# QUADRAPHONIC MATRIX MATH

This page explains the mathematics behind the quadraphonic matrix systems. The mathematic principles used are based on algebra, complex numbers, and trigonometry.

### General mathematical and engineering principles used in matrix quadraphonic systems:

• Mathematical principle 1: Use of logarithms and exponentials to calculate decibels (dB)

Calculating decibels:

The formulas in this section are in Microsoft Excel ® form.

• Calculating decibels for voltage ratios:

Value_dB = 20 * log10(voltage_ratio)

Procedure: Find the common logarithm of voltage_ratio. Then multiply by 20.

• Calculating decibels for power ratios:

Value_dB = 10 * log10(power_ratio)

Procedure: Find the common logarithm of power_ratio. Then multiply by 10.

• Finding voltage ratios from decibels:

voltage_ratio = 10 ^ (Value_dB /20)

Procedure: Divide Value_dB by 20. Then raise 10 to the power of the result.

• Finding power ratios from decibels:

power_ratio = 10 ^ (Value_dB /10)

Procedure: Divide Value_dB by 10. Then raise 10 to the power of the result.

• Mathematical principle 2: Solutions of systems of simultaneous equations
• For a solution of simultaneous equations in n variables to be unique, n equations are required.

In quadraphonics, this would be 4 equations in 4 unknowns - a discrete quadraphonic system.

• When a system of simultaneous equations in n variables has fewer than n equations, the system normally has multiple solutions.

Such a system can have no solution. But such systems are not used in quadraphonics.

• Encoding matrix quadraphonics uses a system of 2 equations with n variables (representing n input channels). There are 2 channels on the recording.

In quadraphonics, this would be 2 equations in 4 unknowns.

• Decoding matrix quadraphonics uses a system of n equations with 2 variables (representing n output channels).

In quadraphonics, this would be 4 equations in 2 unknowns.

• The system of n equations in 2 variables is used to recover the n channels. But because these systems of equations do not have a unique solution, there is crosstalk between the channels. But the coefficients of the two systems of equations are chosen by design to place the crosstalk where the designer wants it to be.
• Engineering definition 1: The stereophonic record groove

Modulations on the stereo record groove are defined by the RIAA (Recording Industry Association of America).

Here are the stereophonic groove modulations as defined by the RIAA:

1. The stereophonic groove shall contain two channels of information recorded as orthogonal modulations of a single groove.
2. The 45 degree-45 degree system is the standard. In this system the two mutually perpendicular axes of modulation shall be symmetrically disposed with respect to a normal to the disc surface.
3. The right hand information channel, as viewed by the listener, shall appear as modulation of the outside wall of the groove.
4. When equal in-phase signals are recorded in the two channels they shall result in lateral modulation of the stereophonic groove shall produce equal in-phase acoustical signals.
5. The stereophonic groove contour shall be as recommended in RIAA Dimensional Standards for Disc Phonograph Records, Bulletin E 4. These modulations are as viewed from beyond the cartridge end of a normally mounted pickup arm:

• Left and right channels:

The left channel is recorded on the side of the groove closest to the spindle.
- Left channel modulations slope up to the right and down to the left at a 45° angle (cyan).

The right channel is recorded on the side of the groove closest to the rim.
- Right channel modulations slope up to the left and down to the right at a 45° angle (red).

• In-phase (center) channel and reversed-phase recording:

In-phase modulations are lateral (horizontal - olive). They emulate a monaural record.

Reversed phase modulations are vertical (violet).

• Circular modulations:

Clockwise motion (viewing the cartridge end of the tonearm) occurs when the left channel leads the right channel by 90° (black).

Anticlockwise motion (viewing the cartridge end of the tonearm) occurs when the left channel lags the right channel by 90° (brown).

### Basic mathematic principles of matrix quadraphonic systems:

• Postulate 1: A matrix quadraphonic recording must play on and be compatible with stereo players.

- Special equipment (including a special pickup cartridge) must not be needed to play the recording in stereo.

- Very few people will buy a recording not compatible with normal stereo equipment.

Only quadraphonic enthusiasts would be able to use it. So it is essential to make recordings fully playable on stereo players.

These rules ensure compatibility:

• Corollary 1 to Postulate 1: Quadraphonic recordings must have left-right symmetry in stereo play.

- This is necessary to keep from overloading one side of the stereo recording.

• Corollary 2 to Postulate 1: Many parts should locate similarly in stereo and quadraphonic play.

- The encoding and decoding methods should accomplish this.

• Front center parts belong in the front center of both renderings.
• Parts on the left belong on the left of both renderings.
• Parts on the right belong on the right of both renderings.
• Corollary 3 to Postulate 1: Low bass parts must be recorded in phase in both stereo channels.

- This is so the two stereo speakers will aid each other in reproducing the bass.

- At least two of the quadraphonic speakers should be in phase to aid each other in reproducing the bass.

- In addition, if the recording is on a phonograph record, the bass must be recorded laterally (in phase) in the groove.

Bass recorded in other orientations must be recorded at much lower levels to avoid overcutting the groove.

• Corollary 4 to Postulate 1: Most matrix encoding systems encode center back sounds so they disappear in mono play.

- This is a problem with most encoding systems.

- Usually the recording engineer is advised to not pan important material to center back.

Center back is a good place to hide some extra reverb.

• Corollary 5 to Postulate 1: Avoid Large phase differences between the left front, center front, and right front channels.

- Large phase differences make a weird "phasiness" sensation when playing the recording in stereo.

- This effect makes image location of musical parts difficult.

- This unwanted effect happens with the UMX (BMX), Matrix H, Matrix E, Matrix G, and Phase Location.

- Note that while the matrix decoder removes phase anomalies when decoding the record, the stereo player does not.

- It is not advisable to use a matrix encoding system that shifts the phase of the front channels.

• Corollary 6 to Postulate 1: Standard radio station methods must work properly with quadraphonic recordings.

- A radio-frequency carrier in the groove causes a swooping sound when a phonograph record is static-cued or slip-cued.

Radio-frequency carriers should not be used.

• Postulate 2: Linear algebra with complex numbers is sufficient to represent any quadraphonic matrix system.

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:

• Powers and roots
• Exponentials and logarithms
• Trigonometric functions*
• Other Transcendental functions
• Calculus

* Except where the trigonometric value is a constant used in a polynomial expression.

• Corollary 1 to Postulate 2: The mathematical matrix can be used to calculate any quadraphonic matrix.

