Pulse-density modulation
Passband modulation |
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Analog modulation |
Digital modulation |
Hierarchical modulation |
Spread spectrum |
See also |
Pulse-density modulation, or PDM, is a form of
Description
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
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.
Examples
A single
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.
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. 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 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
Algorithm
The following digital model of pulse-density modulation can be obtained from a digital model of a 1st-order 1-bit
where is the frequency-domain
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
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
Rearranging to solve for yields:
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, real qe = 0) // initial running error is zero var int[0..s] y for n from 0 to s do qe := qe + x[n] if qe > 0 then y[n] := 1 else y[n] := −1 qe := qe - y[n] return y, qe // return output and running error
Applications
PDM is the encoding used in Sony's Super Audio CD (SACD) format, under the name Direct Stream Digital.
PDM is also the output of some
Some systems transmit PDM
See also
References
- ^ Fried, Limor (2018-01-10). "Adafruit PDM Microphone Breakout". Adafruit Learning System. Archived from the original on 2022-12-08. Retrieved 2023-06-30.
- ^ Thomas Kite. "Understanding PDM Digital Audio" (PDF). 2012. The "PDM Microphones" section on p. 6.
- ^ Maxim Integrated. "PDM Input Class D Audio Power Amplifier" (PDF). 2013. Figure 1 on p. 5; and the "Digital Audio Interface" section on p. 13.
- ^ Knowles. "SPK0641 Digital, CMOS MEMS Microphone" (PDF).
Further reading
- 1-bit A/D and D/A Converters – Discusses delta modulation, PDM (also known as Sigma-delta modulation or SDM), and relationships to Pulse-code modulation (PCM)
- Kite, Thomas (2012). "Understanding PDM Digital Audio" (PDF). Audio Precision. Retrieved 19 January 2017.