Unravelling the limitations of DSPism

The believers of the cult of DSPism adhere to the idea that eventually there will be essentially no analogue circuits and that all circuitry will be either purely digital or a mix of mixed-technology ADCs and DACs with microcontrollers (MCUs) or other digital processing elements in between them. This article explores the possibilities and limitations on this viewpoint.

Before MCUs became circuit components, there was a clear divide between digital and analogue. Digital circuits consisted of the kinds of functions found in 7400-series TTL or 4000-series CMOS data books: flops, shift registers, decoders, and gates of all kinds. Analogue data books have quite different parts: mostly op-amps, voltage references, combinations of them such as voltage regulators, and various borderline parts such as comparators, which are analogue in, digital out; the 555 timer, which has mostly analogue pins but also has digital out and reset in; analogue switches, which have digital in; and multipliers, which can have a digital input in some applications. On closer inspection, much of what is in the analogue data books has been partly digital all along. A comparator is a one-bit ADC, and a 555 timer is a kind of voltage-to-time converter, which is itself a kind of ADC.

So what exactly is digital and what is analogue? The best way to define these words is in reference to waveforms, which are signals when they are encoded with a message in communications systems. Waveforms are electrical functions of time: v(t) or i(t) or even p(t). The definition could be extended to any dynamic (time-dependent) physical quantity—and even to social variables such as company cash flow or the U.S. debt as functions of time. Analogue waveforms are simply continuous functions of time. The definition ultimately goes back to mathematical definitions of continuity and analysis. In math, analysis refers to continuous functions and is "analogue math". Wherever waveforms are continuous in time, we have analogue electronics.

In contrast, digital is mathematically synonymous with discrete. Discrete functions have discontinuities in their numerical values and are not associated with the continuous number line of real numbers but with the integers, which leave gaping holes along the number line to be filled by the irrational numbers. Digital computers have waveforms that are discrete in both value and in time. Therefore, to simplify the definitions to their minimalist essence:

Analogue means continuous, digital means discrete
Binary digital waveforms are limited to two values, {0, 1} (which make digital waveforms Boolean functions) but this digital limitation can be partially overcome by grouping multiple scalar digital variables together into a vector quantity that can represent a number in base 2. The earliest MCUs did this with four bits (Intel's 4004, the first MCU), and then progressed quickly to 8 bits, which became a dominant grouping of bits for processors in the 1970s and 1980s. Now 32bit ARM processors are becoming commonplace.

As an aside, the groupings of bits have informally become associated with names that have stuck. The most common is the byte for 8 bits, and of lesser usage, the nibble for 4 bits. I would like to propose a fuller set of neologisms, retaining the ingestion metaphor of the language that has become associated with bit groupings:

As the number of bits increases, numerical resolution increases and approaches the continuum in the limit. At the same time that functions represented in software are gaining in resolution because of increasing MCU bit groupings, MCUs are also increasing in clock rate, causing their discrete-time characteristic to approach continuity. Both trends cause MCU apabilities to approach that of analogue functions. Thus it might seem reasonable to suppose that MCUs with ADCs and DACs are all that should be necessary for electronics in the future.