Consider an LTI continuous-time system with rational system function Hc(s). The input xc...

Consider an LTI continuous-time system with rational system function H_c(s). The input x_c(t) and the output y_c(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 H_c(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 y_c(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 y_c(nT ) in terms of y_c(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] = x_c(nT ) and y[n] = y_c(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 H_c(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.)