# PANNING AND SPATIALIZATION

## Simple Stereo Panning

Csound provides a large number of opcodes designed to assist in the distribution of sound amongst two or more speakers. These range from opcodes that merely balance a sound between two channel to ones that include algorithms to simulate the doppler shift that occurs when sound moves, algorithms that simulate the filtering and inter-aural delay that occurs as sound reaches both our ears and algorithms that simulate distance in an acoustic space.

First we will look at some methods of panning a sound between two speakers based on first principles.

The simplest method that is typically encountered is to multiply one channel of audio (aSig) by a panning variable (kPan) and to multiply the other side by 1 minus the same variable like this:

aSigL = aSig * kPan aSigR = aSig * (1 – kPan) outs aSigL, aSigR

kPan should be a value within the range zero and 1. If kPan is 1 all of the signal will be in the left channel, if it is zero, all of the signal will be in the right channel and if it is 0.5 there will be signal of equal amplitude in both the left and the right channels. This way the signal can be continuously panned between the left and right channels.

The problem with this method is that the overall power drops as the sound is panned to the middle.

One possible solution to this problem is to take the square root of the panning variable for each channel before multiplying it to the audio signal like this:

aSigL = aSig * sqrt(kPan) aSigR = aSig * sqrt((1 – kPan)) outs aSigL, aSigR

By doing this, the straight line function of the input panning variable becomes a convex curve so that less power is lost as the sound is panned centrally.

Using 90º sections of a sine wave for the mapping produces a more convex curve and a less immediate drop in power as the sound is panned away from the extremities. This can be implemented using the code shown below.

aSigL = aSig * sin(kPan*$M_PI_2) aSigR = aSig * cos(kPan*$M_PI_2) outs aSigL, aSigR

(Note that '$M_PI_2' is one of Csound's built in macros and is equivalent to pi/2.)

A fourth method, devised by Michael Gogins, places the point of maximum power for each channel slightly before the panning variable reaches its extremity. The result of this is that when the sound is panned dynamically it appears to move beyond the point of the speaker it is addressing. This method is an elaboration of the previous one and makes use of a different 90 degree section of a sine wave. It is implemented using the following code:

aSigL = aSig * sin((kPan + 0.5) * $M_PI_2) aSigR = aSig * cos((kPan + 0.5) * $M_PI_2) outs aSigL, aSigR

The following example demonstrates all three methods one after the other for comparison. Panning movement is controlled by a slow moving LFO. The input sound is filtered pink noise.

** EXAMPLE 05B01_Pan_stereo.csd**

<CsoundSynthesizer> <CsOptions> -odac ; activates real time sound output </CsOptions> <CsInstruments> sr = 44100 ksmps = 10 nchnls = 2 0dbfs = 1 instr 1 imethod = p4 ; read panning method variable from score (p4) ;---------------- generate a source sound ------------------- a1 pinkish 0.3 ; pink noise a1 reson a1, 500, 30, 1 ; bandpass filtered aPan lfo 0.5, 1, 1 ; panning controlled by an lfo aPan = aPan + 0.5 ; offset shifted +0.5 ;------------------------------------------------------------ if imethod=1 then ;------------------------ method 1 -------------------------- aPanL = aPan aPanR = 1 - aPan ;------------------------------------------------------------ endif if imethod=2 then ;------------------------ method 2 -------------------------- aPanL = sqrt(aPan) aPanR = sqrt(1 - aPan) ;------------------------------------------------------------ endif if imethod=3 then ;------------------------ method 3 -------------------------- aPanL = sin(aPan*$M_PI_2) aPanR = cos(aPan*$M_PI_2) ;------------------------------------------------------------ endif if imethod=4 then ;------------------------ method 4 -------------------------- aPanL = sin((aPan + 0.5) * $M_PI_2) aPanR = cos((aPan + 0.5) * $M_PI_2) ;------------------------------------------------------------ endif outs a1*aPanL, a1*aPanR ; audio sent to outputs endin </CsInstruments> <CsScore> ; 4 notes one after the other to demonstrate 4 different methods of panning ; p1 p2 p3 p4(method) i 1 0 4.5 1 i 1 5 4.5 2 i 1 10 4.5 3 i 1 15 4.5 4 e </CsScore> </CsoundSynthesizer>

An opcode called pan2 exists which makes it slightly easier for us to implement various methods of panning. The following example demonstrates the three methods that this opcode offers one after the other. The first is the 'equal power' method, the second 'square root' and the third is simple linear. The Csound Manual describes a fourth method but this one does not seem to function currently.

** EXAMPLE 05B02_pan2.csd**

<CsoundSynthesizer> <CsOptions> -odac ; activates real time sound output </CsOptions> <CsInstruments> sr = 44100 ksmps = 10 nchnls = 2 0dbfs = 1 instr 1 imethod = p4 ; read panning method variable from score (p4) ;----------------------- generate a source sound ------------------------ aSig pinkish 0.5 ; pink noise aSig reson aSig, 500, 30, 1 ; bandpass filtered ;------------------------------------------------------------------------ ;---------------------------- pan the signal ---------------------------- aPan lfo 0.5, 1, 1 ; panning controlled by an lfo aPan = aPan + 0.5 ; DC shifted + 0.5 aSigL, aSigR pan2 aSig, aPan, imethod; create stereo panned output ;------------------------------------------------------------------------ outs aSigL, aSigR ; audio sent to outputs endin </CsInstruments> <CsScore> ; 3 notes one after the other to demonstrate 3 methods used by pan2 ;p1 p2 p3 p4 i 1 0 4.5 0 ; equal power (harmonic) i 1 5 4.5 1 ; square root method i 1 10 4.5 2 ; linear e </CsScore> </CsoundSynthesizer>

In the next example we will generate some sounds as the primary signal. We apply some delay and reverb to this signal to produce a secondary signal. A random function will pan the primary signal between the channels, but the secondary signal remains panned in the middle all the time.

** EXAMPLE 05B03_Different_pan_layers.csd**

<CsoundSynthesizer> <CsOptions> -o dac -d </CsOptions> <CsInstruments> ; Example by Bjorn Houdorf, March 2013 sr = 44100 ksmps = 32 nchnls = 2 0dbfs = 1 seed 0 instr 1 ktrig metro 0.8; Trigger frequency, instr. 2 scoreline "i 2 0 4", ktrig endin instr 2 ital random 60, 72; random notes ifrq = cpsmidinn(ital) knumpart1 oscili 4, 0.1, 1 knumpart2 oscili 5, 0.11, 1 ; Generate primary signal..... asig buzz 0.1, ifrq, knumpart1*knumpart2+1, 1 ipan random 0, 1; ....make random function... asigL, asigR pan2 asig, ipan, 1; ...pan it... outs asigL, asigR ;.... and output it.. kran1 randomi 0,4,3 kran2 randomi 0,4,3 asigdel1 delay asig, 0.1+i(kran1) asigdel2 delay asig, 0.1+i(kran2) ; Make secondary signal... aL, aR reverbsc asig+asigdel1, asig+asigdel2, 0.9, 15000 outs aL, aR; ...and output it endin </CsInstruments> <CsScore> f1 0 8192 10 1 i1 0 60 </CsScore> </CsoundSynthesizer>

## 3-d Binaural Encoding

3-D binaural simulation is available through a number of opcodes that make use of spectral data files that provide information about the filtering and inter-aural delay effects of the human head. The oldest one of these is hrtfer. Newer ones are hrtfmove, hrtfmove2 and hrftstat. The main parameters for control of the opcodes are azimuth (the horizontal direction of the source expressed as an angle formed from the direction in which we are facing) and elevation (the angle by which the sound deviates from this horizontal plane, either above or below). Both these parameters are defined in degrees. 'Binaural' infers that the stereo output of this opcode should be listened to using headphones so that no mixing in the air of the two channels occurs before they reach our ears (although a degree of effect is still audible through speakers).

The following example take a monophonic source sound of noise impulses and processes it using the *hrtfmove2* opcode. First of all the sound is rotated around us in the horizontal plane then it is raised above our head then dropped below us and finally returned to be level and directly in front of us. For this example to work you will need to download the files hrtf-44100-left.dat and hrtf-44100-right.dat and place them in your SADIR (see setting environment variables) or in the same directory as the .csd.

** EXAMPLE 05B04_hrtfmove.csd**

<CsoundSynthesizer> <CsOptions> -odac ; activates real time sound output </CsOptions> <CsInstruments> ; Example by Iain McCurdy sr = 44100 ksmps = 10 nchnls = 2 0dbfs = 1 giSine ftgen 0, 0, 2^12, 10, 1 ; sine wave giLFOShape ftgen 0, 0, 131072, 19, 0.5,1,180,1 ; U-shape parabola instr 1 ; create an audio signal (noise impulses) krate oscil 30,0.2,giLFOShape ; rate of impulses ; amplitude envelope: a repeating pulse kEnv loopseg krate+3,0, 0,1, 0.05,0, 0.95,0,0 aSig pinkish kEnv ; noise pulses ; -- apply binaural 3d processing -- ; azimuth (direction in the horizontal plane) kAz linseg 0, 8, 360 ; elevation (held horizontal for 8 seconds then up, then down, then horizontal kElev linseg 0, 8, 0, 4, 90, 8, -40, 4, 0 ; apply hrtfmove2 opcode to audio source - create stereo ouput aLeft, aRight hrtfmove2 aSig, kAz, kElev, \ "hrtf-44100-left.dat","hrtf-44100-right.dat" outs aLeft, aRight ; audio to outputs endin </CsInstruments> <CsScore> i 1 0 24 ; instr 1 plays a note for 24 seconds e </CsScore> </CsoundSynthesizer>

## Going Multichannel

So far we have only considered working in 2-channels/stereo but Csound is extremely flexible at working in more that 2 channels. By changing nchnls in the orchestra header we can specify any number of channels but we also need to ensure that we choose an audio hardware device using -odac that can handle multichannel audio. Audio channels sent from Csound that do not address hardware channels will simply not be reproduced. There may be some need to make adjustments to the software settings of your soundcard using its own software or the operating system's software but due to the variety of sound hardware options available it would be impossible to offer further specific advice here.

