  The exponential growth in data generation and large-scale data analysis
creates an unprecedented need for inexpensive, low-latency, and high-density
information storage. This need has motivated significant research into
multi-level memory systems that can store multiple bits of information per
device. Although both the memory state of these devices and much of the data
they store are intrinsically analog-valued, both are quantized for use with
digital systems and discrete error correcting codes. Using phase change memory
as a prototypical multi-level storage technology, we herein demonstrate that
analog-valued devices can achieve higher capacities when paired with analog
codes. Further, we find that storing analog signals directly through
joint-coding can achieve low distortion with reduced coding complexity. By
jointly optimizing for signal statistics, device statistics, and a distortion
metric, finite-length analog encodings can perform comparable to digital
systems with asymptotically infinite large encodings. These results show that
end-to-end analog memory systems have not only the potential to reach higher
storage capacities than discrete systems, but also to significantly lower
coding complexity, leading to faster and more energy efficient storage.
