fft – Cross-correlation in time or frequency domain (FPGA)

I need to compute the complex cross-correlation between two signals a and b and the take only the maximum value of the cross-correlation, regardless of the lag time where this occurs. I can do it either in time domain or in frequency domain. This needs to be later incorporated in a FPGA, so I want to be efficient.

An alternative is to go to frequency domain correlation, but I would need to do

which is probably computationally costly.

Is there a way to avoid going back and forth twice to the frequency and time domain?

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