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\chapter{Synthesizers}
A synthesizer is an electronic musical instrument that allows to generate audio signals. It generally presents as a piano keyboard, along with many knobs for tuning parameters, grouped in sections that correspond to the different stages of shaping the sound of the synthesizer. The first stage is to generate a sound wave, and is called an \textbf{oscillator}. This can be done in several ways which are described in section \ref{synth_types}. The wave is then sent to the next stages which are filters changing its shape. Generally, these filters include an \textbf{equalizer (EQ)} filter, an \textbf{amplifier} filter, and \textbf{effects (FXs)}. These filters are described in section \ref{synth_filters}. Additionally, a \textbf{ low frequency oscillator (LFO)} can be used to automatically control any parameter of each of the stages described above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\center
\begin{dspBlocks}{.7}{.6}
% 1 2 3 4 5
\BDfilterMulti{\textbf{Oscillator}\\- Synthesis\\- Pitch\\- Envelope} & \BDfilterMulti{\textbf{EQ filter}\\- Type\\- Cutoff\\- Envelope} & \BDfilterMulti{\textbf{Amplifier}\\- Distortion\\- Panning\\- Envelope} & \BDfilterMulti{\textbf{FXs}\\- FX type} & ~~sound out\\
\BDfilterMulti{\textbf{LFO}\\- Waveform\\- Target} & \BDsplit & \BDsplit & &
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\ncline{->}{1,3}{1,4}\ncline{->}{1,4}{1,5}
\ncline{2,1}{2,4}\ncline{->}{2,1}{1,1}\ncline{->}{2,2}{1,2}\ncline{->}{2,3}{1,3}\ncline{->}{2,4}{1,4}
\end{dspBlocks}
\caption{Pipeline of the stages of sound processing in a synthesizer.}
\label{synth_pipeline}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{History}
At the beginning of the 20th century, music started making used of electricity. Early electrical instruments include the theremin or the ondes Martenot, although they never achieved widespread success, due to their complexity to use and experimental sound. The first successful electrical music instrument is often considered to be the \textbf{Hammond organ}, introduced in 1935. The Hammond organ used a pile of rotating metal wheels called "tonewheels", through which an electrical current was sent. The resistance encountered in the wheel would generate electromagnetic waves within the metal that would be picked up by a microphone. The Hammond organ is however not an electronic music instrument, since the sound is generated using a physical phenomenon, and not created purely virtually. The warm sound of the Hammond organ would stay popular throughout the 20th century, with bands such as \textit{Emerson Lake and Palmer}, or \textit{Deep Purple}.\\
\begin{figure}[h]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{figures/theremin.jpg}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{figures/hammond.jpg}
\end{minipage}
\caption{Moog's etherwave theremin plus (left), and Hammond organ C-3 (right).}
\end{figure}
Although some experimental virtual sound generators using punched paper tape were developed in the 1950s, the first state-of-the-art synthesizer was the \textbf{Moog synthesizer}, designed in 1964, and first produced in 1967. It was the first instrument to feature a keyboard which could generated purely synthetic sounds tunable in live while playing. It was analog and monophonic, meaning that only one note could be played at a time. This first version of the synthesizer was modular, that is each stage of the sound processing was located in a different box, that could be connected together using cables. The Moog synthesizer was not mass produced and only available on command. Early popular Moog musicians from the 1960s include classical organ player \textit{Wendy Carlos} or psychedelic rock band \textit{The Doors}'s keyboardist \textit{Ray Manzarek}.\\
The Moog synthesizer introduced a set of synthesizer components that became standards in synthesizer design, including the envelopes, noise generators, or EQ filters. In 1970, Moog introduced a smaller and portable synthesizer, available to the large public, the \textbf{MiniMoog}. Over the 1970s, other brands started marketing cheaper synthesizers, although this instrument still staid a minor instrument until the 1980s, as it was still expensive and difficult to use. By the end of the 1970s, digital synthesizers started to overtake analog synthesizers, as they allowed much cheaper implementations.
