Pulse-density modulation
Encyclopedia
Pulse-density modulation, or PDM, is a form of modulation
used to represent an analog signal
with digital data. In a PDM signal, specific amplitude
values are not encoded into pulses of different size as they would be in PCM
. Instead, it is the relative density
of the pulses that corresponds to the analog signal's amplitude. Pulse-width modulation
(PWM) is the special case of PDM where all the pulses corresponding to one sample are contiguous in the digital signal.
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:
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
ing the bipolar PDM bitstream.
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
or
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.
d from an analog signal through the process of 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.
a PDM signal into an analog one is simple: one only has to pass the PDM signal through a low-pass filter
. 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.
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.
A digital model of pulse-density modulation can be obtained from a digital model of the delta-sigma modulator. Consider a signal 
in the discrete time
domain as the input to a first-order delta-sigma modulator, with
the output. In the discrete frequency
domain, the delta-sigma modulator's operation is represented by
Rearranging terms, we obtain
Here,
is the frequency-domain quantization error
of the delta-sigma modulator. The factor
represents a high-pass filter
, so it is clear that
contributes less to the output 
at low frequencies, and more at high frequencies. This demonstrates the noise shaping
effect of the delta-sigma modulator: the quantization noise is "pushed" out of the low frequencies up into the high-frequency range.
Using the inverse Z-transform
, we may convert this into a difference equation relating the input of the delta-sigma modulator to its output in the discrete time
domain,
There are two additional constraints to consider: first, at each step the output sample
is chosen so as to minimize the "running" quantization error 
. Second, 
is represented as a single bit, meaning it can take on only two values. We choose 
for convenience, allowing us to write
This, finally, gives a formula for the output sample
in terms of the input sample 
. The quantization error of each sample is fed back
into the input for the following sample.
The following pseudo-code implements this algorithm to convert a pulse-code modulation
signal into a PDM signal:
// Encode samples into pulse-density modulation
// using a first-order sigma-delta modulator
function pdm(real[0..s] x)
var int[0..s] y
var real[-1..s] qe
qe[-1] := 0 // initial running error is zero
for n from 0 to s
if x[n] >= qe[n-1]
y[n] := 1
else
y[n] := -1
qe[n] := y[n] - x[n] + qe[n-1]
return y, qe // return output and running error
(SACD) format, under the name Direct Stream Digital
.
Modulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a high-frequency periodic waveform, called the carrier signal, with a modulating signal which typically contains information to be transmitted...
used to represent an analog signal
Analog signal
An analog or analogue signal is any continuous signal for which the time varying feature of the signal is a representation of some other time varying quantity, i.e., analogous to another time varying signal. It differs from a digital signal in terms of small fluctuations in the signal which are...
with digital data. In a PDM signal, specific amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
values are not encoded into pulses of different size as they would be in PCM
Pulse-code modulation
Pulse-code modulation is a method used to digitally represent sampled analog signals. It is the standard form for digital audio in computers and various Blu-ray, Compact Disc and DVD formats, as well as other uses such as digital telephone systems...
. Instead, it is the relative density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of the pulses that corresponds to the analog signal's amplitude. Pulse-width modulation
Pulse-width modulation
Pulse-width modulation , or pulse-duration modulation , is a commonly used technique for controlling power to inertial electrical devices, made practical by modern electronic power switches....
(PWM) is the special case of PDM where all the pulses corresponding to one sample are contiguous in the digital signal.
Description
In a pulse-density modulation bitstreamBitstream
A bitstream or bit stream is a time series of bits.A bytestream is a series of bytes, typically of 8 bits each, and can be regarded as a special case of a bitstream....
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
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
ing the bipolar PDM bitstream.
Examples
A single periodPeriodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
of the trigonometric sine function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
, sampled 100 times and represented as a PDM bitstream, is:
0101011011110111111111111111111111011111101101101010100100100000010000000000000000000001000010010101
0101101111111111111101101010010000000000000100010011011101111111111111011010100100000000000000100101
or
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 encodeEncode
Encode may refer to:* Can be related to "Code"* Encode ApS, a Danish software company* Encode SA, a Greek information security company* ENCODE, the ENCyclopedia Of DNA Elements...
d from an analog signal through the process of delta-sigma modulation
Delta-sigma modulation
Delta-sigma modulation is a method for encoding high-resolution or analog signals into lower-resolution digital signals. The conversion is done using error feedback, where the difference between the two signals is measured and used to improve the conversion...
