Hello,

I have a question concerning the conversion of a stereo recording to 
ambisonics. In the Wireless World of 1977 articles there are equations for the 
conversion of stereo to W,X,Y equivalent signals. The use of j*Diff term 
doesn't seem to make sense. As I found no background info on them, maybe 
someone in the group has.

The WW1977 article states:
W=0.71*Sum-0.291*j*Diff
X=0.71*Sum+0.291*j*Diff
Y=0.583*Diff
where Sum=L+R and Diff=L-R.

I was puzzled with the addition of the j*Diff, so I tried to find some 
mathematical support for it. The reasoning I did is as follows:

1. As W,X,Y are a representation of a supposed plane wave with amplitude S 
coming from a certain angle phi (phi=0 if straight ahead), we need to suppose 
a analoguous encoding for stereo. I supposed a the sin-like law of encoding 
f.e.:

(L-R)/(L+R)=sin(phi)/sin(30) if loudspeakers are placed at -30 and 30 degrees.

2. As a starting point a naive conversion of the L,R signals (projection of 
L,R on X,Y) to ambisonic-like format gives:
W=Sum
X=Sum*cos(30)
Y=Diff*sin(30)

If we write Sum and Diff in relation to the supposed direction encoding in (1) 
as function of S and phi, then
W=S
X= S*cos(30)
Y= S*sin(phi)

X is independent of the direction, Y correctly encodes the direction neatly. X 
is only correct for the center direction (phi=0) but for all other directions 
the magnitude of the encoded vector is to large (sqrt(X²+Y²)>W).

3. The adding of the j*Diff terms makes af first sight everything even worse, 
it makes the magnitude of X and W  higher and it also introduces a phase 
differences between X,Y and W which is in theory not possible if one encodes a 
plane wave directly.

The only reason I could think of is that by introducing phase differences once 
you construct the loudspeaker signals by adding weighted versions of the W, X 
and Y signal together these phase differences introduce lower amplitudes (than 
adding signals in phase). I did not go through with this calculation yet.

So if someone have some idea's on it, 

Johan





 




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