Consider an LTI continuous-time system with rational system function Hc(s). The input xc(t) and the output yc(t) satisfy an ordinary linear differential equation with constant coefficients. One approach to simulating such systems is to use numerical techniques to integrate the differential equation. In this problem, we demonstrate that if the trapezoidal integration formula is used, this approach is equivalent to transforming the continuous-time system function Hc(s) to a discrete-time system function H(z) using the bilinear transformation.
To demonstrate this statement, consider the continuous-time system function
where A and c are constants. The corresponding differential equation is
Where
(a) Show that yc(nT ) can be expressed in terms of as
The definite integral in this equation represents the area beneath the function for the interval from (nT − T ) to nT . Figure P7.49 shows a function and a shaded trapezoid-shaped region whose area approximates the area beneath the curve. This approximation to the integral is known as the trapezoidal approximation. Clearly, as T approaches zero, the approximation improves. Use the trapezoidal approximation to obtain an expression for yc(nT ) in terms of yc(nT − T ),
(b) Use the differential equation to obtain an expression for and substitute this expression into the expression obtained in part (a).
(c) Define x[n] = xc(nT ) and y[n] = yc(nT ).With this notation and the result of part (b), obtain a difference equation relating x[n] and y[n], and determine the system function H(z) = Y (z)/X (z) of the resulting discrete-time system.
(d) Show that, for this example,
i.e., show that H(z) can be obtained directly from Hc(s) by the bilinear transformation. (For higher-order differential equations, repeated trapezoidal integration applied to the highest order derivative of the output will result in the same conclusion for a general continuous-time system with rational system function.)
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