Systems & Signals: Notes & Study Guide

Systems and Signals(FA 2020_EEL3135.01FX ENGR)

I will take all notes on the class here and then build a study guide at the end

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Overview

Semester: Fall 2020

School: Florida Polytechnic University

Textbook: Digital Design and Computer Architecture, Second Edition

Continuous-time and discrete-time systems analysis, focusing on linear time-invariant (LTI) systems and the classification of these systems, is presented in this course.

Convolution and its application to LTI systems, the Laplace, Fourier, and Z transforms, the Fourier series and their application to the analysis of LTI systems will also be presented.

Industry applications will be a specific focus.

After successfully completing the course with a grade of C (2.0/4.0) or better, students should be familiar with the structure, representation, computational methods and notation of signals and systems, and be able to apply them to the analysis of communications and control systems.

Outcomes:

1. Determine different type of signals
2. Describe linear time invariant (LTI) systems
3. 3.Apply mathematical tools (differential/difference equations and convolution) to characterize LTI systems and their impulse response
4. Analyze LTI signals and systems using the Fourier series and Fourier transform
5. Determine LTI system response using the Laplace & Z-Transform

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Notes

Signals: A signal is a set of data or information.

Systems: A system is an entity that processes a set of signals (inputs) to yield another set of signals (output).

Size of a Signal

• The size of an entity is a number that indicates the largeness or strength of that entity.

Energy of a Signal

In general, signal energy is defined as:

If x(t) is a real-valued signal:

Signal Power

In general, signal power is defined as:

If x(t) is a real-valued signal

Now in physics, energy is the total amount of work done, and power is energy per unit of time. (Power is watts. Energy is watt-hours.)

Signal Energy and Signal Power are different units that are derived differently.

Signal energy is an indication of the energy capability of the signal not the actual energy

Similarly, signal power is indicative of the power capability of the signal not the actual power

Energy and Power Signals

• A signal with finite energy is an energy signal
• A signal with finite and nonzero power is a power signal
• A signal cannot both be an energy signal and a power signal.

Units depend on the nature of the signal x(t)

If x(t) is a voltage signal:

Ex has units of V²s

Px has units of V

Time Shifting

A signal can be delayed or advanced

Advance: ϕ(t+T)=x(t)

&

Delay: ϕ(t)=x(t-T)

When shifting, replace t with the starting variable

Time Scaling

1. write equation(s) for signal
2. Decide if compressing or expanding

Time Reversal

To time-reverse a signal we replace t with −t, and the time reversal of signal x(t) results in a signal x(−t).

Combined Operations

The most general operation involving all the three operations is x(at-b)which is realized in two possible sequences of operation

1. Time Shift x(t) by b to obtain x(t-b) Now time scaling the shifted signal x(t-b) by a
2. Time-scale x(t) by a to obtain x(at). Now time-shift x(at) by b/a

Classification of signals

Continuous-Time and Discrete-Time Signals

Continuous →Specified for a continuum of values of t

Discrete →Specified only at discrete values of t

Amplitude can take on any value in a continuous range: Analog

Amplitude can take on only a finite number of values: Digital

Periodic and Aperiodic Signals

Causal Signals

x(t) is a causal signal if

• A signal that starts before t=0 , it is a non causal signal
• A signal that is zero for all t≥0 , is called an anti causal signal.

Deterministic and Random Signals

• A signal whose physical description is known completely, either in a mathematical form or a graphical form, is a deterministic signal
• A signal whose values cannot be predicted precisely but are known only in terms of probabilistic description is a random signal.

Even and Odd Functions

A real function xₑ(t) is said to be an even function of t if

•A real function x₀(t) is said to be an odd function of t if

The impulse response of a system is the derivative of the step response. Given the unit step response of a system, yγ(t) the unit impulse response of the system is simply the derivative. yδ(t)=dy γ(t)dt.

