Multi-Carrier and Spread Spectrum Systems: From OFDM and MC ...

Synchronization 143resulting SNR can be written asSNR ICI ≈N∑c −1n=0n≠m|d m | 2 sinc 2 (f error T s )|d n | 2 sinc 2 (n − m + f error T s ) + P N, (4.24)where P N is the power of the noise n ′ .IfE s is the average received energy of theindividual sub-carriers and N 0 /2 is the noise power spectral density of the AWGN, thenand the SNR can be expressed asE s= |d m| 2(4.25)N 0 P NSNR ICI ≈ E s sinc 2 (f error T s ). (4.26)N 01 + E ∑ssinc 2 (n − m + f error T s )N 0N c −1n=0n≠mThis equation shows that a frequency error can cause a significant loss in SNR. Furthermore,the SNR depends on the number of sub-carriers.4.2.2.2 Analysis of the SNR in the Presence of a Clock ErrorHere, we consider only the effect of a clock error, i.e. f error = 0 in the above expressions.If the clock error is within the guard time, i.e. |τ error | T g (or early synchronization), thetiming error is absorbed and hence there is no ISI and no ICI. It results only in a phaseshift at a given sub-carrier which can be compensated for by the channel estimation (seeSection 4.3).However, if the timing error exceeds the guard time, i.e. |τ error | >T g (or late synchronization),both ISI and ICI appear. As Equations (4.18) to (4.21) show, the clock erroralso introduces an amplitude reduction and a phase rotation which is proportional to thesub-carrier index. In a similar manner as above, the expression of the SNR can be derivedas [73]|d m | 2 (1 − τ error /T s ) 2SNR ICI +ISI ≈⎛⎞(τ error /T s ) 2 ⎝1 + 2 N ∑c−1|d n | 2 sinc 2 [(n − m)τ error /T s ] ⎠ + P N≈n=0n≠m⎛E s(τ error /T s ) 2 ⎜⎝1 + 2N 0E sN 0(1 − τ error /T s ) 2N∑c −1n=0n≠m⎞sinc 2 ⎟[(n − m)τ error /T s ] ⎠ + 1. (4.27)It can be observed again that a clock error exceeding the guard time will introduce areductioninSNR.