# GNSS Acquisition in time domain: Why is conjugate of FFT replica used for the mixer?

Pretty new on all this GNSS stuff, I'm just studying from scratch an acquisition unit based in time domain using a parallel FFT structure. Just like the image got from "Acquisition Strategies of GNSS Receiver" from Rafiullah Khan et al.

My main concern here is, why should we apply the complex conjugate of the FFT replica to the mixer with the FFT of the signal and not only do the mixing?

You remember how convolution of two signals $$x(t)$$ and $$h(t)$$ in time domain is equivalent to point-wise multiplication in the frequency domain, $$\mathcal F\{x\}(f)\cdot \mathcal F\{h\}(f) = \mathcal F\{x*h\}(f)$$. (For the DFT, the same is true; the convolution $$*$$ is circular.)
The FFT of the time-reversed sequence of length $$N$$, $$x[N-t]$$ is the frequency-reversed $$X[N-f] = \mathcal F\{x\}[N-f]$$. Since $$x$$ is real-valued (because GNSS systems use bipodal spreading sequences applied to real symbols), it's frequency domain representation is conjugate symmetrical, $$\mathcal F\{x\}[t] = \mathcal F\{x\}[N-t]^*$$, and thus, the time-reversal is identical to a complex conjugation.