Pulse-density modulation (PDM) is a form of modulation used to represent an analog signal with a binary signal. In a PDM signal, specific amplitude values are not encoded into codewords of pulses of different weights as they would be in pulse-code modulation (PCM); rather, the relative density of the pulses corresponds to the analog signal's amplitude. 1-bit DAC is an example of a PDM application.
Description In a pulse-density modulation bitstream, a 1 {\displaystyle 1} corresponds to a pulse of positive polarity ( + A {\displaystyle +A} ), and a 0 {\displaystyle 0} corresponds to a pulse of negative polarity ( − A {\displaystyle -A} ). Mathematically, this can be represented as
x [ n ] = − A ( − 1 ) a [ n ] , {\displaystyle x[n]=-A(-1)^{a[n]},}
where x [ n ] {\displaystyle x[n]} is the bipolar bitstream (either − A {\displaystyle -A} or + A {\displaystyle +A} ), and a [ n ] {\displaystyle a[n]} is the corresponding binary bitstream (either 0 {\displaystyle 0} or 1 {\displaystyle 1} ). A run consisting of all 1 {\displaystyle 1} s would correspond to the maximum (positive) amplitude value, all 0 {\displaystyle 0} s would correspond to the minimum (negative) amplitude value, and alternating 1 {\displaystyle 1} s and 0 {\displaystyle 0} s would correspond to a zero amplitude value. The continuous amplitude waveform is recovered by low-pass filtering the bipolar PDM bitstream.
Examples A single period of the trigonometric sine function, sampled 100 times and represented as a PDM bitstream, is: 0101011011110111111111111111111111011111101101101010100100100000010000000000000000000001000010010101
Two periods of a higher frequency sine wave would appear as: 0101101111111111111101101010010000000000000100010011011101111111111111011010100100000000000000100101
In pulse-density modulation, a high density of 1s occurs at the peaks of the sine wave, while a low density of 1s occurs at the troughs of the sine wave.
Analog-to-digital conversion
A PDM bitstream is encoded from an analog signal through the process of a 1-bit delta-sigma modulation. This process uses a one-bit quantizer that produces either a 1 or 0 depending on the amplitude of the analog signal. A 1 or 0 corresponds to a signal that is all the way up or all the way down, respectively. Because in the real world, analog signals are rarely all the way in one direction, there is a quantization error, the difference between the 1 or 0 and the actual amplitude it represents. This error is fed back negatively in the ΔΣ process loop. In this way, every error successively influences every other quantization measurement and its error. This has the effect of averaging out the quantization error, while noise shaping it to push most of the quantization error into higher frequencies, which for audio signals would be ultrasonic.
PDM-to-PCM conversion Decimation is needed to convert a PDM signal from its very high sampling rate (e.g. some PDM mics may sample between 1 MHz to 3.25 MHz) to the much lower PCM sampling rate (which for audio may range between 16 kHz to 48 kHz).
Digital-to-analog conversion The frequency components of interest, for example the audio frequency range, are much lower than the PDM's very high sampling rate. So, the process of converting a PDM signal into an analog one is simple: one only has to pass the PDM signal through a low-pass filter. Because the delta-sigma modulator had pushed most quantization noise into higher frequencies, low-pass filtering removes the high-frequency quantization noise while keeping the lower-frequency signal of interest.
Relationship to PWM Pulse-width modulation (PWM) is a special case of PDM where the switching frequency is fixed and all the pulses corresponding to one sample are contiguous in the digital signal. The method for demodulation to an analogue signal remains the same, but the representation of a 50% signal with a resolution of 8 bits, a PWM waveform will turn on for 128 clock cycles and then off for the remaining 128 cycles. With PDM and the same clock rate the signal would alternate between on and off every other cycle. The average obtained by a low-pass filter is 50% of the maximum signal level for both waveforms, but the PDM signal switches more often. For 100% or 0% level, they are the same, with the signal permanently on or off respectively.
Relationship to biology Notably, one of the ways animal nervous systems represent sensory and other information is through rate coding whereby the magnitude of the signal is related to the rate of firing of the sensory neuron. In direct analogy, each neural event – called an action potential – represents one bit (pulse), with the rate of firing of the neuron representing the pulse density.
Algorithm
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