Pulse-density modulation

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Pulse-density modulation, or PDM, is a form of modulation used to represent an analog signal in the digital domain. In a PDM signal, specific amplitude values are not encoded into pulses as they would be in PCM. Instead it is the relative density of the pulses that corresponds to the analog signals 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.

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[edit] Basics

In a pulse-density modulation 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:

x[n] = -A (-1)^{a[n]} \
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 filtering the bipolar PDM bitstream.

[edit] Analog-to-digital conversion

A PDM bitstream is encoded 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.

[edit] Digital-to-analog conversion

The process of decoding a PDM signal into an analog one is amazingly simple. One only has to pass that signal through an analog low-pass filter. This works because the function of a low-pass filter is essentially to average the signal. The density of pulses is measured by the average amplitude of those pulses over time, thus a low pass filter is the only step required in the decoding process.

[edit] Examples

A single period of the trigonometric sine function, sampled 100 times and represented as a PDM bitstream, is:

0101011011110111111111111111111111011111101101101010100100100000010000000000000000000001000010010101

An example of PDM of 100 samples of one period a sine wave. 1s represented by blue, 0s represented by white, overlaid with the sine wave.
An example of PDM of 100 samples of one period a sine wave. 1s represented by blue, 0s represented by white, overlaid with the sine wave.

Two periods of a higher frequency sine wave would appear as: 0101101111111111111101101010010000000000000100010011011101111111111111011010100100000000000000100101 or Image:Pulse-density modulation 2 periods.gif

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.

[edit] Algorithm

Pulse-density modulation of a sine wave using this algorithm.
Pulse-density modulation of a sine wave using this algorithm.

A digital model of pulse-density modulation can be obtained from a digital model of the delta-sigma modulator. Consider a signal x[n] in the discrete time domain as the input to a first-order delta-sigma modulator, with y[n] the output. In the discrete frequency domain, the delta-sigma modulator's operation is represented by

Y(z)=E(z) + \left[ X(z)-Y(z)z^{-1} \right] \left( \frac{1}{1-z^{-1}} \right).

Rearranging terms, we obtain

Y(z)=X(z)+E(z)\left(1-z^{-1}\right)

Here, E(z) is the frequency-domain quantization error of the delta-sigma modulator. The factor 1 − z − 1 represents a high-pass filter, so it is clear that E(z) contributes less to the output Y(z) 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,

y[n] = x[n] + e[n] − e[n − 1]

There are two additional constraints to consider: first, at each step the output sample y[n] is chosen so as to minimize the "running" quantization error e[n]. Second, y[n] is represented as a single bit, meaning it can take on only two values. We choose y[n]=\pm 1 for convenience, allowing us to write

y[n] = \begin{cases} +1 & x[n]\geq e[n-1] \\ -1 & x[n]<e[n-1]\end{cases}
e[n] = y[n] − x[n] + e[n − 1]

This, finally, gives a formula for the output sample y[n] in terms of the input sample x[n]. 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 first-order noise shaping

function pdm(real[0..s] x)
  var int[0..s] y
  var int[-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

[edit] Applications

PDM is the encoding used in Sony's Super Audio CD (SACD) format, under the name Direct Stream Digital.

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[edit] See Also

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