This page explains the mathematics behind the quadraphonic matrix systems. The mathematic principles used are based on algebra, complex numbers, and trigonometry.
Nothing in the encoding and decoding system may change the basic waveforms that are encoded. This means that the following mathematical operations must not be used on the signals:
Except where the trigonometric value is a constant used in a polynomial expression.
The process used for quadraphonic matrix is called matrix multiplication.
The following are the matrices used for one encode and one decode operation in matrix quadraphonics:
Here is an example of this use of the matrix. The ElectroVoice Stereo4 System is shown. But it works for any matrix system.
ENCODER  DECODER  


The procedure used to calculate this is found at the matrix multiplication page.
The rules of complex mathematics;





The Poincaré Sphere (also called the Stokes Sphere, the Foucault Sphere, and the Fresnel Sphere) was originally conceived by Henri Poincaré in 1892 to describe polarized light, and by Foucault to describe freeswinging pendulum motion. Stokes and Fresnel independently discover its use for polarized light. Scheiber and Robinson independently discovered it for use in quadraphonic signals.
The Poincaré Sphere, representing the two channels of a stereo recording or phono stylus modulations, represents stylus motions on a phonograph record and signal relationships in other media. It is shown in the diagram at right as follows:
Any plane passing through the center of the Poincaré Sphere produces a great circle where it intersects the Poincaré Sphere.
The following matrix systems are examples of GreatCircle Matrix systems. The order of the listed colors in the table shows the movement on the Poincaré sphere (shown at right) of a sound panned clockwise around the listener starting at the front. Large spots are on the far side of the sphere.
MATRIX  ORIENTATION  COLORS ON DIAGRAM AT RIGHT  

FRONT  RIGHT  BACK  LEFT  FRONT  
QS, Stereo4, Dynaquad  Equator  olive  red  violet  cyan  olive 
UMX and BMX  Opposite Meridians  black  red  brown  cyan  black 
BBC Matrix H (see below)  45° diagonal  orange  red  blue  cyan  orange 
Matrix HR (see below)  45° opposite diagonal  pink  red  yellow  cyan  pink 
Phase Location (Denon experiment)  Opposite Meridians  olive  brown  violet  black  olive 
Most other matrix systems are not greatcircle systems.
The relative phase angles are as follows:
The following formula calculates the geodesic distance between two points on the surface of the sphere. This geodesic distance gives a central angle, which is used in the formulas below:
The variables:
The calculations:
Examples of orthogonal pairs (see diagram above):
Proof for the following three orthogonal pairs:
These are the original left and right encoded channels, recorded in the left wall and the right of the stereo record groove using the Westrex 45/45 process. Since no signal processing was done before making the recording, they are still the original unchanged L and R signals from the quadraphonic encoder.
L' = L  R' = R 
The created orthogonal signals are signals from a sum and difference matrix:  
H = 0.71 L + 0.71 R  V = 0.71 L − 0.71 R 
This orthogonal pair is used for FM multiplex stereo.  
The original signals are recovered using a sum and difference matrix:  
L' = 0.71 H + 0.71 V  R' = 0.71 H − 0.71 V 
L' = 0.71 (0.71 L + 0.71 R) + 0.71 (0.71 L − 0.71 R)  R' = 0.71 (0.71 L + 0.71 R) − 0.71 (0.71 L − 0.71 R) 
L' = 0.5 L + 0.5 R + 0.5 L − 0.5 R  R' = 0.5 L + 0.5 R − 0.5 L + 0.5 R 
L' = L  R' = R 
The created orthogonal signals are signals from a quadrature matrix:  
A = 0.71 L + 0.71 Rj  B = 0.71 R + 0.71 Lj 
The original signals are also recovered using a quadrature matrix:  
L' = 0.71 A − 0.71 Bj  R' = 0.71 B − 0.71 Aj 
L' = 0.71 (0.71 L + 0.71 Rj) − 0.71j (0.71 R + 0.71 Lj)  R' = 0.71 (0.71 R + 0.71 Lj) − 0.71j (0.71 L + 0.71 Rj) 
L' = 0.5 L + 0.5 Rj + 0.5 L − 0.5 Rj  R' = 0.5 R + 0.5 Lj + 0.5 R − 0.5 Lj 
L' = L  R' = R 
Proof: Any monophonic signal taken from 2channel program material must be at some location on the Poincaré Sphere. Because of this, any material at the point orthogonal to the point used must completely disappear from the monophonic signal.
The usual way to create a mono signal from a stereo signal is to use this sum:
M = 0.71 L + 0.71 R
But the following signal is entirely absent from the mono signal:
0.71 L  0.71 R
Sansui used the sum of all of the decoder outputs, which reproduced the clockwise stylus motion and other motions that put sound into the clockwise reproduction
The anticlockwise stylus motion fully disappears from this signal.
The following are true of any two points on the Poincaré Sphere:
Note that when angle α is 0°, cos(α/2) = 1.
Note that when angle α is 180°, cos(α/2) = 0.
The separation between points A and B is 20 log10(cos(α/2)).
Here are some of the commonly used values:
Sphere Angle  0  30  39  45  54  60  82  90  109  120  133  135  143  150  151  160  169  176  179  180 

