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On the rear side the function given by the AES paper has the same end points, but is
different in-between.

The mathematical method in the AES paper provides the following equations for the Left
Rear matrix elements over the range 22.5 < lr < 45: (remember that t = 45-lr)

LRL = cos(45-lr)*sin(4*(45-lr))-sin(45-lr)*cos(4*(45-lr)) = sra(lr)

LRR = -(sin(45-lr).*sin(4*(45-lr))+cos(45-lr).*cos(4*(45-lr))) = -srac(lr)

If cs <= 22.5, lr can still vary from 0 to 45. The AES paper defines LRL and LRR when lr
has the range 0 < lr < 22.5 of: - see Figure 6 in the AES paper.

LRL = cos(lr) = sra(lr)

LRR = -sin(lr) = -srac(lr)

The two functions sra(x) and srac(x) – are defined for 0 < lr < 45.

March 1997

In March 1997 the following technique is used to fix the discontinuity across the cs=0
boundary. In the AES paper near cs=0 LRL and LRR are both functions of a single
variable. To fix the lack of continuity along the cs=0 boundary we add a function of a
composite of lr and cs. The new variable is lr_bounded, the bounded difference between
lr and cs. The definition of this variable is sufficiently complicated that I will present it in
MATLAB.

lr_bounded = lr - cs; % find the difference

if (lr_bounded <0) % % only if lr > cs

lr_bounded = 0;

if (45-|cs| < lr_bounded) % use the smaller of the two values

lr_bounded = 45-cs;

We define a new function which is equal to the difference between equation sra(x) and
(1-GSL(x)) when cs=0. This is rear_active_correct(lr_bounded).

For 0 < x< 45

Rear_active_correct(x) = sra(x) – (1 - GSL(x))

LRL = (sri(cs) + sra(lr) - rear_active_correct(lr_bounded) -1 ) * tvcorr(|lr|+|cs|)

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Summary of Contents for 5-2-5 Matrix Systems

Page 1: ...ey are capable of much more A preliminary design for a new matrix topology has been tested by the IRT in Munich as a 5 2 5 codec with excellent results on a wide range of broadcast material Although there were audible differences the differences were perceived as small changes in localization and were sometimes preferred to the original We have extensively tested the new matrix with ordinary stere...

Page 2: ...sound spacious after stereo reverberation is added In popular music such recordings are perhaps the rule rather than the exception It is easy to show that two ordinary stereo loudspeakers in a small room cannot reproduce the spatial diffusion of a large hall or concert space However if the reverberant portion of the stereo signal can be reproduced through an array of loudspeakers at the sides of t...

Page 3: ...rix decoder that can be described by the elements of a two by n matrix where n is the number of output channels Each output can be seen as a linear combination of the two inputs where the coefficients of the linear combination are given by the elements in the matrix In this paper the elements are identified by a simple combination of letters The previous paper described a five channel and a seven ...

Page 4: ... is possible for the sum to exceed 45 but we try to minimize this overrun The mathematics presented here for the matrix elements is well behaved during overruns However when we graph the matrix elements we arbitrarily zero the values when the legal sum of the input variables is exceeded This allows us to directly view the behavior of the element along the boundary trajectory the trajectory followe...

Page 5: ...f the of the non zero matrix elements are for any given channel is one In practice the elements are usually scaled so the maximum value of each element is one or less In any case in a final product the scaling of the elements is additionally varied in the calibration procedure The matrix elements presented here should be assumed be scalable by appropriate constants The left front matrix elements i...

Page 6: ... constant regardless of the direction of the steering Mathematically this means that the root mean square sum of the lfl and lfr matrix elements should be a constant It was pointed out in the paper that this goal should be relaxed in the direction of the steering that is when the steering is full left the sum of the squares of the matrix elements should rise by 3dB Figure 3 shows that the above ma...

Page 7: ...rected the amplitude errors in figure 3 by replacing the function G x in the matrix equations with sines and cosines See Figure 4 For the left front quadrant LFL cos cs 41 G lr LFR sin cs For the right front quadrant LFL cos cs LFR sin cs For the left rear quadrant LFL cos cs 41 G lr LFR sin cs For the right rear quadrant LFL cos cs LFR sin cs Figure 4 The square root of the sum of the squares of ...