\begin{figure}[h]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{figures/1c.jpg}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{figures/YAMAHA_DX7.jpg}
\end{minipage}
\caption{Modular synthesizer Moog 1C (1967, left), and FM synthesizer Yamaha DX7 (1982, right).}
\end{figure}
By the early 1980s, the technological improvements allowed to make synthesizers for much cheaper prices, and allowed new features such as polyphony and sampling. Polyphony is the ability to play many notes at the same time on the keyboard, and sampling is the ability to record a sound with a microphone, and later play it back at different pitches on the keyboard. These new advantages quickly turned the synthesizer into one of the emblematic instrument of the 1980s. As many companies started making affordable synthesizers, such as \textit{Yamaha}, \textit{Korg}, or \textit{Akai}, many pop music bands integrated synthesizers as a lead instrument in their compositions, such as \textit{New Order}, or \textit{The Human League}.\\
The versatility of synthesizers soon gave birth to a whole new music genre, \textbf{electronic music}, generated mainly using synthesizers, samplers, and other electronic instruments. In 1997, software synthesizers were introduced, allowing to generate sounds directly from a DAW on a producer's computer and getting rid of all physical device. They are generally contained in a VSP plugin, and can be controlled by MIDI.
\section{Waveforms}\label{waveforms_section}
An important tool in sound generation are waveforms. From an electronic music point of view, a waveform is a periodic signal with tunable parameters: typically the amplitude, frequency, and phase. It can hence mathematically written in the form
\begin{align*}
wave[A, \omega_c, \phi; n]
\end{align*}
where $A$ is the amplitude, $\omega_c$ is the angular frequency, $\phi$ is the phase, and $n$ is the sample number.\\
Waveforms have many uses in a synthesizers. They can be used to directly generate an audio signal in an oscillator, as well as to control the parameters of any component of the synthesizer. For instance, a waveform can be used to control the volume of the synthesizer, hence creating fluctuation of the volume over time. Waveforms can basically be used to control any parameter set in a range. In the next sections, we will see the main usages of waveforms in the different stages of sound making.\footnote{See this page for nice animations of how to build basic waveforms from sines: \url{https://commons.wikimedia.org/wiki/File:Fourier_series_square_wave_circles_animation.gif}.}
\subsection{Sine wave}
The sine wave is the most fundamental waveform that an oscillator can generate. In an analog oscillator, all waveforms are generated as a (large but) finite sum of sines. It is composed of only one frequency and hence has a rather quiet sound. Its shape being smooth and lacking edges, the sine waveform has a rather natural, colorless sound.
\begin{align*}
sine[A, \omega_c, \phi; n] = A \sin (\omega_c n - \phi)
\end{align*}
\begin{figure}[h]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/sine.pdf}
\end{minipage}%
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/sine_fft.pdf}
\end{minipage}
\caption{Waveform and log-domain FFT of a sine wave.}
\label{sine_wave}
\end{figure}
In the figure \ref{sine_wave}, one can see the graphical waveform and the corresponding FFT of a sine wave. Note that on the FFT plot, the amplitude axis is in log domain, and the frequency axis corresponds to the angular frequency $\omega_c$.
\subsection{Square wave}
The square wave is simply as signal alternating between $-A$ and $+A$. Its equation can be simply written as
\begin{align*}
square[A, \omega_c, \phi;n] = A \sgn \left( \sin (\omega_c n -\phi \right) ).
\end{align*}
This implementation is very simple and can be efficiently coded for digital synthesizers. In the old days of analog synthesizers, this wave was generated using an additive oscillator, adding a fixed number of sine waves to approximated the square wave. The corresponding equation reads
\begin{align*}
square[A, \omega_c, \phi;n] = \frac{4A}{\pi} \sum_{k=1}^{\infty} \frac{ \sin ((2k-1)\omega_c n - \phi)}{2k - 1} ,
\end{align*}
where the infinite sum is in practice approximated with a high number.