## Sending Multichannel Sound to the Loudspeakers

In order to send multichannel audio we must use opcodes designed for that task. So far we have used outs to send stereo sound to a pair of loudspeakers. (The 's' actually stands for 'stereo'.) Correspondingly there exist opcodes for quadophonic (outq), hexaphonic (outh), octophonic (outo), 16-channel sound (outx) and 32-channel sound (out32).

For example:

outq a1, a2, a3, a4

sends four independent audio streams to four hardware channels. Any unrequired channels still have to be given an audio signal. A typical workaround would be to give them 'silence'. For example if only 5 channels were required:

nchnls = 6 ; --snip-- aSilence = 0 outh a1, a2, a3, a4, a5, aSilence

These opcodes only address very specific loudspeaker arrangements (although workarounds are possible) and have been superseded, to a large extent, by newer opcodes that allow greater flexibility in the number and routing of audio to a multichannel output.

outc allows us to address any number of output audio channels, but they still need to be addressed sequentially. For example our 5-channel audio could be design as follows:

nchnls = 5 ; --snip-- outc a1, a2, a3, a4, a5

outch allows us to direct audio to a specific channel or list of channels and takes the form:

**outch** kchan1, asig1 [, kchan2] [, asig2] [...]

For example, our 5-channel audio system could be designed using outch as follows:

nchnls = 5 ; --snip-- outch 1,a1, 2,a2, 3,a3, 4,a4, 5,a5

Note that channel numbers can be changed at k-rate thereby opening the possibility of changing the speaker configuration dynamically during performance. Channel numbers do not need to be sequential and unrequired channels can be left out completely. This can make life much easier when working with complex systems employing many channels.

## Flexibly Moving Between Stereo and Multichannel

It may be useful to be able to move between working in multichannel (beyond stereo) and then moving back to stereo (when, for example, a multichannel setup is not available). It won't be sufficient to simple change nchnls = 2. It will also be necessary to change all *outq*, *outo*, *outch* etc to *outs.* In complex orchestras this could laboursome and particularly so if it is required to go back to a multichannel configuration later on. In this situation conditional outputs based on the nchnls value are useful. For example:

if nchnls==4 then outq a1,a2,a3,a4 elseif nchnls==2 then outs a1+a3, a2+a4 endif

Using this method it will only be required to change nchnls = ... in the orchestra header. In stereo mode, if nchnls = 2, at least all audio streams will be monitored, even if the results do not reflect the four channel spatial arrangement.

## Rendering Multichannel Audio Streams as Sound Files

So far we have referred to outs, outo etc. as a means to send audio to the speakers but strictly speaking they are only sending audio to Csound's output (as specified by nchnls) and the final destination will be defined using a command line flag in <CsOptions></CsOptions>. -odac will indeed instruct Csound to send audio to the audio hardware and then onto the speakers but we can alternatively send audio to a sound file using -oSoundFile.wav. Provided a file type that supports multichannel interleaved data is chosen (wav will work), a multichannel file will be created that can be used in some other audio applications or can be re-read by Csound later on by using, for example, diskin2. This method is useful for rendering audio that is too complex to be monitored in real-time. Only single interleaved sound files can be created, separate mono files cannot be created using this method. Simultaneously monitoring the audio generated by Csound whilst rendering will not be possible when using this method; we must choose one or the other.

An alternative method of rendering audio in Csound, and one that will allow simulatenous monitoring in real-time, is to use the fout opcode. For example:

```
fout "FileName.wav", 8, a1, a2, a3, a4
outq a1, a2, a3, a4
```

will render an interleaved, 24-bit, 4-channel sound file whilst simultaneously sending the quadrophonic audio to the loudspeakers.

If we wanted to de-interleave an interleaved sound file into multiple mono sound files we could use the code:

a1, a2, a3, a4 soundin "4ChannelSoundFile.wav" fout "Channel1.wav", 8, a1 fout "Channel2.wav", 8, a2 fout "Channel3.wav", 8, a3 fout "Channel4.wav", 8, a4

## VBAP

Vector Base Amplitude Panning^{1} can be described as a method which extends stereo panning to more than two speakers. The number of speakers is, in general, arbitrary. You can configure for standard layouts such as quadrophonic, octophonic or 5.1 configuration, but in fact any number of speakers can be positioned even in irregular distances from each other. If you are fortunate enough to have speakers arranged at different heights, you can even configure VBAP for three dimensions.

### Basic Steps

First you must tell VBAP where your loudspeakers are positioned. Let us assume you have seven speakers in the positions and numberings outlined below (M = middle/centre):

The opcode vbaplsinit, which is usually placed in the header of a Csound orchestra, defines these positions as follows:

vbaplsinit 2, 7, -40, 40, 70, 140, 180, -110, -70

The first number determines the number of dimensions (here 2). The second number states the overall number of speakers, then followed by the positions in degrees (clockwise).

All that is required now is to provide vbap with a monophonic sound source to be distributed amongst the speakers according to information given about the position. Horizontal position (azimuth) is expressed in degrees clockwise just as the initial locations of the speakers were. The following would be the Csound code to play the sound file "ClassGuit.wav" once while moving it counterclockwise:

** EXAMPLE 05B05_VBAP_circle.csd**

<CsoundSynthesizer> <CsOptions> -odac -d ;for the next line, change to your folder --env:SSDIR+=/home/jh/Joachim/Csound/FLOSS/audio </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 0dbfs = 1 nchnls = 7 vbaplsinit 2, 7, -40, 40, 70, 140, 180, -110, -70 instr 1 Sfile = "ClassGuit.wav" iFilLen filelen Sfile p3 = iFilLen aSnd, a0 soundin Sfile kAzim line 0, p3, -360 ;counterclockwise a1, a2, a3, a4, a5, a6, a7, a8 vbap8 aSnd, kAzim outch 1, a1, 2, a2, 3, a3, 4, a4, 5, a5, 6, a6, 7, a7 endin </CsInstruments> <CsScore> i 1 0 1 </CsScore> </CsoundSynthesizer> ;example by joachim heintz

In the CsOptions tag, you see the option *--env:SSDIR+= ...* as a possibility to add a folder to the path in which Csound usually looks for your samples (SSDIR = Sound Sample Directory) if you call them only by name, without the full path. To play the full length of the sound file (without prior knowledge of its duration) the filelen opcode is used to derive this duration, and then the duration of this instrument (p3) is set to this value. The p3 given in the score section (here 1) is overwritten by this value.

The circular movement is a simple k-rate line signal, from 0 to -360 across the duration of the sound file (in this case the same as p3). Note that we have to use the opcode *vbap8* here, as there is no vbap7. Just give the eighth channel a variable name (a8) and thereafter ignore it.

### The Spread Parameter

As VBAP derives from a panning paradigm, it has one problem which becomes more serious as the number of speakers increases. Panning between two speakers in a stereo configuration means that all speakers are active. Panning between two speakers in a quadro configuration means that half of the speakers are active. Panning between two speakers in an octo configuration means that only a quarter of the speakers are active and so on; so that the actual perceived extent of the sound source becomes unintentionally smaller and smaller.

To alleviate this tendency, Ville Pulkki has introduced an additional parameter, called 'spread', which has a range of zero to hundred percent.^{2} The 'ascetic' form of VBAP we have seen in the previous example, means: no spread (0%). A spread of 100% means that all speakers are active, and the information about where the sound comes from is nearly lost.

As the *kspread* input to the *vbap8* opcode is the second of two optional parameters, we first have to provide the first one*. kelev* defines the elevation of the sound - it is always zero for two dimensions, as in the speaker configuration in our example. The next example adds a spread movement to the previous one. The spread starts at zero percent, then increases to hundred percent, and then decreases back down to zero.

