# Frequency modulation implementation without trig functions

https://dsp.stackexchange.com/a/1087 This answer suggests generating a sine wave using complex phasor multiplication. At each step the phasor is multiplied by another complex number (omega) to rotate it by the desired step. Then the imaginary part of the phasor is a sine wave. Can this be extended further to implement frequency modulation without having to use a trig function as part of updating the angular frequency? I've played with it a bit, but haven't figured out a clever way to remove the trig function when updating the frequency.

My initial idea: the phasor would be updated like so:

phasor = phasor * (omega * modulator)

The modulator would sweep back and forth between exp(j*freq_mod_max) and exp(j*freq_mod_min). I haven't figured out how the modulator would be implemented.

Side note: why would you want to do this? I did some quick and dirty performance tests and using this phasor multiplication method is quicker for generating a sine wave (about 3-4x quicker). Though calling the sine function directly is still of course quick enough so this is a somewhat academic curiosity at this point.

This is what an NCO (Numerically Controlled Oscillator) does. See this post for more details on the implementation and design considerations Numerically Controlled Oscillator (NCO) for phasor implementation?

By combining this with a D/A converter for an analog output we get what is commonly referred to as a Direct Digital Synthesizer (DDS).

With an NCO you can it you can easily do FM, PM and if you add an amplitude control at the output AM as well! To do FM specifically, you update the Frequency Control Word on each cycle according to the frequency vs time desired.

Since your question indicated a concern with efficiency; there has been a lot of work done in optimizing this approach to be highly efficient as well in terms of minimizing ROM storage. This includes more obvious approaches such as storing only 1/4 cycle of a sine wave, but also interpolation approaches are very common and other approaches include the Hutchinson Algorithm and Sunderland Algorithm.

The Hutchinson Algorithm uses a coarse ROM for a major phase step and fine ROM for a smaller increment and makes use of the small angle approximation:

$$\theta =\theta_F+\theta_C$$ $$sin(\theta) = sin(\theta_F)cos(\theta_C)+cos(\theta_F)sin(\theta_C)$$

for small anlge $$\theta_F$$:

$$sin(\theta) = \theta_Fcos(\theta_C)+ sin(\theta_C)$$

Thanks Hutchinson!

Also if you have time to spare in the implementation (meaning do not require a final update on each cycle), the CORDIC rotator is also quite interesting in terms of highly efficient approaches for implementing an NCO (or any other applications requiring phase rotation or trigonometric solutions).

References on the CORDIC:

Andraka, Ray, “A survey of CORDIC algorithms for FPGA based computers”, Proceedings of the 1998 ACM/SIGDA sixth international sympsosium on Field programmable gate arrays, pp191-200, 1998 Available at http://www.adraka.com/files/crdcsrvy.pdf

The CORDIC FAQ at dspguru.com http://dspguru.com/dsp/faqs/cordic

Maharatna, K, “Virtually scaling-free adaptive CORDIC rotator”  IEE Proceedings on Computers and Digital Techniques, 151, (6), 448-456. Available at http://eprints.soton.ac.uk/263504/

Original foundation work on the CORDIC Algorithm: Jack E. Volder, "The CORDIC Trigonometric Computing Technique," IRE Transactions on Electronic Computers, Vol. EC-8, pp.330-334, 1959. John S. Walther, "A Unified Algorithm for Elementary Functions," Spring Joint Computer Conference Proceedings, Vol. 38, pp.379-385, 1971. D. H. Daggett, "Decimal-Binary Conversion in CORDIC," IRE Transactions on electronic Computers, Vol. EC-8, No. 3, pp. 335-339, 1959. M. A. Liccardo, "An Interconnect Processor with Emphasis on CORDIC Mode Operation," Masters Thesis EE Dept, University of California at Berkeley, September 1968. J. E. Volder, "Binary Computation Algorithms for Coordinate Rotation and Function Generation," Convair Report, IAR-1 148, Aeroelectronics Group, June 1956.

en.wikipedia.org/wiki/CORDIC: Matlab/Octave code for v=cordic(beta,n)