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:

•   F ... Original 4 Channels to Encode (column vector LF, RF, LB, RB or L, R, F, B)
•   E ... Encoder Matrix
•   C ... Encoded Program for Record or Tape (column vector L, R)
•   D ... Decoder Matrix
•   S ... Decoded Output to Speakers (column vector LF, RF, LB, RB or L, R, F, B)
•   C = E F ... Encoding Equation
•   S = D C ... Decoding Equation
•   S = D E ... Combined Encode-Decode Effects Matrix
•   S = D (E F) ... Entire Encode-Decode Process

Here is an example of this use of the matrix. The Electro-Voice Stereo-4 System is shown. But it works for any matrix system.

ENCODER DECODER
 ⌈||⌊ LF ⌉||⌋ input matrix RF LB C = E F RB × product ⌈⌊ 1.0 0.3 1.0 −0.5 ⌉⌋ ⌈⌊ LF + 0.3 RF + LB − 0.5 RB ⌉⌋ 0.3 1.0 -0.5 1.0 0.3 LF + RF − 0.5 LB + RB encoder matrix encoded output
 ⌈⌊ L ⌉⌋ encoded input S = D C R × product ⌈||⌊ 1.0 0.2 ⌉||⌋ ⌈||⌊ L + 0.2 R ⌉||⌋ 0.2 1.0 0.2 L + R 1.0 −0.8 L − 0.8 R −0.8 1.0 − 0.8 L + R decoder matrix decoded output

The procedure used to calculate this is found at the matrix multiplication page.

• Definition 1: Complex numbers represent relative phase angles:
• A reference signal x represented by +x.
• A signal leading the reference signal x by 90° is represented by +j x.
• A signal lagging the reference signal x by 90° is represented by −j x.
• A signal at 180° from the reference signal x is represented by −x.
• A signal at an odd angle θ is represented by x cos(θ) + x j sin(θ)

The rules of complex mathematics;

 j can be treated as a variable in polynomial expressions. +j × +1 = +j +j × +j = −1 +j × −1 = −j +j × −j = +1 +1 × +j = +j −j × −j = −1 −1 × +j = −j −j × +j = +1 −j × −j × −j = +j +j × +j × +j × −j = −1 +j × +j × +j = −j +j × +j × +j × +j = +1
• Definition 2: The Poincaré Sphere The Poincaré Sphere (also called the Stokes Sphere, the Foucault Sphere, and the Fresnel Sphere) was originally conceived by Henri Poincaré ("on-REE pwan-car-RAY") in 1892 to describe polarized light, and by Foucault ("foo-CO") to describe free-swinging pendulum motion. Stokes and Fresnel ("freh-NELL") independently discovered its use for polarized light. Scheiber and Robinson independently discovered it for use in matrix quadraphonic signals. The quadraphonic stylus motions are analogous to the Foucault pendulum motions.

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. The diagrams at the right are as follows:

• The upper diagram shows the stylus motions in the matrix quadraphonic phonograph record groove (as seen from the cartridge end of the tonearm). Note that the six fundamental orthogonal pairs are at right angles to each other or rotate in opposite directions.
• The lower diagram shows the Poincaré Sphere representations of these same stylus motions. Note that the six fundamental orthogonal pairs are dots on diametrically opposite sides of the sphere.
• The colors on each diagram show the six fundamental orthogonal pairs. The same color shows the same modulation in both diagrams. These colors apply to only these two diagrams. These are the descriptions of the six orthogonal motions and their positions on the Poincaré Sphere:

• All modulations are on the surface of the sphere.
• The colors in this list apply to only these two diagrams.
• Points on the side of the sphere away from the viewer have large dots.
• Points on the near side of the sphere or on the limb have small dots.
• Lateral (horizontal or mono) stylus motion (0° phase) is at the right side of the sphere (olive) in this diagram
• Vertical stylus motion (180° phase) is at the left side of the sphere (violet).
• The left diagonal (left signal alone) is centered on the far side of the sphere (cyan).
• The right diagonal (right signal alone) is centered on the near side of the sphere (red).
• Clockwise motion (left signal leads right by 90°) is at the top of the sphere (black).
• Anticlockwise motion (right signal leads left by 90°) is at the bottom of the sphere (brown).
• Definition 3: A great circle on the Poincaré Sphere 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 Great-Circle Matrix systems. The order of the listed colors in the table shows the movement on this 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, Stereo-4, Dynaquad Equator oliveredvioletcyanolive
UMX and BMX Opposite Meridians blackredbrowncyanblack
BBC Matrix H (see below) 45° diagonal orangeredbluecyanorange
Matrix HR (see below) 45° opposite diagonal pinkredyellowcyanpink
Phase Location (Denon experiment) Opposite Meridians olivebrownvioletblackolive

Most other matrix systems are not great-circle systems.

• Postulate 3: The Poincaré Sphere can be used to represent the relative phase between any two signals:

The relative phase angles are as follows:

• The colors in this list apply to only the diagram above.
• Points on the far side of the sphere have large dots, points on the near side of the sphere or on the limb have small dots.
• Lateral (horizontal or mono) is at the right of the sphere (olive); vertical (violet) is at the left.
• The left pole diagonal (cyan) is centered on the far side of the sphere.
• The right pole diagonal (red) is centered on the near side of the sphere.
• Clockwise motion (black) is at the top of the sphere, anticlockwise (brown) is at the bottom.
• Corollary 1 to Postulate 3: Finding the great-circle distance between two points on the Poincaré Sphere:

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 poles are the L (cyan) and R (red) points on the sphere (see diagram).
• The equator is the visible circle in the diagram.
• The prime meridian crosses the equator at the 0° phase point (olive).
• The first point has latitude φ1 has and longitude λ1
• The second point has latitude φ2 has and longitude λ2
• κ is the central angle along the equator.
• σ is the central angle between the two points.
• d is the length of the geodesic subtending the central angle.

The calculations:

• κ = |λ1 − λ2|
• σ - arccos( sin(φ1) sin(φ2) + cos(φ1) cos(φ2) cos(κ) )
• d = rσ (sigma is in radians)
• Since the radius of the Poincaré Sphere is 1:
d = σ (in radians), but is also the length of the geodesic arc.
• Definition 4: Signals are said to be orthogonal (or to be an orthogonal pair) if they are located at opposite points (180° apart) on the Poincaré Sphere.

