# Matrix decoder

Matrix decoding is an audio technology where a finite number of discrete audio channels (e.g., 2) are decoded into a larger number of channels on play back (e.g., 5). The channels are generally, but not always, arranged for transmission or recording by an encoder, and decoded for playback by a decoder.

The function is to allow multichannel audio, such as quadraphonic sound or surround sound to be encoded in a stereo signal, and thus played back as stereo on stereo equipment, and as surround on surround equipment – this is "compatible" multichannel audio.

## Process

Matrix encoding does not allow one to encode several channels in fewer channels without losing information: one cannot fit 5 channels into 2 (or even 3 into 2) without losing information, as this loses dimensions: the decoded signals are not independent. The idea is rather to encode something that will both be an acceptable approximation of the surround sound when decoded, and acceptable (or even superior) stereo.

## Notation

The notation for matrix encoding consists of the number of original discrete audio channels separated by a colon from the number of encoded and decoded channels. For example, four channels encoded into two discrete channels and decoded back to four-channels would be notated:

4:2:4


A simpler situation would be to derive extra channels from the existing ones, but with no special encoding at the origin. For example, five discrete channels decoded to six channels would be notated:

5:5:6


Many matrix decoders take advantage of the Haas effect[citation needed], as well as audio cues inherent in the source channels.

The various encoding matrixes are described below.

## Dynaquad matrix (2:2:4) / (4:2:4)

Mostly used as a simple method for deriving back channels out of normal stereo recording (2:2:4), this matrix was also used for a specific encoding of 4 sound channels in some albums (4:2:4).[1]

### Encoding matrix

Matrix Left Front Right Front Left Back Right Back
Left Total 1.0 0.25 1.0 -0.5
Right Total 0.25 1.0 -0.5 1.0

### Decoding matrix

Matrix[2] Left Front Right Front Left Back Right Back
Left Total 1.0 0.0 0.64 -0.36
Right Total 0.0 1.0 -0.36 0.64

## Electro-Voice Stereo-4 matrix (2:2:4) / (4:2:4)

The first matrix system on the market, it was invented by Leonard Feldman and Jon Fixler, and sold by Electro-Voice. This matrix was used for specific encoding of 4 sound channels in many albums (4:2:4).[3]

### Encoding matrix

Matrix Left Front Right Front Left Back Right Back
Left Total 1.0 0.3 1.0 -0.5
Right Total 0.3 1.0 -0.5 1.0

### Decoding matrix

Matrix[2] Left Front Right Front Left Back Right Back
Left Total 1.0 0.2 1.0 -0.8
Right Total 0.2 1.0 -0.8 1.0

## SQ matrix, "Stereo Quadraphonic", CBS SQ (4:2:4)

Matrix Left Front Right Front Left Back Right Back
Left Total 1.0 0.0 k0.7 0.7
Right Total 0.0 1.0 -0.7 j0.7

$j = +90^\circ$ phase-shift, $k = -90^\circ$ phase-shift

The basic SQ matrix had mono/stereo anomalies as well as encoding/decoding problems, heavily criticized by Michael Gerzon and others.[4]

An attempt to improve the system lead to the use of other encoders or sound capture techniques, yet the decoding matrix remained unchanged.

### Position Encoder

An N/2 encoder that encoded every position in a 360° circle - it had 16 inputs and each could be dialed to the exact direction desired, generating an optimized encode.

### Forward-Oriented encoder

Matrix Left Front Right Front Left Back Right Back
Left Total 1.0 0.0 0.7 k0.7
Right Total 0.0 1.0 j0.7 0.7

$j = +90^\circ$ phase-shift, $k = -90^\circ$ phase-shift

The Forward-Oriented encoder caused Center Back to be encoded as Center Front and was recommended for live broadcast use for maximum mono compatibility - it also encoded Center Left/Center Right and both diagonal splits in the optimal manner. Could be used to modify existing 2-channel stereo recordings and create 'synthesized SQ' that when played through a Full-Logic or Tate DES SQ decoder, exhibited a 180° or 270° synthesized quad effect. Many stereo FM radio stations broadcasting SQ in the 70's used their Forward-Oriented SQ encoder for this. For SQ decoders, CBS designed a circuit that produced the 270° enhancement using the 90° phase shifters in the decoder. Sansui's QS Encoders and QS Vario-Matrix Decoders had a similar capability.

