CMPE 312: Midterm Topic 2

Continuous-Time Signal

A continuous-time signal is defined for all values of time \(t\): x(t)

Discrete-Time Signal

A discrete-time signal is defined only at discrete instants of time: x[n]

A discrete-time signal can be obtained by sampling a continuous-time signal:

x[n]=x(nT_s)

Even Signals

A signal is even if:

x(-t)=x(t)

• Symmetric about the vertical axis.

• The left side mirrors the right side.

ex: x(t)=t² because (-t)²=t²

Odd Signal

A signal is odd if:

x(-t)=-x(t)

• Antisymmetric about the time origin.

• The signal changes sign when time is reversed.

Ex: x(t)=t³ because (-t)³ = -t³

Problem: expressing a signal as the sum of Even and Odd Components

Periodic Signals

A periodic signal repeats itself after a fixed time interval T.

x(t) = x(t+T)

Aperiodic Signal

An aperiodic signal does not repeat itself at regular intervals.

There is no positive period \(T\) that satisfies the periodicity condition.

Examples:

• Periodic: sine wave, cosine wave, repeating square wave.

• Aperiodic: a single rectangular pulse or a decaying exponential.

The 4 elementary signals

1. exponential signal

2. sinusoidal signal

3. unit-step function

4. unit-impulse function

Exponential Signal

x(t) = Be^at

Where:

• B = amplitude constant

• a = real parameter

• a<0 means decaying exponential

• a>0 means growing exponential

Sinusoidal Signal

Unit-step function

Unit-Impulse Function

Basic Operations on Signals

Time scaling

y(t) = a(at)

• a > 1: signal is compressed

• 0 < a < 1: signal is expanded

ex: x(2t) → compressed by 2

x(t/2) = expanded by 2

Time shifting

y(t) = x(t-t_0)

• t_0 > 0: shift right

• t_0 < 0: shift left

ex: x(t - 3) → right by 3

x(t + 2_ → lefr by 2

Reflection

y9t) = x(-t)

Reflection reverses the signal about the vertical axis (t=0).

Combining Time Shifting and Scaling

y(t) = x(at - b)

• First, perform time shifting: \(v(t)=x(t-b)\).

• Then, perform time scaling: \(y(t)=v(at)=x(at-b)\).

Formula Cheat Sheet