Consider a causal LTI system S whose input x[n] and output y[n] are related by the diffe...

Consider a causal LTI system S whose input x[n] and output y[n] are related by the difference equation

(a) Verify that S may be considered a cascade connection of two causal LTI systems S₁ and S₂ with the following input-output relationship:

(b) Draw a block diagram representation of .

(c) Draw a block diagram representation of S₂.

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

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

(f) Show that the two unit-delay elements in the block diagram representation of S obtained in part (e) may be collapsed into one unit-delay element. The resulting block diagram is referred to as a Direct Form II realization of S, while the block diagrams obtained in parts (d) and (e) are referred to as Direct Form I realizations of S.