** EXAMPLE 05B06_VBAP_spread.csd**

<CsoundSynthesizer> <CsOptions> -odac -d ;for the next line, change to your folder --env:SSDIR+=/home/jh/Joachim/Csound/FLOSS/audio </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 0dbfs = 1 nchnls = 7 vbaplsinit 2, 7, -40, 40, 70, 140, 180, -110, -70 instr 1 Sfile = "ClassGuit.wav" iFilLen filelen Sfile p3 = iFilLen aSnd, a0 soundin Sfile kAzim line 0, p3, -360 kSpread linseg 0, p3/2, 100, p3/2, 0 a1, a2, a3, a4, a5, a6, a7, a8 vbap8 aSnd, kAzim, 0, kSpread outch 1, a1, 2, a2, 3, a3, 4, a4, 5, a5, 6, a6, 7, a7 endin </CsInstruments> <CsScore> i 1 0 1 </CsScore> </CsoundSynthesizer> ;example by joachim heintz

### New VBAP Opcodes

As a response to a number of requests, John fFitch has written new VBAP opcodes in 2012 whose main goal is to allow more than one loudspeaker configuration within a single orchestra (so that you can switch between them during performance) and to provide more flexibility in the number of output channels used. Here is an example for three different configurations which are called in three different instruments:

** EXAMPLE 05B07_VBAP_new.csd**

<CsoundSynthesizer> <CsOptions> -odac -d ;for the next line, change to your folder --env:SSDIR+=/home/jh/Joachim/Csound/FLOSS/audio </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 0dbfs = 1 nchnls = 7 vbaplsinit 2.01, 7, -40, 40, 70, 140, 180, -110, -70 vbaplsinit 2.02, 2, -40, 40 vbaplsinit 2.03, 3, -70, 180, 70 instr 1 aSnd, a0 soundin "ClassGuit.wav" kAzim line 0, p3, -360 a1, a2, a3, a4, a5, a6, a7 vbap aSnd, kAzim, 0, 0, 1 outch 1, a1, 2, a2, 3, a3, 4, a4, 5, a5, 6, a6, 7, a7 endin instr 2 aSnd, a0 soundin "ClassGuit.wav" kAzim line 0, p3, -360 a1, a2 vbap aSnd, kAzim, 0, 0, 2 outch 1, a1, 2, a2 endin instr 3 aSnd, a0 soundin "ClassGuit.wav" kAzim line 0, p3, -360 a1, a2, a3 vbap aSnd, kAzim, 0, 0, 3 outch 7, a1, 3, a2, 5, a3 endin </CsInstruments> <CsScore> i 1 0 6 i 2 6 6 i 3 12 6 </CsScore> </CsoundSynthesizer> ;example by joachim heintz

Instead of just one loudspeaker configuration, as in the previous examples, there are now three configurations:

vbaplsinit 2.01, 7, -40, 40, 70, 140, 180, -110, -70 vbaplsinit 2.02, 2, -40, 40 vbaplsinit 2.03, 3, -70, 180, 70

The first parameter (the number of dimensions) now has an additional fractional part, with a range from .01 to .99, specifying the number of the speaker layout. So *2.01* means: two dimensions, layout number one, *2.02* is layout number two, and *2.03* is layout number three. The new vbap opcode has now these parameters:

ar1[, ar2...]vbapasig, kazim [, kelev] [, kspread] [, ilayout]

The last parameter *ilayout* refers to the speaker layout number. In the example above, instrument 1 uses layout 1, instrument 2 uses layout 2, and instrument 3 uses layout 3. Even if you do not have more than two speakers you should see in Csound's output that instrument 1 goes to all seven speakers, instrument 2 only to the first two, and instrument 3 goes to speaker 3, 5, and 7.

In addition to the new vbap opcode, vbapg has been written. The idea is to have an opcode which returns the gains (amplitudes) of the speakers instead of the audio signal:

`k1[, k2...] `**vbapg** kazim [,kelev] [, kspread] [, ilayout]

## Ambisonics

Ambisonics is another technique to distribute a virtual sound source in space.

There are excellent sources for the discussion of Ambisonics online^{3 }and the following chapter will give a step by step introduction. We will focus just on the basic practicalities of using the Ambisonics opcodes of Csound, without going into too much detail of the concepts behind them.

Ambisonics works using two basic steps. In the first step you **encode** the sound and the spatial information (its localisation) of a virtual sound source in a so-called **B-format**. In the second step you **decode** the B-format to match your loudspeaker setup.

It is possible to save the B-format as its own audio file, to preserve the spatial information or you can immediately do the decoding after the encoding thereby dealing directly only with audio signals instead of Ambisonic files. The next example takes the latter approach by implementing a transformation of the VBAP circle example to Ambisonics.

** EXAMPLE 05B08_Ambi_circle.csd**

<CsoundSynthesizer> <CsOptions> -odac -d ;for the next line, change to your folder --env:SSDIR+=/home/jh/Joachim/Csound/FLOSS/Release01/Csound_Floss_Release01/audio </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 0dbfs = 1 nchnls = 8 instr 1 Sfile = "ClassGuit.wav" iFilLen filelen Sfile p3 = iFilLen aSnd, a0 soundin Sfile kAzim line 0, p3, 360 ;counterclockwise (!) iSetup = 4 ;octogon aw, ax, ay, az bformenc1 aSnd, kAzim, 0 a1, a2, a3, a4, a5, a6, a7, a8 bformdec1 iSetup, aw, ax, ay, az outch 1, a1, 2, a2, 3, a3, 4, a4, 5, a5, 6, a6, 7, a7, 8, a8 endin </CsInstruments> <CsScore> i 1 0 1 </CsScore> </CsoundSynthesizer> ;example by joachim heintz

The first thing to note is that for a counterclockwise circle, the azimuth now has the line 0 -> 360, instead of 0 -> -360 as was used in the VBAP example. This is because Ambisonics usually reads the angle in a mathematical way: a positive angle is *counter*clockwise. Next, the encoding process is carried out in the line:

aw, ax, ay, az bformenc1 aSnd, kAzim, 0

Input arguments are the monophonic sound source *aSnd*, the xy-angle *kAzim*, and the elevation angle which is set to zero. Output signals are the spatial information in x-, y- and z- direction (*ax, ay, az*), and also an omnidirectional signal called *aw*.

Decoding is performed by the line:

a1, a2, a3, a4, a5, a6, a7, a8 bformdec1 iSetup, aw, ax, ay, az

The inputs for the decoder are the same *aw, **ax, ay, az*, which were the results of the encoding process, and an additional *iSetup* parameter. Currently the Csound decoder only works with some standard setups for the speaker: *iSetup = 4* refers to an octogon.^{4} So the final eight audio signals *a1, ..., a8* are being produced using this decoder, and are then sent to the speakers in the same way using the outch opcode.

### Different Orders

What we have seen in this example is called 'first order' ambisonics. This means that the encoding process leads to the four basic dimensions w, x, y, z as described above.^{5} In "second order" ambisonics, there are additional "directions" called r, s, t, u, v. And in "third order" ambisonics again the additional k, l, m, n, o, p, q. The final example in this section shows the three orders, each of them in one instrument. If you have eight speakers in octophonic setup, you can compare the results.

** EXAMPLE 05B09_Ambi_orders.csd**

<CsoundSynthesizer> <CsOptions> -odac -d ;for the next line, change to your folder --env:SSDIR+=/home/jh/Joachim/Csound/FLOSS/Release01/Csound_Floss_Release01/audio </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 0dbfs = 1 nchnls = 8 instr 1 ;first order aSnd, a0 soundin "ClassGuit.wav" kAzim line 0, p3, 360 iSetup = 4 ;octogon aw, ax, ay, az bformenc1 aSnd, kAzim, 0 a1, a2, a3, a4, a5, a6, a7, a8 bformdec1 iSetup, aw, ax, ay, az outch 1, a1, 2, a2, 3, a3, 4, a4, 5, a5, 6, a6, 7, a7, 8, a8 endin instr 2 ;second order aSnd, a0 soundin "ClassGuit.wav" kAzim line 0, p3, 360 iSetup = 4 ;octogon aw, ax, ay, az, ar, as, at, au, av bformenc1 aSnd, kAzim, 0 a1, a2, a3, a4, a5, a6, a7, a8 bformdec1 iSetup, aw, ax, ay, az, ar, as, at, au, av outch 1, a1, 2, a2, 3, a3, 4, a4, 5, a5, 6, a6, 7, a7, 8, a8 endin instr 3 ;third order aSnd, a0 soundin "ClassGuit.wav" kAzim line 0, p3, 360 iSetup = 4 ;octogon aw, ax, ay, az, ar, as, at, au, av, ak, al, am, an, ao, ap, aq bformenc1 aSnd, kAzim, 0 a1, a2, a3, a4, a5, a6, a7, a8 bformdec1 iSetup, aw, ax, ay, az, ar, as, at, au, av, ak, al, am, an, ao, ap, aq outch 1, a1, 2, a2, 3, a3, 4, a4, 5, a5, 6, a6, 7, a7, 8, a8 endin </CsInstruments> <CsScore> i 1 0 6 i 2 6 6 i 3 12 6 </CsScore> </CsoundSynthesizer> ;example by joachim heintz

In theory, first-order ambisonics need at least 4 speakers to be projected correctly. Second-order ambisonics needs at least 6 speakers (9, if 3 dimensions are employed). Third-order ambisonics need at least 8 speakers (or 16 for 3d). So, although higher order should in general lead to a better result in space, you cannot expect it to work unless you have a sufficient number of speakers. Of course practice over theory may prove to be a better judge in many cases.

## Ambisonics UDOs

### Usage of the ambisonics UDOs

This chapter gives an overview of the UDOs explained below.

The channels of the B-format are stored in a zak space. Call zakinit only once and put it outside of any instrument definition in the orchestra file after the header. zacl clears the za space and is called after decoding. The B format of order n can be decoded in any order <= n.

The text files "ambisonics_udos.txt", "ambisonics2D_udos.txt", "AEP_udos.txt" and "utilities.txt" must be located in the same folder as the csd files or included with full path.

