Abstract
Let $S = \{z=x+yj|x$ and $y$ real, $-\pi \lt y \leq \pi \}$ and let $S^* = S \cup \{ - \infty \}$. If $z = x_1 + y_1j$, $w = x_2 + y_2j \in S$, then define a cylindrical addition, $\oplus$, on $S^*$ by $z \oplus w = (x_1 + x_2) + [(y_1 + y_2) \pmod{2\pi}]j$ and $z \oplus - \infty = - \infty$ for all $z \in S^*$. Define a subaddition, $\odot$, on $S^*$ by $z \odot w = \log(e^z + e^w)$. Then $(C, +, \cdot) \cong (S^*, \oplus, \odot)$.
Subaddition has an application in signal processing.
A channel's fading can be modeled as the product of a slowly varying component and the transmitted signal. An amplitude-modulated signal can also be represented by a product of a carrier signal and envelope function.
The logarithmic function will transform a system modeled on a product to a conventional linear system that will yield to a classical attack. It is shown here that the logarithm, as a generalized superposition, will also transform a conventional linear system into another linear system, and therefore, nothing need be known about the original system before applying a logarithmic transformation.
Citation
David Choate. "Subaddition." Missouri J. Math. Sci. 9 (3) 170 - 177, Fall 1997. https://doi.org/10.35834/1997/0903170
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