Laws@SRI-STRIPE.ARPA (Ken Laws) (11/06/86)
From: nsc!amdahl!apple!turk@hplabs.hp.com (Ken "Turk" Turkowski) Message-Id: <267@apple.UUCP> *Increasing the sampling rate beyond this "Nyquist rate" cannot result in higher fidelity*. >>... Losses of information in processing analog signals tend to >>be worse, and for an analog transformation to be exactly invertible, it >>*must* preserve all the information in its input. Including the exclusion of noise. Once noise is introduced, the signal cannot be exactly inverted. To pick a couple of nits: Sampling at the Nyquist rate preserves information, but only if the proper interpolation function is used to reconstruct the continuous signal. Often this function is nonphysical in the sense that it extends infinitely far in each temporal direction and contains negative coefficients that are difficult to implement in some types of analog hardware (e.g., incoherent optics). One of the reasons for going to digital processing is that [approximate] sinc or Bessel functions are easier to deal with in the digital domain. If a sampled signal is simply run through the handiest speaker system or other nonoptimal reconstruction, sampling at a higher rate may indeed increase fidelity. The other two quotes are talking about two different things. No transformation (analog or digital) is invertible if it loses information, but adding noise to a signal may or may not degrade its information content. An analog signal can be just as redundant as any coded digital signal -- in fact, most digital "signals" are actually continuous encodings of discrete sequences. To talk about invertibility one must define the information in a signal -- which, unfortunately, depends on the observer's knowledge as much as it does on the degrees of freedom or joint probability distribution of the signal elements. Even "degree of freedom" and "probability" are not well defined, so that our theories are ultimately grounded in faith and custom. Fortunately the real world is kind: our theories tend to be useful and even robust despite the lack of firm foundations. Philosophers may demonstrate that engineers are building houses of cards on shifting sands, but the engineers will build as long as their houses continue to stand. -- Ken Laws -------