How Does Partial Differentiation Affect This Summation Function?

[vaibhavtewari]

I have a function

\displaystyle\sum_{n=0}^{N-1}\left(x_n-\sum_{i=0}^{L-1}a_iS_{i+n}\right)^2

if I do a partial differentiation with respect to S_k what would I get ?

thanks

 

[CompuChip]

Will it help you if I note that you can write it as
\sum_{n = 0}^{N - 1} (C - a_{k - n} S_k)^2
where C is a constant with respect to S_k.

 

[vaibhavtewari]

variable S_k is not a constant, so there are basically N+L variables...thanks for trying

 

[CompuChip]

I assume that you only differentiate with respect to one value of k at the time?

 

[vaibhavtewari]

that is true but then I solve these simultaneous equations...so I was hoping for something structured..