Height=Unit Impulse response(τ) • Input function (time-τ)

Width = d(τ)

so area = Unit Impulse response(τ) • Input function (time-τ)•d(τ)

Total output in this moment in time (when t=τ) is:

∫ Unit Impulse response(τ) • Input function (time-τ)•d(τ)

Laplace Transform

L{∫ Unit Impulse response(τ) • Input function (time-τ)•d(τ)}=

L{Unit Impulse response(τ)} • L{Input function}

Transfer Function of a system =H(s)=

L{∫ Unit Impulse response(τ) • Input function (time-τ)•d(τ)}

Find zero input response of an LTIC system described by D+5 y(t)=x(t) if the initial condition is y(0)=5

• y₀ (5)=0 is the zero input (x(t)=0), the solution of (D+5)y₀(t)=0
• The characteristic polynomial is λ+5=0
• The characteristic root is λ=-5
• The characteristic mode is e⁻⁵ᵗ
• The zero input response of the characteristic mode is y₀(t)=C₁• e⁻⁵ᵗ

Use the initial condition to solve

5=C₁• e⁻⁵ᵗ →5 =C₁

• The zero input response is y(t)= 5e⁻⁵ᵗ

Find the zero input response of a second order system letting

y₀(0)=1, y₀’(0)=4

(D²+2D)y₀(t)=0

(λ+2)(λ+0)=0

characteristic modes = e⁻²ᵗ and e⁰ᵗ

The zero impulse response of mode

y₀(t)=c₁e⁰ᵗ+c₂e⁻²ᵗ

y₀’(t)=0–2c₂e⁻²ᵗ

=-2e⁻²ᵗ

Determine the unit impulse response of LTIC systems described by the following equations:

(D+2)y(t)=(3D+5)x(t)

The general form of a linear differential system is

Q(D)y(t)=P(D)x(t)Q(D)=Dⁿ+a₁Dⁿ⁻¹+…+ a_n

and

P(D)=b₀Dⁿ+b₁Dⁿ⁻¹+…+ b_n

h(t)=b₀δ(t)+[P(D)•y_n(t)]•u(t)

Given (D+2)y(t)=(3D+5)x(t)

as an order of P(D),

b₀=3

order of the system is:

N=1 then y(0)=1

Thus the characteristic equation is:

λ+2=0

The characteristic root is

λ=-2

Then y_n(t)=ce⁻²ᵗ

Set t equal to 0

0=ce⁻²ᵗ

substitute initial condition to get

c=1

y_n(t)=e⁻²ᵗ

Then P(D)=3D+5

P(D)y_n(t)=3D•y_n(t)+5y_n(t)

P(D)y_n(t)=3D(e⁻²ᵗ)+5(e⁻²ᵗ)

P(D)y_n(t)=3(-2e⁻²ᵗ)+5(e⁻²ᵗ)

P(D)y_n(t)=-6e⁻²ᵗ+5e⁻²ᵗ

h(t)= 3δ(t)-e⁻²ᵗu(t)

Find the unit impulse response of an LTI system specified by the equation (D2+6D+9)y(t)=(2D+9)x(t)

h(t)=b_n•δ(t)+[P(D)y_n(t)]u(t)

Characteristic equation is (λ²+bλ+9)=0

λ₁=-3

λ₂=-3

y_n(t)=a₁e⁻³ᵗ+a₂te⁻³ᵗ

y_n(0)=a₁=0

y_n(0)=-3•a•₁e⁻³ᵗ +a₂•e⁻³ᵗ-a₂te⁻³ᵗ| t=0

→a₁=0

→a₂=1

y_n(t)=te⁻³ᵗ

P(D)y_n(t)=(2D+9)y_n(t)=2y_n(t)+9y_n(t)

D(D+2)y(t)=(D+4)x(t)

(D²+2D+1)y(t)=Dx(t)

Study Guide

Exam 1:

Modules 1,2, & 3 (Chapters 1 & 2 of the textbook) — Linear Systems and Signals

Chapter 1:

In this chapter we shall discuss basic aspects of signals and systems. We shall also introduce fundamental concepts and qualitative explanations of the hows and whys of systems theory, thus building a solid foundation for understanding the quantitative analysis in the remainder of the book.

