Sub-Nyquist Sampling Achieves Optimal Rate-Distortion

The minimal sampling frequency required to achieve the rate-distortion function of a Gaussian stationary process is analyzed. Although the Nyquist rate is the minimal sampling frequency that allows perfect reconstruction of a bandlimited signal from its samples, relaxing perfect reconstruction to a prescribed distortion may allow a lower sampling frequency to achieve the optimal rate-distortion trade-off. We consider a combined sampling and source coding problem in which an analog Gaussian source is reconstructed from its rate-limited sub-Nyquist samples. We show that each point on the distortion-rate curve of the source corresponds to a sampling frequency f(DR) smaller than the Nyquist rate, such that this point can be achieved by sampling at frequency f(DR) or above. This can be seen as an extension of the sampling theorem in the sense that it describes the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples, and provides the minimal sampling frequency required in order to achieve that distortion.