\begin{figure}[h]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/square.pdf}
\end{minipage}%
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/square_fft.pdf}
\end{minipage}
\caption{Waveform and log-domain FFT of a square wave generated using the digital and additive methods.}
\label{square_wave}
\end{figure}
One can observe the square waveform and its FFT in figure \ref{square_wave}. The square wave has a lot of harmonics that decrease slowly in amplitude. This give the square wave a sharper, brighter sound than the sine wave. The square wave was used a lot in early videogames consoles, such as the Commodore 64\footnote{An example of Commodore 64 music: \url{https://youtu.be/EEbFlL_jZ5s}}, and is now often associated with retro games music, or \textit{chiptune}. In the figure are compared a digital square wave, generated directly using the exact equation described above, as well as its additive synthesis approximation, with $k=10$. One can observe in the FFT plot that only the 5 first harmonics of the square wave are generated.\\
An issue that is met when generating a square wave using additive synthesis is the\textbf{ Gibbs phenomenon}\footnote{\url{https://en.wikipedia.org/wiki/Gibbs_phenomenon}}. This phenomenon corresponds to the apparition of small vertical spikes at the corners of the square wave, as observed in the left plot of figure \ref{square_wave}.\footnote{An interactive animation from Dr. Prandoni's introductory DSP class highlights the Gibbs phenomenon for classical waveforms: \url{https://www.sp4comm.org/gibbs/gibbs.html}.} The phenomenon highlights the fact that Fourier series can describe perfectly any smooth function, but meet limitations with non smooth functions, such as the square wave. Indeed, the additive synthesis square wave does not converge to the digital square wave, even for $k \rightarrow \infty$, and A spike measuring approximately $18\%$ of the amplitude $A$ is always observed for arbitrary great values of $k$. This however does not generate problems from an audio point of view, as the sound of a square wave with spikes is generally considered similar to the sound of a digital square wave. Most additive synthesis synthesizer hence simply use this formula for generating square waves.\footnote{A nice animation of how to build a square wave from a sum of sines can be found here: \url{https://www.youtube.com/watch?v=k8FXF1KjzY0}.}
\subsection{Sawtooth wave}
The sawtooth wave first increases linearly from $-A$ to $+A$ and then instantly jumps back to $-A$. This is digitally written as
\begin{align*}
saw[A, \omega_c, \phi;n] = \frac{2A}{\pi} \arctan ( \tan (\omega_c n/2 - \phi) ) .
\end{align*}
Similarly as for the square wave, analog synthesizer approximate this wave using additive synthesis as
\begin{align*}
saw[A, \omega_c, \phi;n] = -\frac{2A}{\pi} \sum_{k=1}^{\infty} \frac{(-1)^k}{k}\sin (k \omega_c n - \phi) .
\end{align*}
\begin{figure}[h]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/saw.pdf}
\end{minipage}%
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/saw_fft.pdf}
\end{minipage}
\caption{Waveform and log-domain FFT of a sawtooth wave generated using the digital and additive methods.}
\label{saw_wave}
\end{figure}
In figure \ref{saw_wave}, one can observe a sawtooth waveform and its FFT, where $k=15$. One can remark that the Gibbs phenomenon is present as for the square wave, since the sawtooth contains a vertical discontinuity. The sawtooth wave has a very crispy, aggressive sound and is for that reason quite popular in Trance and House electronic music. A common use of the sawtooth wave, called the \textbf{supersaw}, consists in adding several sawtooth waves with slightly detuned frequencies (+/- 1Hz) together. This gives the sawtooth a more dense sound, similar to what a chorus effect would do. Famous artists using supersaws include Avicii or Martin Garrix\footnote{See notably the song \url{https://youtu.be/ScNsBpQ8KIg?t=60}}.
\subsection{Triangle wave}
The triangle wave is an alternatively increasing then decreasing linear signal between $-A$ and $+A$. It is digitally written as
\begin{align*}
tri[A, \omega_c, \phi ;n] = \frac{2A}{\pi} \arcsin ( \sin (\omega_c n - \phi) ).
\end{align*}
The corresponding additive synthesis implementation reads
\begin{align*}
tri[A, \omega_c, \phi;n] = \frac{8A}{\pi^2} \sum_{k=0}^\infty \frac{ (-1)^k }{ (2k + 1)^2} \sin ((2k + 1) \omega_c n - \phi).
