Reverse summation of a complex exponential

Question

In the equation below: $$s_l(t)=\sum_{k=-\lfloor{N_{sc}^{RB}/2}\rfloor}^{\lceil{N_{sc}^{RB}/2}\rceil-1}a_{k^{(-)},l}^\,.e^{j2\pi(k+1/2)\Delta f(t-N_{CP,l}T_s)}$$

where: $0\le t\lt (N_{CP,l}+N)\times T_s$ , $k^{(-)}=k+\lfloor{N_{sc}^{RB}/2\rfloor}$, $N = 2048$, $\Delta f=15kHz$

I know the values of: $N_{sc}^{RB}$, $N_{CP,l}$ and $T_s$.

Simply put, I want to solve for $a_{k^{(-)},l}$ which is a vector of values. And I know all the values of the other parameters. $s_l(t)$ is a vector as well which values I know.

Is it possible to reverse the order of the equation to calculate for $a_{k^{(-)},l}$ ?

Answer 1

This is a long comment, that's why I put it as an answer. By the looks of it, this is some sort of Discrete Fourier Transform. To reverse it, you would just use the Inverse Discrete Fourier Transform.

$$X_k = \sum_{n=0}^{N-1} x_n\cdot e^{-\frac {2\pi i}{N}kn} \ ,$$

$$x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k\cdot e^{i 2 \pi k n / N} \ .$$

Also, based on the facts that you used $j$ for the unit of imaginary numbers, and having units of $Hz$; I believe the context of this equation is Electrical Engineering. And Fourier Transforms are widely used in EE.