Consider a causal LTI system S whose input x(t) and output y(t) are related by the diffe...

Consider a causal LTI system S whose input x(t) and output y(t) are related by the differential equation

(a) Show that

and express the constants A, B, and C in terms of the constants a₀, a₁, b₀ and b₁ .

(b) Show that S may be considered a cascade connection of the following two causal LTI systems:

(c) Draw a block diagram representation of S.

(d) Draw a block diagram representation of S₂.

(e) Draw a block diagram representation of S as a cascade connection of the block diagram representation of S₁ followed by the block diagram representation of S₂.

(f) Draw a block diagram representation of S as a cascade connection of the block diagram representation of S₂ followed by the block diagram of representation S₁.

(g) Show that the two integrators in your answer to part (f) may be collapsed into one. The resulting block diagram is referred to as a Direct Form II realization of S, while the block diagrams obtained in parts (e) and (f) are referred to as Direct Form I realizations of S.