# converting complex sum of exponential signal to real signal

I have the following Signal model, that generates a discrete-time complex signal, where $$a_k$$ - amplitudes, $$\phi_k$$ - phases, $$\alpha_k$$ - damping factors and $$f_k$$ - frequencies are the parameters of the model and $$n$$ is the time index and $$\Delta t$$ is the time interval.

$$$$x_n = \sum_{k=1}^{K} (a_k e^{j\phi_k})(e^{\{(- \alpha_k + j2\pi f_k )\Delta t\}n}), \quad n = 0,1,...,N-1$$$$

For what values of the model parameters, does it generate a real signal? I worked out to write the phase $$\phi$$ as a function of $$n$$ and $$f$$ as: $$\phi_k = n\pi (1-2f_k \Delta t)$$ or simply, $$\phi_k = -2n\pi f_k \Delta t$$

But can phase be a function of time index? When I set this value, then the frequency has no role to play. Is there any other values of the parameter, that makes this signal sequence real?

If you make the sum run from $$-K$$ to $$+K$$ and set $$a_k = a_{-k}, \alpha_k = \alpha_{-k}, \phi_k = -\phi_{-k}$$, the result will indeed be real. But that's more or less equivalent by replacing the complex exponential with a cosine (phase shifted by $$\phi_k$$).
With the damping you may be able to enforce real values for some samples simply by solving for $$\Im\{x[n]\} = 0$$, but you only have $$3K$$ model parameters, so you can't do it for all samples of the infinitely long signal