These files can be downloaded together with the entire examples (some of them for CsoundQt) from:http://www.icst.net/research/downloads/

zakinit isizea, isizek (isizea = (order + 1)^2 in ambisonics (3D); isizea = 2·order + 1 in ambi2D; isizek = 1) ;#include "ambisonics_udos.txt" (order <= 8) k0 ambi_encode asnd, iorder, kazimuth, kelevation (azimuth, elevation in degrees) k0 ambi_enc_dist asnd, iorder, kazimuth, kelevation, kdistance a1 [, a2] ... [, a8] ambi_decode iorder, ifn a1 [, a2] ... [, a8] ambi_dec_inph iorder, ifn f ifn 0 n -2 p1 az1 el1 az2 el2 ... (n is a power of 2 greater than 3·number_of_spekers + 1) (p1 is not used) k0 ambi_write_B "name", iorder, ifile_format (ifile_format see fout in the csound help) k0 ambi_read_B "name", iorder (only <= 5) kaz, kel, kdist xyz_to_aed kx, ky, kz ;#include "ambisonics2D_udos.txt" k0 ambi2D_encode asnd, iorder, kazimuth (any order) (azimuth in degrees) k0 ambi2D_enc_dist asnd, iorder, kazimuth, kdistance a1 [, a2] ... [, a8] ambi2D_decode iorder, iaz1 [, iaz2] ... [, iaz8] a1 [, a2] ... [, a8] ambi2D_dec_inph iorder, iaz1 [, iaz2] ... [, iaz8] (order <= 12) k0 ambi2D_write_B "name", iorder, ifile_format k0 ambi2D_read_B "name", iorder (order <= 19) kaz, kdist xy_to_ad kx, ky #include "AEP_udos.txt" (any order integer or fractional) a1 [, a2] ... [, a16] AEP_xyz asnd, korder, ifn, kx, ky, kz, kdistance f ifn 0 64 -2 max_speaker_distance x1 y1 z1 x2 y2 z2 ... a1 [, a2] ... [, a8] AEP asnd, korder, ifn, kazimuth, kelevation, kdistance (azimuth, elevation in degrees) f ifn 0 64 -2 max_speaker_distance az1 el1 dist1 az2 el2 dist2 ... (azimuth, elevation in degrees) ;#include "ambi_utilities.txt" kdist dist kx, ky kdist dist kx, ky, kz ares Doppler asnd, kdistance ares absorb asnd, kdistance kx, ky, kz aed_to_xyz kazimuth, kelevation, kdistance ix, iy, iz aed_to_xyz iazimuth, ielevation, idistance a1 [, a2] ... [, a16] dist_corr a1 [, a2] ... [, a16], ifn f ifn 0 32 -2 max_speaker_distance dist1, dist2, ... (distances in m) irad radiani idegree krad radian kdegree arad radian adegree idegree degreei irad kdegree degree krad adegree degree arad

**Introduction**

In the following introduction we will explain the principles of ambisonics step by step and write an opcode for every step. The opcodes above combine all of the functionality described. Since the two-dimensional analogy to Ambisonics is easier to understand and to implement with a simple equipment, we shall fully explain it first.

Ambisonics is a technique of three-dimensional sound projection. The information about the recorded or synthesized sound field is encoded and stored in several channels, taking no account of the arrangement of the loudspeakers for reproduction. The encoding of a signal's spatial information can be more or less precise, depending on the so-called order of the algorithm used. Order zero corresponds to the monophonic signal and requires only one channel for storage and reproduction. In first-order Ambisonics, three further channels are used to encode the portions of the sound field in the three orthogonal directions x, y and z. These four channels constitute the so-called first-order B-format. When Ambisonics is used for artificial spatialisation of recorded or synthesized sound, the encoding can be of an arbitrarily high order. The higher orders cannot be interpreted as easily as orders zero and one.

In a two-dimensional analogy to Ambisonics (called Ambisonics2D in what follows), only sound waves in the horizontal plane are encoded.

The loudspeaker feeds are obtained by decoding the B-format signal. The resulting panning is amplitude panning, and only the direction to the sound source is taken into account.

The illustration below shows the principle of Ambisonics. First a sound is generated and its position determined. The amplitude and spectrum are adjusted to simulate distance, the latter using a low-pass filter. Then the Ambisonic encoding is computed using the sound's coordinates. Encoding mth order B-format requires n = (m+1)^2 channels (n = 2m + 1 channels in Ambisonics2D). By decoding the B-format, one can obtain the signals for any number (>= n) of loudspeakers in any arrangement. Best results are achieved with symmetrical speaker arrangements.

If the B-format does not need to be recorded the speaker signals can be calculated at low cost and arbitrary order using so-called ambisonics equivalent panning (AEP).

**Ambisonics2D **

Introduction We will first explain the encoding process in Ambisonics2D. The position of a sound source in the horizontal plane is given by two coordinates. In Cartesian coordinates (x, y) the listener is at the origin of the coordinate system (0, 0), and the x-coordinate points to the front, the y-coordinate to the left. The position of the sound source can also be given in polar coordinates by the angle ψ between the line of vision of the listener (front) and the direction to the sound source, and by their distance r. Cartesian coordinates can be converted to polar coordinates by the formulae:

r = and ψ = arctan(x, y),

polar to Cartesian coordinates by

x = r·cos(ψ) and y = r·sin(ψ).

The 0th order B-Format of a signal S of a sound source on the unit circle is just the mono signal: W0 = W = S. The first order B-Format contains two additional channels: W1,1 = X = S·cos(ψ) = S·x and W1,2 = Y = S·sin(ψ) = S·y, i.e. the product of the Signal S with the sine and the cosine of the direction ψ of the sound source. The B-Format higher order contains two additional channels per order m: Wm, 1 = S·cos(mψ) and Wm, 2 = S·sin(mψ).

W0 = S

W1,1 = X = S·cos(ψ) = S·x W1,2 = Y = S·sin(ψ) = S·y

W2,1 = S·cos(2ψ) W2,2 = S·sin(2ψ)

...

Wm,1 = S·cos(mψ) Wm,2 = S·sin(mψ)

From the n = 2m + 1 B-Format channels the loudspeaker signals pi of n loudspeakers which are set up symmetrically on a circle (with angle ϕi) are:

pi = 1/n(W0 + 2W1,1cos(ϕi) + 2W1,2sin(ϕi) + 2W2,1cos(2ϕi) + 2W2,2sin(2ϕi) + ...)

= 2/n(1/2 W0 + W1,1cos(ϕi) + W1,2sin(ϕi) + W2,1cos(2ϕi) + W2,2sin(2ϕi) + ...)

(If more than n speakers are used, we can use the same formula)

In the Csound example udo_ambisonics2D_1.csd the opcode ambi2D_encode_1a produces the 3 channels W, X and Y (a0, a11, a12) from an input sound and the angle ψ (azmuth kaz), the opcode ambi2D_decode_1_8 decodes them to 8 speaker signals a1, a2, ..., a8. The inputs of the decoder are the 3 channels a0, a11, a12 and the 8 angles of the speakers.

* EXAMPLE 05B10_udo_ambisonics2D_1.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 8 0dbfs = 1 ; ambisonics2D first order without distance encoding ; decoding for 8 speakers symmetrically positioned on a circle ; produces the 3 channels 1st order; input: asound, kazimuth opcode ambi2D_encode_1a, aaa, ak asnd,kaz xin kaz = $M_PI*kaz/180 a0 = asnd a11 = cos(kaz)*asnd a12 = sin(kaz)*asnd xout a0,a11,a12 endop ; decodes 1st order to a setup of 8 speakers at angles i1, i2, ... opcode ambi2D_decode_1_8, aaaaaaaa, aaaiiiiiiii a0,a11,a12,i1,i2,i3,i4,i5,i6,i7,i8 xin i1 = $M_PI*i1/180 i2 = $M_PI*i2/180 i3 = $M_PI*i3/180 i4 = $M_PI*i4/180 i5 = $M_PI*i5/180 i6 = $M_PI*i6/180 i7 = $M_PI*i7/180 i8 = $M_PI*i8/180 a1 = (.5*a0 + cos(i1)*a11 + sin(i1)*a12)*2/3 a2 = (.5*a0 + cos(i2)*a11 + sin(i2)*a12)*2/3 a3 = (.5*a0 + cos(i3)*a11 + sin(i3)*a12)*2/3 a4 = (.5*a0 + cos(i4)*a11 + sin(i4)*a12)*2/3 a5 = (.5*a0 + cos(i5)*a11 + sin(i5)*a12)*2/3 a6 = (.5*a0 + cos(i6)*a11 + sin(i6)*a12)*2/3 a7 = (.5*a0 + cos(i7)*a11 + sin(i7)*a12)*2/3 a8 = (.5*a0 + cos(i8)*a11 + sin(i8)*a12)*2/3 xout a1,a2,a3,a4,a5,a6,a7,a8 endop instr 1 asnd rand .05 kaz line 0,p3,3*360 ;turns around 3 times in p3 seconds a0,a11,a12 ambi2D_encode_1a asnd,kaz a1,a2,a3,a4,a5,a6,a7,a8 \ ambi2D_decode_1_8 a0,a11,a12, 0,45,90,135,180,225,270,315 outc a1,a2,a3,a4,a5,a6,a7,a8 endin </CsInstruments> <CsScore> i1 0 40 </CsScore> </CsoundSynthesizer> ;example by martin neukom

The B-format of all events of all instruments can be summed before decoding. Thus in the example udo_ambisonics2D_2.csd we create a zak space with 21 channels (zakinit 21, 1) for the 2D B-format up to 10th order where the encoded signals are accumulated. The opcode ambi2D_encode_3 shows how to produce the 7 B-format channels a0, a11, a12, ..., a32 for third order. The opcode ambi2D_encode_n produces the 2(n+1) channels a0, a11, a12, ..., a32 for any order n (needs zakinit 2(n+1), 1). The opcode ambi2D_decode_basic is an overloaded function i.e. it decodes to n speaker signals depending on the number of in- and outputs given (in this example only for 1 or 2 speakers). Any number of instruments can play arbitrary often. Instrument 10 decodes for the first 4 speakers of an 18 speaker setup.