Crossfeed  1.00  0.97  0.94  0.92  0.89  0.87  0.75  0.71  0.58  0.50  0.40  0.38  0.32  0.26  0.25  0.18  0.10  0.03  0.01  0.00 
Separation dB  0.0  0.3  0.5  0.7  1.0   1.2   2.5  3.0  4.7  6.0  8.0  8.3  10.0  11.7  12.0  15.0  20.0  30.0  40.0  60.0 
Front separation:
Back separation:
Front to back separation:
Front separation:
Back separation:
Front separation:
Back separation:
Mono to back separation:
Back attenuation:
Any position attenuated more than 10 dB effectively disappears from mono playback.
Here are separations calculated using these methods.
Procedure:
C = cos(α/2)
Procedure:
S = 20 log10(C)
Procedure:
S = 20 log10(cos(α/2))
Procedure:
C = 10^(S/20)
Procedure:
α = 2 arccos(C)
Procedure:
α = 2 arccos(10^(S/20))
The colors in this list apply to only the diagram at right.
The angle θ (theta) is the angle from the L pole (cyan) of the sphere to the desired point C.
The angle θ is also the angle from the R pole (red) of the sphere to the corresponding desired point C'.
C' is 180°θ from the L pole.
The angle ø (phi) is the phase angle between L and R on the equator of the sphere, where:
The angle ρ (rho) is a phase shift added so the encoded signal has a wanted phase relationship to other signals.
The angles are converted into encoding sinecosine variable pairs:
The leftside left and right encoding outputs for point C, using signal x, is:
L = atx R = bftx
The corresponding mirrorimage rightside left and right encoding outputs for point C', using signal y, is:
L = bfty R = aty
Obtain the same angles used in the encoding equations above.
The angle θ (theta) is the angle from the L pole (cyan) of the sphere to the desired point C.
The angle θ is also the angle from the R pole (red) of the sphere to the corresponding desired point C'.
C' is 180°θ from the L pole.
The angle ø (phi) is the phase angle between L and R on the equator of the sphere, where:
The angle ρ (rho) is a phase shift added so the encoded signal has a wanted phase relationship to other signals.
The angles are converted into decoding sinecosine variable pairs:
The decoding output for point C, using signals L and R, is:
U = atL + bftR
The corresponding mirrorimage decoding output for point C', using signals L and R, is:
V = bftL + btR
In these equations, a, b, c, and d are values between 1 and 1.
In these equations, s and t are sign values of +1 or 1. S goes with X, and t goes with Y.
In these equations, u and v are sign values of +1 or 1. u goes with L, and v goes with R.
In these equations, m and n are sign values of +1 or 1. m goes with Lj, and v goes with Rj.
For the encoding of any pair of symmetric signals, the following equations are used:
For the decoding of any pair of symmetric signals, the following equations are used:
Examples include the following matrix systems:
PAIR  a  b  c  d  s  t  u  v  m  n 

lf and rf  0.92  0.00  0.38  0.00  +1  +1  +1  +1  +1  +1 
lb and rb  0.00  0.92  0.00  0.38  +1  +1  +1  1  +1  +1 
Encoding equations:
Decoding equations:
PAIR  a  b  c  d  s  t  u  v  m  n 

lf and rf  1.00  0.00  0.00  0.00  +1  +1  +1  +1  +1  +1 
lb and rb  0.00  0.71  0.71  0.00  1  +1  +1  +1  +1  +1 
Encoding equations:
Decoding equations:
PAIR  a  b  c  d  s  t  u  v  m  n 

lf and rf  0.85  0.35  0.35  0.15  +1  +1  +1  +1  +1  1 
lb and rb  0.35  0.85  0.15  0.85  +1  +1  +1  +1  1  +1 
Encoding equations:
Decoding equations:
Note the following special values in the above matrices:
No matter which greatcircle matrix is used, there is at least one point where panning must be interrupted for a phase change as the sound is panned completely around the listener:
No matter which great circle is used, either the phase is continually changed as the signal is panned, or the phase must be changed at some point. If all of the phase changes are at zero crossings, then the final phase of the signal after the panning will be reversed from the original signals before the panning, as shown here:
DIR  F  RF  R  RB  B  LB  L  LF  F 

L  0.71  0.38  0.00  −0.38  −0.71  −0.92  −1.00  −0.92  −0.71 
R  0.71  0.92  1.00  0.92  0.71  0.38  0.00  −0.38  −0.71 
Examples:
The red bar shows the encoding phase hole.
DIR  F  RF  R  RB  B  B  LB  L  LF  F  

L  0.71  0.38  0.00  −0.38  −0.71  0.71  0.92  1.00  0.92  0.71  
R  0.71  0.92  1.00  0.92  0.71  −0.71  −0.38  0.00  0.38  0.71 
The orange bars are where the bus selector is changed when panning.
DIR  F  RF  R  R  RB  B  LB  L  L  LF  F  

