Using 2:1 Shannon mapping for joint source-channel coding

The Archimedes' spiral can be used as a 2:1 bandwidth reducing mapping in a joint source-channel coding (JSCC) system. The combined point of two iid Gaussian sources (the source space) is mapped, or approximated, onto a double Archimedes' spiral (the codebook), and the squared angle from the origin to the mapped point is transmitted as an analogue channel symbol (the channel space), e.g. PAM. It is shown that the total distortion of this JSCC system is minimised when the distortion contributions from the approximation noise and channel noise are equal. The given system produces a channel input distribution close to a Laplace probability density function (pdf) instead of the optimal Gaussian pdf. The loss when using this mismatched pdf is shown to be approximately equal to the relative entropy of the two pdf.