* EXAMPLE 05B11_udo_ambisonics2D_2.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 4 0dbfs = 1 ; ambisonics2D encoding fifth order ; decoding for 8 speakers symmetrically positioned on a circle ; all instruments write the B-format into a buffer (zak space) ; instr 10 decodes ; zak space with the 21 channels of the B-format up to 10th order zakinit 21, 1 ;explicit encoding third order opcode ambi2D_encode_3, k, ak asnd,kaz xin kaz = $M_PI*kaz/180 zawm asnd,0 zawm cos(kaz)*asnd,1 ;a11 zawm sin(kaz)*asnd,2 ;a12 zawm cos(2*kaz)*asnd,3 ;a21 zawm sin(2*kaz)*asnd,4 ;a22 zawm cos(3*kaz)*asnd,5 ;a31 zawm sin(3*kaz)*asnd,6 ;a32 xout 0 endop ; encoding arbitrary order n(zakinit 2*n+1, 1) opcode ambi2D_encode_n, k, aik asnd,iorder,kaz xin kaz = $M_PI*kaz/180 kk = iorder c1: zawm cos(kk*kaz)*asnd,2*kk-1 zawm sin(kk*kaz)*asnd,2*kk kk = kk-1 if kk > 0 goto c1 zawm asnd,0 xout 0 endop ; basic decoding for arbitrary order n for 1 speaker opcode ambi2D_decode_basic, a, ii iorder,iaz xin iaz = $M_PI*iaz/180 igain = 2/(2*iorder+1) kk = iorder a1 = .5*zar(0) c1: a1 += cos(kk*iaz)*zar(2*kk-1) a1 += sin(kk*iaz)*zar(2*kk) kk = kk-1 if kk > 0 goto c1 xout igain*a1 endop ; decoding for 2 speakers opcode ambi2D_decode_basic, aa, iii iorder,iaz1,iaz2 xin iaz1 = $M_PI*iaz1/180 iaz2 = $M_PI*iaz2/180 igain = 2/(2*iorder+1) kk = iorder a1 = .5*zar(0) c1: a1 += cos(kk*iaz1)*zar(2*kk-1) a1 += sin(kk*iaz1)*zar(2*kk) kk = kk-1 if kk > 0 goto c1 kk = iorder a2 = .5*zar(0) c2: a2 += cos(kk*iaz2)*zar(2*kk-1) a2 += sin(kk*iaz2)*zar(2*kk) kk = kk-1 if kk > 0 goto c2 xout igain*a1,igain*a2 endop instr 1 asnd rand p4 ares reson asnd,p5,p6,1 kaz line 0,p3,p7*360 ;turns around p7 times in p3 seconds k0 ambi2D_encode_n asnd,10,kaz endin instr 2 asnd oscil p4,p5,1 kaz line 0,p3,p7*360 ;turns around p7 times in p3 seconds k0 ambi2D_encode_n asnd,10,kaz endin instr 10 ;decode all insruments (the first 4 speakers of a 18 speaker setup) a1,a2 ambi2D_decode_basic 10,0,20 a3,a4 ambi2D_decode_basic 10,40,60 outc a1,a2,a3,a4 zacl 0,20 ; clear the za variables endin </CsInstruments> <CsScore> f1 0 32768 10 1 ; amp cf bw turns i1 0 3 .7 1500 12 1 i1 2 18 .1 2234 34 -8 ; amp fr 0 turns i2 0 3 .1 440 0 2 i10 0 3 </CsScore> </CsoundSynthesizer> ;example by martin neukom

**In-phase Decoding**

The left figure below shows a symmetrical arrangement of 7 loudspeakers. If the virtual sound source is precisely in the direction of a loudspeaker, only this loudspeaker gets a signal (center figure). If the virtual sound source is between two loudspeakers, these loudspeakers receive the strongest signals; all other loudspeakers have weaker signals, some with negative amplitude, that is, reversed phase (right figure).

To avoid having loudspeaker sounds that are far away from the virtual sound source and to ensure that negative amplitudes (inverted phase) do not arise, the B-format channels can be weighted before being decoded. The weighting factors depend on the highest order used (M) and the order of the particular channel being decoded (m).

gm = (M!)^2/((M + m)!·(M - m)!)

The decoded signal can be normalised with the factor gnorm(M) = (2M + 1) !/(4^M (M!)^2)

The illustration below shows a third-order B-format signal decoded to 13 loudspeakers first uncorrected (so-called basic decoding, left), then corrected by weighting (so-called in-phase decoding, right).

Example udo_ambisonics2D_3.csd shows in-phase decoding. The weights and norms up to 12th order are saved in the arrays iWeight2D[][] and iNorm2D[] respectively. Instrument 11 decodes third order for 4 speakers in a square.

* EXAMPLE 05B12_udo_ambisonics2D_3.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 4 0dbfs = 1 opcode ambi2D_encode_n, k, aik asnd,iorder,kaz xin kaz = $M_PI*kaz/180 kk = iorder c1: zawm cos(kk*kaz)*asnd,2*kk-1 zawm sin(kk*kaz)*asnd,2*kk kk = kk-1 if kk > 0 goto c1 zawm asnd,0 xout 0 endop ;in-phase-decoding opcode ambi2D_dec_inph, a, ii ; weights and norms up to 12th order iNorm2D[] array 1,0.75,0.625,0.546875,0.492188,0.451172,0.418945, 0.392761,0.370941,0.352394,0.336376,0.322360 iWeight2D[][] init 12,12 iWeight2D array 0.5,0,0,0,0,0,0,0,0,0,0,0, 0.666667,0.166667,0,0,0,0,0,0,0,0,0,0, 0.75,0.3,0.05,0,0,0,0,0,0,0,0,0, 0.8,0.4,0.114286,0.0142857,0,0,0,0,0,0,0,0, 0.833333,0.47619,0.178571,0.0396825,0.00396825,0,0,0,0,0,0,0, 0.857143,0.535714,0.238095,0.0714286,0.012987,0.00108225,0,0,0,0,0,0, 0.875,0.583333,0.291667,0.1060601,0.0265152,0.00407925,0.000291375,0,0,0,0,0, 0.888889,0.622222,0.339394,0.141414,0.043512,0.009324,0.0012432, 0.0000777,0,0,0,0, 0.9,0.654545,0.381818,0.176224,0.0629371,0.0167832,0.00314685, 0.000370218,0.0000205677,0,0,0, 0.909091,0.681818,0.41958,0.20979,0.0839161,0.0262238,0.0061703, 0.00102838,0.000108251,0.00000541254,0,0, 0.916667,0.705128,0.453297,0.241758,0.105769,0.0373303,0.0103695, 0.00218306,0.000327459,0.0000311866,0.00000141757,0, 0.923077,0.725275,0.483516,0.271978,0.12799,0.0497738,0.015718, 0.00392951,0.000748478,0.000102065,0.00000887523,0.000000369801 iorder,iaz1 xin iaz1 = $M_PI*iaz1/180 kk = iorder a1 = .5*zar(0) c1: a1 += cos(kk*iaz1)*iWeight2D[iorder-1][kk-1]*zar(2*kk-1) a1 += sin(kk*iaz1)*iWeight2D[iorder-1][kk-1]*zar(2*kk) kk = kk-1 if kk > 0 goto c1 xout iNorm2D[iorder-1]*a1 endop zakinit 7, 1 instr 1 asnd rand p4 ares reson asnd,p5,p6,1 kaz line 0,p3,p7*360 ;turns around p7 times in p3 seconds k0 ambi2D_encode_n asnd,3,kaz endin instr 11 a1 ambi2D_dec_inph 3,0 a2 ambi2D_dec_inph 3,90 a3 ambi2D_dec_inph 3,180 a4 ambi2D_dec_inph 3,270 outc a1,a2,a3,a4 zacl 0,6 ; clear the za variables endin </CsInstruments> <CsScore> ; amp cf bw turns i1 0 3 .1 1500 12 1 i11 0 3 </CsScore> </CsoundSynthesizer> ;example by martin neukom

**Distance**

In order to simulate distances and movements of sound sources, the signals have to be treated before being encoded. The main perceptual cues for the distance of a sound source are reduction of the amplitude, filtering due to the absorbtion of the air and the relation between direct and indirect sound. We will implement the first two of these cues. The amplitude arriving at a listener is inversely proportional to the distance of the sound source. If the distance is larger than the unit circle (not necessarily the radius of the speaker setup, which does not need to be known when encoding sounds) we can simply divide the sound by the distance. With this calculation inside the unit circle the amplitude is amplified and becomes infinite when the distance becomes zero. Another problem arises when a virtual sound source passes the origin. The amplitude of the speaker signal in the direction of the movement suddenly becomes maximal and the signal of the opposite speaker suddenly becomes zero. A simple solution for these problems is to limit the gain of the channel W inside the unit circle to 1 (f1 in the figure below) and to fade out all other channels (f2). By fading out all channels except channel W the information about the direction of the sound source is lost and all speaker signals are the same and the sum of the speaker signals reaches its maximum when the distance is 0.