Examples of orthogonal pairs (see diagram above):

• Lateral stylus motion (olive) and vertical stylus motion (violet)
• The left diagonal (cyan) and the right diagonal (red)
• Clockwise motion (black) and anticlockwise motion (brown)
• Note that ANY pair of signals represented by points diametrically opposite on the Poincaré Sphere is an orthogonal pair.
• Theorem 1: Any orthogonal pair on the Poincaré Sphere can be a pair of signals that can transmit the entire matrix quadraphonic program.

Proof for the following three orthogonal pairs:

• The left diagonal (left channel) and the right diagonal (right channel):

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
• Lateral stylus motion and vertical stylus motion:  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
• Clockwise stylus motion and anticlockwise motion  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
• In the general case, the recovery equations are always the complex conjugate of the pair-creating equations.
• Theorem 2: If program material covers the entire Poincaré Sphere, then it is impossible to recover from that material a monophonic signal that contains all of it.

Proof: Any monophonic signal taken from 2-channel 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 left channel and right channel:

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

• The mono reconstruction Sansui used for mono playback of a QS record:

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 (not usually used) fully disappears from this signal.

• This cancellation does not fully occur in a surround field (below) because every sound is picked up by several mics with different delays. At least one version of each sound will be picked up at an angle where it will be plainly decoded.
• Theorem 3: The separation between two points on the Poincaré Sphere is calculated using the cosine of half the included angle.

The following are true of any two points on the Poincaré Sphere:

• The straight line connecting any two points on the sphere is a chord.
• The diagram at right is a plane passing through two pairs of points on the sphere for separation measurement and the center of the sphere.
• The length of a chord of a circle is 2 r sin(θ/2) for radius r and subtended angle θ.
• The radius r of the Poincaré Sphere is 1. Example 1:

1. Points A and B are the two points to be calculated.
2. Line segment AB is a chord subtending ∠α.
3. length(AB) = 2 sin(α/2)
4. length(AC) = sin(α/2)
5. Δ ACO is a right triangle with ∠COA = α/2.
6. length(CO) / length(AO) = cos(α/2)
7. Because hypotenuse AO has length 1, cos(α/2) = length(CO)
8. length(CO) is the crosstalk between A and B

- Note that when ∠α is 0°, cos(α/2) = 1.

- Note that when ∠α is 180°, cos(α/2) = 0.

9. The separation between points A and B is 20 log10(cos(α/2)).

Example 2:

1. Points Z and Y are the two points to be calculated.
2. Line segment ZY is a chord subtending ∠β.
3. length(ZY) = 2 sin(β/2)
4. length(ZX) = sin(β/2)
5. Δ ZXO is a right triangle with ∠XOZ = β/2.
6. length(XO) / length(ZO) = cos(β/2)
7. Because hypotenuse ZO has length 1, cos(β/2) = length(XO)
8. length(XO) is the crosstalk between A and B

- Note that when ∠β is 0°, cos(β/2) = 1.

- Note that when ∠β is 180°, cos(β/2) = 0.

9. The separation between points Y and Z is 20 log10(cos(α/2)).

Here are some of the commonly used values:

 Sphere Angle Crossfeed Separation dB 0 30 39 45 54 60 82 90 109 120 133 135 143 150 151 160 169 176 179 180 1 0.97 0.94 0.92 0.89 0.87 0.75 0.71 0.58 0.5 0.4 0.38 0.32 0.26 0.25 0.18 0.1 0.03 0.01 0 0 -0.3 -0.5 -0.7 -1 - 1.2 - 2.5 -3 -4.7 -6 -8 -8.3 -10 -11.7 -12 -15 -20 -30 -40 -60
• Corollary 1 to Theorem 3: The following separations can be calculated using this method:
• The separation between any two encoding points can be calculated from the positions of those points on the Poincaré Sphere.
• The separation between any two decoding points can be calculated from the positions of those points on the Poincaré Sphere:

Front separation:

• Between the left front and right front decoded positions

Back separation:

• Between the left back and right back decoded positions

Front to back separation:

• Between the left front and left back decoded positions
• Between the right front and right back decoded positions
• The stereo playback separation of a quadraphonic recording can be calculated by finding the following separations:

Front separation:

• Between the left pole and the right front encoded position
• Between the right pole and the left front encoded position

Back separation:

• Between the left pole and the right back encoded position
• Between the right pole and the left back encoded position
• The quadraphonic playback of a stereo recording can be calculated by finding the following separations:

Front separation:

• Between the left pole and the right front decoded position
• Between the right pole and the left front decoded position

Back separation:

• Between the left pole and the right back decoded position
• Between the right pole and the left back decoded position

Mono to back separation:

• Between the mono (lateral) pole and the left back decoded position
• Between the mono (lateral) pole and the right back decoded position
• The separation between any encoding point and any decoding point can be calculated from their positions on the Poincaré Sphere.
• The mono playback of any quadraphonic recording can be calculated by finding the following separations:

Back attenuation:

• Between the mono (lateral) pole and the center back encoded position
• Between the mono (lateral) pole and any other encoded point

Any position attenuated more than 10 dB effectively disappears from mono playback.

Here are separations calculated using these methods.

• Calculation 1: Calculate the crosstalk between A and B from positions A and B on the Poincaré Sphere:

Procedure:

1. Divide the subtended angle between A and B by 2.
2. Find the cosine of the result.

C = cos(α/2)

• Calculation 2: Calculate the separation between A and B in dB from the crosstalk between A and B:

Procedure:

1. Take the common logarithm of the crosstalk.
2. Multiply the result by 20.

S = 20 log10(C)

• Calculation 3: Calculate the separation between A and B in dB from positions A and B on the Poincaré Sphere:

Procedure:

1. Divide the subtended angle between A and B by 2.
2. Find the cosine of the result.
3. Take the common logarithm of the new result.
4. Multiply the newest result by 20.

S = 20 log10(cos(α/2))

• Calculation 4: Calculate the crosstalk between A and B from the separation between A and B in dB:

Procedure:

1. Divide the separation in dB by 20.
2. Raise 10 to the power of the result.

C = 10^(S/20)

• Calculation 5: Calculate positions A and B on the Poincaré Sphere from the crosstalk between A and B:

Procedure:

1. Find the arc cosine of the crosstalk.
2. Multiply the result by 2.

α = 2 arccos(C)

• Calculation 6: Calculate positions A and B on the Poincaré Sphere from the separation between A and B in dB:

Procedure:

1. Divide the separation in dB by 20.
2. Raise 10 to the power of the result.
3. Find the arc cosine of the new result.
4. Multiply the newest result by 2.

α = 2 arccos(10^(S/20))

• Calculation 7: Calculate L and R signal outputs to encode a desired position C on the Poincaré sphere: 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:

• 0° is where L and R are in phase (olive).
• 90° is where L leads R by 90° (+j or black)
• 180° is where L is in opposite phase to R (violet)
• -90° is where L lags R by 90° (-j or brown)

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 sine-cosine variable pairs:

• a = cos(θ/2)       b = sin(θ/2)
• f = cos(ø) - j sin(ø)
• t = cos(ρ) + j sin(ρ)

The left-side left and right encoding outputs for point C, using signal x, is:

L = atx            R = bftx

The corresponding mirror-image right-side left and right encoding outputs for point C', using signal y, is:

L = bfty          R = aty

• Calculation 8: Calculate a mix of L and R signals to decode a desired position D on the Poincaré sphere:

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 sine-cosine variable pairs:

• a = cos(θ/2)       b = sin(θ/2)
• f = cos(ø) + j sin(ø)
• t = cos(ρ) - j sin(ρ)

The decoding output for point C, using signals L and R, is:

U = atL + bftR

The corresponding mirror-image decoding output for point C', using signals L and R, is:

V = bftL + btR

• Theorem 4: To decode original encoding points, decode equations must use complex conjugates of encode equations.

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. Variable s goes with X, and t goes with Y.

In these equations, u and v are sign values of +1 or -1. Variable u goes with L, and v goes with R.

In these equations, m and n are sign values of +1 or -1. Variable m goes with Lj, and v goes with Rj.

For the encoding of any pair of symmetric signals, the following equations are used:

• L = u(saX + smbXj + tcY + tmdYj)
• R = v(taY + tnbYj + scX + sndXj)

For the decoding of any pair of symmetric signals, the following equations are used:

• X = s(uaL - umbLj + vcR - vndRj)
• Y = t(vaR - vnbRj + ucL - umdLj)

Examples include the following matrix systems:

1. Sansui QS:
PAIR abcd stuvmn
lf and rf 0.920.000.380.00 +1+1+1+1+1+1
lb and rb 0.000.920.000.38 +1+1+1-1+1+1

Encoding equations:

• L = .92lf + .38rf + .92lbj + .38rbj
• R = .92rf + .38lf - .92rbj - .38lbj

Decoding equations:

• lf = .92L + .38R
• rf = .38L + .92R
• lb = -.92Lj + .38Rj
• rb = -.38Lj + .92Rj
2. CBS SQ:
PAIR abcd stuvmn
lf and rf 1.000.000.000.00 +1+1+1+1+1+1
lb and rb 0.000.710.710.00 -1+1+1+1+1+1

Encoding equations:

• L = lf - .71lbj + .71rb
• R = rf - .71lb + .71rbj

Decoding equations:

• lf = L
• rf = R
• lb = .71Lj - .71R
• rb = .71L - .71Rj
3. BBC Matrix H:
PAIR abcd stuvmn
lf and rf 0.850.350.350.15 +1+1+1+1+1-1
lb and rb 0.350.850.150.85 +1+1+1+1-1+1

Encoding equations:

• L = .85(lf - lbj) + .35(lfj + rf + lb - rbj) + .15(rfj + rb)
• R = .85(rf + rbj) + .35(lf - rfj + lbj + rb) - .15(lfj - lb)

Decoding equations:

• lf = .85L - .35(Lj - R) + .15Rj
• rf = .85R + .35(L + Rj) - .15Lj
• lb = .85Lj + .35(L - Rj) + .15R
• rb = -.85Rj + .35(Lj + R) + .15L

Note the following special values in the above matrices:

• 0.92 = cos(22.5°)
• 0.83 = cos(67.5°)
• 0.71 = cos(45.0°)
• 0.15 = cos(67.5°) × cos(67.5°)
• 0.35 = cos(67.5°) × cos(22.5°)
• 0.85 = cos(22.5°) × cos(22.5°)
• Theorem 5: It is impossible to pan a signal all the way around any great circle on the Poincaré Sphere without encountering a phase discontinuity (usually called a hole).

No matter which great-circle 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:

• Phase reversal without hole:
DIR FRFR RBB LBL LFF
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:

• Scheiber, Dynaquad, Stereo-4:

The red bar shows the encoding phase hole.

DIR FRFR RBB   BLBL LFF
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
• Uniquad Mixer-Bus RM Encoding:

The orange bars are where the bus selector is changed when panning.

DIR FRFR   RRBB LBL   LLFF
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
• Sansui QS:

The orange bars show the encoding phase holes, automatically adjusted by phasors in the encode equations.

DIR FRFR   RRBB LBL   LLFF
L 0.71 0.38 0.00  0.00j0.38j0.71j0.92j1.00j 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
• Denon BMX:

The orange bars show the encoding phase holes. It is unknown how they were implemented in practice.

DIR FRFR   RRBB LBL   LLFF
L 0.50 0.26 0.00  0.00 0.26 0.50  0.65 0.71  0.71 0.65 0.50
Lj 0.50j0.26j0.00j −0.00j−0.26j−0.50j −0.65j−0.71j 0.71j0.65j0.50j
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 0.71j0.65j0.50j0.26j0.00j −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.

• Theorem 6: Different matrix definitions use different coefficients to define the same matrix encodings. The coefficients differ only in total level.
• To go from one definition to another, multiply all coefficients by the same value.

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:

1. Find which coefficient in the original system has the value 1.00.
2. Use the coefficient for the same value from the resulting system as the multiplier.
• The following table shows the same matrix encoding in several different ways:
Percent
Blend
Major is 1.00 Result is 1.00Encode Angle
MinorResultMajorMinor Stylus θ
100%1.001.410.710.71 45.0°
80%0.801.280.780.62 38.7°
50%0.501.120.890.46 26.6°
41%0.411.080.920.38 22.5°
32%0.321.050.950.30 22.5°
30%0.301.040.960.29 16.7°
25%0.251.030.970.24 14.0°
20%0.201.020.980.20 11.3°

Each row is the same matrix encoding, shown using different coefficients.

