Extremum seeking control (ESC) is a non-model real-time for tuning parameters to optimize an unknown nonlinear map. Gradient extremum seeking control and bounded update rate extremum seeking control (or shortly bounded extremum seeking control) are two main structures of extremum seeking control. In gradient extremum seeking control, sinusoidal signals are used to estimate the gradient of the map to be optimized. In bounded extremum seeking control, unknown function that is being optimized appears as the argument of a sine/cosine function. The stability analysis of both structures are carried out by appropriate change of variables and using the continuous time version averaging theorem. Also, in each of these structures, the convergence rate to extremum point is depend on the Hessian matrix of unknown function. Therefore, the convergence rate is not adjustable. Both of extremum seeking control structures have been modified such that, to be able to estimate the Hessian of unknown function. Therefore, Newton-based extremum seeking control is proposed. On the other hand, discrete time extremum seeking control structures have been considered by researchers. In this regard, the discrete time form of gradient extremum seeking is proposed. In this research, the discrete time form of bounded extremum seeking control has been proposed. The stability analysis of the proposed scheme is carried out using the discrete time version of the averaging theorem, that in comparison with stability analysis continuous time bounded extremum seeking is much more complicated. This complexity comes from difficulty in choosing the change of variables transforming the proposed scheme to the standard form of averaging theorem. While for the continuous time bounded extremum seeking, it is easy to reach the standard form of the continuous time version of averaging theorem. In the following, the discrete time quasi Newton extremum seeking is proposed. This structure has the ability to estimate the Hessian of the unknown mapping, which makes the convergence rate to the extremum point independent of the Hessian. The stability analysis of this structure also has its complexity. At the end, as an application of the proposed discrete time bounded extremum seeking, a solution to the LQ problem is proposed. Key Words: Extremum Seeking Control, Discrete Time Bounded Extremum Seeking Control, Discrete time Averaging, Stability Analysis