Asymptotically Optimal Joint Sampling and Compression for Timely Status Updates: Age–Distortion Tradeoff

We consider a joint sampling and compression system for timely status updates. Samples are taken, quantized and encoded into binary sequences, which are sent to the destination. According to the timeliness requirement represented by an age penalty function at the destination, we formulate an optimization problem to jointly design sampler, quantizer and encoder, optimizing data freshness on the basis of satisfying a mean-squared error (MSE) distortion constraint of the samples. We use average nonlinear age (ANA) to measure data freshness and study linear, squared and exponential penalty functions. We prove that the zero-wait sampling, the uniform quantization, and the ANA-optimal coding policies together provide an asymptotically optimal solution, in the sense that, as the average distortion approaches zero, the solution achieves the minimum ANA asymptotically, for the linear and squared penalty functions. When the probability density function (pdf) is weakly unimodal, we also prove that the zero-wait sampling, the K-L quantization which is asymptotically optimal for the quantization with Rényi entropy constraint, and the ANA-optimal coding policies are asymptotically optimal for the exponential penalty function. Furthermore, we prove that the optimal ANA is asymptotically linear with respect to the log MSE distortion with a slope of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$-\frac{3}{4}$</tex-math></inline-formula>, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$-\sqrt{\frac{7}{12}}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$-1$</tex-math></inline-formula>, corresponding to the linear, squared and exponential penalty functions, respectively.