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Discrete Time Systems

ELEC 3004: Systems: Signals & ControlsDr. Surya Singh(Some material adapted from courses by Russ Tedrake, MIT)

Lecture 15

elec3004@itee.uq.edu.auhttp://robotics.itee.uq.edu.au/~elec3004/

© 2013 School of Information Technology and Electrical Engineering at The University of Queensland

April 26, 2012

• Properties of Discrete Time [DT] Signals

• DT Signal Models

• DT Signal Operations

• DT Convolution

• DT Systems Friday

• (also for Lab 3 / Exp 4): Introduce FIR Filters Friday

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Goals for the Week

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Today…

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Week Date Lecture Title

127-FebIntroduction1-MarSystems Overview

26-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 Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals

26-AprDiscrete-Time Systems9

1-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

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

Recap: AliasingAnother view of this

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• Aliasing - through sampling, two entirely different analogsinusoids take on the same “discrete time” identity

For f[k]=cosΩk, Ω=ωT:

The period has to be less than Fh (highest frequency):

Thus:ωf: aliased frequency:

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Recap: Alliasing

Basic Signal Processing: Continuous & Discrete

Continuous

Discrete

x(t)

h(t,T)

y(t,T)

x(n) y(n,k)

h(n,k) Delay

( )0

tdT

( )k

n

0

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4

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Discrete LTI

• Circulant matrices are the “finite” equivalent of LTI systems.• Their properties are very similar to those of LTI systems.• The mathematics are a bit simpler, just “standard” linear algebra.• In particular, they have a nice and simple factorization structure.

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Discrete LTI Circulant Systems

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• In general, matrices do not commute (AB ≠ BA)• Any two circulant matrices do commute!

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Circulant Systems

• A special circulant matrix

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Shift matrix Z

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• The shift Z acts on signals like a unit delay (but it “wraps around”)

• Recall time-invariance: “shifting the input shifts the output”

• A natural notion: linear shift-invariant systems

or equivalently, DZ = ZD (“system commutes with the shift”)

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Shift invariant systems

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Shift invariant systems are circulants

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• Eigenvalues (λ) and Characteristic Functions:

• For DT LTI the eigenfunctions are complex exponentials

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Eigenvalues of DT-LTI Systems

• Every circulant matrix has an eigenvalue decomposition:

eigenvectors:• Notice that the eigenvectors T do not depend on entries ai!• This change-of-base matrix T is called the Fourier matrix.

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Back to Circulant Matrices…Their eigenvectors do not depend on ai

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• They do depend on the values of ai (a0, a1, …, an-1).(you’d hope)

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Eigenvalues of Circulant Matrices

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Ex:

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• Ex: Kinematic (constantly accelerating body) systems:

• Discrete Time Convolution in Matrix Formvia the Toeplitz Matrix

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WHY?:These Systems Show Up Everywhere

• Remember: Flip Shift Slide SumDT Convolution: Vector Sums

x(n) = 1 2 3 4 5 h(n) = 3 2 1

0 0 1 2 3 4 5

1 2 3 0 0 0 0

0 0 1 2 3 4 5

0 1 2 3 0 0 0

0 0 1 2 3 4 5

0 0 1 2 3 0 0

x(k)

h(n,k)

3 2 6 1 4 9y(n,k)

e.g. convolution

y(n) 3 8 14

Sum over all k

Notice the gain

h(n-k)

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DT Convolution: Matrix Formulation of Convolution

3

8

14

20

26

14

5

1 2 3 0 0 0 0

0 1 2 3 0 0 0

0 0 1 2 3 0 0

0 0 0 1 2 3 0

0 0 0 0 1 2 3

0 0 0 0 0 1 2

0 0 0 0 0 0 1

0 0

0 0

0 0

0 0

0 0

3 0

2 3

0

0

1

2

3

4

5

0

0

.

y H x

Toeplitz Matrix

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• Impulse Response:Imaging a discrete-time, LTI System F . It’s impulse response is given by: • Arbitrary Response:An arbitrary input x[n] can be written: (Sampling theorem)

So:

∴ a DT LTI is completely characterized by its impulse responseELEC 3004: Systems 26 April 2013 - 20

Response of a Discrete-Time LTI System & Convolution Sum

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• As with the continuous domain, commutatively, associativity, and distributivity hold…

• Commutativity gives a nice result:

• Makes Step Responses “Easy”:

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Convolution Response

• Causality:

» or

• Input is Causal if:

• Then output is Causal:

• And, DT LTI is BIBO stable if:

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DT Causality & BIBO Stability

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Impulse Response (Graphically)

∞ matrix × ∞ vector?

• First let’s multiply circulant matrices…– A circulant matrix can be descibed completely by its first row or column

• Multiply by u[k]

∴ For circulant matrices, matrix multiplication reduces to a weighted combination of shifted impulse responses

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How do you multiply an infinite matrix?

Z: Shift operator

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Two Types of Systems• Linear shift-invariant:

Z: Shift operator

• Linear time-invariant system

R: Unit delay operator

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Impulse Response of Both Types

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Impulse Response of Both Types

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Next Time in Linear Systems ….

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Week Date Lecture Title

127-FebIntroduction1-MarSystems Overview

26-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 Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals

26-AprDiscrete-Time Systems9

1-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

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