recognize as and by integrating.. which is an analytical solution to the continuous model. Now we want to discretise the above expression. We assume that u is constant during each timestep. We recognize the bracketed expression as , and the second term can be simplified by substituting with the function . Note that . We also assume that is constant during the integral, which in turn yields which is an exact solution to the discretization problem. When is singular, the latter expression can still be used by replacing by its Taylor expansion, This yields which is the form used in practice. Approximations Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate discrete model, based on that for small timesteps . The approximate solution then becomes: This is also known as the Euler method, which is also known as the forward Euler method. Other possible approximations are , otherwise known as the backward Euler method and , which is known as the bilinear transform, or Tustin transform. Each of these approximations has different stability properties. The bilinear transform preserves the instability of the continuous-time system. Discretization of continuous features In statistics and machine learning, discretization refers to the process of converting continuous features or variables to discretized or nominal features. This can be useful when creating probability mass functions. Discretization of smooth functions In generalized functions theory, discretization arises as a particular case of the Convolution Theorem on tempered distributions where is the Dirac comb, is discretization, is periodization, is a rapidly decreasing tempered distribution (e.g. a Dirac delta function or any other compactly supported function), is a smooth, slowly growing ordinary function (e.g. the function that is constantly or any other band-limited function) and is the (unitary, ordinary frequency) Fourier transform. Functions which are not smooth can be made smooth using a mollifier prior to discretization. As an example, discretization of the function that is constantly yields the sequence which, interpreted as the coefficients of a linear combination of Dirac delta functions, forms a Dirac comb. If additionally truncation is applied, one obtains finite sequences, e.g. . They are discrete in both, time and frequency. See also Discrete event simulation Discrete space Discrete time and continuous time Finite difference method Finite volume method for unsteady flow Smoothing Stochastic simulation Time-scale calculus References Further reading External