Now, we are looking for gain functions that are smoother at d = 1. The functions should be differentiable and the slope of f1 at distance d = 0 should be 0. For distances greater than 1 the functions should be approximately 1/d. In addition the function f1 should continuously grow with decreasing distance and reach its maximum at d = 0. The maximal gain must be 1. The function atan(d·π/2)/(d·π/2) fulfills these constraints. We create a function f2 for the fading out of the other channels by multiplying f1 by the factor (1 – E^(-d)).

In example udo_ambisonics2D_4 the UDO ambi2D_enc_dist_n encodes a sound at any order with distance correction. The inputs of the UDO are asnd, iorder, kazimuth and kdistance. If the distance becomes negative the azimuth angle is turned to its opposite (kaz += π) and the distance taken positive.

*EXAMPLE 05B13_udo_ambisonics2D_4.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 8 0dbfs = 1 #include "ambisonics2D_udos.txt" ; distance encoding ; with any distance (includes zero and negative distance) opcode ambi2D_enc_dist_n, k, aikk asnd,iorder,kaz,kdist xin kaz = $M_PI*kaz/180 kaz = (kdist < 0 ? kaz + $M_PI : kaz) kdist = abs(kdist)+0.0001 kgainW = taninv(kdist*1.5707963) / (kdist*1.5708) ;pi/2 kgainHO = (1 - exp(-kdist)) ;*kgainW kk = iorder asndW = kgainW*asnd asndHO = kgainHO*asndW c1: zawm cos(kk*kaz)*asndHO,2*kk-1 zawm sin(kk*kaz)*asndHO,2*kk kk = kk-1 if kk > 0 goto c1 zawm asndW,0 xout 0 endop zakinit 17, 1 instr 1 asnd rand p4 ;asnd soundin "/Users/user/csound/ambisonic/violine.aiff" kaz line 0,p3,p5*360 ;turns around p5 times in p3 seconds kdist line p6,p3,p7 k0 ambi2D_enc_dist_n asnd,8,kaz,kdist endin instr 10 a1,a2,a3,a4, a5,a6,a7,a8 ambi2D_decode 8,0,45,90,135,180,225,270,315 outc a1,a2,a3,a4,a5,a6,a7,a8 zacl 0,16 endin </CsInstruments> <CsScore> f1 0 32768 10 1 ; amp turns dist1 dist2 i1 0 4 1 0 2 -2 ;i1 0 4 1 1 1 1 i10 0 4 </CsScore> </CsoundSynthesizer> ;example by martin neukom

In order to simulate the absorption of the air we introduce a very simple lowpass filter with a distance depending cutoff frequency. We produce a Doppler-shift with a distance dependent delay of the sound. Now, we have to determine our unit since the delay of the sound wave is calculated as distance divided by sound velocity. In our example udo_ambisonics2D_5.csd we set the unit to 1 metre. These procedures are performed before the encoding. In instrument 1 the movement of the sound source is defined in Cartesian coordinates. The UDO xy_to_ad transforms them into polar coordinates. The B-format channels can be written to a sound file with the opcode fout. The UDO write_ambi2D_2 writes the channels up to second order into a sound file.

* EXAMPLE 05B14_udo_ambisonics2D_5.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 8 0dbfs = 1 #include "ambisonics2D_udos.txt" #include "ambisonics_utilities.txt" ;opcodes Absorb and Doppler /* these opcodes are included in "ambisonics2D_udos.txt" opcode xy_to_ad, kk, kk kx,ky xin kdist = sqrt(kx*kx+ky*ky) kaz taninv2 ky,kx xout 180*kaz/$M_PI, kdist endop opcode Absorb, a, ak asnd,kdist xin aabs tone 5*asnd,20000*exp(-.1*kdist) xout aabs endop opcode Doppler, a, ak asnd,kdist xin abuf delayr .5 adop deltapi interp(kdist)*0.0029137529 + .01 ; 1/343.2 delayw asnd xout adop endop */ opcode write_ambi2D_2, k, S Sname xin fout Sname,12,zar(0),zar(1),zar(2),zar(3),zar(4) xout 0 endop zakinit 17, 1 ; zak space with the 17 channels of the B-format instr 1 asnd buzz p4,p5,50,1 ;asnd soundin "/Users/user/csound/ambisonic/violine.aiff" kx line p7,p3,p8 ky line p9,p3,p10 kaz,kdist xy_to_ad kx,ky aabs absorb asnd,kdist adop Doppler .2*aabs,kdist k0 ambi2D_enc_dist adop,5,kaz,kdist endin instr 10 ;decode all insruments a1,a2,a3,a4, a5,a6,a7,a8 ambi2D_dec_inph 5,0,45,90,135,180,225,270,315 outc a1,a2,a3,a4,a5,a6,a7,a8 ; fout "B_format2D.wav",12,zar(0),zar(1),zar(2),zar(3),zar(4), ; zar(5),zar(6),zar(7),zar(8),zar(9),zar(10) k0 write_ambi2D_2 "ambi_ex5.wav" zacl 0,16 ; clear the za variables endin </CsInstruments> <CsScore> f1 0 32768 10 1 ; amp f 0 x1 x2 y1 y2 i1 0 5 .8 200 0 40 -20 1 .1 i10 0 5 </CsScore> </CsoundSynthesizer> ;example by martin neukomThe position of a point in space can be given by its Cartesian coordinates x, y and z or by its spherical coordinates the radial distance r from the origin of the coordinate system, the elevation δ (which lies between –π and π) and the azimuth angle θ.

The formulae for transforming coordinates are as follows:

The channels of the Ambisonic B-format are computed as the product of the sounds themselves and the so-called spherical harmonics representing the direction to the virtual sound sources. The spherical harmonics can be normalised in various ways. We shall use the so-called semi-normalised spherical harmonics. The following table shows the encoding functions up to the third order as function of azimuth and elevation Ymn(θ,δ) and as function of x, y and z Ymn(x,y,z) for sound sources on the unit sphere. The decoding formulae for symmetrical speaker setups are the same.

In the first 3 of the following examples we will not produce sound but display in number boxes (for example using CsoundQt widgets) the amplitude of 3 speakers at positions (1, 0, 0), (0, 1, 0) and (0, 0, 1) in Cartesian coordinates. The position of the sound source can be changed with the two scroll numbers. The example udo_ambisonics_1.csd shows encoding up to second order. The decoding is done in two steps. First we decode the B-format for one speaker. In the second step, we create a overloaded opcode for n speakers. The number of output signals determines which version of the opcode is used. The opcodes ambi_encode and ambi_decode up to 8th order are saved in the text file "ambisonics_udos.txt".

* EXAMPLE 05B15_udo_ambisonics_1.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 1 0dbfs = 1 zakinit 9, 1 ; zak space with the 9 channel B-format second order opcode ambi_encode, k, aikk asnd,iorder,kaz,kel xin kaz = $M_PI*kaz/180 kel = $M_PI*kel/180 kcos_el = cos(kel) ksin_el = sin(kel) kcos_az = cos(kaz) ksin_az = sin(kaz) zawm asnd,0 ; W zawm kcos_el*ksin_az*asnd,1 ; Y = Y(1,-1) zawm ksin_el*asnd,2 ; Z = Y(1,0) zawm kcos_el*kcos_az*asnd,3 ; X = Y(1,1) if iorder < 2 goto end i2 = sqrt(3)/2 kcos_el_p2 = kcos_el*kcos_el ksin_el_p2 = ksin_el*ksin_el kcos_2az = cos(2*kaz) ksin_2az = sin(2*kaz) kcos_2el = cos(2*kel) ksin_2el = sin(2*kel) zawm i2*kcos_el_p2*ksin_2az*asnd,4 ; V = Y(2,-2) zawm i2*ksin_2el*ksin_az*asnd,5 ; S = Y(2,-1) zawm .5*(3*ksin_el_p2 - 1)*asnd,6 ; R = Y(2,0) zawm i2*ksin_2el*kcos_az*asnd,7 ; S = Y(2,1) zawm i2*kcos_el_p2*kcos_2az*asnd,8 ; U = Y(2,2) end: xout 0 endop ; decoding of order iorder for 1 speaker at position iaz,iel,idist opcode ambi_decode1, a, iii iorder,iaz,iel xin iaz = $M_PI*iaz/180 iel = $M_PI*iel/180 a0=zar(0) if iorder > 0 goto c0 aout = a0 goto end c0: a1=zar(1) a2=zar(2) a3=zar(3) icos_el = cos(iel) isin_el = sin(iel) icos_az = cos(iaz) isin_az = sin(iaz) i1 = icos_el*isin_az ; Y = Y(1,-1) i2 = isin_el ; Z = Y(1,0) i3 = icos_el*icos_az ; X = Y(1,1) if iorder > 1 goto c1 aout = (1/2)*(a0 + i1*a1 + i2*a2 + i3*a3) goto end c1: a4=zar(4) a5=zar(5) a6=zar(6) a7=zar(7) a8=zar(8) ic2 = sqrt(3)/2 icos_el_p2 = icos_el*icos_el isin_el_p2 = isin_el*isin_el icos_2az = cos(2*iaz) isin_2az = sin(2*iaz) icos_2el = cos(2*iel) isin_2el = sin(2*iel) i4 = ic2*icos_el_p2*isin_2az ; V = Y(2,-2) i5 = ic2*isin_2el*isin_az ; S = Y(2,-1) i6 = .5*(3*isin_el_p2 - 1) ; R = Y(2,0) i7 = ic2*isin_2el*icos_az ; S = Y(2,1) i8 = ic2*icos_el_p2*icos_2az ; U = Y(2,2) aout = (1/9)*(a0 + 3*i1*a1 + 3*i2*a2 + 3*i3*a3 + 5*i4*a4 + 5*i5*a5 + 5*i6*a6 + 5*i7*a7 + 5*i8*a8) end: xout aout endop ; overloaded opcode for decoding of order iorder ; speaker positions in function table ifn opcode ambi_decode, a,ii iorder,ifn xin xout ambi_decode1(iorder,table(1,ifn),table(2,ifn)) endop opcode ambi_decode, aa,ii iorder,ifn xin xout ambi_decode1(iorder,table(1,ifn),table(2,ifn)), ambi_decode1(iorder,table(3,ifn),table(4,ifn)) endop opcode ambi_decode, aaa,ii iorder,ifn xin xout ambi_decode1(iorder,table(1,ifn),table(2,ifn)), ambi_decode1(iorder,table(3,ifn),table(4,ifn)), ambi_decode1(iorder,table(5,ifn),table(6,ifn)) endop instr 1 asnd init 1 ;kdist init 1 kaz invalue "az" kel invalue "el" k0 ambi_encode asnd,2,kaz,kel ao1,ao2,ao3 ambi_decode 2,17 outvalue "sp1", downsamp(ao1) outvalue "sp2", downsamp(ao2) outvalue "sp3", downsamp(ao3) zacl 0,8 endin </CsInstruments> <CsScore> ;f1 0 1024 10 1 f17 0 64 -2 0 0 0 90 0 0 90 0 0 0 0 0 0 i1 0 100 </CsScore> </CsoundSynthesizer> ;example by martin neukomExample udo_ambisonics_2.csd shows in-phase decoding. The weights up to 8th order are stored in the arrays iWeight3D[][].

