Signal Fundamentals

Question 1

Signal Classifications:

Pairs/Axes (7)

Answer 1

Continuous or Discrete

Analog or Digital

Periodic or Aperiodic

Deterministic or Random/Probabilistic

Energy or Power

Causal or Non-Causal

Even or Odd

Question 2

What is a Signal?

Answer 2

A set of data or information.

Typically represented as a function of the independent variable time, but not always.

Can include anything that ‘transmits’ information, energy, power:

Examples:

• Telephone signal
• A companies’ monthly sales
• Power delivery from an engine to wheels
• Stock prices

Question 3

Signal Classifications:

Analog Signals

Answer 3

Analog signals have a signal amplituded

which can take an infinite number of possible values, bounded by some range.

Question 4

Signal Classifications:

Digital Signals

Answer 4

In a digital signal, the amplitude is restricted to a set of distinct possible values.

The most common digital signal is binary, taking only the values 0 or 1

Question 5

Signal Classifications:

Periodic Signals

Answer 5

A signal is periodic if it repeats in constant intervals.

Formally:

For some positive constant, T₀ - Called the period,

x(t) = x(t + T₀) for all t

The signal repeats with a period of T₀

The smallest value of T₀ that satisfies this is called the Fundamental Period of the signal.

Question 6

Signal Classifications:

What is the Fundamental Period

of a Periodic Signal?

Answer 6

The smallest change of time that the signal repeats itself in.

Formally, the smallest value T₀ that satisfies:

x(t) = x( t + T₀ ) for all t

Question 7

Signal Classifications:

Aperiodic Signal

Answer 7

A signal that is not periodic.

It either has no repeating values, such as an exponential function, or it’s values do not repeat in a regular fashion.

Question 8

Signal Classifications:

Analog vs Digital

Answer 8

Analog signals can take infinite values within a range.

Digital signals can only take specific values, usually 0 or 1

Question 9

Signal Classifications:

Periodic Signals

vs

Aperiodic Signals

Answer 9

Periodic Signals have a repeating pattern.

Aperiodic signals do not.

Question 10

Signal Classifications:

Continous Time Signal

Answer 10

A signal where there are an infinite number of values within a portion of the domain.

There is a value of x(t) for every value of t in the domain.

Example: t can be 0, 0.1, 0.0000000001, etc

Question 11

Signal Classifications:

Discrete Time Signals

Answer 11

In Discrete Time Signals,

the domain is separated into discrete steps.

The function is only defined at these steps.

For a function x(t), only some values of t have a corresponding x(t).

The steps are usually very small and spread regularly.

Example: A signal recorded by taking regular samples

Question 12

Signal Classifications:

Continuous vs Discrete Time Signals

Answer 12

Continous Time Signals

have a value for any possible t in the domain

Discrete Time Signals

are only defined at values of t within a particular set

Question 13

Signal Classifications:

Energy Signals

Answer 13

Refers to how the “size” of the signal is measured.

An Energy Signal

is one that has finite energy, basically one that “dies out”

Formally:

x(t) → 0 as | t | → infinity

Question 14

Signal Classifications:

Power Signals

Answer 14

Refers to how the “size” of the signal is measured

A Power Signal

is one that doesn’t “die out” over time

Formally:

x(t) does not approach 0 as | t | → infinity

All Periodic Signals are Power Signals

Trying to measure the Energy would yield an infinite result, so the signal size is measured as “Power”

Question 15

Signal Classfications:

Energy vs Power Signals

Answer 15

Energy Signals

The amplitude approaches 0 over time, so the signal size can be measured as “Energy”

Power Signals

The amplitude does not approach 0 over time, signal size must be measured as “Power”

Important Note:

A signal belongs in neither classification if it’s Power, P_x is calculated to be infinite

Question 16

Signal Classifications:

Causal

vs

Non-causal

Answer 16

Causal Signals

• Begin at or after t=0
• Do not extend to t= -inf
• Signal amplitude is not affected by “future” values
• Implies that changes are “caused” by something immediate

Non-Causal Signals

• Continuos towards t= -inf
• “Future” values may affect “current” values

Question 17

Signal Classfications:

