14.md

EE 120: Signals and Systems

March 6, 2012.

Sampling Cont'd

We are still in the first of three blocks (where we take a continuous-time signal and create a discrete-time signal).

This end-to-end system is effectively a continuous-time system.

This kind of processing we refer to (for obvious reasons) as discrete-time processing of continuous-time signals. The opposite is also possible, where you start out with a discrete-time signal, process in continuous-time, and spit out a discrete-time signal.

Last time, we opened up the first box ($C \to D$). We didn't even talk about the entire box -- there's still some stuff to discuss.

So, consider an impulse train. We then take this through a Dirac $\to$ Kr"onecker block to produce $x_d(n)$.

Question: How is $X_q(\omega)$ related to $X_d(\Omega)$?

etc, work with moving around between coords.

All of this assumes that there has been no aliasing. For us to have had no aliasing, remember the Nyquist sampling theorem.

Considerations of LTI for $Y_c(\omega) \equiv X_c\parens{\frac{\omega T_r}{T_s}}$.

The only way your end-to-end system will have an LTI-equivalent is if you fulfill the conditions of the Nyquist sampling theorem (no aliasing), and your reconstruction period is the same as your sampling period.

Then you know that your output is equal $\frac{G}{T} H_d\parens{ \omega T} X_c(\omega)$. The equivalent LTI filter is simply $\frac{G}{T} nH_d \parens{ \omega T}$