Djuro G. Zrilic Functional Processing of Delta-Sigma Bit-Stream,Springer,Cham,Switzerland,2020

What is delta-sigma bit-stream: the main idea. Nowadays, sensors can measure different quantities reasonably accurately and very frequently. In many cases –e.g., for sensors on a spaceship or on a drone –the main limitation is communication. Because of this limitation, sending measurement results sometimes takes much longer than the measurement itself.

It is therefore desirable to compress the data, so that the same information can be described by fewer bits and thus, it would be possible to send a larger amount of information within the same communication bandwidth. There exist many efficient compression techniques –e.g., wavelet-based techniques are usually very efficient –but many of these techniques require a large amount of data processing to compress, which is beyond the ability of many currently used simple and cheap sensors.

One of the natural ways to compress the data without overtaxing the sensor’s computational abilities is based on the fact that most measured quantities have a limited rate of change. As a result, if we measure the corresponding quantity sufficiently frequently, the change from one measurement result to the next one does not exceed the measurement accuracy ɛ. If we take this accuracy into account, then each measurement result x is indistinguishable from one of the values 0, ɛ, 2ɛ, …, k · ɛ, …In other words, we can safely replace each value x with the nearest value of the type k · ɛ; we will denote this nearest value by [x]. For a sufficiently slow-changing quantity, values [x (t)] and [x (t + Δt)] are either equal to each other or differ by ±ɛ. So, in principle, at each moment of time, instead of sending the whole measurement result x (t), it is sufficient to send one of the three signals 0, 1, or -1, indicating whether the difference [x (t + Δt)] - [x (t)] is equal to 0, to ɛ, or to -ɛ.

This idea drastically reduces the number of bits that need to be transmitted. And we can still reconstruct the original signal –to be precise, each value [x (t)] –by simply adding up the initial measurement result [x (t₀)] and all the previous differences [x (t₀ + Δt)] - [x (t₀)], [x (t₀ + 2Δt)] - [x (t₀ + Δt)], …, [x (t)] - [x (t - Δt)].

Comment. What we just described is a simplified version of the main idea. Since modern computing and communication are based on binary signals, the actual scheme uses not three but two possible signals: crudely speaking, it uses only 1 and -1. This makes the compression scheme somewhat more complicated, but it makes communication easier.

The main topic of the book. Traditionally, practitioners would reconstruct the signals upon receiving them, and then process the reconstructed signals. It turned out, however, that in many cases, it is possible (and efficient) to process compressed signals –and to reconstruct only the final results of data processing.

Many such algorithms are presented in the book, many of them pioneered by the book’s author himself. For example, it is possible to add, multiply, differentiate, and integrate compressed signals directly –and thus, e.g., to design a controller in which all the processing is performed only on the modulated signals, and even only the difference between the current and the previous control values is sent to the actuator.

Open problems. The book’s designs are mostly oriented towards traditional PID-type controllers. Since the proposed methods are very efficient, it is desirable to extend them to intelligent control algorithms.

The need for such extension is especially acute for deep learning, where the number of input neurons is limited and thus, compression may be needed. For example, one of the main motivations behind convolutional neural networks is that the usual signal or image contains too many bits, many of them irrelevant; so, convolution decreases the number of bits. If the original data comes from a sensor in a modulated form, this means that we first add extra bits by reconstructing the signal –and then delete the extra bits by applying convolution. Maybe it is possible to start training directly with the modulated signal –this will decrease the need for convolution (and also decrease the distortion caused by convolution)?