Deterministic

vs

Random(Probabilistic)

Answer 17

Deterministic Signals

• The signal x(t) is a function of t

Random or Probabilistic Signals

• The signal x(t) is NOT a function of t
• But it may be related with probabilities

Question 18

Signal Classification:

Even Signals

vs

Odd Signals

Answer 18

Refers to the type of Symmetry a signal/function may have

Even Signals/Functions

• Reflect across the Y-axis, so the “left” is a mirror image of the “right”
• No sign change
• x(-t) = x(t)

Odd Signals/Functions

• Reflect across the Y-axis AND the X-axis
• Sign change
• x(-t) = - x(t)

Question 19

Even and Odd Functions:

What is their importance in analyzing signals?

Answer 19

Every signal x(t) can be expressed as a sum of Even and Odd component functions:

x(t) =

(1/2)[x(t) + x(-t)] (Even component)

+ (1/2)[x(t) - x(-t)] (Odd component)

Question 20

Even and Odd Functions:

Multiplication Rules

Answer 20

Apply when multiplying two functions together

Even X Odd = Odd

Odd X Odd = Even

Even X Even = Even

Note: It sort of parallels the rules of multiplying positive and negative values

Question 21

Signal Size:

Two ways to measure the

size of a signal

Answer 21

Signal Energy

Used when signal exists for finite time

Signal Power

Used when signal exists for infinite(in effect) time, such as a periodic signal

Question 22

Signal Size:

Why is signal size NOT just the integral of the signal?

Answer 22

The integral of the signal would calculate the area under the curve.

BUT, in most signals, this includes both negative and positive areas that cancel each other out.

This would show a signal size that is much smaller than the actual signal in many instances.

Question 23

Signal Size:

What conditions are required for

Energy to be an appropriate measure of

signal size?

Answer 23

Signal energy must be finite

in order to be a useful measure.

So:

The signal amplitude must approach 0 as time approaches infinity and negative infinity:

lim_|t|→inf x(t) = 0

If this condition is not met, Power of the signal should be used to measure the signal size

Question 24

Signal Size:

Energy of a Signal

Formula

Answer 24

The Energy of the Signal, E_x

is the Integral of the square of the amplitude, wrt time

E_x = INT( x(t)² )dt

Note: More generally, use the absolute value of x(t) for complex valued signals

Question 25

Signal Size:

Relationship between

Energy and Power

Answer 25

A signal with Finite Energy, E_x = c,

has Zero Power: P_x = 0

A signal with Finite Power, P_x = c

has Infinite Energy: E_x = INF

Question 26

Signal Size:

Power of a Signal

Basic Idea

Answer 26

The energy of a signal is infinite. Presumably because it is periodic.

Power is a measure of the time average of the energy over a period.

Question 27

Signal Size:

Power of a Signal

Formula

Answer 27

For a Real signal:

P_x =

lim_T→∞ (1/T) ∫_-T/2^T/2 x²(t) dt

Note: If period T is specified, ignore the limit

Complex Signal:

P_x =

lim_T→∞ (1/T) ∫_-T/2^T/2 |x²(t)| dt

Question 28

Signal Size:

What is the Root Mean Square (RMS)

of a Signal?

Answer 28

The RMS is simply the square of the Power of the signal.

rms = P_x²

Question 29

Fundamental Signals:

List of Basic Functions/Signals

Answer 29

Unit Step Function, u(t)

Unit Impulse Function, δ(t)

Unit Ramp Signal, u_r(t)

Exponential Signal, e^st

Pulse Signals

Question 30

Fundamental Signals:

Unit Step Function

Answer 30

u(t) = 1 when t ≥ 0

u(t) = 0 when t < 0

• Amplitude is zero before t=0, then immediately jumps to 1 and stays there
• Useful to describe signals that begin at t=0
• Multiply the function by u(t) to “get rid of” everything before 0, creating a causal form of the signal

Question 31

Fundamental Signals:

Unit Impulse Function

δ(t)

Answer 31

Represents an “impulse” at exactly zero

δ(t) = 0 when t ≠ 0

and ∫ δ(t)dt = 1

Very useful in determing system responses.

