If an LTI continuous$-$time system has a rational system function, then its input and output sat$-$ isfy 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 ye$($f$).$

dye $(1)$ di T

it seems plausible that if T is "small enough," we should obtain a good approximation if we replace dye $(1)/$dt by $[$ye$(1)-$yelt$-$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

H$(3)$ A

where A and e are constants

$($a$)$ Show that the input xe $(0)$ and the output y$,(7)$ of the system satisfy the differential equa tion dy $(1)$ dt Ax$($t$).$

$($b$)$ Evaluate the differential equation at $7,$ and substitute dye$($t$)$ YnT$)-$yeonT$-$T$)$ Le$.,$ replace the first derivative by the first backward difference.

$($c$)$ Definex$(7)$ and eye$($nT$).$ With this notation and the result of part $($b$).$ obtain a difference equation relating all and yiel, and determine the system function H$(2)$Y$(2)/$X $(2)$ of the resulting discrete$-$time system.

$($d$)$ Show that, for this example.