# Apply timeshift to undersampled signal after bandpass after sampling?

I have sampled my continuous signal $x(t)$ at $x[n]=x(n/f_{\rm samp})$. The signal is undersampled, and I know I will not be able to recover my original signal anymore. After sampling a bandpass is applied to $x[n]$. The signal is now bandlimited, but is affected by aliasing. I now have

$$x[n]=\textrm{Bandpass}\left[x\left(\frac{n}{f_{\rm samp}}\right)\right]$$

However, can I somehow reconstruct

$$z[n]=\textrm{Bandpass}\left[x\left(\frac{n}{f_{\rm samp}}+\Delta\tau\right)\right]\quad\text{from my}\quad x[n]\quad?$$

$z[n]$ would also be an undersampled version of $x(t)$ but sampled with a timeshift. I have no intentions of recovering $x(t)$, but I need my signal shifted to synchronize it. Even though it is undersampled.

Any ideas or is that a lost case?

• Does the band pass filter have a linear phase? – Matt L. Sep 5 '16 at 7:30
• Yes, the band pass filter has a linear phase – torpedo Sep 6 '16 at 13:52