basic concepts in ct and dt systems

8/12/2019 Basic Concepts in CT and DT Systems

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ELEC 310-31

Basic concepts in CT and DT

systems


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Readings and exercise

Textbook: Sections 1.5, 1.6

Suggested exercise:

pp. 57-58:

1.15

1.16

1.18

1.19


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Course outline

Some examples of systems

System properties:

Causality

Stability

Linearity

Invariance


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What is a system?


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Examples of systems (1)

Example 1. RLC circuit


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Examples of systems (2)

A rudimentary edge detector


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Observations

A very rich class of systems (but by no means allsystems of interest to us) are described bydifferential and difference equations

Such an equation, by itself, does not completelydescribe the input-output behaviour of thesystem: we need auxiliary conditions (initialconditions)

In some cases the system of interest has time asthe natural independent variable and is causal.However, that is not always the case.


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System interconnections

Serial

Parallel

Feedback

example


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feedback connection

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System properties

Causality, linearity, stability, time-invariance etc.

Why?

Important practical/physical implications

They provide us with structure that we can exploitfor both system analysis and system design


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Causality

A system is causal if the output does not anticipatefuture values of the input, i.e. if the output at any timedepends only on the values of the input up to that time

All real-time physical systems are causal. Time only

moves forward, and effect occurs after the cause. Causality does not apply to systems processing spatially

varying signals (we can move both left and right, up anddown)

Causality does not apply to systems processingrecorded signals (e.g. taped sports games versus livebroadcast)


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Causal or non causal?

y(t)=x2(t-1)

y(t)=x(t+1)

y[n]=x[-n]

y[n]=(1/2)n+1 x3[n-1]


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Stability

A stable system is one in which small inputs lead

to responses that do not diverge

Ex: model for the balance of a bank accountfrom month to month

y[n]=1.01y[n-1]+x[n], x[n]>0 for all n>=0.

http://www.youtube.com/watch?v=MWJHcI7UcuE
http://www.youtube.com/watch?v=MWJHcI7UcuEhttp://www.youtube.com/watch?v=MWJHcI7UcuE


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Stable or unstable?

y[n]=nx[n]

y[n]=x[4n+1]

Strategy:

to prove that a system is unstable, we need to

find a specificbounded input that leads to an

unbounded output. If such an example does not exist, then the

system is stable


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Time invariance

Conceptually: A system is time invariant (TI) if its

behaviour does not depend on a particular

moment in time.

Mathematically: for a CT time-invariant system:

for a DT time-invariant system:


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Time invariant or time-varying?

y[n]=nx[n]

y[n]=sin(x[n])

Strategy: we must determine whether the time invariance

property holds for any input and any time shift

When a system is suspected of being time-varying, we can

seek a counterexample (a specific input signal for whichthe condition of time-invariance would be violated)


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Now we can deduce something!

If the input to a TI system is periodic, then the

output is periodic with the same period.


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Linear and non-linear systems

Many systems are non-linear (ex: circuit

elements: diodes, dynamics of aircraft etc.)

In ELEC 310 we focus exclusively on linear

systems Why?

We can often linearize models to examine small

signal perturbations around operating points

Linear systems are analytically tractable,

providing basis for important tools used in DSP.


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Linearity

A system is called linear if it has two mathematicalproperties:

Let us consider x1(t)y1(t) and x2(t)y2(t)

Additivity: x1(t)+ x2(t)y1(t) + y2(t)

Homogeneity (scaling): ax1(t)ay1(t),

ais any complex number

We can combine these two properties into one:

ax1(t)+ bx2(t)ay1(t) + by2(t)

ax1[n]+ bx2[n]ay1[n] + by2[n]


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Properties of linear systems

Superposition

For linear systems, zero inputzero output

A linear system is causal if and only if it satisfiesthe condition of init ial rest


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Linear Time-Invariant (LTI) Systems

Our focus for most of this course

A basic fact: If we know the response of an LTI

system to some inputs, we actually know its

response to manyinputs


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Example: DT LTI system

Given

Compute the response of the system to


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Example: DT LTI system (contd)


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Example of a complete analysis of

system properties

Determine which properties (time-invariance,

linearity, causality, stability) hold for the following

system:

=

=

nk

kxny ][][1


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System invertibility

A system is invertible if distinct inputs lead to

distinct outputs

A serialinterconnection of an invertible system

with its inverse leads to an output w[n] equal tothe input x[n].

Invertibility is important in signal transmission

(lossless signal coding)


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Invertibility analysis

Determine if the following systems are invertible

or not. If yes, construct their corresponding

inverse systems

y[n]=nx[n]

y[n]=x[1-n]


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Learning outcome

the primary focus in this class is on linear, time-invariantLTI systems in the DT domain

LTI systems are defined in a similar way in both CT andDT domains

How to compute the global input-output function ofinterconnected systems

How to determine whether a system is: Causal

Stable

Time-invariant

Linear

Invertible

basic concepts in ct and dt systems

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