Page 8: ... cos cs 0 41 G lr boost1 cs LFR sin cs boost1 cs For the right front quadrant LFL cos cs boost1 cs LFR sin cs boost1 cs For the left rear quadrant LFL cos cs 0 41 G lr boost cs LFR sin cs boost cs For the right rear quadrant LFL cos cs boost cs LFR sin cs boost cs The function G x is the same as the one in the 89 patent When expressed with angles as an input it can be shown to be equal to G x 1 ta...

Page 9: ...left to the center it can be seen in figure 5 that the value of the lfl matrix element increases to a maximum half way between left and center This increase in value is an unintended consequence of the deliberate increase in level for the left and right main outputs as a center signal is added to stereo music When a stereo signal is panned forward it is desirable that the left and right front outp...

Page 10: ...s 1 this limits the maximum value bcs 1 end else bcs 47 cs lr 1 if bcs 1 bcs 1 end end If cs 22 5 and lr 0 In matlab convention cs 24 and lr 1 bcs is equal to cs However as lr increases bcs will decrease to zero If cs 22 5 as lr increases bcs also decreases Now to find the correction function needed we find the difference between the boosted matrix elements and the non boosted ones along the lr 0 ...

Page 11: ...plain sine and cosine and the boosted sine and cosine We now define LFL cos cs 41 G lr cos_tbl_plus bcs LFR sin cs sin_tbl_plus bcs LFL and LFR in the front right quadrant are similar but without the 41 G term These new definitions lead to the matrix element in Figure 6 Figure 6 The new left front left matrix element has the correct amplitude along the left to center boundary as well as along the ...

Page 12: ... is sufficient for good performance of the matrix but it is not as good as it could be This performance can be improved by altering the LFL and LFR matrix elements in the left rear quadrant Notice that here we are concerned with how the matrix elements vary along the boundary between left and rear The mathematical method given in the AES paper can be used to find the behavior of the elements along...

Page 13: ...e LFL cos cs LFR sin cs We need a method of smoothly transforming the above equations into the equations along the boundary as lr and cs approach the boundary A linear interpolation could be used In the processor used in Lexicon products where multiplies are expensive a better strategy is to define a new variable the minimum of lr and cs new find the boundary parameter bp x if bp y bp y end and a ...

Page 14: ...r Note the large correction along the left rear boundary This causes the front left output to go to zero when steering goes from left to left rear The output remains zero as the steering progresses to full rear However along the lr 0 axis and in the right rear quadrant the function is identical to the 89 matrix Figure 9 the left front right matrix element Note the large peak in the left to rear bo...

Page 15: ...gn For this plot we deleted the 1 sin cs cos cs correction in the rear quadrant so we could see how accurately the resulting sum came to unity Note the 3dB peak in the left direction and the somewhat lesser peak as a signal goes from unsteered to 22 5 degrees in the center direction This second peak is a result of the deliberate boost of the left and right outputs during half front steering Note t...

Page 16: ...uises the way these decoders are actually used We derive all the matrix elements with a relatively arbitrary scaling In most cases the elements are presented as if they had a maximum value of 1 41 In fact for technical reasons the matrix elements are all eventually scaled so they have a maximum value of less than one In addition when the decoder is finally put to use the gain of each output to the...

Page 17: ...patent are used the same calibration procedure results in 3dB less sound power from the rear when the decoder inputs are uncorrelated The issue of how loud the rear channels should be when the inputs are decorrelated ends up as a matter of taste When a surround encoded recording is being played one would like to have the balance as the producer intended However with standard stereo material one mu...

Page 18: ... fully steered i e 3dB less than the standard calibration then the center matrix elements should have the shape shown in Figure 12 We can do the same thing for the standard calibration and the results in Figure 13 emerge Figure 12 solid curve the center matrix value as a function of CS in dB assuming sound power ratios identical to stereo and using Dolby matrix elements with 3dB less power in the ...

