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We report the observation of the frequency shift of modulation instability sidelobes due to the synchronization mismatch between the pulsed pump period and the resonator's round-trip time. We show that a very small synchronization mismatch leads to a significant frequency shift of the modulation instability sidelobes. We develop a theory originating from absolute and convective instabilities to confirm experimental observations and numerical simulations. The temporal drifts of the Turing patterns induced by the synchronization mismatch are measured in real time using a time-lens system and confirm theoretical predictions.