If an LTI continuous-time system has a rational system function, then its input and outp...

If an LTI continuous-time system has a rational system function, then its input and output satisfy an ordinary linear differential equation with constant coefficients. A standard procedure in the simulation of such systems is to use finite-difference approximations to the derivatives in the differential equations. In particular, since, for continuous differentiable functions y_c(t),

it seems plausible that if T is “small enough,” we should obtain a good approximation if we replace dy_c(t)/dt by [y_c(t) − y_c( t − T )]/T .

While this simple approach may be useful for simulating continuous-time systems, it is not generally a useful method for designing discrete-time systems for filtering applications. To understand the effect of approximating differential equations by difference equations, it is helpful to consider a specific example. Assume that the system function of a continuous time system is

where A and c are constants.

(a) Show that the input x_c(t) and the output y_c(t) of the system satisfy the differential equation

(b) Evaluate the differential equation at t = nT, and substitute

i.e., replace the first derivative by the first backward difference

(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 mapping

(It can be demonstrated that if higher-order derivatives are approximated by repeated application of the first backward difference, then the result of part (d) holds for higher order systems as well.)

(e) For the mapping of part (d), determine the contour in the z-plane to which the jΩ- axis of the s-plane maps. Also, determine the region of the z-plane that corresponds to the left half of the s-plane. If the continuous-time system with system function H_c(s) is stable, will the discrete-time system obtained by first backward difference approximation also be stable? Will the frequency response of the discrete-time system be a faithful reproduction of the frequency response of the continuous-time system? How will the stability and frequency response be affected by the choice of T?

(f) Assume that the first derivative is approximated by the first forward difference; i.e.,

Determine the corresponding mapping from the s-plane to the z-plane, and repeat part (e) for this mapping.