Pulse modulation: PDM vs PFM vs Delta-Sigma

[tooki]

So I've been trying to wrap my head around the practical differences between them and what the relationship/hierarchy of them is. (Question popped in my mind while experimenting with PWM LED control, where flicker easily becomes a problem even at frequencies that seem sufficient on paper, like 400Hz. PDM reduces flicker appreciably at lower light levels, by increasing the LED's pulse frequency.)

What exactly would the difference between Pulse Density Modulation (PDM) and Pulse Frequency Modulation (PFM) be? Both use fewer pulses to encode low values and frequent pulses to encode higher values. All I can come up with is that PDM perhaps requires the pulses to align to a clock signal, while PFM doesn't? Is delta-sigma a subset of PDM or vice-versa?

I'm hoping you guys can educate me! Thanks!

[ade]

Frequency vs. Density.

In PFM, different values are encoded as different frequencies.  Imagine a system where a signal level of 0% is represented by a square-wave signal modulating at 8 Hz, and 100% is represented by the square-wave signal modulating at 10 Hz.  (I picked the frequencies arbitrarily).   With this system you can represent digital 0s and 1s as a stream of square-wave signals at 8 Hz and 10 Hz respectively.   Or, you can represent the value of 0.5 (50%) with a 9 Hz signal.  You can even represent a -1 with a stream at 6 Hz.

So in this contrived square-wave PFM encoding:

6 Hz: -1
7 Hz: -0.5
8 Hz: 0
9 Hz: 0.5
10 Hz: 1

In PDM, however, the values are not represented by frequency, but by density: the ratio when the signal is "high" (on) vs. "low" (off) over a period of time.

Notice that all of the PFM square-waves above have the same density: 50% on : 50% off.  So all represent the same 50% value in PDM (given the range from -1 to 1, 50% might represent the value of 0).  Increasing or decreasing the frequency has no effect.  To reduce the value you need to make the signal more sparse.  Conversely to increase the value you need the signal more dense.


(PDM example from Wikipedia)

If you fix the PDM frequency, and vary the density by changing the duty-cycle, then you have PWM (Pulse Width Modulation).

Delta-Sigma is a method of modulation.   The result of Delta-Sigma is a PDM stream.

[tooki]

That all makes perfect sense. I think what threw me off is this sentence from the wiki article on PFM: "The amplitude and width of the pulses is kept constant." As I pictured it, if the pulse width (not duty cycle!) is kept constant, and the frequency changes, then the duty cycle must change in proportion and thus becomes equivalent to PDM. You're saying that in PFM, the duty cycle remains fixed, so the wiki article must be incorrect about the pulse width being fixed. Correct?

[ade]

In PFM the duty cycle does not have to be fixed.  It can be (as in the simple example above), but they key aspect is that the pulse frequency is being modulated, that's all.

The Wikipedia article isn't wrong per se.  In fact it is common to see PFM with a fixed, short pulse width.  Here's an example:


(Example from digikey.com)

Here what's changing is the "gap" between the pulses, increasing or decreasing the frequency over time.  In this implementation PFM is similar -- but not exactly the same -- as PDM.

[tooki]

How is PFM with a fixed pulse width different in any way from PDM, then?!? The frequency and the "gap between pulses"+pulse width (i.e. the period) are by definition reciprocal, are they not? Not trying to be argumentative, just not seeing any difference.

[tggzzz]

For PWM vs delta-sigma the answer is the power at different frequencies, and the ease of filtering out the high frequency noise with a low pass filter.

Consider two signals with equal mean values: 111111110000000 and 1010101010100.

[tooki]

Indeed, but my question wasn't about PWM, which is widely documented. PDM is much more sparsely documented, and PFM even less. (I bumped into PFM in Forrest Mims' communications projects engineer's notebook I got at radio shack a few decades ago!)

[ade]

If the input is held high (100%) then the PDM output simply follows: a flat line high (1) with frequency of zero.

If the input is held low (0%) then the PDM output also follows: a flat line low (0) with frequency of zero.

This is clearly different from PFM where the frequency varies according to the input level, as described previously.


(PDM from Wikipedia)

From the above you can see that for PDM the frequency varies to the derivative of the signal.  For sine waves this means the frequency varies by the cosine function, or shifted 90 degrees from the original signal.

[tooki]

From the above you can see that for PDM the frequency varies to the derivative of the signal.  For sine waves this means the frequency varies by the cosine function, or shifted 90 degrees from the original signal.

Oh dear. I'm afraid this exceeds my mathematics skills! (If I ever knew this, it's long forgotten!)