1. The problem

A causal LTI system is described by the difference equation .

Determine the

(a)homogeneous( for all ),

(b)impulse,

(c)step

responseof the system.

2. Relevant equations and definitions

2.1Causalare systems for which the output depends only on the input samples , for

2.2For the(a)part of the problem:

Equation is called thehomogeneous difference equationand the homogeneous solution. The sequence is in fact a member of a family of solutions of the form

where the coefficients can be chosen to satisfy a set of auxiliary conditions on .

3. The attempt at a solution

First of all, there is no additional information to be gained from knowing that the LTI system is also "causal", right? That is, being causal ( only depends on , ) is already in the given equation, correct?

(a)homogeneous response, y_h[n]

The official solution is given in the form of Eq. :

I have no idea how one should arrive at this solution. Neither writing it out for particular 's (and hoping to glimpse a pattern), nor writing the system in the general form of and then finding the relevant coefficients (N=2: ; M=1: ), gives way toward the solution.

(b)impulse response, h[n]

I do get the official solution ( ), but I'm exploiting the Z-transform, which is introduced only after this chapter ... so there must be other way without Z-transform:

(c)step response, s[n]

As in the(a)part, I'm completely stuck here. The solution should be , which smells of Z-transform, but I'm also interested if/how could one do without it?

All I can think of for(c)is using the result from(b)(the impulse response), but I get stuck: