Some "experts" complained that the encoding and decoding equations in my website are not the equations specified in the original definitions in the matrix systems. This is because the equations on my page are the normalized versions of the equations.

And as we shall see, the normalized equations are the actual ones that end up being used when encoding and decoding are done.

- What is normalization?
Normalization is the process of making the entire system unity gain. It makes the outputs of the entire encode-decode process have the same audio level that the inputs have. The vector sum for any unity gain encoded channel or decoded channel is 1.

- What happens when the equations are normalized?
When the encode and decode equations are used, no changes in signal level occur during either the encode or the decode process. The encoder and decoder are inserted into the audio chain with no changes in level.

- What happens when equipment uses equations that are not normalized?
When the equipment equations are not normalized, then the output of the device requires a change of level (or volume) to return the signal to the normal level used. This causes changers in fader levels on mixers or inputs to record cutters, recorders, or transmitters. It also can change the total noise or distortion figures in the entire audio chain.

If the unnormalized encoder output is not compensated for, it could overcut a phonograph record.

- What is an example of a normalized matrix system?
The Scheiber system as defined is normalized. The encoding equations are:

- L = .924lf + .383rf + .924lb - .383rb
- R = .924rf + .383lf + .924rb - .383lb

The decoding equations are:

- lf = .924L + .383R
- rf = .924R + .383L
- lb = .924L - .383R
- rb = .924R - .383L

When a left front signal is encoded, the encoder outputs are:

- L = .924lf
- R = .383lf

The vector sum of L and R for lf is

√(.924^{2}+ .383^{2}) = √(.854 + .146) = √(1.000) = 1.000The encoded lf signal is not higher than normal.

When signals are decoded, the decoder outputs are as follows:

- lf = .924L + .383R = .924(.924) + .383(.383) = .854 + .146 = 1.00
- rf = .924R + .383L = .924(.383) + .383(.924) = .354 +. 354 =.708
- lb = .924L - .383R = .924(.924) - .383(.383) = .854 - .146 =.708
- rb = .924R - .383L = .924(.383) - .383(.924) = .354 - .354 =.0

The decoded lf signal is not higher than normal.

- What are examples of unnormalized matrix systems?
The Electro-Voice Stereo-4 system as defined is not normalized. The encoding equations are:

- L = lf + .3rf + lb - .5rb
- R = rf + .3lf + rb - .5lb

The decoding equations are:

- lf = L + .2R
- rf = R + .2L
- lb = L - .8R
- rb = R - .8L

When a left front signal is encoded, the encoder outputs are:

- L = 1lf
- R = .3f

The vector sum of L and R for lf is

√(1^{2}+ .3^{2}) = √(1 + .09) = √(1.09) = 1.044 (4% higher)When signals are decoded, the decoder outputs are as follows:

- lf = 1L + .2R = 1(1) + .2(.3) = 1 - .06 = 1.06
- rf = 1R + .2L = 1(.3) + .2(1) = .3 + .2 = .5
- lb = 1L - .8R = 1(1) - .8(.3) = 1 - .24 = .76
- rb = 1R - .8L = 1(.3) - .8(1) = .3 - .8 = -.5

The decoded lf signal is 6% higher than normal.

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The Sansui QS system as defined is not normalized. The encoding equations are:

- L = lf + .414rf + lbj + .414rbj
- R = rf + .414lf - rbj - .414lbj

The decoding equations are:

- lf = L + .414R
- rf = R + .414L
- lb = -Lj + .414Rj
- rb = Rj - .414L

When a left front signal is encoded, the encoder outputs are:

- L = 1lf
- R = .414rf

The vector sum of L and R for lf is

√(1^{2}+ .414^{2}) = √(1 + .171) = √(1.71) = 1.082 (8% higher)When signals are decoded, the decoder outputs are as follows:

- lf = 1L + .414R = 1(1) + .414(.414) = 1 + .172 = 1.172
- rf = 1R + .414L = 1(.414) + .414(1) = .414 + .414 = .828
- lb = -1Lj + .414Rj = -1(1)j + .414(.414)j = -1j +.172j = -.828j
- rb = 1Rj - .414Lj = 1(.414)j - .414(1)j = .414j - .414j = 0j

The decoded signals are 17% higher than normal.

- How do you normalize matrix equations?
I am using the Sansui QS equations.

For each channel, do:

- First, calculate the vector sum of the encoder or decoder coefficients for one channel.
- Then, divide all of the coefficients for that channel by that vector sum.

The QS encoding equations are:

- L = lf + .414rf + lbj + .414rbj
- R = rf + .414lf - rbj - .414lbj

Calculate the vector sum of the encoder coefficients for each channel.

- The vector sum L(lf) and R(lf) is
√(1
^{2}+ .414^{2}) = √(1 + .171) = √(1.71) = 1.082 - The vector sum L(rf) and R(rf) is
√(.414
^{2}+ 1^{2}) = √(.171 + 1) = √(1.71) = 1.082 - The vector sum L(lb) and R(rb) is
√(1
^{2}+ .414^{2}) = √(1 + .171) = √(1.71) = 1.082 - The vector sum L(rb) and R(lb) is
√(1
^{2}+ .414^{2}) = √(1 + .171) = √(1.71) = 1.082

They are all the same, making it easier. Now divide all of the coefficients by 1.082:

- L = lf + .414rf + lbj + .414rbj is converted to L = .924lf + .383rf + .924lbj + .383rbj
- R = rf + .414lf - rbj - .414lbj is converted to R = .924rf + .383lf - .924rbj - .383lbj

If they are not all the same, divide each channel's coefficients by its own vector sum.

Notice that the coefficients are identical to the Scheiber system, except for the j factors.

The QS decoding equations are:

- lf = L + .414R
- rf = R + .414L
- lb = -Lj + .414Rj
- rb = Rj - .414L

Calculate the vector sum of the decoder coefficients for each channel.

Notice that the vector sums are the same for the decode equations as they are for the encode equations, so we don't have to do them again.

They are again all the same, making it easier. Now divide all of the coefficients by 1.082:

- lf = L + .414R is converted to lf = .924L + .383R
- rf = R + .414L is converted to rf = .924R + .383L
- lb = -Lj + .414Rj is converted to lb = -.924Lj + .383Rj
- rb = Rj - .414L is converted to rb = .924Rj - .383Lj

Again the coefficients are identical to the Scheiber system, except for the j factors.

Thus, the Scheiber and QS systems produce identical encodings, except for how they handle the phase hole.

- How can a normalized matrix be the same as the unnormalized matrix when the coefficients are
different?
Both versions produce the same encodings and decodings. The only difference between the versions is output level. But since this can be adjusted with a level or volume control, there is no actual difference.

In most cases the normalized versions were used in practice anyway because the gains of previous or following stages was adjusted to keep the level correct.

- How does this affect ownership of rights for each matrix?
The different sets of equations were developed to keep the systems apart for ownership rights. But it did not work.

- Peter Scheiber ended up with all of the rights except trademark for all systems by patenting first.
- All of the original matrix systems are now out of patent protection, but not trademark protection.

The tricks to gain separate rights did not work. But it is all moot now.