1 Data Reconstruction

Most of the control systems have analog controlled processes which are inherently driven by analog inputs. Thus the outputs of a digital controller should first be converted into analog signals before being applied to the systems. Another way to look at the problem is that the high frequency components f(t) should be removed before applying to analog devices. A low pass filter or a data reconstruction device is necessary to perform this operation.

In control system, hold operation becames the most popular way of reconstruction due to its simplicity and low cost. Problem of data reconstruction can be formulated as: `` Given a sequence of numbers, $f(0),f(T),f(2T),\cdots,f(kt),\cdots$ , a continuous time signal $f(t),\; t\geq 0$ , is to be reconstructed from the information contained in the sequence.''

Data reconstruction process may be regarded as an extrapolation process since the continuous data signal has to be formed based on the information available at past sampling instants. Suppose the original signal f(t) between two consecutive sampling instants kT and (k + 1)T is to be estimated based on the values of f(t) at previous instants of kT, i.e., (k - 1)T, (k - 2)T.....0.

Power series expansion is a well known method of generating the desired approximation which yields

Since the only available information about f(t) is its magnitude at the sampling instants, the derivatives of f(t) must be estimated from the values of f(kT), as