I have a discrete-time signal as:
$\alpha_l = \sum_k a_k g(lT/2 -k(T+\Delta T))$ where $a_k \in \lbrace\pm 1\rbrace$, $g(t)=\frac {sin(2\pi t/T)}{2\pi t/T}$, $T$ is the bit period, and $\Delta T$ is a constant. Simply put, $\alpha_l$ are the convolution result and because they are sampled at half the bit period, they will be twice as long.
I want to reconstruct the stream of $a_k$'s at twice the bit rate, i.e. the original $a_k$'s and the interpolated symbols between every two $a_k$.
My first and foremost attempt is: $y_l = \sum_k \alpha_k q(l(T+\Delta T)/2 -kT/2)$ where $q(t) = \frac{sin(\pi t/T)}{\pi t/T}$. I want $y_l = a_{l/2}$, however, the implementation result in MATLAB does not quite agree.
Could anyone please share their thoughts/solution?