Phew!!! Firstly, many thanks for the detailed response. I'll admit I find the
math's related to UHJ & Ambisonics somewhat confusing at times, but 'm
persevering.
Again, thanks
This thread got dropped as far as I can see, but has been nagging at
me, and I've finally got round to wading through the calculations.
The equations match the ones I have found, apart from the
multiplication of W by sqrt(2). The 90 degree phase shift is on W &
X, not X & Y.
It doesn't seem possible to solve the encoding and decoding equations
back to W,X and Y, due to j turning up in all sorts of places.
Gerzon's maths was far in advance of mine, but I suspect that the
numbers may have been arrived at through trial and error.
Working with the numbers given and using W' as W*sqrt(2), I get
(though I may have made the odd error)
W'' = W (1.442) + X( 0.0986) + j*Y(0.1075)
X'' = W( 0.133) +X(0.43) - j*Y(0.543)
Y'' = j* W(0.143) + j*X(0.461) + Y(0.5)
where W'', X'' & Y'' are the B-Format recoding of the the UHJ coding
of the original X,Y,Z This is arrived at by taking j*j= -1, and -*-
as +.
None come back exactly to the original. Notably W'' is pretty near
sqrt(2)* the original. X and Y are about half the original in the
real component and in the imaginary component. Generally unwanted
components are about 10% of their original value
So in X'' the Y component is rotated by -90 degrees relative to its
value in Y'',
In Y'' the X component is rotated by +90 degrees relative to its
value in X'', Which sort of makes sense.
I suggest that W should not be multiplied by sqrt(2) in the encoding.
If so the above equations become
W"" = W (1.019) + X( 0.0986) + j*Y(0.1075)
X"" = W( 0.094) +X(0.43) - j*Y(0.543)
Y"" = j* W(0.101) + j*X(0.461) + Y(0.5)
which looks a bit better, though still not a perfect reconstruction.
This sqrt(2) factor is an endless source of confusion. It seems
silly that W should be divided by sqrt(2) in recoding to restore its
value, which was multiplied by this in the encode to UHJ.
I wonder if anyone knows what version of W was used in any encoding
of available UHJ recordings ??
Ciao,
Dave Hunt
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