\end{align*}
\begin{figure}[h]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/triangle.pdf}
\end{minipage}%
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=\linewidth]{figures/triangle_fft.pdf}
\end{minipage}
\caption{Waveform and log-domain FFT of a triangle wave generated using the digital and additive methods.}
\label{triangle_wave}
\end{figure}
On figure \ref{triangle_wave}, a triangle waveform and its FFT are plotted, with $k=2$. The triangle wave having no vertical jump, the Gibbs phenomenon is not significant. The additive approximation converges faster than the sawtooth and the square wave towards the digital implementation. One can notice that with $k=2$, the approximation is already quite close to the digital wave. The triangle wave has a sound in between the smoothness of sine and the brightness of a sawtooth.
\section{Envelope generator}
An Envelope Generator (EG) is a circuit controlling the evolution of a played note over time. An EG can be set to control several note's properties, such as its amplitude, EQ filter, or pitch. The property controlled by the EG but have only one parameter that can be tuned within a range. The EG applies an \textbf{envelope} to the desired parameter of the note. An envelope a function of time that outputs the value of the parameter. When the note is not played, the parameter associated to the EG is set to 0. As soon as a key is pressed to play the note, the EG activates the envelope independently for each note played.\\
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.6\linewidth]{figures/envelope.png}
\caption[Caption for LOF]{Diagram of an EG, as presented in Korg's MicroKORG manual\protect\footnotemark, featuring\\ (1) attack time, (2) decay time, (3) sustain level, (4) release time.}
\label{envelope}
\end{minipage}
\end{figure}
\footnotetext{\url{https://cdn.korg.com/us/support/download/files/8f226053113b3be59753dcce14e74cca.pdf}}
An envelope is generally characterized by 4 elements:
\begin{itemize}
\item The \textbf{attack time} determines how fast the note parameter reaches its maximum value (the attack level) after the key is pressed. A longer attack time will generate smoother notes while short attack times will generate sharp, percussive notes.
\item The \textbf{decay time} determines how fast the parameter transitions from the attack level to the sustain level.
\item The \textbf{sustain level} represents the level of the parameter while the key is being hold.
\item The \textbf{release time} represents the time the parameter takes to go back to 0 after the key is released. Long release times give the note a tail, similarly to the sustain pedal on a piano. Short release times result in a precise, dry sound.
\end{itemize}
The 4 above elements are often abbreviated as \textbf{ADSR}. Figure \ref{envelope} shows the typical graphical representation of an EG. As said before, envelopes are mostly used for controlling a note's pitch (in the oscillator stage), filter's gain (in the EQ filter stage), or volume (in the amplifier stage).
\section{Synthesis}\label{synth_types}
One key implementation specification of a synthesizer is how it creates waves. Over the years, a multitude of synthesis techniques have been proposed, and we here focus on 4 of them: additive synthesis, subtractive synthesis, FM synthesis, and wavetable synthesis.
\subsection{Subtractive synthesis}
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.7\textwidth]{figures/gaia.jpg}
\caption[Caption for LOF]%
{Roland's GAIA SH-01 subtractive synthesizer, introduced in 2010\protect\footnotemark.} The subtractive synthesis is done in the OSC section.
\label{GAIA}
\end{minipage}
\end{figure}
\footnotetext{\url{https://www.roland.com/global/products/gaia_sh-01/}}
Subtractive synthesis is by far the most widespread type of synthesis in synthesizers. In a subtractive synthesizer, the oscillator is a waveform generator. In next steps, the filters remove frequencies from the original waveform, hence the name subtractive synthesis. The layout of the oscillator section generally contains a knob for choosing the waveform to start with, along with a pitch knob, and sometimes more parameters. \\
In figure \ref{GAIA}, one can observe the section "OSC" in the top left of the synthesizer. In this section, a button allows to choose among different waveforms, and a know allows to set its pitch. The other sections are common with all types of synthesizers.
\subsection{Additive synthesis}
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.7\textwidth]{figures/additive.JPG}
\caption[Caption for LOF]%
{Solcito Musica's Arido 3 additive synthesizer.\protect\footnotemark}
\label{additive}
\end{minipage}
\end{figure}
\footnotetext{\url{https://solcitomusica.blogspot.com/p/virtuales.html}}
Additive synthesis is based on Fourier theory, taking advantage of the fact that any audio signal can be generated by summing different sine waves. An additive synthesizer's oscillator hence simply generates many sine waves with different amplitudes, frequencies, and phases, and then sums them together to obtain the desired wave shape, hence the name. Additive synthesizers are not quite common, as shaping the sound they produce is trickier than using a subtractive synthesizer. However they allow a very fine control of the wave shape.\\
An example of additive synthesizer is shown in figure \ref{additive}. In the oscillator 1 section, the user can tune the amplitude of 10 different harmonic sine waves, and set a different amp envelope to each. The others sections include a LFO, an EQ filter, and FXs, which are common to all synthesizers.
\subsection{FM synthesis}
\begin{figure}[h]
\centering
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.7\textwidth]{figures/yamahadx7.png}
\end{minipage}\\
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/DX7_algo.jpg}
\end{minipage}
\caption[Caption for LOF]%
{FM synthesizer Yamaha DX7 (top) and detail of the featured modulation networks (bottom).}
\label{FM}
\end{figure}
FM synthesis relies on Frequency Modulation (FM) as used in radio broadcast. An oscillator produces a waveform, the carrier, that is then modulated with another waveform, the modulator. The carrier is generally a sine wave, while the modulator can have any shape. In FM broadcasting, the carrier has a much higher frequency than the modulator. In FM synthesis, both the carrier and modulator signals have similar, audible frequencies. The carrier wave determines the pitch of the produced note, and the modulator determines its timbre. The equation of modulating a sine carrier with amplitude $A_C$, angular frequency $\omega_C$ and zero phase with an arbitrary modulator wave $x[n]$ reads
\begin{align*}
y[n] = A_C \cdot \sin \left( \omega_C n + 2 \pi K_f \cdot x[n] \right)
\end{align*}
where $K_f$ is the modulation index, the sensitivity of the frequency modulator.\\
The output of the modulation can be used as the input of a subsequent modulation (or itself!), and different dispositions of modulator networks can generate different sounds. Modulations can be done in series and parallel and each output can be connected to any input. The behaviour of a modulation network is quite hard to predict, and generally the user experiments with different networks and parameters until they find a satisfying sound. In figure \ref{FM}, the Yamaha DX7 is shown as well as a detail of several of its available modulation networks\footnote{A nice introduction to FM synthesis can be found in Yamaha DX7's user manual: \url{http://www.synthfool.com/docs/Yamaha/DX_Series/Yamaha\%20DX7\%20Operating\%20Manual.pdf}.}. The diagram of a network is to be read from top to bottom. Each rectangle represents a modulation. Rectangles at the top are pure sine wave generators, while rectangles below receive the signal from their respective top rectangles as the modulator signal, and use a new sine wave as the carrier. Horizontal and diagonal connections mean that the different signals coming from the top are averaged together before being fed to the bottom rectangle.
\subsection{Wavetable synthesis}
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.7\textwidth]{figures/halion3.jpg}
\caption[Caption for LOF]%
{Steinberg HALion 3's ANIMA wavetable synthesizer.\protect\footnotemark}
\label{halion3}
\end{minipage}
\end{figure}
\footnotetext{\url{https://new.steinberg.net/vst-instruments/halion/}}
Wavetable synthesis was introduced in the late 1970s by Palm Products GmbH but has mostly gained popularity recently due to its easiness of implementation and use on a computer as a VST plugin. The principle is very simple: the synthesizer keeps in memory a table whose entries contain each a single cycle of a different waveform (preferably all of the same length so that they have the same frequency when played back). The user can then select (or even import) a waveform and play it periodically at different speeds, resulting in different frequencies. \\
A key feature of wavetables is the ability to quickly and automatically switch from a waveform to another, while playing. For instance, the user can load 10 waveforms at the same time and use a LFO to periodically alternate between them. Similarly, an EG can be set to switch from a harsh sounding to a more mellow waveform as during the decay time. Hence, multiple similar waveforms often are grouped together in order to be switched from and to, like in figure \ref{halion3}. Generally, wavetable synths implement a simple linear interpolation between waveforms, so that the switch from a waveform to another sounds smooth. This is called morphing. Figure \ref{morpn} shows a simple example of morphing between 2 basic waveforms, square and sine. The user can tune the "morphing amount" to shift the generated waveform closer to the sine or the square wave. \\
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.6\textwidth]{figures/morph.pdf}
\caption[Caption for LOF]%
{Wavetable morphing between a square wave and a sine wave. The generated wave is a linear interpolation of the square and the sine, and the quantity of each can be tuned.}
\label{morpn}
\end{minipage}
\end{figure}
Dealing with many waveforms in parallel is better understood visually, which requires a high quality display, hence the growing popularity of wavetables as graphical VST plugins in the recent years, such as the HALion 3 ANIMA VST shown in figure \ref{halion3}.
