the spectra of the folding operation for a sine input of $100 \mathrm{Hz}$ at different gains. With increasing gain, partials are added at the odd integer multiples of the fundamental frequency $f_m = 100 \mathrm{Hz}\ (2 m -1)$:

[ 100  300  500  700  900 1100 1300 1500 1700 1900] ...


Using APIs with Python

Python & APIs

With the modules requests and json it is easy to get data from APIs with Python. Using the above introduced methods for sequencing, the following example requests a response from https://www.boredapi.com/:

#!/usr/bin/env python3

import requests
import json

response = requests.get("https://www.boredapi.com/api/activity")
data     = response.json()

print(json.dumps(data, sort_keys=True, indent=4))

# print(data["activity"])


Combining Nodes in SuperCollider

Creating and Connecting Nodes

Audio buses can be used to connect synth nodes. In this example we will create two nodes - one for generating a sound and one for processing it. First thing is an audio bus:

~aBus = Bus.audio(s,1);


The ~osc node generates a sawtooth signal and the output is routed to the audio bus:

~osc = {arg out=1; Out.ar(out,Saw.ar())}.play;

~osc.set(\out,~aBus.index);


The second node is a simple filter. Its input is set to the index of the audio bus:

~lpf = {arg in=0; Out.ar(0, LPF.ar(In.ar(in),100))}.play;

~lpf.set(\in,~aBus.index);


Warning

Although everything is connected, there is no sound at this point. SuperCollider can only process such chains if the nodes are arranged in the right order. The filter node can be moved after the oscillator node:

Moving Nodes

The moveAfter() function is a quick way for moving a node directly after a node specified as the argument. The target node can be either referred to by its node index or by the related name in sclang:

~lpf.moveAfter(~osc)


Pure Data: Send-Receive & Throw-Catch

Control Rate

Send and receive objects allow a wireless connection of both control and audio signals. The objects are created with send and receive or short s and r for control rate signals and get one argument - a string labeling the connection.

Prepending a $0- to a send label turns it into a local connection. These are only valid inside a patch and its subpatches but not across different abstractions. The example send-receive-help.pd shows the difference between local and global sends when used in both cases. It relies on the additional abstraction send-receive.pd which needs to be in the same directory: Send and receive of control signals with subpatch and abstraction. The inside of both the subpatch and the abstraction are identical: Inside of send-receive and the subpatch. Audio Rate Audio send and receives follow the same rules as control ones. They are created with an additional ~, as usual for audio objects. The example send-receive-audio.pd shows the use of these buses: Send and receive of audio signals with subpatch and abstraction. Throw & Catch Throw and catch are bus extensions of the above introduced send-receive method, only for audio signals. Unlike with s~ and r~, it is possible to send multiple signals to one catch~. This allows a flexible audio routing and grouping without a lot of lines. The example throw-catch.pd throws four sine waves to a common bus for a minimal additive synthesis: Using throw and catch to merge four signals. Digital Waveguides: Discrete Wave Equation Wave Equation for Ideal Strings The ideal string results in an oscillation without losses. The differential wave-equation for this process is defined as follows. The velocity $c$ determines the propagation speed of the wave and this the frequency of the oscillation. \begin{equation*} \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} \end{equation*} A solution for the different equation without losses is given by d'Alembert (1746). The oscillation is composed of two waves - one left-traveling and one right traveling component. \begin{equation*} y(x,t) = y^+ (x-ct) + y^- (x+ct)$ \end{equation*}
• $y^+$ = left traveling wave

• $y^-$ = right traveling wave

Tuning the String

The velocity $c$ depends on tension $K$ and mass-density $\epsilon$ of the string:

\begin{equation*} c^2 = \sqrt{\frac{K}{\epsilon}} = \sqrt{\frac{K}{\rho S}} \end{equation*}

With tension $K$, cross sectional area $S$ and density $\rho$ in ${\frac{g}{cm^3}}$.

Frequency $f$ of the vibrating string depends on the velocity and the string length:

\begin{equation*} f = \frac{c}{2 L} \end{equation*}

Make it Discrete

For an implementation in digital systems, both time and space have to be discretized. This is the discrete version of the above introduced solution:

\begin{equation*} y(m,n) = y^+ (m,n) + y^- (m,n) \end{equation*}

For the time, this discretization is bound to the sampling frequency $f_s$. Spatial sample distance $X$ depends on sampling-rate $f_s = \frac{1}{T}$ and velocity $c$.

