Introduction to signals and systems/basic signal properties

EG1110 Signals and Systems

WHERE DOES IT FIT?

Dr. Matt Turner - [email protected]

EG1110 Signals and Systems

EG1050 Circuits and Systems

• To analyse signals and systems in

Course aims:

– Time domain

• To introduce concepts of signals and systems

– ‘Laplace domain’ - useful for solving Control Theory

equations

– What are they? Why are they used?

– Frequency domain - useful for uncov-

• To discuss properties of signals and sys-

ering important properties

tems, with reference to simple systems

EG2030 EG3020 EG3110

COURSE INTRODUCES GENERIC TOOLS FOR SYSTEM ANALYSIS

Modelling/ Simulation

Signal Processing

EG3170 EG3220

Electronics/ Communication

EG1200 EG1120 EG2055 EG3040

• Concepts from “signals and systems” are heavily used in subsequent modules

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2

By the end of the course you should be able to

Recommended textbooks

1. Appreciate basic concepts of signals and systems (convolution, transfer functions etc.) 2. Understand Laplace Transforms and their use in the analysis of signals and systems.

Book

Pros

Cons(?)

3. Work out time/frequency domain properties of simple first and second order systems.

Signals and Systems

Comprehensive

Hard to follow in places

4. Understand basic concepts of stability and feedback.

Oppenheim, Willsky and Young

More general than necessary A classic text

Learning: • Lectures - slides available on-line (http://www.le.ac.uk/eg/mct6/teaching/eg1110.html)

Elements of Signals and Systems

Emphasis different to Oppenheim Some material out of scope

Poularikas and Seely

More modelling Good rigorous book

Fundemental of Signals and

Similar to Oppenheim

• Seminars (and some labs)

Systems using the web and MATLAB New book, well laid out

• Private study - reading books, lecture notes and attempting problems.

Kamen and Heck

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More discrete, more state-spa

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More general than necessary

Why Signals and systems?

Motivational Example: flight control system Aircraft Dynamics:

Commands

Position

Mechanical System Control System: • Used to COMBINE and UNIFY different types of process

Electrical system

Pilot

Pilot:

– electrical and mechanical systems e.g. power systems, automotive systems, aircraft systems

θ,φ,ψ Control System

Biological system

Aircraft Dynamics y

Sensors • Elucidates the MAIN (MOST USEFUL?) parts of a physical/virtual process

Mixed

• Abstract SYSTEM/SIGNAL representation more COMPACT.

system

Sensors

electrical/mechanical

ANALYSIS OF “SYSTEM” REQUIRES DIFFERENT DISCIPLINES

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Motivational Example: flight control system (cont.) Pilot Control System

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Aircraft Dynamics Nonlinear differential equations:

Neuromuscular model:

“Difference” equations

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Syllabus

• Frequency response of systems, Bode plots • Response of special systems: second order systems

• Introduction to signals and systems:

• Block diagrams, feedback, stability

– definitions – basic properties • Time-domain response of general linear systems – Convolution integrals – free response – forced respose • Laplace Transforms and transfer functions – Laplace solution of differential equations – Laplace transforms and inverse – Input-output behaviour of systems in Laplace domain 9

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What is a signal?

What is a system?

Some notions:

Some notions:

• Something which carries some sort of information.

• Something which processes (in some way) a signal.

• They arise from a response to an ‘event’, or they may play a role in the cause of an ‘event’. • Something which produces a reaction in response to a stimulus. • Can be (in principle) measured - often corresponds to some physical quantity. • Examples: voltage, current, position, velocity. • More examples: heart-rate, computer variables, salary... • Need not be a “physical quantity” - but often is.

• Something which produces an output signal from an input signal. • Examples: operational amplifier, DC motor, speedometer. • More examples: heart, computer, bank account. • Need not be something “physical” - but often is.

• Often something which can be plotted. Our notion of a signal is something which is a function of an independent variable (normally time).

Our notion of a system is one of an operator which produces an output y(t) in response to an input

i.e. x(t) would denote a signal.

u(t).

such a signal generally (but not always) would vary with time.

Many systems are defined by their input-output behaviour (to a certain extent).

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Basic signal properties

Important signals - Periodic signals

Continuous time signals: f (t) is a continuous

Discrete time signals: f (k) is a sequence de-

function of time i.e. it is defined at every point

fined only at integer values of the variable k.

in time.

• Often encounter periodic signals • A periodic signal f (t) has the property that f (t) = f (t ± T ) for some T > 0

f(k)

f(t)

The smallest T > 0 for which this holds is called the fundemental period • Important periodic signals k

t

sin(ωt) = sin(ωt ± T )

T = 2π

cos(ωt) = cos(ωt ± T )

T = 2π

Normally these integers are spaced equidistant on a corresponding continuous time axis

• Can always “build” any periodic signal from a series of sine and cosine signals.

- but not always We mainly deal with continuous signals - discrete signals mainly used for comparison and illustration. 13

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• Often the complex form is used:

Important signals - Non-periodic signals

ejωt = cos(ωt) + j sin(ωt)

... ? 1

e−jωt = cos(ωt) − j sin(ωt)

1. Unit-step: u(t)

... ? 2     

Hence

u(t) =    

cos(ωt) = Re(ejωt)

1

t>0

0

t t0

(t − t0) > 0)

0

t < t0

(t − t0) < 0) t t0

2j sin(ωt) = ejωt − e−jωt

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2. Unit Impulse

3. Unit Pulse

A signal with several peculiar properties.....

δ(t − t0) = Z t 2

t1

        

t1

pa(t) =  

0 t 6= t0

 

∞ t = t0

x(t)δ(t − t0)dt = x(t0),

Z t 2

    

x(t)δ(t − t0) = 1,

if

|t| > a |t| < a

or

t 1 < t0 < t2     

x(t) = 1

pa(t − t0) =    

i.e. the impulse is an infinitely tall signal with an infinitessimal width. Its “weight” is proportional to x(t) - the weight of a unit impulse is unity.

0

|t − t0| > a

1

|t − t0| < a

Homework: prove pa(t − t0) can be written as a sum of step functions.

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4. Sinc function (often encountered in Fourier analysis)

sinc(ωt) =

0 1

5. Exponentially weighted sine waves

sin(ωt) ωt

x(t) = Ceatsin(ωt) a > 0 - growing sinusoid

1

a < 0 - decaying sinusoid

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Amplitude − sinc(5 π t)

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(What about a = 0? What if a has imaginary parts?) 0.4

Encountered in stability analysis and analysis of linear systems.

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0

−0.2

−0.4 −5

−4

−3

−2

−1

0 1 Time − seconds

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5

Useful for expressing power spectra

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Main points of lecture 1. Signals are normally time varying quantities representing (normally physical or measurable) quantities. 2. Systems are mathematical operators which produce output signals in response to input signals 3. Signals can be classed as either periodic or non-periodic 4. Sines, impulses, steps, exponentials are common types of signals.

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