1. ## Convolution

These are my equations
x[n] = δ[n] + 2δ[n-1] - δ[n-3]
h[n] = 2δ[n+1] + 2δ[n-1]

They want to know what
y[n] = x[n] * h[n] (where * is the convolution)

I am getting so lost on this problem. I do not understand how to combine these two to get y[n]. I understand the definition of convolution, just I do not understand how to go about finding y[n]. Can someone please help me out on this problem.

I looked in the back and the answer is this...
y[n] = 2δ[n+1] + 4δ[n] + 2δ[n-1] + 2δ[n-2] - 2δ[n-4]
No idea how they get that... Can someone please show me the steps to get that answer?

2. It's actually quite simple, first let us recall the shifting property of the delta function:

$\displaystyle \delta [n - k]*f[n] = f[n - k]$

$\displaystyle y[n] = x[n]*h[n] = \left( {\delta \text{[n] + 2}\delta \text{[n - 1] - }\delta \text{[n - 3]}} \right)*\left( {\text{ 2}\delta \text{[n + 1] + 2}\delta \text{[n - 1]}} \right) =$

$\displaystyle = 2\delta \text{[n]*}\delta \text{[n + 1] + 2}\delta \text{[n]*}\delta \text{[n - 1]} $$\displaystyle \text{ + 4}\delta \text{[n - 1]*}\delta \text{[n + 1] + 4}\delta \text{[n - 1]*}\delta \text{[n - 1] - 2}\delta \text{[n - 3]*}\delta \text{[n + 1] - 2}\delta \text{[n - 3]*}\delta \text{[n - 1] = } \displaystyle \text{ = 2}\delta \text{[n + 1] + 2}\delta \text{[n - 1] + 4}\delta \text{[n] + 4}\delta \text{[n - 2] - 2}\delta \text{[n - 2] - 2}\delta \text{[n - 4] = } \displaystyle \text{ = 2}\delta \text{[n + 1] + 2}\delta \text{[n - 1] + 4}\delta \text{[n] + 2}\delta \text{[n - 2] - 2}\delta \text{[n - 4]} 3. For the second line \displaystyle = 2\delta \text{[n]*}\delta \text{[n + 1] + 2}\delta \text{[n]*}\delta \text{[n - 1]}$$\displaystyle \text{ + 4}\delta \text{[n - 1]*}\delta \text{[n + 1] + 4}\delta \text{[n - 1]*}\delta \text{[n - 1] - 2}\delta \text{[n - 3]*}\delta \text{[n + 1] - 2}\delta \text{[n - 3]*}\delta \text{[n - 1] = }$

How do you get this is it that you take the function and then multiply it out?

4. I'm not sure I understand your question, but maybe you're refering to the distributive property of the convolution:

$\displaystyle f[n]*\left( {g[n] + w[n]} \right) = f[n]*g[n] + f[n]*w[n]$

5. ## One more question!

If we were given a piece wise function such as...

x(t) =
t+1, 0 <= t <= 1

2-t, 1 < t <= 2

0, elsewhere

And give that
$\displaystyle h(t) = \delta \text{(t+2) + 2}\delta \text{(t+1)}$