L  0.71  0.38  0.00  −0.00  −0.38  −0.71  −0.92  −1.00  1.00  0.92  0.71  
R  0.71  0.92  1.00  1.00  0.92  0.71  0.38  0.00  0.00  0.38  0.71 
The orange bars show the encoding phase holes, automatically adjusted by phasors in the encode equations.
DIR  F  RF  R  R  RB  B  LB  L  L  LF  F  

L  0.71  0.38  0.00  0.00j  0.38j  0.71j  0.92j  1.00j  1.00  0.92  0.71  
R  0.71  0.92  1.00  −1.00j  −0.92j  −0.71j  −0.38j  −0.00j  0.00  0.38  0.71 
The orange bars show the encoding phase holes. It is unknown how they were implemented in practice.
DIR  F  RF  R  R  RB  B  LB  L  L  LF  F  

L  0.50  0.26  0.00  0.00  0.26  0.50  0.65  0.71  0.71  0.65  0.50  
Lj  0.50j  0.26j  0.00j  −0.00j  −0.26j  −0.50j  −0.65j  −0.71j  0.71j  0.65j  0.50j  
R  0.50  0.65  0.71  0.71  0.65  0.50  0.26  0.00  0.00  0.26  0.50  
Rj  −0.50j  −0.65j  −0.71j  0.71j  0.65j  0.50j  0.26j  0.00j  −0.00j  −0.26j  −0.50j 
Most matrix systems that are not great circle matrix systems also have at least one point where panning must be interrupted for a phase change as the sound is panned completely around the listener.
Example: Scheiber system and Sansui QS:
Value  Original system  ×  Resulting System 
Value  Original system  ×  Resulting System 


System  Scheiber  QS  System  QS  Scheiber  
Major Value  0.92  × 1.08 =  1.00  Major Value  1.00  × 0.92 =  0.92  
Minor Value  0.38  × 1.08 =  0.41  Minor Value  0.41  × 0.92 =  0.38  
Resultant Value  1.00  × 1.08 =  1.08  Resultant Value  1.08  × 0.92 =  1.00 
The procedure to find out what to multiply the original system coefficients by to get the resulting system is:
Percent Blend  Major is 1.00  Result is 1.00  Encode Angle  

Minor  Result  Major  Minor  Sphere α  Stylus α/2  
20%  0.20  1.02  0.98  0.20  22.6°  11.3° 
25%  0.25  1.03  0.97  0.24  28.1°  14.0° 
30%  0.30  1.04  0.96  0.29  33.4°  16.7° 
41%  0.41  1.08  0.92  0.38  45.0°  22.5° 
50%  0.50  1.12  0.89  0.46  53.2.4°  26.6° 
80%  0.80  1.28  0.78  0.62  77.4°  38.7° 
100%  1.00  1.41  0.71  0.71  90.0°  45.0° 
Each row is the same matrix encoding, shown using different coefficients.
Hans Wallach, Edwin Newman, and Mark Rosenzweig defined the precedence effect in 1949. They showed that when a sound reaches the ears, followed by an identical sound from a different direction, the listener hears only one sound where the first sound came from. The delay between the sounds can be as long as 5 ms for clicks, and 40 ms for complex sounds (speech or music). Larger delays are heard as echoes.
Human hearing uses these time and level differences in sounds reaching each ear to indicate the direction each sound comes from. A special microphone array is used to encode a quadraphonic matrix signal with the time delays included to give additional clues to correctly locate the sound. No separation enhancement is needed.
Here is a table of the relative levels and delays (in microseconds) received on the input microphones l, r, and s of such a recording for different directions of sound sources:
Angle  Level l  Delay l 
Level r  Delay r 
Level s  Delay s 
Direction  1st  2nd  3rd  
0  0.866  0  0.866  0  ~ 0.200  885  Straight Ahead  l r  s  
30  0.707  369  0.966  0  0.259  951  r  l  s  
60  0.500  639  1.000  0  0.500  762  Right Mic  r  l s  
90  0.259  737  0.966  0  0.707  369  Straight Right  r  s  l  
120  ~ 0.200  762  0.866  123  0.866  0  r s  l  
180  0.500  885  0.500  885  1.000  0  Straight Back Mic  s  l r  
240  0.866  123  ~ 0.200  762  0.866  0  l s  r  
270  0.966  0  0.259  737  0.707  369  Straight Left  l  s  r  
300  1.000  0  0.500  639  0.500  762  Left Mic  l  r s  
330  0.966  0  0.707  369  0.259  951  l  r  s  
360  0.866  0  0.866  0  ~ 0.200  885  Straight Ahead  l r  s 
~ Approximate value  Spill sound from room reflections into mics.
The stereo recording is matrixed from these three channels:
Decoding works with Dolby Surround or with any regular matrix (RM or QM) decoder.
The separations in a quadraphonic program that the listener notices the most are:
The other separations are not as noticeable. The separation between left back and right back is usually the least noticed.