Adaptive equalizer for BPSK signals with frequency offset

Question

I'm trying to equalize a BPSK signal which has a residual frequency offset on it. This frequency offset is dealt with at a later stage in the DSP chain, along with phase estimation.

I have implemented a FIR filter which updates its filter taps using gradient descent CMA:

The equalized signal (y) is a discrete-time convolution of the filter taps (W) and the input signal (X):

$$ y = W^HX $$

The update error is: $$ \epsilon = |y|^2-1 $$

Finally I compute the update as: $$ W_{i+1} = W_i - \mu \epsilon y^*X $$ Where $\mu$ is the step size = 0.001.

My problem is that due to the frequency offset, my input signal X is complex, which is resulting in complex filter taps in W. This in turn introduces a huge phase error into my signal. One possible solution I've looked at so far is putting the equalizer behind the frequency offset compensation stage. However, since the signal still contains some phase error, it's not entirely constrained to the real axis in the constellation diagram and I have the same problem. My workaround so far has been to update the filter taps as: $$ W_{i+1} = W_i - real(\mu \epsilon y^*X) $$ My reasoning being that after frequency offset compensation and phase estimation, there should only be noise in the quadrature part of the signal, and I can safely discard it.

I'm unhappy with this solution for 2 reasons:

1. I'm not entirely sure it's theoretically sound, and at the very least it's rather unorthodox.

2. I would prefer to equalize the signal upstream in the DSP chain.

I would appreciate any help or insights!

Answer 1

It sounds like your biggest issue is dealing with a phase offset and otherwise you have no other concerns with your equalization approach? In that case, the issue is resolved by simply having a phase correction prior to making your decisions on the real signal. If done correctly your carrier recovery implementation should remove any residual long term phase offset. This post may help give you further insight: Phase synchronization in BPSK. I duplicated the final block diagram below where in your case "I data demod" would be your BPSK output. I want to point out using this diagram, that any residual phase error would accumulate in the PI Loop Filter which would then adjust the NCO to rotate the signal to have zero error (on average). Note that the Loop as shown if completely blind will lock in quadrature or 180° phase positions - if that is your remaining issue then another question on carrier recovery implementations would be more appropriate specific to any issue you have there, but a simple solution is to derive the phase error itself from $\phi \propto Q \text{sign}(I)$ since the objective in this case is to drive any signal on Q down to the noise. The assumption is the signal has been AGC'd prior to this point so such an approach to phase detection would result in a consistent loop gain. Here is another related post with regards to what gain constants you should use (what loop BW should be used): Loop bandwidth for symbol timing recovery

Also notice the similarity of the above diagram to a classical Costas Loop with the following figure from http://michaelgellis.tripod.com/mixerscom.html. This is because the Decision Directed CR Loop is a Costas Loop. The referenced link also diagrams the classical Costas Loop for BPSK specifically, which also shows how the digital implementation such as shown in the diagram above can be modified for the BPSK only case (which is modified the phase error metric as I described earlier). This solution will still have a 0/180° phase ambiguity which is typically resolved with minimum knowledge of the message (header or some other recognized feature in the data message or encoding in the data message).