Derivation of new linear matrix for 3 loudspeakers preserving energy and maximising channel separation

The presented optimal matrix has two important properties

- Preserves energy
- Maximises channel separation

The 3 loudspeakers are arranged in front of the listener as pictured below.

The loudspeaker signals Ls, Cs and Rs are generated by the linear matrix from the stereo signals L and R.

The general matrix equation for the loudspeaker signals Ls, Cs and Rs is presented below.

The same equation can also be written in maybe more intuitive form as presented below.

For any reasonable matrix the matrix coefficients are set under the following restrictions. The first line indicates the "direct" signal will dominate, and together with the second line it indicates the cross-talk is less than the direct signal. The last line indicates the center mono signal has positive phase.

The matrix preserves the original stereo signal energy if and only if the following equation is fulfilled [1].

To analyse the matrix, we select the coefficient p as an independent variable and derive equations for remaining coefficients n and k as a function of p. The eguations are given below.

Now the coefficients k and n can be plotted as a function of coefficient p as pictured below. For any value of p (p0) there exists one and only one value of k (k0) and n (n0) which fullfill the energy preservation requirement.

"Good" old stereo. It is only 2 speakers so center speaker has no signal.

MS stereo presented through 3 speakers.

Gerzon matrix with coefficient phi = 35.24 degrees [1].

Gerzon matrix with coefficient phi = 54.74 degrees [1].

Some interesting things happen when one inserts value of 0.2679 for parameter m in Miles matrix [2]. While Miles does not recommend using such a low value of m, but instead gives optimal range between about 0.45...0.7. However here is presented the Miles matrix with m = 0.2679.

Now if the above matrix is multiplied with an arbitrary scaling term of 0.7887, one achieves the matrix below.

Comparing this matrix to the Gerzon matrix with phi = 35.26 degrees reveals they appear to be exactly the same !

The above presented matrixes are placed in the (p, k, n) plot as given below.

Now looking at the above plot several things can be observed.

Firstly, it can be seen that 2 speaker stereo and 3 speaker MS stereo are at the opposite sides of the curve. This would indicate they sound maximally different, which is interesting to say at least !

Also included are Gerzon matrixes for parameter phi = 35.26 and 54.74 degrees.

There are three Miles matrixes plotted. Because Miles matrix is not energy preserving they are plotted as linear lines originating from origo (which equals to arbitrary scaling). The Miles recommended range of parameter m = 0.45...0.7 is plotted as two lines correcponding to the limits of the range. There is only one value of parameter m = 0.2679 when Miles matrix is energy preserving, and as can seen from the figure it coindices the Gerzon matrix of phi = 35.26 exactly.

It is important that the amplitude separation between the loudspeaker signals is maximised when signal is panned through stereo space. This requirement is important if the sweet spot area is to be as large as possible, because otherwise the listener not located at the sweet spot could easily hear only the nearest speaker.

In the center panned case the optimal channel separation is achieved when the ratio of speaker signals of the center speaker and side speaker is maximised.

In this case the ratio is

Cs/Ls (or Cs/Rs)

This can be expressed in dB as

20*log(Cs/Ls)

In the side panned case the optimal channel separation is achieved when the ratio of speaker signals of the panned side speaker and the center speaker is maximised. And also simultaneously maximising the ratio of speaker signals of the panned side and the signal of the opposite side.

In this case the first ratio is

Ls/Cs

The second ratio is

Ls/Rs

These can be expressed in dB as

20*log(Ls/Cs)

and

20*log(Ls/Rs)

Here is presented the loudspeaker signal ratios as a function of matrix parameter p for center and side panned cases in dB scale. The dark blue curve is center panned Cs/Ls. The violet curve is side panned Ls/Rs. The light blue curve is side panned Ls/Cs.

The optimal matrix, which is energy preserving and which maximises the channel separation, can be found when all the three curves are simultaneously in a global maximum. This is presented with green dots in the figure below. It can be seen there is only one solution for such case. This is the new optimal matrix.

After one finds the global maximum for loudspeaker signal ratios from the above figure one immediately finds the matrix parameter p for the optimal matrix. Then the remaining matrix parameters k and n can be calculated using the equation presented before.

The new optimal linear matrix thus has the form given below in its exact values.

And the same with approximate desimal values.

This matrix is energy preserving.

The maximum channel separations are given below.

For the center panned case L=1 R=1

Cs/Ls = 3.98 dB

For the side panned case L=1 R=0

Ls/Cs = 3.98 dB

Ls/Rs = 14.0 dB

A new linear stereo matrix for 3 speakers has been derived. The matrix is energy preserving and maximises the loudspeaker signal separations.

- [1] Michael Gerzon: Optimal Reproduction Matrices for Multispeaker Stereo, AES, 1991
- [2] Michael Miles: An Optimum Linear-Matrix Stereo Imaging System, AES, 1996

Last modified 10.8.2013

© Elias Pekonen