### PUTTING IT ALL TOGETHER

Input Values  ΔXEF (XE=1)  Angle  ΔXDO (XO=1)  Poincaré Sphere (r = 1) Sphere Separation Stylus Motion
Matrix
used with
%
Blend
dB tan(θ)
FE
opp
sec(θ)
XF
hyp
xsc(θ)
OF
hyp
csc(θ)
XC
hyp
xsc(θ)
BC
hyp
∠EXF
ZXY
θ
∠ZOY
cos(θ)
XD
sin(θ)
OD
opp
∠XOZ
α
XZ
Chord
OD
⊥ XZ
Chord
OM
⊥ YZ
Chord
OU
⊥ VW
Chord
OD
dB
OM
dB
OU
dB
to opp
chan
∠PXE
chan
∠AXE
Calc:bl %vdB = blsec(θ)XF-1 csc(θ)XC-1 atn(FE) cos(θ)sin(θ) 180-2θ2sin(α/2) cos(α/2)cos(θ) sin(2θ) vdBvdBvdB 90-θ θ
SQ-b,HA-b100.00.0 1.0001.4140.4141.4140.414 45.090.0 0.7070.707 90.01.414 0.7070.7071.000 -3.01-3.010.00 45.045.0
EV4-bd80.0-1.9 0.8001.2810.2811.6010.601 38.777.3 0.7810.625 102.71.562 0.6250.7810.976 -4.09-2.15-0.21 51.338.7
Nar67.0-3.5 0.6701.2040.2041.7970.797 33.867.6 0.8310.557 112.41.662 0.5570.8310.925 -5.09-1.61-0.68 56.233.8
Nar260.0-4.4 0.6001.1660.1661.9440.944 31.061.9 0.8570.514 118.11.715 0.5140.8570.882 -5.77-1.34-1.09 59.031.0
DQ-bd57.7-4.8 0.5771.1550.1552.0011.001 30.060.0 0.8660.500 120.01.732 0.5000.8660.866 -6.03-1.25-1.25 60.030.0
EV4-be,DQ-be50.0-6.0 0.5001.1180.1182.2361.236 26.653.1 0.8940.447 126.91.789 0.4470.8940.800 -6.99-0.97-1.94 63.426.6
QS,UM,H,HA-f41.4-7.7 0.4141.0820.0822.6141.614 22.545.0 0.9240.383 135.01.848 0.3830.9240.707 -8.35-0.69-3.01 67.522.5
SQB-bd,EVU-bd40.0-8.0 0.4001.0770.0772.6931.693 21.843.6 0.9280.371 136.41.857 0.3710.9280.690 -8.60-0.64-3.23 68.221.8
UQE31.8-10.0 0.3181.0490.0493.3002.300 17.635.3 0.9530.303 144.71.906 0.3030.9530.578 -10.4-0.42-4.77 72.417.6
EV4-fe30.0-10.5 0.3001.0440.0443.4802.480 16.733.4 0.9580.287 146.61.916 0.2870.9580.550 -10.8-0.37-5.19 73.316.7
DQ-fe25.0-12.0 0.2501.0310.0314.1233.123 14.028.1 0.9700.243 151.91.940 0.2430.9700.471 -12.3-0.26-6.55 76.014.0
EV4-fd,EVU-fd20.0-14.0 0.2001.0200.0205.0994.099 11.322.6 0.9810.196 157.41.961 0.1960.9810.385 -14.2-0.17-8.30 78.711.3
SQB-fd10.0-20.0 0.1001.0050.00510.0509.050 5.711.4 0.9950.100 168.61.990 0.1000.9950.198 -20.0-0.04-14.0 84.35.7
Wide5.0-26.0 0.0501.0010.00120.02519.025 2.95.7 0.9990.050 174.31.998 0.0500.9990.100 -26.0-0.01-20.0 87.12.9
DQ-fd0.0-80.0 0.0001.0000.00099.00098.001 0.00.0 1.0000.000 180.02.000 0.0001.0000.000 -80.00.00-80.0 90.00.0
MatrixBlenddB FEXF OF XCBC θ XDOD αXZ ODOMOU OD dBOM dB OU dB ∠PXE∠AXE  Here are all of the items above put into one table.

The variables in the table are:

MATRIX COEFFICIENTS

USING ΔXEF

• FE = the blending between the channels used to encode or decode
• With Excel, I had to enter the % Blend value 0.0 as 0.01 to prevent #NUM! errors.
• FE = tan(θ)
• XE = the fixed unity value (1) the blend is added to
• XF = sec(theta;) = the resultant sum of 1 and the blend
• OF = xsc(theta;) = sec(θ) − 1 = the exsecant of θ
• θ   = the angle ∠EXF from the blend - the arctangent of FE/1
• θ   = ∠EXF = ∠AXE = ∠DXO

USING ΔCEX

• CE = cot(θ)
• XC = csc(θ)
• BC = xcs(θ) = csc(θ) − 1 = the excosecant of θ

USING ΔXAE and ΔXDO

• XD = XA = cos(θ) = calculate major coefficient and stylus vector
• OD = EA = sin(θ) = calculate minor coefficient and stylus vector
• OX = EX =the fixed unity value (1) for stylus motion resultants

TRIGONOMETRIC AND GEOMETRIC IDENTITIES

• XE = OX = 1
• FE = tan(θ) = sin(θ) / cos(θ)
• XF = sec(θ) = 1 / cos(θ)
• CE = cot(θ) = cos(θ) / sin(θ)
• XC = csc(θ) = 1 / sin(θ)
• XE = XB = OX = OZ = OY = OW = OV = 1
• θ   = ∠AXE = ∠DXO = ∠DZO = ∠MOZ = ∠MOY = ∠PEX
• θ   = ∠EXF = ∠ZXY = ∠PCE = ∠ECX = ∠XCF = ∠AEF
• ΔAXE ≅ ΔDXO ≅ Δ;DZO ≅ Δ;MOZ ≅ Δ;MOY ≅ Δ;PEX
• ΔEXF ∼ ΔAXE ∼ ΔZXY ∼ ΔPCE ∼ ΔECX ∼ ΔXCF ∼ ΔAEF

STYLUS VECTORS (Lower diagram for proper orientation to a record groove)

• XA ≅ XD = actual major stylus vector (both diagrams)
• EA ≅ OD = actual minor stylus vector (both diagrams)
• EX ≅ OX = actual stylus resultant (=1 both diagrams))
• ΔXDO ≅ ΔXAE ≅ ΔOMY ≅ ΔOMZ
• ΔXAE (lower diagram) shows an encoding of a Regular Matrix right front signal.

