Transform Coding of Densely Sampled Gaussian Data
With mean-squared error D as a goal, it is well known that one may approach the rate-distortion function R(D) of a nonbandlimited, continuous-time Gaussian source by sampling at a sufficiently high rate, applying the Karhunen-Loeve transform to sufficiently long blocks, and then independently coding the transform coefficients of each type. In particular, the coefficients of a given type are ideally encoded with performance attaining a suitably chosen point on the first-order rate-distortion function of that type of coefficient. This paper considers a similar sample-and-transform coding scheme in which ideal coding of coefficients is replaced by coding with some specified family of quantizers, whose operational rate-distortion function is convex. A prime example is scalar quantization with entropy-coding and, if needed for convexity, time sharing. It is shown that when the sampling rate is large, the operational rate-distortion function of such a scheme comes within a finite constant of R(D). Applied to the scalar quantization family, the finiteness of this bound contrasts with a recent result showing that direct scalar quantization of samples (without a transform) has unbounded rate when distortion is held constant and sampling rate becomes large, even when the quantized samples are compressed to their entropy-rate. Thus, at high sampling rates, the transform reduces the loss due to scalar quantization from something infinite to something finite.