Lecture 4 - Notes

January 12, 2016

Energy and Power of Signals

The notion of power and energy were initially only applicable to signals produced by physical systems only (i.e. v(t), i(t) across a resistor of resistance R). We create a generalization to allow power and energy to characterize any type of signal.

We calculate the energy of a discreet time signal $x[n]$ over $[n_1,n_2]$ as

$$E_x = \sum_{-\infty}^{\infty} \left|x[n]\right|^2$$

Example

Find the energy contained in the signal,

$$x[n] = \left(\frac{1}{2}\right)^n u[n]$$

Solution

Since $x[n]$ contains a unit step function, it will be causal, so

$$\begin{align} E_x &= \sum_{-\infty}^{\infty} \left|x[n]\right|^2 \newline &= \sum_{-\infty}^{\infty} \left|\left(\frac{1}{2}\right)^n u[n]\right|^2 \newline &= \sum_{0}^{\infty} \left|\left(\frac{1}{2}\right)^n \right|^2 \newline &= \sum_{0}^{\infty} \left(\frac{1}{4}\right)^n \newline &= \frac{1}{1 - \frac{1}{4}} \newline &= \frac{4}{3} \newline \end{align}$$

Growing and Decaying

Given an exponential signal,

$$x[n] = \left(\frac{a}{b} \right)^n u[n]$$

A growing exponential is where $a \gt b$ and the energy will non-finite. A decaying exponential is where $a \lt b$ and will (possibly?) have a finite energy.

Average Power

A signal with finite energy is called an energy signal. Many Discreet Time signals don't have finite energy. For these signals we consider their average power.

$$\begin{align} P_x = \lim_{N \to \infty} \frac{1}{2N} \sum_{-N}^{N - 1} \left|x[n]\right|^2 \end{align}$$

Discreet Time Systems

We represent a discrete-time system as a transformation that maps an input sequence $x[n]$ into a unique output sequence $y[n]$. We can classify these systems as,

• Memoryless systems
• Linear systems (examples 2.5, 2.6)
• Time-invariant systems (example 2.7, 2.8)
• Causal systems (example 2.9)
• Stable systems (example 2.10)

Linear Systems

definition: A system is linear if the principle of superposition holds.

The Property of Superposition

If we consider two inputs to a system $x_1[n]$ and $x_2[n]$, we get two outputs $y_1[n]$ and $y_2[n]$ such that,

$$\begin{align} x_1[n] \to y_1[n] \newline x_2[n] \to y_2[n] \end{align}$$

now if we create any linear combination of $x_1[n]$ and $x_2[n]$, denoted $x_3[n]$. Where,

$$\begin{align} x_3[n] = a_1 x_1[n] + a_2 x_2[n] \end{align}$$

and $a_1, a_2 \in \mathbb R$. Then if $y_3[n]$, the system output for $x_3[n]$, is the same linear combination of $y_1[n]$ and $y_2[n]$, i.e., if

$$\begin{align} x_3[n] &\to y_3[n] \newline &\text{and} \newline y_3[n] &= a_1 y_1[n] + a_2 y_2[n] \end{align}$$

the system is linear.

Time Invariant Systems

definition: A system is time invariant if for any input signal $x[n]$ with output signal $y[n]$, i.e.,

$$\begin{align} x[n] \to y[n] \end{align}$$

then for any time shift $n_0$ where $n_0 \in \mathbb Z$,

$$\begin{align} x[n - n_0] \to y[n - n_0] \end{align}$$