POINCARÉ SPHERE VALUES

SPHERE ANGLES

• ∠ZOY = 2θ = 2(∠ZXY) ... Central angle theorem
• ∠XYZ ≅ ∠XFE ≅ ∠XOD ≅ ∠XEA
• α = ∠XOZ = 180°-2θ = Sphere angle opposite
• α/2 = 90°-θ
• θ = 90°-α/2
• FE = tan(θ) = blend

SPHERE ANGLE OPPOSITE

• α = Sphere angle for encoded signal to opposite stereo channel
• XZ = Sphere chord = tool for calculating opposite Sphere separation.
• OD = Perpendicular bisector of XZ ... gives separation value of opposite sphere angle.
• OD = EA = sin(θ) = Separation to opposite stereo channel.
• OD in dB is found by multiplying the common logarithm of OD by 20.

• 2θ = ∠YOZ = 180°-α = Sphere angle adjacent
• ∠YOZ = 2θ = Sphere angle for encoded signal to adjacent stereo channel
• YZ = Sphere chord = tool for calculating adjacent Sphere separation.
• OM = Perpendicular bisector of YZ and gives separation value of adjacent sphere angle.
• OM = AX = cos(θ) = Separation to adjacent stereo channel.
• OM in dB is found by multiplying the common logarithm of OM by 20.

SPHERE ANGLE CODED TO CODED

• ∠VOW = 180°-4θ = Sphere angle between left and right identically coded outputs
• ∠VOW = Sphere angle for coded signal to coded signal
• VW = Sphere chord = tool for calculating coded Sphere separation.
• OU = Perpendicular bisector of VW and gives separation value of sphere angle.
• OU = sin(2θ) = cos(90°-2θ) = Separation between paired encoded or decoded channels.
• OU in dB is found by multiplying the common logarithm of OU by 20.

### CALCULATIONS FROM THE ABOVE FACTS

• Calculation 9: Derive resistors for an output of L and R signals to matrix encode a desired position (diagram 1):
• Two resistors, G and F, form a voltage divider for the major side of encoding.
Two more resistors, H and F, form another voltage divider for the minor side.
The page author used a 1K ohm at 1/8 watt for resistor F.
• G = 1K (xsc(θ)) = 1K (sec(θ) − 1) = 1K ((1 ⁄ cos(θ)) − 1)  H = 1K (xcs(θ)) = 1K (csc(θ) − 1) = 1K ((1 ⁄ sin(θ)) − 1)
• The phase of either signal can be changed before this mixer.  The decode angle is θ from the table above.
• Calculation 10: Derive resistors for an output of L and R signals to matrix decode a desired position (diagram 2):
• Two resistors, A and B, form a mixer to matrix decode a channel.  The page author used a 1K ohm resistor at 1/8 watt for resistor A.
• B = 1K (cot(θ)) = 1K (1 ⁄ tan(θ))
• The phase of either signal can be changed before this mixer.  The decode angle is θ from the table above.
• Calculation 11: Derive resistors for a mix of L and R signals to matrix decode a desired channel (diagram 2):
• Two resistors, A and B, form a mixer to matrix decode a channel.  The page author used a 10K ohm sum at 1/8 watt.
• A = 10K (cos(θ) ⁄ (cos(θ) + sin(θ))  B = 10K (sin(θ) ⁄ (cos(θ) + sin(θ))  are the decode resistors.
• The phase of either signal can be changed before this mixer.  The decode angle is θ from the table above.
• Calculation 12: Derive a potentiometer setting for a mix of L and R signals to matrix decode a desired channel (diagram 3):
• A potentiometer P is the matrix parameter control and mixer.  The page author used 10K ohms at 2 watt.
• T = P (cos(θ) ⁄ (cos(θ) + sin(θ))  is the decode setting of the tap.
• The phase of either signal can be changed before this mixer.  The decode angle is θ from the table above.
• Calculation 13: Derive resistors for a blend of L and R signals to encode or decode a desired matrix blend coefficient (diagram 4):
• Two fixed input resistors ɪ achieve the blend. The page author used 1K ohms at 1/8 watt.
• A potentiometer W is the Width control. It should be at least 15 × ɪ.  The page author used 20K ohms at 2 watt.
• W = (ɪ ⁄ width_factor) − ɪ  is the width setting.  W = (1000 ⁄ width_factor) − 1000  is the page author's setting.
• The phase of either signal can be changed before this mixing.  The width_factor is tan(θ) from the table above.
• Calculation 14: Derive resistor for a blend of L and R signals to decode a desired speaker matrix width coefficient (diagram 4):
• Two fixed input resistors ɪ achieve the blend. The page author used 3 ohms at amplifier power.
• A rheostat W is the Width control. It should be at least 15 × ɪ.  The page author used 50 ohms at amplifier power.
• W = (ɪ ⁄ width_factor) − ɪ  is the width setting.  W = (3 ⁄ width_factor) − 3  is the page author's setting.
• The width_factor is tan(θ) from the table above.
• Calculation 15: Derive resistor for an antiblend of L and R signals to decode a desired speaker matrix depth coefficient (diagram 5):
• The speakers themselves are the input resistances S. The page author used his 8-ohm speakers, so S is 8 ohms.
• A rheostat is the Depth control D. Its resistance should be at least 6 × S. The page author used 50 ohms at amplifier power.
• D = (S ⁄ (1 − depth_factor)) − S  is the depth setting.  D = (8 ⁄ (1 − depth_factor)) − 8  is the page author's setting.
• The depth_factor is tan(θ) from the table above.

### Features of matrix quadraphonic systems:

• Theorem 7: A surround field uses level and time differences between channels to locate the sound, using sound location properties of the human hearing system.

Hans Wallach, Edwin Newman, and Mark Rosenzweig defined the precedence effect in 1949. Helmut Haas later investigated the subject more 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
- 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 create a surround field. It encodes a quadraphonic matrix signal with the time delays already included to give additional clues to correctly locate the sound. No separation enhancement is needed in playback.

Surround field page

Why surround fields work

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 Levell Delayl Levelr Delayr Levels Delays 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:

• L = 0.98 l + 0.20 r - 0.71 s
• R = 0.98 r + 0.20 l + 0.71 s

Decoding works with Dolby Surround or with any regular matrix (RM or QM) decoder.

Notice that this idea works only with a great-circle matrix. It won't work with phase matrix systems.

