2. DiscreteTime Simulation of FirstOrder Response 

2.1 Background 

The first order lead/lag transfer function is a commonly used mathematical model for representing simple, continuous, dynamic response. A first order lead/lag coupling of two variables implies the relationship 

_{} 

where x is the independent variable, y is the dependent variable, t is time, and the coefficients t_{D} and t_{N} are coupling parameters (not necessarily constant). In the continuous domain this relationship uniquely determines the output response y(t) for any given initial conditions and input function x(t). However, there exists a variety of algorithms for simulating (2.11) in a digital computer, and these algorithms differ both in their computational complexity and in their computed response characteristics. 

There are two reasons for the diversity of digital simulations. First, according to equation (2.11) the change in the output during any interval of time depends on the value of the input continuously throughout the interval, but in digital systems the input is specified only by a sequence of discrete values. Hence the relationship is underspecified in the discretetime domain, and can only be solved by making an assumption as to the behavior of the input variable between the discrete values. Second, practical constraints on execution time, execution frequency, and memory storage often lead to the use of modified solutions based on various simplifying assumptions. 

In Section 2.2 we describe the optimum assumption and the corresponding algorithm. Section 2.3 contains a discussion of several common simplifications and modified methods. 
