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While reading Estimation of Symbol Timing Offset, i came across an equation like multiplying a signal with its conjugate.

$$ \hat{\delta} = \arg \max_{\delta} \sum_{i=\delta}^{N_g-1+\delta} \left|y^*(n+i) y(n+N+i)\right|^2 $$ I know multiplying

(5+4i) X (5-4i) = 25 - 20i + 20i + 16 = 25 + 16 = 41.

But am not getting the meaning of this 41. What it represents ???

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  • $\begingroup$ If it would have summed 42 it would be the answer to"the Ultimate Question of Life, the Universe, and Everything" but being 41 is just the natural number that precedes it. See this reference news.mit.edu/2019/… $\endgroup$
    – VMMF
    Feb 24, 2023 at 16:17

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Multiplying a complex number with it's conjugate yields the squared magnitude of that complex number.

However, in your formula, you are multplying the signal with the conjugate of the N samples delayed signal, take the squared magnitude of that (thus dropping phase differences between the signal and delayed signal, which would result from a frequency offset) and average it over Ng sample times. This means you are calculating the autocorrelation of the signal at a time delay of N sample times.

I assume the formula is from an algorithm for coarse symbol timing estimation for OFDM signals. The pseudo-random pilot sequence of length N is repeated twice. It's autocorrelation thus has a peak at N. Now, if you want to estimate the timing offset, you just take the time index of this peak.

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