Discrete Approximation of Continuous Transfer Functions

The basic strategy to accomplish this goal is to develop discrete approximations for integration (like rectangular or trapezoidal) in terms of functions of z. These are called z forms, and they are different than z transforms. Then we substitute the z form for 1/s in G(j,. So the first thing we must do is develop discrete approximations for integration (1/s). 

RECTANGULAR. Rectangular integration uses the present value of the input times the sampling period to approximate the integral of the input. 

Rearranging to get the input-output transfer function gives 

Example 18.11. Suppose we want to find a discrete approximation for a first-order lag. This would be called a first-order digital filter. Let x, be the output of the filter and m ) be the input. 

Dividing by s gives an equation that contains 1/s terms. 

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