Looking at the encoding and decoding formulae, let's work through the effect of encoding and decoding the source which is 45 degrees (positive) with respect to the front-back axis (X) as shown. Forty-five degrees is simple from a mathematics point of view because cos 45 degrees and sin 45 degrees are the same and equal to 1/ sqrt 2 = 0.707.
In this example, I have assumed the recording of a real source of sound in an acoustic environment. A Soundfield microphone "encodes" the signals according to these mathematical rules by virtue of its physical properties and characteristics. But the encoding equations given in the figure could equally be used to encode a virtual source in a virtual soundfield by means of three-dimensional Ambisonic pan-pots. The sound is shown arriving at a point where the axis of particle velocity intersects the positive response lobe of the front-back microphone (the X signal) and the positive response lobe of the left-right microphone (the Y signal) and where the response has fallen by 0.707 compared with its respective on-axis response. In this particular circumstance, the B-Format signals will therefore be:
(Incidentally, you will see in the Ambisonics literature that W is multiplied by 0.707 before transmission. This is not part of the encoding; rather it is a factor which is included to improve signal to noise ratio in the transmission system and a complementary multiplier (1.414) is applied prior to decoding. In order to simplify the explanation, I have ignored this manipulation.)
If we now decode these signals according to the decode equations for the loudspeakers in the positions shown in the figure (Gerzon 1977), the results will be:
(Remember that the factor root2. cosA can be ignored for this example because they cancel out.) Which means that the vector field is re-created as shown here.
The figure may be something of a surprise, because all loudspeakers are shown sounding. But it's quite correct (according to the theory), because the loudspeakers are contributing to an overall vector-quantity particle velocity.
Unfortunately, this principle of Ambisonics is really only valid at a single point and, even the most self-effacing of us, would claim to have a vanishingly small head! As soon as the diameter of the head starts to be an appreciable proportion of the wavelength of the reproduced sound, it is not the velocity of the sound which is detected, but two independent pressures at the two ears. Perceptions are therefore ambiguous at HF.
Now that Ambisonics is no longer protected by patents (GB1 550 628 having now expired), it will be interesting to see if parts of this intriguing idea will be more widely adopted. This seems unlikely however, Ambisonics is an elegant theory but is not rooted in psychoacoustics and that is its failing.
Michael Gerzon, 'Multi-System Ambisonic Decoder', Part 1: 'Basic Design Philosophy', Wireless World, vol. 83 no. 1499, pp. 43-47 (1977 July) Part 2: 'Main Decoder Circuits', Wireless World, vol. 83 no. 1500, pp. 69-73 (1977 Aug.)
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