Table Lookup Oscillators
Often it is much more efficient to create a waveform using a lookup table as opposed to calculating the output directly. This is most often the case with the most acoustically simple waveform, the sine wave. One can calculate a sine wave from a phasor by multiplying the phasor by 2π, the number of radians in a circle, such that the phasor now extends from 0 to 2π. That value is then used in a sine operation such that the final equation looks like this:
f(x) = sin(\phi2\pi)or:
sin(phasor(f)*2.0*PI);
where `PI` is the value of π and `f` is the frequency in Hz of the waveform. This, however, means that the sine of the phasor is calculated for every sample. On the other hand, if we populate a table with a single cycle of a precomputed sine wave, all that needs to be done is to index into the table at the rate of the frequency of the sine wave desired.
static double 2PI = 3.14159 * 2.0; // two pi
int tablesize = 1024; // size of the table
double *sinetable = malloc(tablesize); // allocate a table
// fill the table
for(int i = 0; i < tablesize; i++)
sinetable[i] = sin((2PI*i)/tablesize);
Using a tablesize of 1024, the table looks like this:
To index into this table, we need to use integers. If we take a phasor at some desired frequency, we can take the output and multiply it by the size of the table minus one (since in most languages we start counting from 0) and cut off the fractional part to obtain an index:
i = \lfloor \phi * s \rfloorwhere s is the size of the table and phi is the value of the phasor from 0 to 1. In code:
index = int(phasor(f)*tablesize);
This method of simply removing the fractional part of the phase is called phase truncation. We can then use this output to index into the table and get the correct value for the sine wave. Below are the first 512 samples of a 100Hz phasor indexing into a wavetable of size 1024 at an 8kHz samplerate:
