signals & signal models - systems: signals and control
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Signals & Signal Models
ELEC 3004: Systems: Signals & ControlsDr. Surya Singh
Lecture 3
[email protected]://robotics.itee.uq.edu.au/~elec3004/
© 2013 School of Information Technology and Electrical Engineering at The University of Queensland
March 6, 2013
• Equivalence Across Domains
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Recall From Last Time ...
Source: Dorf & Bishop, Modern Control Systems, 12th Ed., p. 73
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• Assignment 0 is online – Please complete by tomorrow!
• Adam Keys is looking for volunteers as part of a study of motion and video simulations.
• Thank you for your participation in the Tutorials ! Based on feedback we’ll run Tutorials on Week 4 as well!!
• Practise Shuffle / Peer Feedback on Friday.
• Assignment 1 comes out next Monday – March 11…
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Announcements:
!
• Space Audit on March 22 (F) and March 28 (W)– They will be going around in black hats– P&F will be checking on us– This effects class funding (read tutor hours!!)– Sounds like an ideal time for a pop-quiz ( just saying )
• Q & A Forum Software– Sadly, there is no “StackOverflow” for classes1. Blackboard Forums – Allows Ranking (**) and is “local”2. Piazza – Yet another IT system3. news:uq.itee.elec3004 -- A classic!
Matlab Transfer Function Example By R. Simpson on Tutorial Page. Thank you!
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Announcements [2]:
!
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Today:Week Date Lecture Title
127-FebIntroduction1-MarSystems Overview
2 6-MarSignals & Signal Models8-MarSystem Models
313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition
420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability
527-MarSignal Representation29-MarHoliday
610-AprFrequency Response & Fourier Transform12-AprAnalog Filters
717-AprIIR Systems19-AprFIR Systems
824-Aprz-Transform26-AprDiscrete-Time Signals
91-MayDiscrete-Time Systems3-MayDigital Filters
108-MayState-Space
10-MayControllability & Observability
1115-MayIntroduction to Digital Control17-MayStability of Digital Systems
1222-MayPID & Computer Control24-MayInformation Theory & Communications
1329-MayApplications in Industry31-MaySummary and Course Review
• Communicates information
• Varies in Time and/or Space
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What is a Signal?
Everywhere:• The type
• The screen
• Your 5 senses
• Telephony
• Interest rates
• Leaf colour
Temperature
Distance
Force
Speed
F(x) Signal
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Abdo RIP
Thorax RIP
Flow
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Signals!
s1
Gevv
s
Echem
ELEC 3004: Systems
Understanding & Classifying SignalsMetrics:
• Size
• Signal Energy
• Signal Power
• Frequency
• Phase
• Entropy– Deterministic signals
– Random signals
Classifications:
• Continuous-Time
• Discrete-Time
• Analog
• Digital
• Even
• Odd
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• 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.
• 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.
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Signal Size
• 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.
• Instead we look at it’s energy or signal energy
• But this can be infinite!
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Signal Energy
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• Generalize this to a finite measure via a RMS (root means square measurand)
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Signal Power:
Finite EnergyFinite Power
∞ EnergyFinite Power
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• Classifications – A “pair-wise” way of characterizing the signal by putting it into
a “descriptive bin”– Ex: How to describe a person – well we could measure them
(weight, height, power, etc.) or we could sort by category (“North/South-side,” Australian, etc.) Neither is perfect
• Some common classifications:1. Continuous-time and discrete-time signals2. Analog and digital signals3. Periodic and aperiodic signals4. Real and complex signals5. Deterministic and probabilistic signals
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Signal Classifications
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• The independent variable (x-axis) – in this case t – is continuous (which may be the eals, but can be others)
• This does not dictate the form of u(t) -- it may be either continuous or discrete.
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Continuous-Time
u(t)
t
t∈R
• The independent variable (x-axis) – in this case t – is continuous (which may be the eals, but can be others)
• This does not dictate the form of u(t) -- it may be either continuous or discrete.
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Continuous Time
u(t)
t
t∈R
Continuous
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• The independent variable is only set for fixed, discrete values• Ex: t ∈ Integers• Written in [Square Brackets]• u[tn] = u[n]• tn={t0, t1, t2, …, tn}• Relationship to continuous time: u[n]=u(nTs)
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Discrete Timeu[t ]n
t
t∈Z (Integers)
Ts = Sample Period = Time between samples
Discrete
• Continuous and Discrete on the y-axis• u(t) = Continuous Analog• u(t) = Discrete Digital
– Number of levels can be many – Simplest case is for two Binary digit (or bit) signal
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Analog and Digital Signals
u(t)
t
Discretedigital levels
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• (a) analog, continuous time | (b) digital, continuous time• (c) analog, discrete time and (d) digital, discrete time
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Analog and Digital Signals
• A signal x(t) is periodic if for some positive constant T0
• For periodic signals x(t) of period T0 : – the area under x(t) over any interval of duration T0 is the same
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Periodic and Aperiodic Signals
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Real and Complex Signals
• Deterministic – are those whose description is
known completely
• Random– Those signals whose descriptions is unknown incompletely via
statistical or probabilistic descriptions.
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Deterministic and Random
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Ergodicity
1. Over time: multiple readings of a quantity over time
• “stationary” or “ergodic” system• Sometimes called “integrating”
2. Over space: single measurement (summed) from multiple sensors each distributed in space
3. Same Measurand: multiple measurements take of the same observable quantity by multiple, related instruments
e.g., measure position & velocity simultaneously
Basic “sensor fusion”
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Classifying Signals: Even and Odd• Even
– Area:
Every signal can be expressed as a sum of even and odd components
• Odd
– Area:
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SymmetryLine
SymmetryLine
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Signal ModelsA “model” of signals
• Key functions include:– Step
– Impulse
– Exponential functions play
• Basis for representing other signals,
• Simplify many aspects of the signals and systems.
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sin( x)/( x)
Signal Models1] Unit Step
• if A=1, t0=0– Heaviside function
2] Unit Impulse
• Can be approximated as:
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3] Sinusoidal & Exponential Signals
• A, ω : Amplitude and Frequency (ω=2πf)• ϕ: phase angle
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Signal Models
• We’ll talk about System Models
• Pleae Review: – B1-B5 of Lathi:
• Complex numbers, sinusoids, partial fraction expansion
– Chapter 1.7 to 2.1 of Lathi• Systems, classifications of systems, and system responses.
• Try the practise assignment – by tomorrow please!
• Assignment 1 comes out Next Monday (March 11).
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Next Time…