Lecture 26 - Notes

March 15, 2016

Sampling

Question: Under what conditions can we reconstruct the original signal, $x(t)$, from its discrete, sampled, version, $x[n]$?

To construct an impulse train we will use a sampling function $s(t)$, where,

$$\begin{aligned} x_s(t) = x_c(t) \cdot s(t) \end{aligned}$$

We can recover the signal $x_c(t)$ using an ideal lowpass filter.

Example

A sinusoidal input signal, $x(t) = \cos{(10t)}$, is sampled with $\Omega _s = 90$. A signal $y(t)$ is generated using the bandpass filter,

$$\begin{aligned} H\left( j\Omega \right) = \begin{cases} T & \text{for}~90 \lt \Omega \lt 180 \\\\ 0 & \text{otherwise} \end{cases} \end{aligned}$$

and a sampling function,

$$\begin{aligned} s(t) = \sum_{k = -\infty}^{\infty} \delta(t - kT) \end{aligned}$$

where $T = \frac{2\pi}{90}$. Find $y(t)$.

Discrete Time Processing of Continuous Time Signals

$$\begin{aligned} X_d\left(e^{j\omega}\right) = \frac{1}{T} \sum_{k=-\infty}^{\infty} X_C \left( j \left( \frac{\omega}{T} - 2k \frac{\pi}{T} \right) \right) \end{aligned}$$