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

Abstract

If f(t) is a band-limited function, with band limit -fl to fl, the result of instantaneously companding f(t) is in general no longer band-limited. Nevertheless, it has been proved that knowledge of merely those frequencies of the compandor output which lie in the band from -fl to (1 is sufficient to recover the original 81,'gnal f( t). A n iteration formula has been proposed that, in theory, performs the desired recovery. In this paper we study in detail some of the practical questions raised by that formula. TVe show that the successive approximations converge to the solution f( t) at a geometric rate, uniformly for all t, and that the iteration procedure is stable. TVe then describe a method of performing the recovery in real time and a successful simulation of it on a general-purpose analog computer. The circuit used in the simulation serves as a first approximation to a practical realization of the recovery scheme. I. INTRODUCTIONWhen a signal, J(t), is transmitted over a channel there is a tendency for the low-amplitude part of J(t) to become masked by the presence of channel noise and for the high-amplitude part of f(t) to become distorted by the nonlinearity of components in those ranges. It would be valuable, therefore, to find a way of assigning to f(t) another signal from which J(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 jet). One relatively simple way of obtaining such a signal is by instantaneous companding: The signal sent is 1Pl((t)], where lP(x) is a monotonic function (to allow recovery of f(t) from 1Pl(t)]) , 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

Robust Unlimited Sampling Beyond Modulo

“…To avoid clipping either instantaneous companding or modulo operations are used before sampling which limits the dynamic range of the signal. Instantaneous companding uses a nonlinear, monotone function G : R → R such that Gf (t) ∈ [−λ, λ] [7]. One can recover f (nT s ) from Gf (nT s ) by sampling at the Nyquist rate.…”

“…In addition, G boosts low amplitudes of the signal to improve the signal to noise ratio (SNR) which helps in accurate recovery. Existing companders are required to be monotone, differentiable, and Gf (t) ∈ L 2 (R) which limits their practical application [7], [8]. An alternative to avoid clipping is to perform a modulo operation prior to sampling, that is, sample M λ f (t) instead of f (t) [14].…”

“…Similar to clipping, companding is a nonlinear operation and increases the bandwidth of the signal. Beurling proved that knowledge of the companded signal over the input signal's bandwidth is sufficient to uniquely identify the signal provided that the compander is a monotone function and its output to finite energy input has finite energy [7]. This result implies that a companded signal can be sampled at the Nyquist rate by applying an antialiasing filter before sampling.…”

“…While the aforementioned companding-based approaches operate at a minimal possible sampling rate, the requirement of monotonicity, differentiability, and finite energy output limits their hardware implementation [7], [8]. Specifically, it is difficult to realize a monotone operator over the entire dynamic range of the signal.…”

LMS Algorithm for Blind Adaptive Nonlinear Compensation

“…• The recovery of a bandlimited signal from nonlinear companding and subsequent bandlimiting based on Beurling's theorem [4], [5], • Extension of the previous technique to stochastic processes [6], • Reconstruction of a noisy and not necessarily bandlimited signal from a known monotonic or non-monotonic nonlinear distortion [7], • Reconstruction of a noisy sampled signal from known nonlinear distortion [8]. In these studies, the prior knowledge of the nonlinear distortion is assumed to be available.…”