• Observation 1: The separations the listeners notice the most

The separations in a quadraphonic program that the listener notices the most are:

1. Left front to right front
2. Center front to center back

The other separations are not as noticeable. The separation between left back and right back is usually the least noticed.

### SIDE LOCATION AND SEPARATION ENHANCEMENT:

• Observation 2: The Side-Location problem - The fact that two speakers placed to the left-front and left-back of the listener cannot produce a sound image between those speakers when sounds are panned between them.

Normal level-differential panning techniques cannot produce an image between these speakers unless the listener turns his head so that one each ear is facing each speaker. But there are several ways to make the side image appear, and some of them use normal panning techniques when the recording is mixed:

1. Speakers directly to the left side and the right side of the listener

Any surround sound system with speakers located directly to the left and directly to the right of the listener does not have the side-location problem.

- The Dynaco Diamond has these speakers and unintentionally fixed this problem.

- The Denon QX Dual Triphonic system also unintentionally fixed this problem.

2. Diagonal Delays

This method uses the precedence effect to make the separations seem larger. It introduced a phase-reversed and delayed version of each decoded channel into the diagonally opposite channel.

- It did increase the apparent separation.

- It fixed the side location problem.

- It also caused an annoying tension feeling from phase reversed sound.

3. Binaural recording.

This uses microphones spaced the same distance apart that human ears are spaced. The microphones are aimed to the sides.

The listener listens to a recording from those microphones through earphones. The effect is very real, but there is no sound from the front or the back even if such sounds were there.

The effect is also audible with speakers. Quadraphonic speakers using an RM or QM matrix will place the images between the side speakers with the listener's head facing the front. This places images between the speakers, but does not produce a quadraphonic recording.

4. Delayed back channels

Dolby Surround plays a normally matrixed Dolby Surround, RM, or QM recording, decoding the dialog, left, right, and surround channels. The surround channels are delayed by the decoder to prevent surround leakage into the front from being noticed.

But this delay also uses the precedence effect to cause the side images to be located by the listener correctly as it was panned.

- It requires the listener to continue facing forward to hear all of the location effects correctly.

5. Enhancement in the Recording

This method uses the precedence effect to make the separations seem larger. The difference is that it is in the recording. Nothing is added to the decoding equipment.

This is a surround (or Spheround) version of binaural recording, with delays in both left-right and front-back directions (and up-down directions too for Spheround). So the position of the listener's head is unimportant.

- A special microphone technique creates a sound field. The human hearing system does the rest.

- This is like binaural recording, but works in a plane or even a solid space.

- It increases the apparent separation.

- No other separation enhancement is needed.

- It fixes the side-location problem.

- It gives a natural "you are there" effect that is a characteristic of sound fields.

6. Many Channels

The effect here is having so many channels that at least one or two pairs of channels are at opposite sides of the listener's head. This removes the side-location problem.

The Octophonic system works well.

• Definition 5: Separation Enhancement - A method to make the separation of a matrix system seem larger than it really is.

There are several different methods:

1. Gain Riding

In this method, the gain of each channel is quickly varied to make the dominant channels louder and the other channels quieter. This seemed to make the separation larger, but it had two disadvantages:

- It reduced the level of any ambience in the recording.

- It had the effect of "pumping" (where changes in signal levels or ambience caused by gain changes are noticed by the listener).

2. Diagonal Delays

This method uses the precedence effect to make the separations seem larger. It introduced a phase-reversed and delayed version of each decoded channel into the diagonally opposite channel.

- It did increase the apparent separation.

- It fixed the side location problem.

- It also caused an annoying tension feeling from phase reversed sound.

3. Automatic Matrix Varying

This method is used in the Sansui Variomatrix, the CBS Variblend, Dolby Pro-Logic, the Electro-Voice separation enhancement, and UniQuad Autovary.

- This looks for where the strongest signal is and momentarily adjusts the matrix to increase separation for that signal.

- The reduced separations for the other channels are not normally noticed.

- The upper diagram shows the normal separation pattern for Dolby Surround.

- The lower diagram shows the instantaneous changes Dolby Surround makes with a dominant front-center soloist.

- Decoding moves away from dominant sources. Front separation widens while back separation narrows with front-center solos.

DECODE EQUATIONS DECODE RESULTS Image
Without Pro-Logic
Left_Speaker = 1.00 Left_Matrix   Left_Speaker = 1.00 Left + .707 Dialog + .707 j Surround Right_Speaker = 1.00 Right_Matrix Right_Speaker = 1.00 Right + .707 Dialog - .707 j Surround
Dialog_Speaker = .707 Left_Matrix + .707 Right_Matrix Dialog_Speaker = 1.00 Dialog + .707 Left + .707 Right
Surround_Speaker = .707 Right_Matrix - .707 Left_Matrix Surround_Speaker = 1.00 Surround + .707 Left - .707 Right
With Pro-Logic (front solo)
Left_Speaker = .924 Left_Matrix - .383 Right_Matrix   Left_Speaker = .924 Left + .383 Dialog + .924 j Surround Right_Speaker = .924 Right_Matrix - .383 Left_Matrix Right_Speaker = .924 Right + .383 Dialog - .924 j Surround
Dialog_Speaker = .707 Left_Matrix + .707 Right_Matrix Dialog_Speaker = 1.00 Dialog + .707 Left + .707 Right
Surround_Speaker = .707 Right_Matrix - .707 Left_Matrix Surround_Speaker = 1.00 Surround - .707 Left - .707 Right

Pumping is inaudible using this method because the signal levels are unchanged. Only the instantaneously perceived directions of the low level signals are changed. These are not normally noticed because human hearing normally notices the direction of the dominant signal over the other signals.

4. Delayed back channels

Dolby Surround plays a normally matrixed Dolby Surround, RM, or QM recording, decoding the dialog, left, right, and surround channels. The surround channels are delayed by the decoder.

This delay prevents surround leakage into the front from being noticed.

This delay prevents front leakage into the surround channels from being noticed.

This delay also uses the precedence effect to cause the side images to be located correctly as panned by the listener.

- It requires the listener to continue facing forward to hear all of the location effects correctly.

5. Enhancement in the Recording

This method uses the precedence effect to make the separations seem larger. The difference is that it is in the recording. Nothing is added to the decoding equipment.

See "Theorem 7: Surround Fields" above and the Surround Fields page.

- A special microphone technique creates a sound field. The human hearing system does the rest.