\section{Components of a synthesizer}\label{synth_filters}
As seen in diagram \ref{synth_pipeline}, a synthesizer is a pipeline of different component shaping the sound in turn. In this section, we review each of them in more details, by each time analysing the \textit{Roland GAIA SH-01} as an example.\footnote{See the SH-01 owner's manual for a detailed description of each component: \url{https://static.roland.com/assets/media/pdf/SH-01_OM.pdf}.}
\subsection{Oscillator}
As said before, the first step of generating a sound is to generate a wave using an oscillator. An oscillator implements one of the synthesis method presented in \ref{synth_types}, before feeding the produced wave to further synthesizer components for refining the sound more.\\
On figure \ref{GAIA_osc}, the user can switch between the basic waveforms using a switch button (remember that this is a subtractive oscillator) and subsequently change the base pitch of the wave using the 2 knobs. The 2 bottom left sliders allow some basic distortion of the waveform, making a cycle more or less symmetrical. The 3 bottom right sliders allow to add an EG to the \textbf{pitch} of the wave, unlike the subsequent amplifier EG which will control the volume of the wave (more details in section \ref{amp_EG}).
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.4\textwidth]{figures/gaia_osc.png}
\caption[Caption for LOF]%
{Detail of the oscillator section on the Roland GAIA SH-01.}
\label{GAIA_osc}
\end{minipage}
\end{figure}
\subsection{EQ filter}
The oscillator often generates complex wavesforms that contain a lot of harmonics. An equalizer (EQ) filter (sometimes simply called \textbf{filter}) is generally positioned after the oscillator in order to filter out the unwanted harmonics and to make the sound more precise in the frequency domain. Generally, the filter can be of 3 types: lowpass, highpass, and bandpass. Lowpass keeps only low frequencies, highpass only high frquencies, and bandpass only mid frequencies. The filter is often parametrized by the range of frequencies it applies to, and by how fast the transition happens between the filterer frequencies and the untouched frequencies. EQ filters are described in details in the section \ref{EQ_section}.\\
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.4\textwidth]{figures/gaia_filter.jpg}
\caption[Caption for LOF]%
{Detail of the filter section on the Roland GAIA SH-01.}
\label{GAIA_filter}
\end{minipage}
\end{figure}
On most synthesizers, the intensity (or gain) of the filter can be controlled by a dedicated envelope generator, called the filter EG. An example of using the filter EG would be to set the filter to lowpass and use a slow attack, hence keeping all the note's frequencies at first and then quickly filtering them. The sound is then quite powerful for a very short time before reducing to a more filtered sound.
\subsection{Amplifier}\label{amp_EG}
The step after the filter is generally the amplifier. The amplifier is in charge in controlling the amplitude of the note. Generally, a single nob controls the general static base level of all notes' amplitude, and an envelope generator is then used to control the amplitude with more precision over time.Some amplifiers also allow to control the panning of the note, as well as the amount of distortion to apply.
