On the Recovery of a Band-Limited Signal, After Instantaneous Companding and Subsequent Band Limiting

When a signal, f(t), is transmitted over a channel there is a tendency for the low-amplitude part of f(t) to become masked by the presence of channel noise and for the high-amplitude part of f i t ) to become distorted by the nonlinearity of components in those ranges. It would be valuable, therefore, to find a way of assigning to fit) another signal from which f(t) could be recovered, but which would have the property that its amplitude lay more nearly in the middle ranges than did that of f(t). This second signal is then transmitted, instead of the original fit). One relatively simple way of obtaining such a signal is by instantaneous companding: The signal sent is

which has a large slope around x = 0 so as to magnify signals of low amplitude, and which approaches a constant value for large x so as to cut down on signals of high amplitude. 351 352 T H E BELL SYSTEM TECHNICAL J O U R N A L , MARCH 1960