- It increases the apparent separation.

- No other separation enhancement is needed.

- It fixes the side-location problem.

- It gives a natural "you are there" effect that is a characteristic of sound fields.

6. Many Channels

Each sound comes from a wall of sound, not a single point. This does the following:
- It makes it useful from more locations in the room.
- It creates a linear wave front instead of multiple spherical wave fronts.
- It keeps the ears from finding a single speaker in the room.
Thus, the ears locate the correct direction of the sound as panned.

How Octophonics enhance separation

### SOUND IMAGE DIAGRAMS:

• Definition 6: Sound Image Diagram - A diagram showing the size and shape of the total sound image of a quadraphonic system.

This procedure was developed by the page author through empirical experimentation. But it seems to work very well:

The page author wanted to create a diagram of room size as shown in Leonard Feldman's book "Four Channel Sound" on page 62.

1. He started with separation figures in dB for the various systems. Value A Separation between front channels Value B Separation between back channels Value C Separation between front and back channels on each side Value D Separation between each front channel and the opposite stereo channel Value E Separation between each back channel and the opposite stereo channel Value F Separation between each back channel and a mono signal Value G Separation between the center back signal and a mono signal Value H Separation between each front channel and the center back channel Value K Separation between each front channel and the opposite back channel Value S Normal stereo separation of the record (assumed 40 dB) Value T Separation between a stereo channel and mono (3 dB) Value U Separation between a stereo channel and the center back signal

For values of 0 dB, he had to use 0.05 dB to keep Excel from giving an error.

2. Next he created a linear version of a separation indication.

He took the common logarithm of each of the dB values above.

This gave a scale of values from -1.3 to 1.6 for values from .05 dB to 40 dB.

He then centered the values by subtracting 0.5 from each value for the final effect.

This gave a scale of values from -1.8 to 1.1 for values from .05 dB to 40 dB.

This would expand the space for large separations while contracting it for small ones. It also gave the wanted appearance.

MatrixABCDE FGHKSTU
QS3338.38.3 8.3408.3404033
-0.02-0.02-0.020.420.42 0.421.100.421.101.10 -0.02-0.02
EV48.30.24.9144.1 19405.15.34033
0.42-1.20.190.650.11 0.781.100.210.221.10 -0.02-0.02
3. He then created a framework to put these in:

This frame is a 4-unit by 4-unit square centered on the origin of the graph.

This plot area is an 8-unit by 8-unit box centered on the origin of the graph.

4. He then empirically found equations to represent the effects the above separations have on the image:
LOCATIONDIRECTIONSAME SIDELET OPP INPUTLETDIAGONALLETMIRROR OFFSETFORMULA
COEFFICIENTS:  1.0  0.1 0.01  +2
Right FrontHorizontalLeft FrontA Left InputDLeft BackKLeft Front +2'W = 1.0 A + 0.1 D + 0.01 K + 2
Left FrontHorizontalRight FrontA Right InputDRight BackKRight Front +2'-W = -1.0 A - 0.1 D - 0.01 K - 2
Right FrontVerticalRight BackC Center BackHLeft BackKLeft Front +2'X = 1.0 C + 0.1 H + 0.01 K + 2
Left FrontVerticalLeft BackC Center BackHRight BackKRight Front +2'X = 1.0 C + 0.1 H + 0.01 K + 2
Right BackHorizontalLeft BackB Left InputELeft FrontKLeft Back +2'Y = 1.0 B + 0.1 E + 0.01 K + 2
Left BackHorizontalRight BackB Right InputERight FrontKRight Back +2'-Y = -1.0 B - 0.1 E - 0.01 K - 2
Right BackVerticalRight FrontC MonoFLeft FrontKLeft Back +2'Z = -1.0 C - 0.1 F - 0.01 K - 2
Left BackVerticalLeft FrontC MonoFRight FrontKRight Back +2'Z = -1.0 C - 0.1 F - 0.01 K - 2

A second set was needed for diamond speaker layouts:

LOCATIONDIRECTIONSAME SIDELET OPP INPUTLETDIAGONALLETMIRROR OFFSETFORMULA
COEFFICIENTS:  1.0  0.1 0.1  +2
FrontHorizontal0         'W = 0
BackHorizontal0         'W = 0
FrontVerticalBackG Left InputTRight InputTBack +2'X = 1.0 G + 0.1 T + 0.1 T + 2
BackVerticalFrontG Left InputURight InputUFront +2'-X = -1.0 G - 0.1 U - 0.1 U - 2
RightHorizontalLeftS MonoTBackULeft +2'Y = 1.0 S + 0.1 T + 0.1 U + 2
LeftHorizontalRightS MonoTBackURight +2'-Y = -1.0 S - 0.1 T - 0.1 U - 2
RightVertical0         'Z = 0
LeftVertical0         'Z = 0
5. The final step was to get Excel to make the diagrams:

Provide a table row for the x coordinates and a row for the y coordinates

The points are in the order lb, lf, rf, rb, lb (for diamond pattern: l, f, r, b, l).

The lb point is repeated to complete the quadrilateral (repeat l for the diamond pattern).

Use the plot option: Scatter with straight lines.

Select the orange values to make the QS plot.

EFFECT VALUES   VALUES lblfrfrblb IMAGE
MatrixAB CDE FGH K STU WYx -Y-W +W+Y -Y
XZy ZX XZ Z
QS3338.38.3 8.3408.3404033 2.032.03x
-2.03
-2.03
2.03
2.03
-2.03 effect-0.02-0.02-0.02 0.420.42 0.421.100.421.10 1.10-0.02-0.02 2.03-2.03y
-2.03
2.03
2.03
-2.03
-2.03
EV48.30.24.9144.1 19405.15.34033 2.490.81x -0.81-2.492.490.81-0.81 effect0.42-1.20.19 0.650.11 0.781.100.210.22 1.10-0.02-0.02 2.21-2.27y -2.272.212.21-2.27-2.27
EFFECT VALUES VALUES lfrbl IMAGE
MatrixAB CDE FGH K STU WYx -Y-W +Y+W -Y
XZy -Z+X +Z-X -Z
DQ3338.38.3 8.3408.3404033 03.10x -3.1003.100-3.10 effect-0.02-0.02-0.02 0.420.42 0.421.100.421.10 1.10-0.02-0.02 3.100y 03.100-3.100
6. See diagrams for each matrix at Quadraphonic Systems