Understanding the Polar Discriminator (Finally)

Gary Schafer, December 2024

This will be one of those, "Well, duh!" posts. Ever since I learned about the polar discriminator circuit used for frequency demodulation in digital signal processing, I've been wondering about it. Who came up with the original idea? What gave them the idea? The concept is straightforward. Take the incoming sample stream, split it, then one part of the stream is delayed by one sample followed by taking the complex conjugate before being fed into one of two inputs of a multiplier. The other half of the split is sent straight to the same multiplier. The output will be a sample whose phase is the difference in phase between the two samples sent into the multiplier (one straight, the other a delayed and conjugated version).

The part that I didn't understand was, "Why is the result always close to the positive real axis?" Then it hit me. I was reading Andreas Schwarzinger's book "Digital Signal Processing in Modern Communication Systems (2nd Edition)". Let me quote you the appropriate part (my emphasis added):

Remember that any complex number C that is multiplied by its complex conjugate C^* will yield the square of the absolute value of that complex number. C‧C^*=|C|² is always a positive real number.

That last sentence was the answer I'd been searching for. Phase is measured based on the angle from the positive real axis. Any complex number times its complex conjugate will give you a positive, real number. This means it will be centered on the positive, real axis. That's precisely where we want it to be. A polar discriminator is not taking C‧C^*; it's taking C(n)‧C(n-1)^*, but it's close enough. At the very least, the product will not cross the +/-π boundary. (NOTE: If that happens, the system has violated Nyquist and you have bigger problems to worry about.)

Here's an Example

Let's start with two, consecutive samples (shown on the right). The first sample to come through (blue dot) is followed by another sample (green dot). The block diagram at the bottom shows the path that each sample is taking through the circuit. We're using two samples because, in order to create a frequency disciminator, we need to know the phase difference. These two samples happen to straddle the +/-π line. If we were to calculate the phase of each sample first (using the arctangent function, for example), then calculate the difference (by subtracting one from the other), we would wind up with a value greater than +/-π. That would not work.

The delayed (older) sample is complex conjugated, a fancy term to mean "we flip the sign of the imaginary component". This might not seem like a big deal, but it has huge implications for the follow-on processing. Now when we multiply the two complex samples together, the product will be centered around the positive real axis.

The last step is to multiply the non-delayed sample with the delayed and conjugated sample. The combination of conjugation and multiplication creates a difference in the phase between the two, consecutive samples. The result is a sample that is relatively close to the positive real axis. This sample represents the frequency demodulated baseband signal. This multiplication of a sample with a conjugated sample when the phase of the two samples are roughly the same, ensures that the resulting product sample will be close to, if not on, the positive real axis.

That was the key. That was the answer. Thanks, Andreas. My brain can move on to other things now.