# What does the phase discriminator portion of the Costas Receiver do mathematically?

What does the phase discriminator portion of the Costas Receiver do mathematically? 1. The output of the I-channel is $$\ \frac{1}{2}A_C \cos \phi m(t) \$$. Which means for small deviation of phase $$\ \phi \$$ , $$\ \frac{1}{2}A_C \cos \phi m(t) \approx \frac{1}{2}A_C m(t) \$$. Or it can be said that $$\ \frac{1}{2}A_C m(t) \$$ would be attenuated by a small amount. However, we have to keep in mind as the text says (as it not might be as innocuous as it seems): 2. The output of the Q-channel is $$\ \frac{1}{2}A_C \sin \phi m(t) \$$. Which means for small deviation of phase $$\ \phi \$$ , $$\ \frac{1}{2}A_C \sin \phi m(t) \approx \frac{1}{2}A_C \phi m(t)\$$.

3. As per the text, the phase discriminator consists of a multiplier followed by a low pass filter.

4. Which means multiplication of $$\ \frac{1}{2}A_C \cos \phi m(t) \$$ and $$\ \frac{1}{2}A_C \phi m(t) \$$ would yield:

$$\ g(t)=\frac{1}{2}A_C \cos \phi m(t) * \frac{1}{2}A_C \phi m(t) \$$

$$\ g(t)= \frac{1}{4}A^2_C \phi \cos \phi m^2(t)\$$

Now, let's say $$\ m(t) \$$ is band limited to $$\ M \$$. then the term $$\ m^2(t) \$$ in the frequency domain would spread across $$\ -2M \$$ to $$\ +2M \$$ centered around $$\ f=0 \$$ and the term $$\ \frac{1}{4}A^2_C \phi \cos \phi \$$ is a constant.

5. Now, if $$\ g(t) \$$ is subjected to a LPF, then the term $$\ \frac{1}{4}A^2_C \phi \cos \phi m^2(t) \$$ would be retained as it is(if the cutoff frequency of the LPF is slightly > $$\ 2M \$$).

6. So, what does phase discriminator in Costas receiver do mathematically?

Also, if the phase error is significant(that is we have no idea at all about the probable phase of the carrier), then how does the analysis change? Because this is a possibility that we might have no idea about the phase of the carrier.

Text Used: Communication Systems By Simon Haykin

• I think this tells you the answer en.wikipedia.org/wiki/Costas_loop. First, the VCO frequency error must be less than the LPF bandwidth to be able to capture quickly then the phase detector gives a voltage proportional to phase error between I (sin) and Q (cos) channel outputs to correct phase of VCO to centre it between +/- deg error – Sunnyskyguy EE75 Feb 15 at 16:47

This circuit is an attempt to reconstruct the suppressed carrier of a DSB signal and use it to modulate the signal back to the baseband ie. to perform fake synchronous detection. I wrote fake, because the carrier isn't the original, but a guessed estimate.

Phase detector can be seen as a product mixer and PI-controller. If the VCO happens to get locked to the missing carrier, then the inputs of the phase detector have all the time 90 degrees phase difference. The integrator of the PI controller gets averagely zero input => the integrator doesn't drift =>VCO frequency stays.

Traditional PLL mumbojumbo doesn't say Phase detector is a product mixer+PI controller. They have named the PI controller as "loop filter". But that's not only a naming convention. It allows also finer control methods than PI, altough I have found in my simulation experiments that PI controller works fine.

What does the phase discriminator portion of the Costas Receiver do mathematically?

It maintains the VCO's output at precisely: -

• The right frequency
• The right phase angle

It does this by comparing the two the quadrature low-pass filter outputs and continuously adjusts the VCO to be "right on song" so that precisely 90 degrees is the phase difference between the two filter outputs. It will even compensate for small errors in the 90 degrees phase shifter.