96 kHz.org Advanced Audio Recording Optimized sine wave generation in digital systems For sound generation in music synthesizers and industrial systems, often a precise sine wave is required. There are several methods to do this like the table based approximation used with classical DDS: The sine wave is stored in a RAM/ROM and directly read with an address counter derived from an accumulator representing the phase. These tables can easily be generated by e.g. an Excel sheet or also with MATLAB. Often only 90 degrees of the sine wave are stored and the values for other angles are derived by angle mapping and eventual level inversion. Here we can find two possibilities to store the values:   Representation of the phase vector in DDS Most people use the borders of the focused region when defining the phase such as 0.0, 1.0, 2.0, 3.0, ... (n/4)-1 und repeat the curve backwards starting from the point n/4. This leads to 32 points in this case with the coordinates [0,1,2,3,4,5,6,7] and [8,7,6,5,4,3,2,1] which are repeated aging with inverted sign for angles beyond 180 degrees. This means the fundamental element starts at the angles 0, 90, 180 and 270 and does not include the last point of the area not to get this point twice. So the phases are p = 0, 1/n, 2/n ... (n-1)/n. It is important to mention that the division required to find the relative angle has to be performed with "n" and not with "n-1", what some people do. The dots generated this way, first cover a full period with 17 values / 9 values  0…180° = π. There are two slightly different quarters which are not symmetrical to each other. Another bit is required to represent the phase vector because of the (0…7) / (1…8) issue. From the point of the ease of calculation and symmetry purposes it seems more appropriate to focus the mid of the ranges defined by the digital virtual phase 0 ... n-1. This leads in (this case) to 8 points in each quarter which are totally identical. The digital value 0 addressing the first dot represents it's region by the average value of the region and the value 7 does the same for the region between point 7 and point 8. To get the right value, a correction of Phase/n has to added to the phase vector. Thus the phase values are p = 0,5 , 1,5/n, 2,5/n ... (n-0,5)/n. The vector has the same size for all quarters and saves one bit.   Representation of the amplitude Depending on the integer range used for the scale of the sine wave, it is necessary to use one bit more for just representing the maximum value, especially with small number of points. With the unsymmetrical representation, one should think about if it is really necessary to use the full "2 by n" range. Sometimes it is better to limit the maximum value, for example to maintain symmetry for a certain value. Especially with method 1, there is a true zero point, and the most negative value e.g. "-128" at 8 bits can be left away. Graphical representation of the two possible declarations starting with the angle as a float value in between 0 ... X ... 16, and it's digital "code" from 0 ... 15, the focused sector of the sine wave (please have a look at the special point 4), the relative phase related to the sector origin, the absolute phase in between 360 degrees and the resulting sine waves. The added values in the right row (int) give an impression of the  integral = energy of the curve. The curves represent the 4 used points. The blue curve represents the angle from 0 ... 90 degrees.   Analytical calculation the sine values Y = x * (1 - x) / 4 optimized equation for 16 bit sine wave with max 60 digit of error thus the sine wave has a final precision of 10 Bits    Read the former version of Sine Wave Generation © 2004 J.S.