I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ where $T_s \leq 1/(2\Delta f)$, $\theta$ is a random variable with uniform probability density in $[0,2 \pi)$, and $v_k$ is a zero-mean complex Gaussian random variable.
I follow the author up to and including the statement
$$ \sum^N_{k=1} \sum^N_{m=1} (k-m) r_k r^*_m e^{-j2\pi \Delta f T_s (k-m)} = 0 $$
the step that loses me is when terms are re-arranged to give
$$ Im \{ \sum^{N-1}_{k=1} k (N-k) R(k) e^{-j 2 \pi \Delta f k T_s} \} = 0 $$
where
$$ R(k) \triangleq \frac{1}{N-k} \sum^N_{i=k+1} r_i r^*_{i -k} , 0\leq k \leq N-1 $$
I can see that one needs to rearrange the order of the summations to get the equation in a form that will allow $R(k)$ to be substituted.
What I think is throwing me off is how the imaginary part and the different indices of $k$ in $R(k)$, $0 \leq k \leq N-1$, affects the expression. I'd appreciate any hints that could set me in the right direction.