Lecture 3 - Notes

January 8, 2016

Basic Discreet Time Signals

The Unit Step

$$\begin{align} u[n] &= 1 &~\text{if}~ n \ge 0 \newline &= 0 &~\text{if}~n \lt 0 \end{align}$$

The Unit Impulse

$$\begin{align} \delta[n] &= 1 &~\text{if}~ n = 0 \newline &= 0 &~\text{if}~ n \neq 0 \end{align}$$

Relationship between Unit Impulse and Step

$$\delta[n] = u[n] - u[n-1]$$

$$u[n] = \sum_{m = -\infty}^{n} \delta[m]$$

Properties

The Sampling Property allows you to fix $x[n]$ if it is multiplied by the unit impulse function,

$$x[n]\delta[n-n_0] = x[n_0]\delta[n-n_0]$$

Compared to Continuous Time

In discreet time $\delta[0] = 1$ compared to $\delta(0) = \infty$ in continuous time.

Exponential Signals

A discreet time exponential function is of the form,

$$x[n] = A\alpha^n$$

where $A,\alpha \in \mathbb{C}$.

Real Exponential Signals

If $A,\alpha \in \mathbb{R}$, we call it a Real Exponential Signal and write it as,

$$x[n] = Ce^{an}$$

where $C,a \in \mathbb{R}$.

Sinusoidal Signals

We can write a Sinusoidal Signal in two ways,

$$\begin{align} x[n] &= Ae^{j \omega_0 n} \newline &= \cos{(\omega_0 n + \varphi)} \end{align}$$

we only consider this signal over a period of $2\pi$ typically $[-\pi,\pi]$

Even and Odd Symbols

A discreet time signal is even if,

$$x[n] = x[-n]$$

for all $n \in \mathbb Z$.

A signal is odd if,

$$x[-n] = -x[n]$$

Any signal can be broken in an even and odd portion, the sum of which is the signal,

$$\begin{align} x_{\text{even}}[n] = \frac{x[n] + x[-n]}{2} \end{align}$$ $$\begin{align} x_{\text{odd}}[n] = \frac{x[n] - x[-n]}{2} \end{align}$$

and,

$$\begin{align} x[n] = x_{\text{even}}[n] + x_{\text{odd}}[n] \end{align}$$

Transformations of Discreet Time Signals

We can do a time shift by $n_0$ by,

$$\begin{align} x_{\text{shifted}}[n] = x[n-n_0] \end{align}$$

We can reverse time with,

$$\begin{align} x_{\text{reversed}}[n] = x[-n] \end{align}$$

We can scale time by $a$ with,

$$\begin{align} x_{\text{scaled}}[n] = x[an] \end{align}$$

Combining Transformations

Recommended order,

1. Apply the shift
2. Apply the scale

To shift then scale, given $x[n]$, find $y[n] = x[an + b]$. So,

1. Shift: $w[n] = x[n + b]$
2. Scale: $y[n] = w[an] = x[an + b]$

take note that, $w[an] \neq x[an + ab]$.