Pulse Density Modulation for software DACs

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Pulse Density Modulation for software DACs
difference between the ideal analogue output and the actual digital output:
Q(t) = D(t) - A(t)
We can then define the quantization error integral, which is the sum of all the quantization errors from the first interrupt interval to the current one:
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Pulse Density Modulation for software DACs
0x4f, 0x4c, 0x49, 0x46, 0x43, 0x41, 0x3e, 0x3b, 0x39, 0x36, 0x34, 0x31, 0x2f, 0x2c, 0x2a, 0x28, 0x25, 0x23, 0x21, 0x1f, 0x1d, 0x1b, 0x19, 0x17, 0x15, 0x14, 0x12, 0x11, 0x0f, 0x0e, 0x0c, 0x0b, 0x0a, 0x09, 0x07, 0x06, 0x05, 0x05, 0x04, 0x03, 0x02, 0x02, 0x01, 0x01, 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01, 0x01, 0x01, 0x02, 0x02, 0x03, 0x04, 0x05, 0x05, 0x06, 0x07, 0x09, 0x0a, 0x0b, 0x0c, 0x0e, 0x0f, 0x11, 0x12, 0x14, 0x15, 0x17, 0x19, 0x1b, 0x1d, 0x1f, 0x21, 0x23, 0x25, 0x28, 0x2a, 0x2c, 0x2f, 0x31, 0x34, 0x36, 0x39, 0x3b, 0x3e, 0x41, 0x43, 0x46, 0x49, 0x4c, 0x4f, 0x52, 0x55, 0x58, 0x5a, 0x5d, 0x61, 0x64, 0x67, 0x6a, 0x6d, 0x70, 0x73, 0x76, 0x79, 0x7c,
PDM-OUTPUT(input): error := error + input
IF error > SIGNAL-MAX: error := error - SIGNAL-MAX RETURN 1
RETURN 0
Here, we keep track of the current value of the integral in the variable error. The analog signal is assumed to have a range of [0..SIGNAL-MAX], and we keep the quantization error to less than the maximum possible over one interrupt period.
static const unsigned char sine_table[] = { 0x80, 0x83, 0x86, 0x89, 0x8c, 0x8f, 0x92, 0x95, 0x98, 0x9b, 0x9e, 0xa2, 0xa5, 0xa7, 0xaa, 0xad, 0xb0, 0xb3, 0xb6, 0xb9, 0xbc, 0xbe, 0xc1, 0xc4, 0xc6, 0xc9, 0xcb, 0xce, 0xd0, 0xd3, 0xd5, 0xd7, 0xda, 0xdc, 0xde, 0xe0, 0xe2, 0xe4, 0xe6, 0xe8, 0xea, 0xeb, 0xed, 0xee, 0xf0, 0xf1, 0xf3, 0xf4, 0xf5, 0xf6, 0xf8, 0xf9, 0xfa, 0xfa, 0xfb, 0xfc, 0xfd, 0xfd, 0xfe, 0xfe, 0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe, 0xfe, 0xfe, 0xfd, 0xfd, 0xfc, 0xfb, 0xfa, 0xfa, 0xf9, 0xf8, 0xf6, 0xf5, 0xf4, 0xf3, 0xf1, 0xf0, 0xee, 0xed, 0xeb, 0xea, 0xe8, 0xe6, 0xe4, 0xe2, 0xe0, 0xde, 0xdc, 0xda, 0xd7, 0xd5, 0xd3, 0xd0, 0xce, 0xcb, 0xc9, 0xc6, 0xc4, 0xc1, 0xbe, 0xbc, 0xb9, 0xb6, 0xb3, 0xb0, 0xad, 0xaa, 0xa7, 0xa5, 0xa2, 0x9e, 0x9b, 0x98, 0x95, 0x92, 0x8f, 0x8c, 0x89, 0x86, 0x83, 0x80, 0x7c, 0x79, 0x76, 0x73, 0x70, 0x6d, 0x6a, 0x67, 0x64, 0x61, 0x5d, 0x5a, 0x58, 0x55, 0x52,
sine_table[i] = sin(i * 2.0 * M_PI / 256) * 127.5 + 127.5;
Now we introduce a function to cyclically step through the table and retrieve appropriate values at each interrupt period. The actual frequency is controlled by the parameter sine_freq.
Daniel Beer dlbeer@ 8 Mar 2011
A common method of generating an analog output signal using a digital microcontroller is to filter a pulse-width modulated digital output. A pulse-width modulator is usually implemented in hardware by having a counter run freely, with triggers set to turn off the output when the counter rolls overflows its maximum value, and turn on when the counter reaches the software-set threshold value.
Qint(t) = integral{0, t} Q(t)
PDM seeks to minimze Qint(t)at each step, or to at least hold it within a small bounded range. A simple algorithm for this is:

/* Sine table. 256 entries representing one complete cycle. The signal * is scaled so that it fits the range representable with an 8-bit * unsigned integer. */
PDM is a method of continuously varying the average value of a digital signal, but without having a fixed modulation period, as in PWM. Here's an example that shows a single cycle of a sine wave rendered as a digital signal using PDM:
Most microcontrollers have hardware built-in to do this, but when it's not available, it can be done in software. Y ou have an interrupt called at the same frequency as the hardware counter would have counted, implement the same logic using a threshold and a counter variable, and switch the output pin on and off accordingly. T he only problem with this is the computing resources used - an 8-bit PWM would require 256 counts per PWM cycle. If you were to produce an output signal at the modest sampling rate of 1 kHz, you'd need to handle 256,000 interrupts per second - quite a lot of work for a small microcontroller.
This table is small enough to fit into most microcontrollers' ROMs, and is scaled and shifted to make PDM easier. The actual formula used to generate the above table is: