> cardioid = 1/2 ( sqrt2 W + X )

> but for Mid-Side decoding from B-Format, where M is the same cardioid 
> component, I often find this equation:

> M = sqrt2 W + X

This is an instance of a patented Microsoft software invention called a Volume 
Control.  There are also ancient hardware implementations of this useful device 
which can be used to avoid clipping by those versed in the art.  

Also to generate clipping where this is a desired effect.  It is sometimes 
called a Gain Control.  Both positive & negative gain is possible.

> writing the equation inside max6 I had several clipping problems with the
decoding of a Soundfield ST450 signal, that has four matched line under single 
knob.

The ST450 uses such a hardware implementation I designed circa 1980 for the Mk4 
Soundfield.  Experiment to find the best settings for you.

It is possible to have several instances (both hardware & software) of these 
devices to optimise the dynamic range and clipping performance of a complex 
signal processing chain at various points.

> I record the b-format in a single quadro file.

If you do all recording and processing in 64b Floating Point, you may be able 
to avoid infringing Microsoft patents on the Volume Control.
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