*EXAMPLE 05B16_udo_ambisonics_2.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 1 0dbfs = 1 zakinit 81, 1 ; zak space for up to 81 channels of the 8th order B-format ; the opcodes used below are safed in "ambisonics_udos.txt" #include "ambisonics_udos.txt" ; in-phase decoding up to third order for one speaker opcode ambi_dec1_inph3, a, iii ; weights up to 8th order iWeight3D[][] init 8,8 iWeight3D array 0.333333,0,0,0,0,0,0,0, 0.5,0.1,0,0,0,0,0,0, 0.6,0.2,0.0285714,0,0,0,0,0, 0.666667,0.285714,0.0714286,0.0079365,0,0,0,0, 0.714286,0.357143,0.119048,0.0238095,0.0021645,0,0,0, 0.75,0.416667,0.166667,0.0454545,0.00757576,0.00058275,0,0, 0.777778,0.466667,0.212121,0.0707071,0.016317,0.002331,0.0001554,0, 0.8,0.509091,0.254545,0.0979021,0.027972,0.0055944,0.0006993,0.00004114 iorder,iaz,iel xin iaz = $M_PI*iaz/180 iel = $M_PI*iel/180 a0=zar(0) if iorder > 0 goto c0 aout = a0 goto end c0: a1=iWeight3D[iorder-1][0]*zar(1) a2=iWeight3D[iorder-1][0]*zar(2) a3=iWeight3D[iorder-1][0]*zar(3) icos_el = cos(iel) isin_el = sin(iel) icos_az = cos(iaz) isin_az = sin(iaz) i1 = icos_el*isin_az ; Y = Y(1,-1) i2 = isin_el ; Z = Y(1,0) i3 = icos_el*icos_az ; X = Y(1,1) if iorder > 1 goto c1 aout = (3/4)*(a0 + i1*a1 + i2*a2 + i3*a3) goto end c1: a4=iWeight3D[iorder-1][1]*zar(4) a5=iWeight3D[iorder-1][1]*zar(5) a6=iWeight3D[iorder-1][1]*zar(6) a7=iWeight3D[iorder-1][1]*zar(7) a8=iWeight3D[iorder-1][1]*zar(8) ic2 = sqrt(3)/2 icos_el_p2 = icos_el*icos_el isin_el_p2 = isin_el*isin_el icos_2az = cos(2*iaz) isin_2az = sin(2*iaz) icos_2el = cos(2*iel) isin_2el = sin(2*iel) i4 = ic2*icos_el_p2*isin_2az ; V = Y(2,-2) i5 = ic2*isin_2el*isin_az ; S = Y(2,-1) i6 = .5*(3*isin_el_p2 - 1) ; R = Y(2,0) i7 = ic2*isin_2el*icos_az ; S = Y(2,1) i8 = ic2*icos_el_p2*icos_2az ; U = Y(2,2) aout = (1/3)*(a0 + 3*i1*a1 + 3*i2*a2 + 3*i3*a3 + 5*i4*a4 + 5*i5*a5 + 5*i6*a6 + 5*i7*a7 + 5*i8*a8) end: xout aout endop ; overloaded opcode for decoding for 1 or 2 speakers ; speaker positions in function table ifn opcode ambi_dec2_inph, a,ii iorder,ifn xin xout ambi_dec1_inph(iorder,table(1,ifn),table(2,ifn)) endop opcode ambi_dec2_inph, aa,ii iorder,ifn xin xout ambi_dec1_inph(iorder,table(1,ifn),table(2,ifn)), ambi_dec1_inph(iorder,table(3,ifn),table(4,ifn)) endop opcode ambi_dec2_inph, aaa,ii iorder,ifn xin xout ambi_dec1_inph(iorder,table(1,ifn),table(2,ifn)), ambi_dec1_inph(iorder,table(3,ifn),table(4,ifn)), ambi_dec1_inph(iorder,table(5,ifn),table(6,ifn)) endop instr 1 asnd init 1 kdist init 1 kaz invalue "az" kel invalue "el" k0 ambi_encode asnd,8,kaz,kel ao1,ao2,ao3 ambi_dec_inph 8,17 outvalue "sp1", downsamp(ao1) outvalue "sp2", downsamp(ao2) outvalue "sp3", downsamp(ao3) zacl 0,80 endin </CsInstruments> <CsScore> f1 0 1024 10 1 f17 0 64 -2 0 0 0 90 0 0 90 0 0 0 0 0 0 0 0 0 0 i1 0 100 </CsScore> </CsoundSynthesizer> ;example by martin neukom

The weighting factors for in-phase decoding of Ambisonics (3D) are:

Example udo_ambisonics_3.csd shows distance encoding.

* EXAMPLE 05B17_udo_ambisonics_3.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 2 0dbfs = 1 zakinit 81, 1 ; zak space with the 11 channels of the B-format #include "ambisonics_udos.txt" opcode ambi3D_enc_dist1, k, aikkk asnd,iorder,kaz,kel,kdist xin kaz = $M_PI*kaz/180 kel = $M_PI*kel/180 kaz = (kdist < 0 ? kaz + $M_PI : kaz) kel = (kdist < 0 ? -kel : kel) kdist = abs(kdist)+0.00001 kgainW = taninv(kdist*1.5708) / (kdist*1.5708) kgainHO = (1 - exp(-kdist)) ;*kgainW outvalue "kgainHO", kgainHO outvalue "kgainW", kgainW kcos_el = cos(kel) ksin_el = sin(kel) kcos_az = cos(kaz) ksin_az = sin(kaz) asnd = kgainW*asnd zawm asnd,0 ; W asnd = kgainHO*asnd zawm kcos_el*ksin_az*asnd,1 ; Y = Y(1,-1) zawm ksin_el*asnd,2 ; Z = Y(1,0) zawm kcos_el*kcos_az*asnd,3 ; X = Y(1,1) if iorder < 2 goto end /* ... */ end: xout 0 endop instr 1 asnd init 1 kaz invalue "az" kel invalue "el" kdist invalue "dist" k0 ambi_enc_dist asnd,5,kaz,kel,kdist ao1,ao2,ao3,ao4 ambi_decode 5,17 outvalue "sp1", downsamp(ao1) outvalue "sp2", downsamp(ao2) outvalue "sp3", downsamp(ao3) outvalue "sp4", downsamp(ao4) outc 0*ao1,0*ao2;,2*ao3,2*ao4 zacl 0,80 endin </CsInstruments> <CsScore> f17 0 64 -2 0 0 0 90 0 180 0 0 90 0 0 0 0 i1 0 100 </CsScore> </CsoundSynthesizer> ;example by martin neukomIn example udo_ambisonics_4.csd a buzzer with the three-dimensional trajectory shown below is encoded in third order and decoded for a speaker setup in a cube (f17).

* EXAMPLE 05B18_udo_ambisonics_4.csd*

<CsoundSynthesizer> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 8 0dbfs = 1 zakinit 16, 1 #include "ambisonics_udos.txt" #include "ambisonics_utilities.txt" instr 1 asnd buzz p4,p5,p6,1 kt line 0,p3,p3 kaz,kel,kdist xyz_to_aed 10*sin(kt),10*sin(.78*kt),10*sin(.43*kt) adop Doppler asnd,kdist k0 ambi_enc_dist adop,3,kaz,kel,kdist a1,a2,a3,a4,a5,a6,a7,a8 ambi_decode 3,17 ;k0 ambi_write_B "B_form.wav",8,14 outc a1,a2,a3,a4,a5,a6,a7,a8 zacl 0,15 endin </CsInstruments> <CsScore> f1 0 32768 10 1 f17 0 64 -2 0 -45 35.2644 45 35.2644 135 35.2644 225 35.2644 -45 -35.2644 .7854 -35.2644 135 -35.2644 225 -35.2644 i1 0 40 .5 300 40 </CsScore> </CsoundSynthesizer> ;example by martin neukom

**Ambisonics Equivalent Panning (AEP)**

If we combine encoding and in-phase decoding, we obtain the following panning function (a gain function for a speaker depending on its distance to a virtual sound source):

P(γ, m) = (1/2+ 1/2 cos γ)^m

where γ denotes the angle between a sound source and a speaker and m denotes the order. If the speakers are positioned on a unit sphere the cosine of the angle γ is calculated as the scalar product of the vector to the sound source (x, y, z) and the vector to the speaker (xs, ys, zs).

In contrast to Ambisonics the order indicated in the function does not have to be an integer. This means that the order can be continuously varied during decoding. The function can be used in both Ambisonics and Ambisonics2D.

This system of panning is called Ambisonics Equivalent Panning. It has the disadvantage of not producing a B-format representation, but its implementation is straightforward and the computation time is short and independent of the Ambisonics order simulated. Hence it is particularly useful for real-time applications, for panning in connection with sequencer programs and for experimentation with high and non-integral Ambisonic orders.

The opcode AEP1 in the example udo_AEP.csd shows the calculation of ambisonics equivalent panning for one speaker. The opcode AEP then uses AEP1 to produce the signals for several speakers. In the text file "AEP_udos.txt" AEP ist implemented for up to 16 speakers. The position of the speakers must be written in a function table. As the first parameter in the function table the maximal speaker distance must be given.

* EXAMPLE 05B19_udo_AEP.csd*

<CsoundSynthesizer> <CsOptions> </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 4 0dbfs = 1 ;#include "ambisonics_udos.txt" ; opcode AEP1 is the same as in udo_AEP_xyz.csd opcode AEP1, a, akiiiikkkkkk ; soundin, order, ixs, iys, izs, idsmax, kx, ky, kz ain,korder,ixs,iys,izs,idsmax,kx,ky,kz,kdist,kfade,kgain xin idists = sqrt(ixs*ixs+iys*iys+izs*izs) kpan = kgain*((1-kfade+kfade*(kx*ixs+ky*iys+kz*izs)/(kdist*idists))^korder) xout ain*kpan*idists/idsmax endop ; opcode AEP calculates ambisonics equivalent panning for n speaker ; the number n of output channels defines the number of speakers (overloaded function) ; inputs: sound ain, order korder (any real number >= 1) ; ifn = number of the function containing the speaker positions ; position and distance of the sound source kaz,kel,kdist in degrees opcode AEP, aaaa, akikkk ain,korder,ifn,kaz,kel,kdist xin kaz = $M_PI*kaz/180 kel = $M_PI*kel/180 kx = kdist*cos(kel)*cos(kaz) ky = kdist*cos(kel)*sin(kaz) kz = kdist*sin(kel) ispeaker[] array 0, table(3,ifn)*cos(($M_PI/180)*table(2,ifn))*cos(($M_PI/180)*table(1,ifn)), table(3,ifn)*cos(($M_PI/180)*table(2,ifn))*sin(($M_PI/180)*table(1,ifn)), table(3,ifn)*sin(($M_PI/180)*table(2,ifn)), table(6,ifn)*cos(($M_PI/180)*table(5,ifn))*cos(($M_PI/180)*table(4,ifn)), table(6,ifn)*cos(($M_PI/180)*table(5,ifn))*sin(($M_PI/180)*table(4,ifn)), table(6,ifn)*sin(($M_PI/180)*table(5,ifn)), table(9,ifn)*cos(($M_PI/180)*table(8,ifn))*cos(($M_PI/180)*table(7,ifn)), table(9,ifn)*cos(($M_PI/180)*table(8,ifn))*sin(($M_PI/180)*table(7,ifn)), table(9,ifn)*sin(($M_PI/180)*table(8,ifn)), table(12,ifn)*cos(($M_PI/180)*table(11,ifn))*cos(($M_PI/180)*table(10,ifn)), table(12,ifn)*cos(($M_PI/180)*table(11,ifn))*sin(($M_PI/180)*table(10,ifn)), table(12,ifn)*sin(($M_PI/180)*table(11,ifn)) idsmax table 0,ifn kdist = kdist+0.000001 kfade = .5*(1 - exp(-abs(kdist))) kgain = taninv(kdist*1.5708)/(kdist*1.5708) a1 AEP1 ain,korder,ispeaker[1],ispeaker[2],ispeaker[3], idsmax,kx,ky,kz,kdist,kfade,kgain a2 AEP1 ain,korder,ispeaker[4],ispeaker[5],ispeaker[6], idsmax,kx,ky,kz,kdist,kfade,kgain a3 AEP1 ain,korder,ispeaker[7],ispeaker[8],ispeaker[9], idsmax,kx,ky,kz,kdist,kfade,kgain a4 AEP1 ain,korder,ispeaker[10],ispeaker[11],ispeaker[12], idsmax,kx,ky,kz,kdist,kfade,kgain xout a1,a2,a3,a4 endop instr 1 ain rand 1 ;ain soundin "/Users/user/csound/ambisonic/violine.aiff" kt line 0,p3,360 korder init 24 ;kdist Dist kx, ky, kz a1,a2,a3,a4 AEP ain,korder,17,kt,0,1 outc a1,a2,a3,a4 endin </CsInstruments> <CsScore> ;fuction for speaker positions ; GEN -2, parameters: max_speaker_distance, xs1,ys1,zs1,xs2,ys2,zs2,... ;octahedron ;f17 0 32 -2 1 1 0 0 -1 0 0 0 1 0 0 -1 0 0 0 1 0 0 -1 ;cube ;f17 0 32 -2 1,732 1 1 1 1 1 -1 1 -1 1 -1 1 1 ;octagon ;f17 0 32 -2 1 0.924 -0.383 0 0.924 0.383 0 0.383 0.924 0 -0.383 0.924 0 -0.924 0.383 0 -0.924 -0.383 0 -0.383 -0.924 0 0.383 -0.924 0 ;f17 0 32 -2 1 0 0 1 45 0 1 90 0 1 135 0 1 180 0 1 225 0 1 270 0 1 315 0 1 ;f17 0 32 -2 1 0 -90 1 0 -70 1 0 -50 1 0 -30 1 0 -10 1 0 10 1 0 30 1 0 50 1 f17 0 32 -2 1 -45 0 1 45 0 1 135 0 1 225 0 1 i1 0 2 </CsScore> </CsoundSynthesizer> ;example by martin neukom

**Utilities**

The file utilities.txt contains the following opcodes:

dist computes the distance from the origin (0, 0) or (0, 0, 0) to a point (x, y) or (x, y, z)

kdist dist kx, ky

kdist dist kx, ky, kz

Doppler simulates the Doppler-shift

ares Doppler asnd, kdistance

absorb is a very simple simulation of the frequency dependent absorption

ares absorb asnd, kdistance

aed_to_xyz converts polar coordinates to Cartesian coordinates

kx, ky, kz aed_to_xyz kazimuth, kelevation, kdistance

ix, iy, iz aed_to_xyz iazimuth, ielevation, idistance

dist_corr induces a delay and reduction of the speaker signals relative to the most distant speaker.

a1 [, a2] ... [, a16] dist_corr a1 [, a2] ... [, a16], ifn

f ifn 0 32 -2 max_speaker_distance dist1, dist2, ... ;distances in m

radian (radiani) converts degrees to radians.

irad radiani idegree

krad radian kdegree

arad radian adegree

degree (degreei) converts radian to degrees

idegree degreei irad

kdegree degree krad

adegree degree arad

## VBAP or Ambisonics?

Csound offers a simple and reliable way to access two standard methods for multi-channel spatialisation. Both have different qualities and follow different aesthetics. VBAP can perhaps be described as clear, rational and direct. It combines simplicity with flexibility. It gives a reliable sound projection even for rather asymmetric speaker setups. Ambisonics on the other hand offers a very soft sound image, in which the single speaker becomes part of a coherent sound field. The B-format offers the possibility to store the spatial information independently from any particular speaker configuration.

The composer, or spatial interpreter, can choose one or the other technique depending on the music and the context. Or (s)he can design a personal approach to spatialisation by combining the different techniques described in this chapter.

- First described by Ville Pulkki in 1997: Ville Pulkki, Virtual source positioning using vector base amplitude panning, in: Journal of the Audio Engeneering Society, 45(6), 456-466
^{^} - Ville Pulkki, Uniform spreading of amplitude panned virtual sources, in: Proceedings of the 1999 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, Mohonk Montain House, New Paltz
^{^} - For instance www.ambisonic.net or www.icst.net/research/projects/ambisonics-theory
^{^} - See www.csounds.com/manual/html/bformdec1.html for more details.
^{^} - Which in turn then are taken by the decoder as input.
^{^}