For simplicity, the focus of this chapter is on continuous-time signals and systems. Chapter 3 presents the same ideas for discrete-time signals and systems.

SIGNALS

A signal is a set of data or information.

Examples include a telephone or a television signal, monthly sales of a corporation, or daily closing prices of a stock market (e.g., the Dow Jones averages). In all these examples, the signals are functions of the independent variable time. This is not always the case, however. When an electrical charge is distributed over a body, for instance, the signal is the charge density, a function of space rather than time.

In this book we deal almost exclusively with signals that are functions of time. The discussion, however, applies equally well to other independent variables.

SYSTEMS

Signals may be processed further by systems, which may modify them or extract additional information from them.

For example, an anti-aircraft gun operator may want to know the future location of a hostile moving target that is being tracked by his radar. Knowing the radar signal, he knows the past location and velocity of the target. By properly processing the radar signal (the input), he can approximately estimate the future location of the target.

Thus, a system is an entity that processes a set of signals (inputs) to yield another set of signals (outputs).

A system may be made up of physical components, as in electrical, mechanical, or hydraulic systems (hardware realization), or it may be an algorithm that computes an output from an input signal (software realization).

Size of a Signal

The size of any entity is a number that indicates the largeness or strength of that entity.

Generally speaking, the signal amplitude varies with time.

How can a signal that exists over a certain time interval with varying amplitude be measured by one number that will indicate the signal size or signal strength?

Such a measure must consider:

• signal amplitude
• duration

For instance, if we are to devise a single number V as a measure of the size of a human being, we must consider not only his or her width (girth), but also the height.

If we make a simplifying assumption that the shape of a person is a cylinder of variable radius r (which varies with the height h), then one possible measure of the size of a person of height H is the person’s volume V, given by

V=π •∫ (from 0 to H) r² (h)dh

Arguing in this manner, we may consider the area under a signal x(t) as a possible measure of its size, because it takes account not only of the amplitude but also of the duration. However, this will be a defective measure because even for a large signal x(t), its positive and negative areas could cancel each other, indicating a signal of small size. This difficulty can be corrected by defining the signal size as the area under |x(t)|2, which is always positive. We call this measure the signal energy Ex, defined as

Eₓ=∫ (from -∞ to ∞) |x(t)|² dt

The energy measure, however, is not only more tractable mathematically but is also more meaningful (as shown later) in the sense that it is indicative of the energy that can be extracted from the signal.

Chapter 7: Fourier Transform

Aperiodic and Periodic Signals

Omega

e^-j omega t

e^j omega t

Laplace is more general than Fournier

Find the compact Fourier series of a periodic signal

by decompose into cosines

cos is the built in unit so it breaks down to 0

• Research Fourier Transform
• Research Eullers Equation

Exam 2

Topics:

1. Fourier Series
2. Fourier Transform
3. Continuous-Time Analysis using Laplace transform

What is a Fourier series? — an infinite series of trigonometric functions which represents an expansion or approximation of a periodic function, used in Fourier analysis. Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

A signal is a periodic signal if it completes a pattern within a measurable time frame, called a period and repeats that pattern over identical subsequent periods. The completion of a full pattern is called a cycle. A period is defined as the amount of time (expressed in seconds) required to complete one full cycle. The duration of a period represented by T, may be different for each signal but it is constant for any given periodic signal.

What does it mean in context of systems and signals? — When these systems generate signals, these signals are essential combinations of sin and cosine functions with respect to time so representing them in simpler terms that are more malleable mathematically can be beneficial for calculations

Signals and Systems/Periodic Signals

What is a Fourier transform? —

What is Continuous-Time Analysis using Laplace transform ?—