• $t = \ nT$

• $x = \ mX$

• $X = cT$

Faust: MIDI

Using MIDI CC

Using MIDI in Faust requires only minor additions to the code and compiler arguments. For first steps it can be helpful to control single synth parameters with MIDI controllers. This can be configured via the UI elements. The following example uses MIDI controller number 48 to control the frequency of a sine wave by adding [midi:ctrl 48] to the hslider parameters.

// midi-example.dsp
//
// Control a sine wave frequency with a MIDI controller.
//
// Henrik von Coler
// 2020-05-17

import("stdfaust.lib");

freq = hslider("frequency[midi:ctrl 48]",100,20,1000,0.1) : si.smoo;

process = os.osc(freq) <: _,_ ;


CC 48 has been chosen since it is the first slider on the AKAI APC mini. If the controller numbers for other devices are not known, they can be found using the PD patch reverse_midi.pd.

Compiling with MIDI

In order to enable the MIDI functions, the compiler needs to be called with an additional flag -midi:

MIDI on Linux

Faust programs use Jack MIDI, whereas MIDI controllers usually connect via ALSA MIDI. In order to control the synth with an external controller, a bridge is nedded:

$a2jmidi_bridge The MIDI controller can now connect to the a2j_bridge input, which is then connected to the synth input. Faust: Splitting and Merging Signals Splitting a Signal To Stereo The <: operator can be used to split a signal into an arbitrary number of branches. This is frequently used to send a signal to both the left and the right channel of a computer's output device. In the following example, an impulse train with a frequency of$5\ \mathrm{Hz}\$ is generated and split into a stereo signal.

import("stdfaust.lib");

// a source signal
signal = os.imptrain(5);

// split signal to stereo in process function:
process = signal <: _,_;


To Many

The splitting operator can be used to create more than just two branches. The following example splits the source signal into 8 signals:

To achieve this, the splitting directive can be extended by the desired number of outputs:

process = signal <: _,_,_,_,_,_,_,_;


Merging Signals

Merging to Single

The merging operator :> in Faust is the inversion of the splitting operator. It can combine an arbitrary number of signals to a single output. In the following example, four individual sine waves are merged:

Input signals are separated by commas and then joined with the merging operator.

import("stdfaust.lib");

// create four sine waves
// with arbitrary frequencies
s1 = 0.2*os.osc(120);
s2 = 0.2*os.osc(340);
s3 = 0.2*os.osc(1560);
s4 = 0.2*os.osc(780);

// merge them to two signals
process = s1,s2,s3,s4 :> _;


Merging to Multiple

Merging can be used to create multiple individual signals from a number of input signals. The following example generates a stereo signal with individual channels from the four sine waves:

To achieve this, two output signals need to be assigned after merging:

// merge them to two signals
process = s1,s2,s3,s4 :> _,_;


Exercise

Exercise

Extend the Merging to Single example to a stereo output with individual left and right channels.

Subtractive Example

The following example uses a continuous square wave generator with different filters for exploring their effect on a harmonic signal.

Pitch (VCF):

Filter Type:

Lowpass Highpass Bandpass Notch (Band Reject)

Cutoff (VFC):

Q (VCF):

Gain (VCA):

Time Domain:

Frequency Domain:

Fourier Series: Triangular

Formula

The triangular wave is a symmetric waveform with a stronger decrease towards higher partials than square wave or sawtooth. Its Fourier series has the following characteristics:

• only odd harmonics

• altering sign

• (squared)

$X(t) = \frac{8}{\pi^2} \sum\limits_{i=0}^{N} (-1)^{(i)} \frac{\sin(2 \pi (2i +1) f\ t)}{(2i +1)^2}$

Interactive Example

Pitch (Hz):

Number of Harmonics:

Output Gain:

Time Domain:

Frequency Domain:

Sampling & Aliasing: Square Example

For the following example, a sawtooth with 20 partials is used without band limitation. Since the builtin Web Audio oscillator is band-limited, a simple additive synth is used in this case. At a pitch of about $2000 Hz$, the aliases become audible. For certain fundamental frequencies, all aliases will be located at actual multiples of the fundamental, resulting in a correct synthesis despite aliasing. In most cases, the mirrored partials are inharmonic and distort the signal and for higher fundamental frequencies the pitch is fully dissolved.

Pitch (Hz):

Output Gain:

Time Domain:

Frequency Domain:

Anti-Aliasing Filters

In analog-to-digital conversion, simple anti-aliasing filters can be used to band-limit the input and discard signal components above the Nyquist frequency. In case of digital synthesis, however, this principle can not be applied. When generating a square wave signal with an infinite number of harmonics, aliasing happens instantaneously and can not be removed, afterwards.

Band Limited Generators

In order to avoid the aliasing, band-limited signal generators are provided in most audio programming languages and environments.