Ok, this is leaving me increasingly frustrated. This builds on a previous question here but I am still not getting the indexing, although I learned from that question all the same.
Case 1: First convolution equation example:
$$ h[k] * h^*[-k] = \sum_{n=-\infty}^{\infty} h[n] \ h^*[n+k] $$
The way I got the right hand side, was:
1) Make the $h$'s have dummy variables, so $h[n]$ and $h^*[-n]$.
2) Flip one of them around, so now I get $h[n]$ and $h^*[n]$
3) Add the delay $k$ into the one that way flipped, so finally I have $\sum_{n=-\infty}^{\infty} h[n] \ h^*[n+k]$.
Great, I have a convolution.
Case 2: Second convolution example: (Unrelated to first)
We have:
$$ \sum_{i=-\infty}^{\infty} h[i-m] \ R_{xx}[i]. $$
Here we want to try and find out what convolution this corresponds to. The way to solve this, (apparently), is to make the assumption that another variable $g[n]=h[-n]$, and call it a day. Then we get:
$$ \sum_{i=-\infty}^{\infty} h[i-m] \ R_{xx}[i] = \sum_{i=-\infty}^{\infty} g[-(i-m)] \ R_{xx}[i] = \sum_{i=-\infty}^{\infty} g[m-i] \ R_{xx}[i] = g[m] * R_{xx}[m] = h[-m] * R_{xx}[m] $$
This works great.
Case3: The problem:
This is my problem. Case 1 showed the steps for how to do a normal convolution. Case 2 showed how the method of substitution works. However, if I try to solve Case 1 using this substitution method, I get the wrong answer, even though it worked on case 2. Why is that?
To wit - where am I going wrong here?
$$ h[k] * h^*[-k] = h[k] * g^*[k] = \sum_{n=-\infty}^{\infty} h[n] \ g^*[k-n] = \sum_{n=-\infty}^{\infty} h[n] \ h^*[-(k-n)] = \sum_{n=-\infty}^{\infty} h[n] \ h^*[n-k] $$
However, this is **clearly* the wrong answer! What am I doing wrong here?
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