### Backwards-Oriented encoder

Matrix Left Front Right Front Left Back Right Back
Left Total -1.0 0.0 k0.7 0.7
Right Total 0.0 1.0 -0.7 j0.7

$j = +90^\circ$ phase-shift, $k = -90^\circ$ phase-shift

The Backwards-Oriented Encoder was the reverse of the Forward-Oriented Encoder - it allowed sounds to be placed optimally in the back half of the room, but mono-compatibility was sacrificed. When used with standard stereo recordings it created "extra wide" stereo with sounds outside the speakers.

Some encoding mixers had channel strips switchable between forward-oriented and backwards-oriented encoding.

### London Box

It encoded the Center Back in such a way that it didn't cancel in mono playback, thus its output was usually mixed with that of a Position Encoder or a Forward Oriented encoder. After 1972, the vast majority of SQ Encoded albums were mixed with either the Position Encoder or the Forward-Oriented encoder.

### Ghent microphone

In addition, CBS created the SQ Ghent Microphone, which was a spatial microphone system using the Neumann QM-69 mic. The signals from the QM-69 were differenced, and then phase-matrixed into 2-channel SQ.[5] With the Ghent Microphone, SQ was transformed from a Matrix into a Kernel and an additional signal could be derived to provide N:3:4 performance.

### Universal SQ

In 1976, Ben Bauer integrated matrix and discrete systems into USQ, or Universal SQ. It was a hierarchical 4-4-4 discrete matrix that used the SQ matrix as the baseband for discrete quadraphonic FM broadcasts using additional difference signals called "T" and "Q". For a USQ FM broadcast, the additional "T" modulation was placed at 38 kHz in quadrature to the standard stereo difference signal and the "Q" modulation was placed on a carrier at 76 kHz. For standard 2-channel SQ Matrix broadcasts, CBS recommended that an optional pilot-tone be placed at 19 kHz in quadrature to the regular pilot-tone to indicate SQ encoded signals and activate the listeners Logic decoder. CBS argued that the SQ system should be selected as the standard for quadraphonic FM because, in FCC listening tests of the various four channel broadcast proposals, the 4:2:4 SQ system, decoded with a CBS Paramatrix decoder, outperformed 4:3:4 (without logic) as well as all other 4:2:4 (with logic) systems tested, approaching the performance of a discrete master tape within a very slight margin.[6] At the same time, the SQ "fold" to stereo and mono was preferred to the stereo and mono "fold" of 4:4:4, 4:3:4 and all other 4:2:4 encoding systems.

### Tate DES decoder

The Directional Enhancement System, also known as the Tate DES, was an advanced decoder that enhanced the directionality of the basic SQ matrix.

It first matrixed the four outputs of the SQ decoder to derive additional signals, then compared their envelopes to detect the predominant direction and degree of dominance. A processor section, implemented outside of the Tate IC chips, applied variable attack/decay timing to the control signals and determined the coefficients of the "B" (Blend) matrices needed to enhance the directionality. These were acted upon by true analog multipliers in the Matrix Multiplier IC's, to multiply the incoming matrix by the "B" matrices and produce outputs in which the directionality of all predominant sounds were enhanced.

Since the DES could recognize all three directions of the Energy Sphere simultaneously, and enhance the separation, it had a very open and 'discrete' sounding soundfield.

In addition, the enhancement was done with sufficient additional complexity that all non-dominant sounds were kept at their proper levels.

Dolby used the Tate DES IC's in their theater processors until around 1986, when they developed the Pro Logic system. Unfortunately, delays and problems kept the Tate DES IC's from the market until the late 70's and only two consumer decoders were ever made that employed them, the Audionics Space & Image Composer and the Fosgate Tate II 101A. The Fosgate used a faster, updated version of the IC, called the Tate II, and additional circuitry that provided for separation enhancement around the full 360 soundfield. Unlike the earlier Full Wave-matching Logic decoders for SQ, that varied the output levels to enhance directionality, the Tate DES cancelled SQ signal crosstalk as a function of the predominant directionality, keeping non-dominant sounds and reverberation in its proper spatial locations at their correct level.

