Let be the output of the Hilbert transform and let be the unwrapped phase of computed using a four-quadrant arctangent on the real and imaginary parts of followed by phase unwrapping. The relationship between instantaneous frequency and phase is given by:
When using sampled data, these relations may be approximated as follows:
where is the sample period and is the sample rate in units of samples per second.
For computer assignment 5 (CA5), you are asked to extract the instantaneous frequency. If the relations above are used to resynthesizes the phase , a constant envelope signal may be re-synthesized using . If you perform these operations, you can listen to the re-synthesized and compare it to the original.