Can be approximated with very short-duration pulse functions, so long as the function integrates to Unity( 1 )

Question 32

Fundamental Signals:

Impulse Function δ(t)

Some approximation functions

Answer 32

The impulse function can be approximated by several different pulse functions, so long as the area under the pulse function is unity.

• Rectangular Function
  
  • f(t) = 1/ε , -ε/2 < t
  • ε → 0
• Exponential
  
  • f(t) = e^-𝛼t , t > 0
  • 𝛼→∞
• Triangular
  
  • f(t) goes from 0 to 1/ε at -ε < t < 0
  • f(t) goes from 1/ε to 0 at 0 < t
  • ε → 0
• Guassian
  
  • some bullshit you’ll look up when you need it

Question 33

Fundamental Signals:

Pulse Signals

Answer 33

A Pulse signal

is a signal that lasts for just a short time.

The simplest way to obtain a pulse signal is to just subtract one step function from another, creating a Rectangular Pulse

Ex:

u(t-2) - u(t-4)

Creates a “rectangular” pulse between t=2 and t=4

Question 34

Fundamental Signals:

Exponential Function

Answer 34

e^st

Very important to analysis of signals and systems.

• S is the complex frequency
  
  • s = σ + jω
  • ω is the real frequency component

Question 35

Fundamental Signals:

Exponential Function:

Representing Complex Frequency S

as a complex sinusoid

Answer 35

e^st

s = σ + jω

e^st = e^{(σ + jω)t} = e^σte^jωt

e^jωt can be represented as the complex sinusoid:

cos(ωt) + jsin(ωt)

So

e^st = e^σt ( cos(ωt) + jsin(ωt) )

Why: Supports finding the conjugate relation

Question 36

Fundamental Signals:

Exponential Function:

Conjugate

and

Conjugate Relation

Answer 36

e^st = e^{(σ + jω)t} = e^σte^jωt

The Conjugate e^s*t = e^{(σ - jω)t} = e^σt / e^jωt

1/e^jωt can be represented as the complex sinusoid:

cos(ωt) - jsin(ωt)

So

e^s*t = e^σt ( cos(ωt) - jsin(ωt) )

We can use this to get the Conjugate Relation for the Exponentional Function:

e^σtcos(ωt) = (1/2)(e^st + e^s*t)

Question 37

Fundamental Signals:

Sampling Property

of the

Unit Impulse Function

Answer 37

For a function f(t), f(t)𝛿(t) = f(0)𝛿(t)

Because for everywhere but 0, 𝛿(t) = 0

Integrating:

∫f(0)𝛿(t) dt = f(0) ∫𝛿(t)dt = f(0)

Because f(0) is a constant and ∫𝛿(t)dt = 1

This winds up being the same as “sampling” the function at t=0.

More generally, to sample at an arbitrary time T, just shift the Impulse function:

∫ f(t) 𝛿(t-T)dt = f(T)

Question 38

Useful Signal Operations:

List of Basic Operations

Answer 38

• Time Shifting
  
  • Delay
  • Advance
• Time Scaling
  
  • Compress
  • Expand
• Time Reversal/Reflection
• Shift
• Scaling
• Sum
• Product

Question 39

Useful Signal Operations:

Time Shifting Operations

Answer 39

Delay signal by T:

x_d = x(t + T)

Advance signal by T:

x_a = x(t - T)

Question 40

Useful Signal Operations:

Time Scaling Operations

Answer 40

Compress signal by a factor of T:

x_c = x(t * T)

Expand signal by a factor of T:

x_e = x(t / T)

Question 41

Useful Signal Operations:

Time Reversal

Answer 41

Reverse the signal, reflecting x(t) across the horizontal axis:

x_r(t) = x(-t)

Question 42

Signal Operations:

Split this function into Even and Odd components:

x(t) = e^-atu(t)

Answer 42

The Even component is the function divided in half, and one half made the reflection:

x_e(t) = (1/2)[e^-atu(t) + e^atu(-t)]

The Odd component is the function divided in half, with one half made a reflection with sign change

x_e(t) = (1/2)[e^-atu(t) - e^atu(-t)]