## QS matrix, "Regular Matrix", "Quadraphonic Sound" (4:2:4)

Matrix Left Front Right Front Left Back Right Back
Left Total 0.92 0.38 j0.92 j0.38
Right Total 0.38 0.92 k0.38 k0.92

$j = +90^\circ$ phase-shift, $k = -90^\circ$ phase-shift

## Matrix H (4:2:4)

Matrix H Matrix[7] Left Front Right Front Left Back Right Back
Left Total -j0.94 -l0.34 +k0.94 +m0.34
Right Total +l0.34 +j0.94 -m0.34 -k0.94

j = 20° phase-shift k = 25° phase-shift l = 55° phase-shift m = 115° phase-shift

## Ambisonic UHJ kernel (3:2:4 or more)

Matrix W (pressure signal) X (front-back signal) Y (left-right signal)
Left Total 0.470 + k0.171 0.093 + j0.255 +0.328
Right Total 0.470 + j0.171 0.093 + k0.255 -0.328

$j = +90^\circ$ phase-shift, $k = -90^\circ$ phase-shift

For more details on this topic, see Ambisonic UHJ format.

## Dolby Stereo and Dolby Surround (matrix 4:2:4)

Dolby Stereo and Dolby Surround are also known as Dolby MP, Dolby SVA and Pro Logic.

Matrix Left Right Center Surround
Left Total $1$ $0$ $\sqrt{\frac{1}{2}}$ $j\sqrt{\frac{1}{2}}$
Right Total $0$ $1$ $\sqrt{\frac{1}{2}}$ $k\sqrt{\frac{1}{2}}$

$j = +90^\circ$ phase-shift, $k = -90^\circ$ phase-shift

Dolby SVA matrix is the original name of the Dolby Stereo 4:2:4 encoding matrix.

The term "Dolby Surround" refers to both the encoding and decoding in the home environment, while in the theater it is known "Dolby Stereo", "Dolby Motion Picture matrix" or "Dolby MP".

The original "Dolby Surround" home implementation omitted the center channel, and only up-mixed stereo to Surround 3.0 (Left, Right, Surround). Yet the matrix used is the same.

"Pro Logic" refers to the decoder used, there is no special Pro Logic encoding matrix.

## Dolby Pro Logic II matrix (5:2:5)

Matrix Left Right Center Rear Left Rear Right
Left Total $1$ $0$ $\sqrt{\frac{1}{2}}$ $j\sqrt{\frac{2}{3}}$ $j\sqrt{\frac{1}{3}}$
Right Total $0$ $1$ $\sqrt{\frac{1}{2}}$ $k\sqrt{\frac{1}{3}}$ $k\sqrt{\frac{2}{3}}$

$j = +90^\circ$ phase-shift, $k = -90^\circ$ phase-shift

The Pro Logic II matrix provides for stereo full frequency back channels. Normally a sub-woofer channel is driven by simply filtering and redirecting the existing bass frequencies of the original stereo track.

## References

1. ^ Feldman, Leonard (1973). Four Channel Sound (1 ed.). Indianapolis IN: Howard W. Sams & C. Inc. pp. 49–51. ISBN 0-672-20966-7.
2. ^ a b http://www.4channelsound.com/encode.htm
3. ^ Feldman, Leonard (1973). Four Channel Sound (1 ed.). Indianapolis IN: Howard W. Sams & C. Inc. pp. 44–49. ISBN 0-672-20966-7.
4. ^ Gerzon, Michael (8 December 1977). "Don't say quad—say psychoacoustics". New Scientist: 634–636. The forward oriented encoder is one of at least six different encoding options offered by SQ so that producers can decide for themselves which collection of faults they prefer.
5. ^ Bauer, Benjamin B.; Louis A. Abbagnaro; Daniel W. Gravereaux; Trevor J. Marshall (January–February 1978). "The Ghent Microphone System for SQ Quadraphonic Recording and Broadcasting". Journal of the Audio Engineeriong Society (AES) 26 (1/2): pp. 2–11.
6. ^ “A subjective evaluation of FM Quadraphonic reproduction systems – Listening tests” Federal Communications Commission, Office of the Chief Engineer, Laboratory Division, Laurel, Maryland. Project Number 2710-1, August 1977