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.

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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?


1 Answer 1


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.)

Correlation and convolution are closely related: Correlation is a convolution with the time-inverse, conjugate sequence.

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.

So, in short: it does a correlation that way. This is a correlation detector.


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