I am occasionally asked by DIY enthusiasts to assist with the design of their own Ambisonic decoders. Here are two e-mail messages and a PDF file which encapsulate my knowledge (and also my ignorance) in this area. Be aware that the W, X and Y channel gains are those inside an Ambisonic decoder and the sqrt(2) boost, applied to X and Y to equalise the energy in the channels, has been removed.
Subject: Re: LEVEL!!! mode select
Date: Sun Oct 4 21:33:35 1998
From: Martin J Leese
There is good news and bad news. The bad news is that the paper distributed does not seem to contain an explanation of how to derive the appropriate shelf filter gains for different speaker layouts, etc. The good news is it does contain enough information to reconcile the apparently different shelf filter gains in the various papers for horizontal four-speaker UHJ and B-Format. For 2-channel UHJ decoding the high-frequency gains are 1.0. Some papers suggest that the low-frequency gains for UHJ should be 1.0, but this turns out to be only because the suggested circuit for switchable gains holds the low-frequency gain at 1.0 and so pushes the required low-frequency gains into a prior resistor matrix. Other minor differences are explained by the shelf filter gains for UHJ differing slightly from its predecessor System 45J. Here are the shelf filter gains. I stress that these are for a horizontal four-speaker layout only. Low High B-Format: W 1.00 1.2247 X,Y 1.00 0.8660 UHJ: W 0.646 1.00 X,Y 1.263 1.00 B 0.775 1.00 In truth, there is still some disagreement between the papers about the gains for the B-channel of UHJ (used for Forward Preference). I have shown above the gains from the Gerzon 1985 reference and suggest you use its matching formulae. The Gerzon 1977d reference states that the gains for Super Stereo are very similar to B-Format. What is still a mystery, at least to me, is what gains are appropriate for five and six speaker layouts. (The gains for B-Format periphony are in Gerzon 1980.) What is also missing is the decoding equation for Super Stereo. There is one in Gerzon 1977d, but it states it is a compromise (also, I don't trust this reference). I list below the various papers that now seem to have been reconciled. Regards, Martin ______________________________________________________________________ Michael A. Gerzon, "Design of Ambisonic Decoders for Multispeaker Surround Sound", 1977a. [This is the paper distributed.] * M.A. Gerzon, "The Optimum Choice of Surround-Sound Encoding Specification", Preprint no. 1199 of the 56th Audio Engineering Society Convention, Paris, (1977b Mar. 1) (About optimising 2-channel encoding for surround-sound - the theory behind UHJ. Very mathematical.) Michael Gerzon, "Multi-System Ambisonic Decoder", Part 1: "Basic Design Philosophy", Wireless World, vol. 83 no. 1499, pp. 43-47 (1977c July) Part 2: "Main Decoder Circuits", Wireless World, vol. 83 no. 1500, pp. 69-73 (1977d Aug.) Later parts never written & published. * M.A. Gerzon, "Practical Periphony", Preprint 1571 of the 65th Audio Engineering Society Convention, London (1980 Feb.) (A relatively non-technical account of practical with-height full-sphere Ambisonic decoding.) *** M.A. Gerzon, "Ambisonics in Multichannel Broadcasting and Video", J. Audio Eng. Soc., vol. 33 no. 11, pp. 859-871 (1985 Nov.) (Possibly the best overview of classic Ambisonics.)
To be clear, use the B-Format shelf filter gains for Super Stereo.
The transition frequency at which the shelf filter gains change from the low- to the high-frequency values is a design compromise. Theory suggests it should be set to 700 Hz so that the distance between your ears is half a wavelength. However, in most Ambisonic decoder designs it is lowered to 400 Hz. This increases the effective listening area, but at the cost of poorer localisation at the sweet spot.
Subject: Decoder equations
Date: Sun Aug 15 20:49:42 1999
From: Martin J Leese
Hi Folks, I am no longer sure who is part of this informal group designing the home build Ambisonic decoder. If I have missed anybody out, please forward this message to them. My chosen role in the group has been simply to feed you references and design equations. As some of you may know, I am shortly moving to California to become a full-time graduate student at Stanford University. This means that my library of Ambisonic papers and my hi-fi are going into storage for several years. Also, I do not know how much free time I will have to devote to stuff like Ambisonics. To date, you have (quite correctly) only concerned yourselves with the wide-band 90-degree phase shifters. However, before disappearing I thought I had better feed you design equations for other parts of the decoder. I have already fed you what information I have on shelf filters. If you would like me to recirculate this then let me know in the next few days. Other parts of the decoder are:
This compensates for the non-planar wavefronts emitted by point speakers placed at a finite distance from the listener. A simple high-pass R-C filter is used with a -3 dB point at a frequency of (53/r) Hz, where r is the distance in metres from the speakers to a central listener. Only the X and Y signals are filtered, not W. Note that it is the *phase* behaviour of the simple R-C filter that is important; the fact that it also filters out stuff below 20 Hz or so is irrelevant.
This is used only with UHJ decoding, not B-Format or Super Stereo. It pushes phasiness to the rear of the sound field where it is less offensive. With no forward preference (a = 0), the phasiness is distributed equally around 360 degrees. The transformation is: new W = W new X = X new Y = Y - a.(jW) where a = a constant j = 90-degree phase shift Now, what value of "a" to use? In theory, 0 < a < 1. Gerzon's patent 4081606 suggests 0.333 < a < 0.5. Gerzon 1977 (refs at end) uses a = 0.3. Gerzon 1985 suggests 0 < k' < 0.7 which is equivalent to 0 < a < 0.49. I would either allow 0 < a < 1.0 and experiment or go with the most recent reference, Gerzon 1985.
This is a "zoom control" that moves the listener forwards or backwards in the sound field. If the listener is moved forward then sounds at the front get louder and sounds at the back get softer. Note that this cannot be accomplished by simply adjusting the volumes of the front and rear speakers because all speakers cooperate to place sounds at the front (or anywhere else). The amount of forward dominance is expressed in terms of the total dB gain at the front relative to the rear. The transformation is: new W = a.W + b.X new X = a.X + 2.b.W new Y = Y where a and b are constants related by: a = 0.5(L + (1/L)) b = (1/(2.sqrt(2))).(L - (1/L)) L*L = (1 + u)/(1 - u) u = 0.1(Gain in dB) (Obviously, this "a" is different from the one used for Forward Preference.) Note that when a = 1 and b = 0 (given by L = 1), the transformation does nothing. Some examples of suitable values of "a" and "b" are given below. I have listed the square of L just to simplify the values in that column; you don't actually need these. Note that the same values occur more than once in the table, thus simplifying the necessary circuit (he said, confidently) :-) Gain (dB) u L*L a b +6 0.6 4 1.25 0.5303 +3.333 0.3333 2 1.0607 0.25 0 0 1 1.0 0.0 -3.333 -0.3333 0.5 1.0607 -0.25 -6 -0.6 0.25 1.25 -0.5303 Well, that's all folks. Best of luck. Regards, Martin ______________________________________________________________________ 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.) Later parts never written & published. *** M.A. Gerzon, "Ambisonics in Multichannel Broadcasting and Video", J. Audio Eng. Soc., vol. 33 no. 11, pp. 859-871 (1985 Nov.) (Possibly the best overview of classic Ambisonics.)
Here is a 1981 reference for the "decoding" equations for stereo sources: