The 101 of analog signal filtering

The conceptually simplest to understand filter for beginners however would be the moving average filter. Just keep a list of the last n samples, and average them. When a new sample comes in, throw out the oldest.

In the end many low pass filters are some variant of this (adding one or more samples from the past to the current sample using certain weights, maybe with feedback thrown into the mix).

in order to get to where you don't have very high frequencies, you need analog filtering on the input and the output, and you need adcs and dacs suitable for the application. you might also need attenuation, amplification, impedance transformation, and/or mixing on the input and output. all this stuff is analog and requires this kind of analysis to understand

but whenever you can do stuff in the digital domain, you do, because you can easily reduce the error introduced by digital computation as small as you want, and you can redesign your signal processing after your space probe has passed neptune so a soldering iron would be inconvenient. and you can fit arbitrarily complex filtering and other signal processing into an arbitrarily small package, near enough; what would be another precision capacitor in an analog circuit is just another 16-bit coefficient loaded from your terabyte microsd card in a dsp setup, if you're processing a slow enough signal

You get ease of prototyping,modification, and excellent performance. In these systems there may be still analog filter components(anti-aliasing,reconstruction), but with digital components so cheap,good, and fast this "mostly digital" technique is almost always the best option.

But if the signal you're measuring is very weak (or poorly matched to the ADC range), you probably need some kind of amplifier for it anyway, and in that case you may as well make a filter out of it at the same time to maximize the signal-to-noise ratio.

Depends what you are getting trained for. An acceptable cobbler who can patch up a broken shoe into something usable would have different pedagogic needs than one who is training to become a designer of the next-gen shoe. A vibrant economy needs both kinds.

That said, a big part of 'hacking' used to be about the pleasure of deeply understanding a topic, not necessarily just as means of earning or employment.

Was it Feynman who said something about whats the utility of physics and why do people engage in it ? I am paraphrasing his comment from faulty memory: sex is useful for reproduction but that's not why we engage in it.

Many here are into it for that other reason of hacking.

I am going to point out that repeatedly asking “why?” will get you into very mathy weeds very quickly, such that “foul” calculus will be the least of your problems. Like here:

> As it turns out, most analog filters are relatively distortion-free only in one special case — a perfectly steady sine waveform[.]

Ever wonder why? And not just in the sense of showing that’s the right answer, in the sense of setting off without the knowledge of the answer and deriving it. Well, the starting point is that the filter is linear and time-translation-invariant (i.e. doesn’t have a clock built into it); the fact that it is causal is somewhat important but not immediately so. But then things get interesting, and if you’re lucky you’ll get to hear such wonderful phrases as “group characters” and “representation theory”.

I don’t blame people who got burned by their math courses, but I do want to emphasize that these kinds of “why?” questions are exactly what math is there for. If the math book in front of you doesn’t look like it works that way, take another one and try again—just don’t expect the specific question you have to be answered immediately.

(I’m assuming I don’t have to explain the virtues of repeatedly asking “why?” on a forum called Hacker News.)

The response of a capacitor or inductor is a differential equation with an exponential solution. The Laplace transform replaces the exponentials with s = iω, where ω just means "a sine wave at some frequency".

(It's actually jω in EE, because reasons, but it means the same thing.)

So if you build a network of capacitors, resistors, and inductors, or springs and masses, or anything else that uses the same maths, you can turn it into a simple algebraic equation in s.

This is called the transfer function. It literally defines the frequency and phase response with some simple algebra.

In practice you don't bother with the calculus and just go straight to s. A single capacitor filter is 1/(1+sRC). Add more stages - the rules are pretty simple - and you get a more complex equation.

Then you can put ω = 200Hz or whatever into the transfer function, and you get a complex number out. This tells how big the output is at that frequency, and how much it's phase shifted.

If you graph ω, you get the complete frequency and phase response.

Really all you're doing is replacing a static DC test voltage with a sine wave "probe" across the frequencies you're interested in.

There's a bit more to it - which is where poles and zeroes appear - but not much.

(It's also related to the Fourier transform, but that's a whole other thing.)

It's incredibly impressive that all of this was invented in the early 19th century by a handful of French aristocrats and a couple of Germans, more or less as a hobby.