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.3\textwidth]{figures/gaia_amp.png}
\caption[Caption for LOF]%
{Detail of the amplifier section on the Roland GAIA SH-01.}
\label{GAIA_amp}
\end{minipage}
\end{figure}
\subsection{LFO}
A Low Frequency Oscillator (LFO) is, as the name suggests, an oscillator which operates at a low frequency, generally within $(0, 20]$Hz. LFOs are generally generated using one of the waveforms presented in section \ref{waveforms_section}, and that for all types of synthesizers. Although FM synthesizers do not use waveform oscillators for generating the source sound, they generally still use waveforms for the LFO. Oppositely to a standard oscillator, an LFO does not generate a sound, but is used to automatically control a parameter of the synthesizer and to make it change over time. Virtually, any parameter of the amplifier can be controlled by an LFO (even the LFO parameters themselves). For instance, an LFO can be used to automatically control the synthesizer's master volume, to create a \textbf{tremolo}, or to control the
phase of the main oscillator to create a \textbf{vibrato}.\\
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.3\textwidth]{figures/gaia_lfo.png}
\caption[Caption for LOF]%
{Detail of the LFO section on the Roland GAIA SH-01.}
\label{GAIA_LFO}
\end{minipage}
\end{figure}
On figure \ref{GAIA_LFO}, one can see the LFO section of the Roland GAIA SH-01. The user can choose among the typical waveforms, and set their frequency (here called rate). The Roland GAIA SH-01 does not allow to send the LFO to any parameter of the synthesizer, but only to 3 of them. Using the bottom sliders, the user can chose how much the LFO is going to control the pitch of the oscillator, the gain of the EQ filter, and the gain of the amplifier, respectively. Some more complex synthesizers allow the user to send the LFO signal to any other component.
\subsection{FXs}
The effects (FXs) section is generally the last section the sound goes through before being set to the master track and being outputed. This section is very versatile and highly depends on the synthesizer, but generally consists in a list of audio effects among which one can be applied to the sound signal. The most popular audio effects are described later in the section \ref{digital_audio_effects_section}.
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.3\textwidth]{figures/gaia_fx.png}
\caption[Caption for LOF]%
{Detail of the FXs section on the Roland GAIA SH-01.}
\label{GAIA_fx}
\end{minipage}
\end{figure}
\section{Other electronic instruments}
\subsection{Sampler}
A sampler is an electronic instrument that allows to record an audio signal, and to play it back at different pitches using a keyboard. Pitch-shifting can be done using the methods shown in the section about time-frequency analysis. Early samplers used a set of looping magnetic tapes that could be played back at different speed to change the pitch. Later implementations would use a RAM storing samples that would be played back at different sampling rates. More complex samplers can use different samples depending on the pitch or velocity of the played notes, instead of pitch-shifting and changing the amplitude.\\
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.7\textwidth]{figures/akai.jpg}
\caption[Caption for LOF]%
{The Akai MPC ONE is a music pad that can be used as sampler and MIDI controller.}
\label{akai}
\end{minipage}
\end{figure}
Most electric piano and keyboards are actually samplers. Nowadays, samplers are mostly virtual and can be used within a DAW. A musician can use any MIDI device, such as a midi keyboard or a pad, to play the samples from the computer. A pad is a electronic devices with large rubber buttons that is generally used to play rhythms using one's finger in real time, or to play special effects sounds. The Akai MPC ONE, pictured in figure \ref{akai} is a popular music pad.
\subsection{Drum machine}
A drum machine is a special type of synthesizer specialized in producing rhythms. It generally does not feature pitch-shifting, as drum elements only have a single tune. Drum machines started being developed from the late 1970ies. Early drum machines generated each drum sound synthetically. As computer memory performance improved in the next years, drum machines started using prerecorded samples instead of synthetic sounds, with additional samples and patches importable by the user. To each sample is associated a loop sequence composed of $2^N$ beats that can each be turned on and off independently using a keyboard. Watch \url{https://www.youtube.com/watch?v=4d89S-jOsfY} for a quick history of drum machines.\\
\begin{figure}[h]
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=.6\textwidth]{figures/TR_808.jpg}
\caption[Caption for LOF]%
{The Roland TR-808.\protect\footnotemark}
\label{tr_808}
\end{minipage}
\end{figure}
\footnotetext{See the official website \url{https://www.roland.com/global/products/rc_tr-808/}.}
In figure \ref{tr_808}, Roland's TR-808's interface is shown. The TR-808 is often considered as one, if not the most, iconic drum machine of all time. It was released in 1980, and used analog synthesis to generated each drum sound. It features 11 different sounds, that all use a dedicated loop sequence composed of 16 beats. The drum machine then plays all the sequences together in a loop. The user can select a sound and then edit its beat sequence live.
\section{Exercises}
\begin{enumerate}
\item Prove that $tri[A, \omega_c, \phi;n] = \int_{0}^n square[A, \omega_c, \phi; u] du$.
\end{enumerate}
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