# Is there an exact solution for this re-sampling problem?

I want to know if there is an exact solution for the following problem and how to approach solving it:

I have a discrete-time signal where the Nyquist theorem is satisfied:

$r_k = \sum_i a_i^{(1)}h^{(1)}(kT-iT-\tau_i^{(1)}) +\sum_i a_i^{(2)}h^{(2)}(kT-iT-\tau_i^{(2)})$, where $a_i^{(1)}$ and $a_i^{(2)}$ are two independent random streams of ${\pm 1}$ and $h^{(1)}(t)$ and $h^{(2)}(t)$ are their corresponding pulses, and $T$ is the bit period, and $\tau_i^{(1)}$ and $\tau_i^{(2)}$ are timing errors.

The problem is that I want to calculate another set of samples while all the parameters of the above equation are known. I want to obtain:

$s_k = \sum_i a_i^{(1)}h^{(1)}(kT-iT) +\sum_i a_i^{(2)}h^{(2)}(kT-iT)$, which looks exactly like the original samples except for that the timing errors do not exist anymore. I know that I cannot achieve this just by resampling. How do I approach this?

Please share your thoughts.

• Are the timing errors subtle enough to allow decoding of the signals? What kind of pulses do they have? – Olli Niemitalo Jun 7 '15 at 8:08
• The timing errors lowers the performance of a decoder/detector, and so they cannot be ignored. The pulses are equalized channel responses; i.e. generally they are short e.g. vectors of 3-4 entries. – Elnaz Jun 8 '15 at 4:41