Optimum sampling frequency for decoding analog PDM stream?

Question

I have an analog value coded as an analog PDM stream, which has a toggling frequency $f_{PDM} \approx$ 100 kHz when the value is at midscale.

I am demodulating this value into a 32 bit result with a simple shift-subtract IIR lowpass filter. Precisely, I am taking the difference between input value $x[n]$ (either 1 or 0) and the filter output value $y[n-1]$. Then I right shift the difference 16 bits and add this to the output:

$$y[n] = y[n-1]+(x[n] - y[n-1]) >>16$$

My first intuition was: Simply sample much faster than the PDM can toggle, and I began with 10 MHz. However, what seemed to happen is that the long consectuive runs of input 1s and 0s appeared to "eat up" the filter dynamic range. Indeed, the result seemed more robust when the PDM was sampled only at 5 MHz.

However, I cannot get too low, because otherwise I would miss pulses when the PDM produces short pulses when the analog value is close to one extreme of its range.

With 5 MHz sampling rate I get about 14 noise-free bits of resolution in the resulting 32 bit number. If I am not mistaken, I should be able to get over 15 noise-free bits (optimally) because I use the 16 lower bits for feeding measurements into the average.

So instead of trial and error I wondered if there is a generic answer to such kind of problem ?

Answer 1

uniform-time ADC-based sampling of PDM

no matter how fast you could sample, your ADC needs an analog anti-aliasing filter. That's the key here:

Your AA filtering inherently "sums up" the past pulses. You don't sample the PDM signal itself, you sample the output of that filter, which will be higher, the more pulses occurred lately. So, pick an anti-aliasing filter that's appropriate for the bandwidth of the analog signal, and filter your PDM with that: the output will be a relatively faithful reconstruction of the original analog signal. Sample that with the ADC, at a sample rate defined by the bandwidth of the analog signal, so, significantly below 100 kHz, not at 10 MHz (wowzers!).

(I'll invite you to model the actual output of this stochastically: Usually, you'll end up saying "OK, this is the sum of the same impulse response sampled at different times, where the times are independently random, so this is a sum of $N$ i.i.d. variables of known variance, and we know that this has $N$ times that variance, and thus, we get, on expectation, something proportional to the PDM-modulated analog signal, added to noise, which is the more Gaussian, the higher $N$ is (thanks to the central limit theorem)", but there's a couple important assumptions you'd be making on the way. Especially, this quantization noise is not white!)

Frequency counter sampling

The above method is of course highly inefficient. What you usually do (and microcontrollers have timer/counter units designed exactly for this kind of thing, and it's trivial to implement in digital logic) is simply have a counter that counts the pulses (e.g. by counting rising edges). You then read that counter periodically. You can then either reset it to 0 when you read, or just generally subtract the last count value, and get a sequence of numbers representing how many pulses occurred since last counter readout.

Now, the number of pulses in a time period is proportional to the analog amplitude prior to PDM, so you then low-pass filter that to get a discrete-time representation of the analog input signal. The quantization noise is differently shaped in spectrum compared to the method above.

Tying it together: $\Delta\Sigma$, friends!

analog signal -> PDM -> Pulse Counting -> count sampling -> digital filtering

order of
magni-
tude of
bandw.: kHz'es   MHz        MHz              kHz               kHz

That's practically a Delta-Sigma converter, which is probably the most prolific ADC architecture in the world! Every midrange microcontroller has one, your phone's microphones are sampled with such, so is its ambient brightness sensor, and so are probably the oscillations in the tiny MEMS accelerometer observed. So, I think, reading up how a Delta-Sigma converter works, how its noise is shaped, and why you'd want that: Your

Answer 2

I'm not familiar with PDM, but the rule of sampling in general is to have a sample rate at least twice faster as the maximum frequency of the signal want to digitalize Nyquist-Shannon Theorem to have zero ambiguity.

You could oversample with high frequency first and decimate at the right time when the signal rises or fall. And maybe it is inconsistent because your samples "slide" on your PDM signal because the sample rate is not sync on the bitrate, wich is accentuated with high sample rate.
I'm not very experimented so maybe the last advice is complete garbage, it was more of a thought or an intuition.