. This process uses a one bit quantizer
Quantization (signal processing)
Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...
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
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
out the quantization error.
Digital-to-analog conversion
The process of decodingDecoding
Decoding is the reverse of encoding, which is the process of transforming information from one format into another. Information about decoding can be found in the following:* Digital-to-analog converter, the use of analog circuit for decoding operations...
a PDM signal into an analog one is simple: one only has to pass the PDM signal through a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
. 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 codingRate coding
The rate coding model of neuronal firing communication states that as the intensity of a stimulus increases the frequency or rate of action potentials, or "spike firing", increases. Rate coding is sometimes called frequency 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
in the discrete timeDiscrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...
domain as the input to a first-order delta-sigma modulator, with
the output. In the discrete frequencyDiscrete frequency
A discrete frequency domain is a frequency domain that is discrete rather than continuous.For example, the discrete Fourier transform maps a function having a discrete time domain into one having a discrete frequency domain...
domain, the delta-sigma modulator's operation is represented by
Rearranging terms, we obtain
Here,
is the frequency-domain quantization errorQuantization error
In analog-to-digital conversion, the difference between the actual analog value and quantized digital value is called quantization error or quantization distortion. This error is either due to rounding or truncation...
of the delta-sigma modulator. The factor
represents a high-pass filterHigh-pass filter
A high-pass filter is a device that passes high frequencies and attenuates frequencies lower than its cutoff frequency. A high-pass filter is usually modeled as a linear time-invariant system...
, so it is clear that
contributes less to the output at low frequencies, and more at high frequencies. This demonstrates the noise shapingNoise shaping
Noise shaping is a technique typically used in digital audio, image, and video processing, usually in combination with dithering, as part of the process of quantization or bit-depth reduction of a digital signal...
effect of the delta-sigma modulator: the quantization noise is "pushed" out of the low frequencies up into the high-frequency range.
Using the inverse Z-transform
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....
, we may convert this into a difference equation relating the input of the delta-sigma modulator to its output in the discrete time
Discrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...
domain,
There are two additional constraints to consider: first, at each step the output sample
is chosen so as to minimize the "running" quantization error . Second, is represented as a single bit, meaning it can take on only two values. We choose for convenience, allowing us to writeThis, finally, gives a formula for the output sample
in terms of the input sample . The quantization error of each sample is fed backNegative feedback
Negative feedback occurs when the output of a system acts to oppose changes to the input of the system, with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...
into the input for the following sample.
The following pseudo-code implements this algorithm to convert a pulse-code modulation
Pulse-code modulation
Pulse-code modulation is a method used to digitally represent sampled analog signals. It is the standard form for digital audio in computers and various Blu-ray, Compact Disc and DVD formats, as well as other uses such as digital telephone systems...
signal into a PDM signal:
// Encode samples into pulse-density modulation
// using a first-order sigma-delta modulator
function pdm(real[0..s] x)
var int[0..s] y
var real[-1..s] qe
qe[-1] := 0 // initial running error is zero
for n from 0 to s
if x[n] >= qe[n-1]
y[n] := 1
else
y[n] := -1
qe[n] := y[n] - x[n] + qe[n-1]
return y, qe // return output and running error
Applications
PDM is the encoding used in Sony's Super Audio CDSuper Audio CD
Super Audio CD is a high-resolution, read-only optical disc for audio storage. Sony and Philips Electronics jointly developed the technology, and publicized it in 1999. It is designated as the Scarlet Book standard. Sony and Philips previously collaborated to define the Compact Disc standard...
(SACD) format, under the name Direct Stream Digital
Direct Stream Digital
Direct-Stream Digital is the trademark name used by Sony and Philips for their system of recreating audible signals which uses pulse-density modulation encoding, a technology to store audio signals on digital storage media which is used for the Super Audio CD .The signal is stored as delta-sigma...
.
External links
- 1-bit A/D and D/A Converters — Discusses delta modulationDelta modulationDelta modulation is an analog-to-digital and digital-to-analog signal conversion technique used for transmission of voice information where quality is not of primary importance. DM is the simplest form of differential pulse-code modulation where the difference between successive samples is encoded...
, PDM (also known as Sigma-delta modulation or SDM), and relationships to Pulse-code modulationPulse-code modulationPulse-code modulation is a method used to digitally represent sampled analog signals. It is the standard form for digital audio in computers and various Blu-ray, Compact Disc and DVD formats, as well as other uses such as digital telephone systems...
(PCM)