Trying to understand difference equation

[OnceMore]

Hello,

I hope someone can help me with a problem I am having. It is neither homework or coursework, but for my own understanding.

I should say from the start, I am one of those people who tend not to be able to see the forset because all the trees are in the way, so I probably will be missing something very obvious to others.

At the minute, I am trying to get better at dealing with difference equations when it comes to designing digital filters. The book I have been reading through gives the following difference equation

h(n) = b₁ . h(n - 1) + δ(n)

With the following table for the results

n δ(n) h(n - 1) h(n)
----------------------------------------------
0 1
1 0
2 0
3 0
4 0

Here, h(n) is the response, and δ(n) is the impulse function.

I hope someone can help me see how the rest of the table is formed. When I understand the process I will be able to apply it better to other problems.

Thanks.
Seán

 

[Ray Vickson]

Hello,

I hope someone can help me with a problem I am having. It is neither homework or coursework, but for my own understanding.

I should say from the start, I am one of those people who tend not to be able to see the forset because all the trees are in the way, so I probably will be missing something very obvious to others.

At the minute, I am trying to get better at dealing with difference equations when it comes to designing digital filters. The book I have been reading through gives the following difference equation

h(n) = b₁ . h(n - 1) + δ(n)

With the following table for the results

n δ(n) h(n - 1) h(n)
----------------------------------------------
0 1
1 0
2 0
3 0
4 0

Here, h(n) is the response, and δ(n) is the impulse function.

I hope someone can help me see how the rest of the table is formed. When I understand the process I will be able to apply it better to other problems.

Thanks.
Seán

Your recursion only makes sense for n = 0 if you have access to h(-1) to start things off; that is, you need an "initial condition" such as h(-1)=c; then, for all n >= 0, the recursion determines all the other h(n) values. Just plug things into the recursion: h(0) = b_1 *c + δ(0), h(1) = b_1*h(0) + δ(1), etc. That's all there is to it!



A more difficult (and more interesting) question would be: find a closed-form formula for h(n) in terms of the initial condition h(-1) = c, the constant b_1 and the given form of {δ(k), k >=0}.

 

[lurflurf]

δ(n) is a function such that
δ(0)=1
δ(n)=0 when n is not 0

thus
δ(n)=δ(n) h(n - 1) h(n)
since h(- 1) h(0)=1