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 ???


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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