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I sent out a complex vector $x_i \in \{x_1,...x_n \}$ and receive a vector $y$.

I use correlation $y^Hx_i$ to detect which $i$ was sent.

If $y$ is corrupted by AWGN channel, the optimum rule should be $$\hat{i} = \text{argmin}_i |y-x_i|^2 = \text{argmax}_i \text{Re}\{y^H x_i\}$$ we compare the real part of correlation results.

But many applications and texts use the absolute value of correlation results, for example Primary Synchronization Signal Detection Algorithm in LTE-A and Cross_Correlation_between_different_PSS.

Why should we use absolute value? Maybe because of fading that rotates the receive vector?

Could you give me some resources that explain this convention in detail?

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The reason we often use absolute value is that the received signal usually has unknown phase.

Consider the case in which $\boldsymbol{y=x_i}\exp(j\theta)$, where $\theta$ is a phase rotation. The phase rotation is due not to "fading" but due to the fact that the received phase in communication systems is generally arbitrary due to the fact that the receiver's oscillator does not have the same phase as that of the received signal. (Note I ignored noise for the sake of simplicity, but this doesn't change the basic point.)

Now $\boldsymbol{y^Hx_i} = \boldsymbol{x_i^H} \exp(-j\theta) \boldsymbol{x_i} = \|\boldsymbol{x_i}\|^2 \exp(-j\theta)$. Depending on the value of $\theta$, the real component of this term may be large or small. That is what Stanley Pawlukiewicz was pointing out. In his example, $\theta = \pi$, so the output of the correlation would be $-\|\boldsymbol{x}\|^2$. The real component would be small, and by using the real component, one would erroneously conclude that the two signals were not aligned, when it fact they were aligned. Using the absolute value corrects that problem.

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  • $\begingroup$ thanks, the random phase rotation is what I said "Maybe because of fading that rotates the receive vector". If you think this is what Stanley Pawlukiewicz meant, how should I understand "One can alternatively have a set of orthogonal $x_i$"?. I upvote your answer anyway. $\endgroup$ – Cath Maillon Jan 21 '19 at 17:58
  • $\begingroup$ Yes, I just wanted to point out that it isn't fading that causes the phase rotation. We'd have a random phase even in a line-of-sight channel with no fading. $\endgroup$ – Ill-Conditioned Matrix Jan 21 '19 at 18:04
  • $\begingroup$ I am hesitant to interpret what someone else said and probably shouldn't have been claiming to do so. But if you sent antipodal signals like $x$ and $-x$, you can't use the absolute value and therefore you can't distinguish the two when the channel has a random phase rotation. On the other hand, if you send orthogonal signals, you can use absolute value to distinguish between two hypotheses even when the phase is random. Thus the type of channel you had to deal with would influence your choice of $x_i$ - channels with random phase rotations would demand orthogonal signal sets. $\endgroup$ – Ill-Conditioned Matrix Jan 21 '19 at 18:10

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