How Do You Solve a First-Order Linear Difference Equation?

[DivGradCurl]

Folks,

I'm a bit rusty on difference eqns. Here's the problem:

$$y_n -0.5 y_{n-1} = x_n$$

Here's what I can get out of it so far:

$$y_1 -0.5 y_0 = x_1$$
$$y_2 -0.5 y_1 = x_2$$
$$y_3 -0.5 y_2 = x_3$$

I just need some sense of direction, not a solution. It seems to me that this kind of problem requires that some boundary conditions be given, but that is not the case. That's all I have! Maybe it's something very simple I can't see right now.

Any help is highly appreciated.

 

[rbj]

so you have a difference equation. what are you trying to do with it?

are you trying to figure out what $y_n$ is? do you know what $x_n$ is?

 

[DivGradCurl]

I assume the goal is to get $$y_n$$ since that was not explicitly stated. The directions are simply "solve the difference equation". That doesn't help much, does it? I do not have $$x_n$$

 

[0rthodontist]

It's already about as solved as it can get if you don't know the x's.

 

[DivGradCurl]

Yes, but the answer wouldn't be that obviuous; it sounds like there is information missing, as rbj pointed out. I'll go talk to my instructor. As soon as I find out the answer or have another question, I'll get back to you guys. Thanks for all the help!

 

[LeBrad]

Have you been learning about z-transforms?

y[n] -> Y(z)
y[n-1] -> Y(z)/z

then solve for Y(z) = f(X(z)), then transform back to y[n] = f(x[n]) and you will have eliminated the y[n-1].

 

[DivGradCurl]

Z-transforms really work on this kind of problem. I've found a book with a very easy-to-follow introduction. Thanks for the hint. Here is my answer:

$$h[n] = -2^{-n}u[n-1]$$

 

[LeBrad]

One thing to be careful about with z-transforms is that some functions, such as $$H(z) = \frac{1}{1-az^{-1}}$$, have multiple inverse transforms. Which to use depends on the region of convergence of H(z), which affects (or is defined by) the stability and causality of the system.