This question is related to white noise representation of WSS sequences using AR(2) (autoregressive) model The function is given as: $$x(k)=\frac{1}{p_1-p_2}(p_1^{k+1}-p_2^{k+1})w[k]u[k]$$ where $w[k]$ is white noise sequence with variance $\sigma_w^2$, and $p_1<1$ and $p_2<1$.
I have to prove its autocorrelation function is given as: $$k_{xx}(k)=\frac{\sigma_w^2}{(p_1-p_2)(1-p_1p_2)}(\frac{p_1^{m+1}}{1-p_1^2}-\frac{p_2^{m+1}}{1-p_2^2})$$
My steps are as follows: I am taking autocorrelation function as:$$k_{xx}(k)=E[x[m]x[m-k]]$$ which leads to:$$(\sum_{m=0}^\infty \frac{1}{p_1-p_2}(p_1^{m+1}-p_2^{m+1})u[m]\frac{1}{p_1-p_2}(p_1^{m-k+1}-p_2^{m-k+1})u[m-k])\sigma_w^2$$ $$(\sum_{m=k}^\infty \frac{1}{(p_1-p_2)^2}(p_1^{2m-k+1}+p_2^{2m-k+1}-p_2^{m+1}p_1^{m-k+1}-p_1^{m+1}p_2^{m-k+1}))\sigma_w^2$$ Taking $m-k=l$ and segregating terms. Also, after using the property that $$\sum_{n=0}^{\infty}a^n=\frac{1}{1-a}$$ when a<1.
Resulting expression is: $$(p_1^{k+1}\frac{1}{1-p_1^2}-p_2^{k+1}p_1\frac{1}{1-p_1p_2}-p_1^{k+1}p_2\frac{1}{1-p_1p_2}+p_2^{k+1}\frac{1}{1-p_2^2})\frac{\sigma_w^2}{(p_1-p_2)^2}$$ which is not same as what needs to be proved.
Where am I wrong? why I do not get this?: $$k_{xx}(k)=\frac{\sigma_w^2}{(p_1-p_2)(1-p_1p_2)}(\frac{p_1^{m+1}}{1-p_1^2}-\frac{p_2^{m+1}}{1-p_2^2})$$