Page 19: ... with the addition of a G lr term in LRR and a GS term in LRL G lr was included to add signals from the B input channel of the decoder to the left rear output to provide some unsteered signal power as the steered signal was being removed We then solved for the function GS lr using the criterion that there should be no signal output with a fully steered signal moving from left to center The formula...

Page 20: ...lr LRR sin cs GR lr We want the sum of the squares to be one along the cs 0 axis 1 GS lr 2 GR lr 2 1 and the output to be zero to a steered signal or as t varies from zero to 45 degrees LRL cos t LRR sin t 0 These two equations result in a messy quadratic equation for GR and GS which is solved numerically in Figure 15 Use of GS and GR as shown results in a large improvement along the lr 0 axis as ...

Page 21: ...29 sin 4 xymin In the front right quadrant LRL cos cs LRR sin cs In Figure 16 these matrix elements are multiplied by the tv matrix correction We will call the correction for TV Matrix tvcorr lr cs Tvcorr lr cs is 3dB at zero and 1 when the argument is 22 5 degrees and higher Figure 16 The square root of the sum of the squares of LRL and LRR using the new values for GR and GS Note that except for ...

Page 22: ...which preserved their behavior along the steering boundaries For the new elements described here these errors have been corrected first by using an interpolation along the cs 0 boundary for LRL where the value is made to match the value of GS lr when cs is zero and smoothly rises to the value given by the previous math as cs increases negatively toward the rear In the newest software LRR interpola...

Page 23: ... is equal to cos x We will use these functions again in defining the Left Rear matrix elements during Left steering Left side rear outputs during rear steering from Right Rear to Rear Now consider the same matrix elements as cs becomes greater than 22 5 degrees As we said in the AES paper and the two patent applications LRL should rise to one or more over this range and LRR should decrease to zero...

Page 24: ...97 the following technique is used to fix the discontinuity across the cs 0 boundary In the AES paper near cs 0 LRL and LRR are both functions of a single variable To fix the lack of continuity along the cs 0 boundary we add a function of a composite of lr and cs The new variable is lr_bounded the bounded difference between lr and cs The definition of this variable is sufficiently complicated that...

Page 25: ...alue of cs over the range of 0 to 15 degrees In other words when cs is zero we employ G lr to find LRR As cs increases to 15 degrees we interpolate to the value of srac lr There is also the possibility of a discontinuity along the lr 0 axis We can solve this by adding a term to LRR which is found by using cs_bounded The term is simply sric cs_bounded This term will insure continuity across the lr ...

Page 26: ... of the corrections for rear loudness For cs 22 5 lr 22 5 LRL sra lr sri cs rboost cs LRR srac lr sric cs_bounded This completes the LRL and LRR matrix elements during left steering The values for right steering can be found by swapping left and right in the definitions Center matrix elements The center matrix elements in v1 11 have major differences with the center elements in the July 1996 paten...

Page 27: ...found that it was important to reduce the unsteered level of the center output and a value 4 5dB less than the Pro Logic level was chosen The boost function 41 G cs was changed to increase the value of the matrix elements back to the Pro Logic value as cs increases toward center The boost function in March of 1997 was chosen relatively arbitrarily In March 1997 the boost function of cs starts at z...

Page 28: ... to center or from right to center along the boundary The value above gives too low an output from the center speaker when the pan is half way between Center channel in the new design The center channel output must be derived from the A and B inputs to the decoder While it is possible to remove a strongly steered signal from the center channel output using matrix techniques any time the steering f...

Page 29: ... the amplitude of the Lin and Rin signals must rise along with the amplitude of the Cin signal However if the listener is not equidistant from each speaker the listener is much more likely to hear the sum of the sound power from each speaker which is equivalent to the sum of the squares of the three front outputs In fact extensive listening has shown that in fact the sum of the powers of ALL the s...

Page 30: ... localization of all sources shifts strongly to the center We tested this curve and found that the center loudspeaker dominated the front sound stage and left right separation was minimal There is a better solution The center attenuation shown in figure 19 is derived assuming the matrix elements previously given for lfl and lfr What if we used different elements Specifically do we need to be aggre...

