Pulse-density modulation, or PDM, is a form of
modulation used to represent an
analog signal with a
binary signal. In a PDM signal, specific
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
values are not encoded into codewords of pulses of different weight as they would be in
pulse-code modulation
Pulse-code modulation (PCM) is a method used to digitally represent sampled analog signals. It is the standard form of digital audio in computers, compact discs, digital telephony and other digital audio applications. In a PCM Stream (comp ...
(PCM); rather, the relative
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of the pulses corresponds to the analog signal's amplitude. The output of a
1-bit DAC A Bitstream or 1-bit DAC is a consumer electronics marketing term describing an oversampling digital-to-analog converter (DAC) with an ''actual'' 1-bit DAC (that is, a simple "on/off" switch) in a delta-sigma loop operating at multiples of the samp ...
is the same as the PDM encoding of the signal.
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. For a 50% voltage 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 is 50% for both waveforms, but the PDM signal switches more often. For 100% or 0% level, they are the same.
Description
In a pulse-density modulation
bitstream
A bitstream (or bit stream), also known as binary sequence, is a sequence of bits.
A bytestream is a sequence of bytes. Typically, each byte is an 8-bit quantity, and so the term octet stream is sometimes used interchangeably. An octet may ...
a 1 corresponds to a pulse of positive polarity (+''A''), and a 0 corresponds to a pulse of negative polarity (−''A''). Mathematically, this can be represented as
:
where ''x''
'n''is the bipolar bitstream (either −''A'' or +''A''), and ''a''
'n''is the corresponding binary bitstream (either 0 or 1).
A run consisting of all 1s would correspond to the maximum (positive) amplitude value, all 0s would correspond to the minimum (negative) amplitude value, and alternating 1s and 0s would correspond to a zero amplitude value. The continuous amplitude waveform is recovered by
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
ing the bipolar PDM bitstream.
Examples
A single
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
of the
trigonometric sine function,
sampled
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of so ...
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
encode
The Encyclopedia of DNA Elements (ENCODE) is a public research project which aims to identify functional elements in the human genome.
ENCODE also supports further biomedical research by "generating community resources of genomics data, software ...
d from an analog signal through the process of
delta-sigma modulation
Delta-sigma (ΔΣ; or sigma-delta, ΣΔ) modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into ...
. 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
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
out the quantization error.
Digital-to-analog conversion
The process of
decoding a PDM signal into an analog one is simple: one only has to pass the PDM signal through a
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
. This works because the function of a low-pass filter is essentially to average the signal. The average amplitude of pulses is measured by the density of those pulses over time, thus a low-pass filter is the only step required in the decoding process.
Relationship to biology
Notably, one of the ways animal nervous systems represent sensory and other information is through
rate coding
Neural coding (or Neural representation) is a neuroscience field concerned with characterising the hypothetical relationship between the stimulus and the individual or ensemble neuronal responses and the relationship among the electrical activity ...
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
A digital model of pulse-density modulation can be obtained from a digital model of the
delta-sigma modulator. Consider a signal