Electronic – Do analog circuits exist that are essentially mathematical functions in DC

I've used a product line called the Electronically Programmable Analog Circuit (EPAC), probably more than ten years ago by now, which claimed to be the analog equivalent of an FPGA, and Cypress has for years produced a line called the PSoC (Programmable System On Chip) which incorporates a switchable arrays of both analog and digital circuitry. Note that in both cases the devices have a moderately small number of functional blocks (3 to 24 or so in the case of the PSoC) with somewhat limited routing options, rather than providing hundreds or thousands of blocks with enough interconnects to allow essentially arbitrary routing.

One reason that analog FPGA's don't offer anywhere near the design flexibility of digital devices is that even if one passes a digital signal through dozens or hundreds of levels of routing and logic circuitry, each of which has a 10dB signal-to-noise ratio (SNR), meaning there's 1/3 as much noise as signal, the resulting signal can be clean. By contrast, getting a clean signal from an analog device requires that every stage the signal goes through must be clean. The more complex the routing, the more difficult it is to avoid picking up stray signals.

In applications that aren't too demanding, having a small amount of analog circuitry combined into a chip can be useful. For example, I've designed a music box which uses a PSoC to drive a piezo speaker directly; the PSoC includes a DAC, a fourth-order low-pass filter, and output amplifier. It wouldn't have been hard to use a separate chip to do the filtering and amplification, but using the PSoC avoided the need for an extra chip.

The moral of the story is digital electronics need an interface to the outside world. Analog electronics are necessary to get the outside world's signals into a form that can be digitized. For example, how do you get that \$\frac{50\mu V}{degree }\$ from a thermocouple into a signal large enough to put into an ADC? Use an instrumentation amplifier.

Here's an example of something I built a while back:

It implements the equation \$V_{out}=1.552V_{in}-0.000560V_{in}^3\$ That is an approximation for sine shaping, the and circuit itself shaped the triangle wave into a sine wave with less than .05% THD. It could have been done in the digital domain, but:

1. the input signal was \$20V_{pp}\$, way to big for immediate conversion without some kind of zero and span, and

2. the signal was already analog and it would have made no sense to convert to digital just to convert back to analog.

On the subject of taking signals digital, an analog anti-aliasing filter is essential before any ADC. This filter is just a low pass filter to ensure \$f_{sig}<f_{Nyquist}\$. That has to be done in the analog domain. Solid analog circuitry is essential to the operation of any embedded system.