In this paper, we present a novel extremum seeking control algorithm based on the two-sided stochastic perturbation-demodulation loop. Almost sure convergence to the extremum point is proved under mild conditions on the measurement noise and without assuming a priori boundedness of the approximation sequence. The introduction of the two-sided stochastic perturbation and the effect of the demodulation using an adaptive vanishing gain are of essential importance for the accomplished convergence results. The perturbation sequence is assumed to be a vanishing martingale difference sequence. The measurement noise is allowed to contain a mean-square bounded deterministic component and a stochastic component which can be a nonstationary colored noise sequence or a state-dependent sequence. The linear input dynamical system is assumed to be strictly positive real.