Hello, I've been trying to figure out how to answer this question.

Represent x[n] = n^2 as a linear combination of time-shifted discrete-time unit impulses delta[n-k]'s

Any help would be appreciated

 

The function

y=x^2

may be integrated between successive intervals 0-1,1-2,2-3 and so on, to give the required equivalent impulse area.

The integral result then becomes

\(Area=[\frac{x^3}{3}]^{L_1}_{L_2}\)

For interval 0 to 1

\(Area=[\frac{x^3}{3}]^{1}_{0}=\frac{1}{3}\)

For interval 1 to 2

\(Area=[\frac{x^3}{3}]^{2}_{1}=\frac{7}{3}\)

For interval 2 to 3

\(Area=[\frac{x^3}{3}]^{3}_{2}=\frac{19}{3}\)

and so on

so

\(x[n]=\frac{1}{3}\mu_0(n)+\frac{7}{3}\mu_0(n-1)+\frac{19}{3}\mu_0(n-2)+\frac{37}{3}\mu_0(n-3)+....\)

or perhaps it should be

\(x[n]=\frac{1}{3}\mu_0(n-1)+\frac{7}{3}\mu_0(n-2)+\frac{19}{3}\mu_0(n-3)+\frac{37}{3}\mu_0(n-4)+....\)

since the function is zero for n=0

or maybe the answer is just

\(x[n]=\mu_0(n-1)+4\mu_0(n-2)+9\mu_0(n-3)+16\mu_0(n-4)+....\)

since the function isn't continuous