Mathematical model of a signal in Compressive Sampling
The model is the Fourier series of a signal up to the W/2th component, with sparse coefficients. That is, most of the set $\{a_k\}_{k=-W/2}^{W/2}$ is zero.


It is because the signal is bandlimited, and thus for a real signal, $a_{W/2}$ will be real. However, if this constraint is not met it is better to use $\pm W/2$
For a real valued signal, the coefficient of the negative frequency is just the conjugate of the positive frequency coefficient. It really just is a mathematical artifact. You could also write the model as 


$$f(t) = \sum_{\omega\in\Omega} b_w \cos(2\pi \omega + \phi_\omega)$$

where $b_\omega$ is real and $\phi_\omega \in [0,2\pi)$ is a real phase value.

That is, the model represents the signal as the sum of sinusoids of different amplitudes and phases.


It will be the Fourier series approximation of the signal using just the first two frequencies.


You should take a look at this paper and the example code:

Towards a mathematical theory of super-resolution