# Difference equation: (1) step response, (2) homogeneous solution, (3) ...

• Sep 30th 2012, 04:30 AM
courteous
Difference equation: (1) step response, (2) homogeneous solution, (3) ...
1. The problem
Consider the difference equation $y[n] - \frac{5}{6}y[n-1] + \frac{1}{6}y[n-2] = \frac{1}{3}x[n-1]$.

What is the
(a) step response for the causal system satisfying this difference equation,
(b) general form of the homogeneous solution of the difference equation?
(c) Consider a different system satisfying the difference equation that is neither causal nor LTI, but that has y[0] = y[1] = 1. Find the response of this system to x[n] = $\delta$[n].

2. Relevant equations and definitions
Oppenheim's definition of (b), the general form of the homogeneous solution:
$y_h[n] = \sum_{m=1}^{N} A_m z_m^{n} \text{ ................... (2.97)}$

3. The attempt at a solution
(a) step response, s[n]
I have calculated the impulse response $h[n] = 2( \frac{1}{2^n} - \frac{1}{3^n}) u[n]$, and put it to use in calculating s[n] (where I get stuck):

$s[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] = 2 \sum_{k=-\infty}^{\infty} u[n-k] ( \frac{1}{2^n} - \frac{1}{3^n}) u[n] = ?$

(b) general form of the homogeneous solution
I have no idea (similar problem as (a) in this question on MHF) ... the solution ought to be $y_h[n] = A_1 \frac{1}{2^n} + A_2 \frac{1}{3^n}$.

(c) response of a particular system to x[n] = $\delta$[n]
Confused as well. I do get particular values for y[n]:
y[-1] = 41/5,
y[0] = 1,
y[1] = 1,
y[2] = 2/3,
...
continuing in this manner (and hoping for a pattern to emerge) surely isn't the way to go?

The solution given is:
Quote:

$y[n] = 4 \frac{1}{2^n} - 3 \frac{1}{3^n} - 2 \frac{1}{2^n} u[-n-1] + 2 \frac{1}{3^n} u[-n-1]$. Other answers are possible.
• Oct 10th 2012, 03:00 AM
courteous
Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...
Bumping in hope of an answer.
• Oct 14th 2012, 06:05 AM
courteous
Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...
Found the method for solving (b); still need help with (a) and (c).
• Oct 14th 2012, 10:25 PM
courteous
Re: Difference equation: (1) step response, (2) homogeneous solution, (3) ...
Got (a), still plead for help with (c).