<aside> 🐟 Discreteness in one domain gives periodicity in the other domain.

</aside>

Problem Set 1:

$$ \text{Some useful formulas:} \newline x_{even} (t) = \frac{x(t)+x(-t)}{2} \newline x_{odd}(t) = \frac{x(t)-x(-t)}{2} $$

• Order of Operations in Signal Manipulation:

  i.e., $f(t) = \sin (4 \pi (t + \frac{1}{4})) + \cos(6 \pi t)$

  1. Factor

     

  2. Scale

     

  3. Shift

     

• Periodicity of CT signals:

  1. Identify the period
     1. i.e., for $\sin (at)$, period is $\frac{2 \pi}{a}$
  2. Calculate the LCM
  3. If LCM exists, then signal composed of these components is periodic, and LCM is the fundamental period of the combined signal
     1. If no LCM exists, signal is not periodic

• Average power per period

• Periodicity of DT signals:

• Plotting DT signals:

CT and DT Signals

Continuous-time (CT) signals

Function of one or more independent variables which range over the real numbers $\R$

Discrete-time (DT) signals

Function of one or more independent variables which range over the integers $\Z$

• $x$ is the signal, we write $x[n]$ for the signal’s value at time $n \in \Z$
• The signal is defined only at integer times $n$, not in-between

Sampling: from CT to DT signals

Process of converting CT signals into DT signals

Continuous signal $x(t)$ is sampled at intervals $T_s$ to produce the discrete samples $x_s [n]$.

• This involved taking regular samples of a CT signal at discrete time intervals. resulting in a sampled DT signal $x_s [n]$
• Rate at which we sample depends on the clock