Sampling theorem review - Rhea

Sampling theorem- As long as a signal is both band-limited and the period is small enough then the signal can be recovered using Xd[n]]X(nT).

Questions you should ask yourself when tackling these problems:

1. Is the signal band-limited?
2. What is the nyquist rate? (2wm)
3. Does it meet the nyquist condition?(ws>2wm)
4. If the nyquist condition is not met, does the signal overlap?

Things you keep in mind when tackling these problems:

1. If a signal does not meet the nyquist condition but the impulse train does not overlap then the signal can still be recovered by applying a filter.
2. A rec in the time domain is equivalent to a sinc in the frequency domain.
3. A sinc in the time domain is equivalent to a rec in the frequency domain.
4. Time shifting (Ex: $ x(t-t_0) \ $->$ e^{-j\omega t_0}X(\omega) $ ) does not affect the nyquist rate.
5. Frequency shifting (Ex: $ e^{j\omega_0 t}x(t) $ -> $ \mathcal{X} (\omega - \omega_0) $) affects the nyquist rate.

ECE301: Sampling Theorem

Random Useful stuff	stuff
CTFT Properties	CTFT
Nyquist Rate	2wm
Sampling Rate	ws=$ 2pi/T $
Nyquist Condition	ws>2wm
Sinc (Time Domain)	Ex: x(t)=$ sin(wt)/t $
Sinc (Frequency Domain)	Ex: X(w)=$ sin(wt)/w $
Rec (Time Domain)	Ex: x(t)=u(t)-u(t-1)
Rec (Frequency Domain)	Ex: X(w)=u(w)-u(w-1)
Modulation 1 (Exponential Carrier)
Demodulation 1 (Exponential Carrier)
Modulation 2 (Sine Carrier)
Demodulation 2 (Sine Carrier)
Modulation 3(Impulse Train Carrier)
Demodulation 3 (Impulse Train Carrier)