Page 31: ...ring CL 42 42 G lr GC cs CR 42 GC cs For rear steering CL 42 42 G lr CR 42 Several functions were tried for GC cs The one given below may not be ideal but seems good enough It is specified in terms of the angle cs in degrees and was obtained by some trial and error In MATLAB center_max 65 center_rate 75 center_max2 1 center_rate2 3 center_rate3 1 if cs 12 gc cs 1 42 10 db center_rate 20 tmp gc cs ...

Page 32: ...x elements which provide this rate of decrease These matrix elements should also provide some boost of the Lin and Rin components and should have the current shape at the left to center boundary as well as the right to center boundary Figure 20 Center attenuation in the new decoder Note the quick rise from 42 followed by a gentle rise followed finally by a steep rise to the value 1 the previous at...

Page 33: ...PC PREAR PT GP 2 GF 2 GC 2 Lin 2 Rin 2 GP GF 2 2 GC 2 Cin 2 PREAR The ratio of Cin power to Lin and Rin power is assume Lin 2 Rin 2 RATIO gp cs gf cs 2 2 gc cs 2 5 cos cs sin cs 2 Cin 2 2 gp cs 2 gc cs 2 gf cs 2 1 Lin 2 RATIO Cin 2 Lin 2 gp cs gf cs 2 2 gc cs 2 5 cos cs sin cs 2 2 gp cs 2 gc cs 2 gf cs 2 1 For normal stereo GC 0 GP 1 and GF 0 The center to LR power ratio is then RATIO Cin 2 Lin 2 ...

Page 34: ...right during strong steering Also GF needs to smoothly interpolate to the previous value along the boundaries In practice all these curves have a negative sign GF gives the shape of the LFR matrix element along the lr 0 axis as cs increases from zero to center We need a method of blending this behavior to that of the previous LFR element which must be preserved along the boundary between left and ...

Page 35: ...dary the cost of course is a divide if y 23 this is the easy way for half the region lfr3d 47 x 47 y sin_tbl y gf_diff bcs else tmp 47 y x 47 y gf_diff y lfr3d 47 x 47 y sin_tbl y tmp end Note that the sign of gf_diff is positive in the equation above Thus gf_diff cancels the value of sin cs reducing the value of the element to zero along the first part of the lr 0 axis See Figure 21 above ...

Page 36: ...ion of cos 2 cs along the left boundary which creates smooth panning between left and center We would like our new center function to have similar behavior along this boundary We can make a correction to the matrix element which will do the job by adding an additional function of xymin center_fix_tbl 8 corr1 1 CL 42 42 G lr GC cs center_fix_table xymin CR 42 GC cs center_fix_table xymin See Figure...

Page 37: ...uture Music recordings where the foreground instruments are placed in the front three channels with primarily reverberation in the rear channels require a different encoding After a series of tests at the IRT and elsewhere it was determined that music recordings of this type were successfully encoded in a stereo compatible form when the surround channels were mixed with 3dB less power than the oth...

Page 38: ...rols the amount of negative cross feed for each surround channel As in the previous encoder when there is only an input to one of the surround channels the A and B outputs have an amplitude ratio of 38 91 which results in a steering angle of 22 5 degrees to the rear As usual the total power in the two output channels is unity that is the sum of the squares of 91 and 38 is one While the output of t...

Page 39: ...evels and when the rear is less by 3dB the value of fc is reduced to a maximum of 3dB The maximum attenuation is reached when the rear channels are 8dB less strong than the front channels This active circuit appears to work well It makes the new encoder compatible with the European standard encoder for classical music However instruments which are intended to be strong in the rear channels are enc...

Page 40: ...stereo recordings Results of the work as shown in listening tests have been highly gratifying While not every detail of original 5 channel recordings is preserved by the matrix the artistic quality of the mix is well preserved Both conventional stereo and film sound mixed for the previous standard matrix are enhanced by the new matrix Figure 24 Solid curve center output level as a steered signal p...

Page 41: ...ntor 1989 2 Patent 5 046 098 A four channel matrix surround decoder Douglas Mandel Inventor 3 Patent 5 109 419 A six channel matrix surround decoder David Griesinger Inventor 1992 4 Multichannel Matrix Surround Decoders for Two Eared Listeners David Griesinger AES preprint 4402 October 1996 ...

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