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Linear Time-InvariantLinear Time-Invariant((“LTI”“LTI”) Systems) Systems

Montek SinghMontek Singh

Thurs., Feb. 7, 2002Thurs., Feb. 7, 2002

3:30-4:45 pm, SN1153:30-4:45 pm, SN115

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What we will learnWhat we will learn

How to represent a circuit as an input-How to represent a circuit as an input-output output system (“black box”)system (“black box”) What are LTI systems?What are LTI systems?

How is their behavior described?How is their behavior described?

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Why treat circuits as I/O systems?Why treat circuits as I/O systems?A system representation …A system representation …

is not bound to a particular input is not bound to a particular input allows us to distill the essence of an arbitrarily complex allows us to distill the essence of an arbitrarily complex

circuit into a concise descriptioncircuit into a concise description e.g., Thevenin and Norton equivalentse.g., Thevenin and Norton equivalents

can incorporate other (non-electrical) technologiescan incorporate other (non-electrical) technologies e.g., acoustic, optical, magnetic etc.e.g., acoustic, optical, magnetic etc.

inp

ut

inp

ut

ou

tpu

tou

tpu

t

inp

ut

inp

ut

ou

tpu

tou

tpu

t

ff

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What are LTI systems?What are LTI systems?LTI systems areLTI systems are linearlinear andand time-invariant:time-invariant:

Linearity:Linearity: output for a sum of inputs = sum of individual outputsoutput for a sum of inputs = sum of individual outputs i.e., i.e.,

Time-Invariance:Time-Invariance: inherent system properties do not change with timeinherent system properties do not change with time delaying the input by time delaying the input by time simply delays the output by simply delays the output by i.e.,i.e.,

))()(()()(

))(()( and ))(()(

2121

2211

tBxtAxftBytAy

txftytxfty

))(()(

))(()(

txfty

txfty

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ExamplesExamplesLTI systems:LTI systems:

Most physical systems when operated at small Most physical systems when operated at small amplitudes:amplitudes:an LCR electrical networkan LCR electrical networka mechanical spring, a glass prism, a loudspeaker …a mechanical spring, a glass prism, a loudspeaker …

Non-linear systems:Non-linear systems: Most physical systems when “stretched to the limit”:Most physical systems when “stretched to the limit”:

a blaring loudspeakera blaring loudspeaker Some systems that are intentionally operated in that Some systems that are intentionally operated in that

mode:mode:diodes, transistors, logic gates, digital systems …diodes, transistors, logic gates, digital systems …

Time-variant systems:Time-variant systems: Systems whose properties change with time:Systems whose properties change with time:

a resistor getting hottera resistor getting hotter the human eyethe human eye

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An LTI system’s behaviorAn LTI system’s behaviorSystem’s behavior = mapping from input to System’s behavior = mapping from input to

outputoutput

How to represent?How to represent? Describe the underlying physical phenomenaDescribe the underlying physical phenomena

goes back to circuit theorygoes back to circuit theory Enumerate all (interesting) input-output pairsEnumerate all (interesting) input-output pairs

unwieldy descriptionunwieldy description Describe output for a select set of inputsDescribe output for a select set of inputs

choose some special inputchoose some special inputcompute output behavior for that inputcompute output behavior for that input infer behavior for arbitrary inputsinfer behavior for arbitrary inputs

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Choosing that special input …Choosing that special input …Unit impulse function: Unit impulse function: ((tt))

Unit impulse = a pulse of:Unit impulse = a pulse of: infinitesimal durationinfinitesimal duration infinite amplitudeinfinite amplitude unit areaunit area

Also known as: Dirac delta functionAlso known as: Dirac delta function

tt

F(t)F(t)

11

11

tt

FF(t)(t)

1/1/

tt

(t)(t)11

)( lim )(0

tFt

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Unit impulse: propertiesUnit impulse: properties

Examples:Examples:

-

1)(

0for ),( undefined is )(

0for ,0)(

dtt

tt

tt

tt

22(t)(t)22

tt

(t-(t-aa))11

aa tt

-½-½(t+(t+11))

-1-1

½½

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Unit impulse: used as a samplerUnit impulse: used as a sampler

Multiplying a signal by Multiplying a signal by (t-a)(t-a) and and integrating has the integrating has the effect of sampling it effect of sampling it at at t = a.t = a.

tt

x(t)x(t)

tt

(t-(t-aa))11

aa tt

x(t)x(t)(t-(t-aa))

11

aa

)(

)()(

)()(

)()(

-

-

-

ax

dtatax

dtatax

dtattx

Sampling Theorem:Sampling Theorem:Sampling Theorem:Sampling Theorem:

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Reconstituting a signal from Reconstituting a signal from samples samples (1)(1)

Swap the roles of Swap the roles of tt and and aa::

)()()(-

axdtattx

Sampling Theorem:Sampling Theorem:Sampling Theorem:Sampling Theorem:

-

-

)()()( or,

)()()(

daataxtx

dataaxtx

x(t)x(t) can be regarded as an can be regarded as an infinite sum of infinitesimal infinite sum of infinitesimal samples, samples, i.e.,i.e., sample sample x(a)x(a) summed over all summed over all a.a.

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Reconstituting a signal from Reconstituting a signal from samples samples (2)(2)

tt

x(t)x(t) (t-a)(t-a)

aa

)(

)()(-

tx

daatax

tt

x(t)x(t)

aa

dada1/da1/da

tt

x(t)x(t)

aa

x(a)x(a)(t-a)da(t-a)da

tt

x(t)x(t)

aa

-

)()( daatax

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Unit impulse: system’s responseUnit impulse: system’s responseOutput of a system when input = Output of a system when input = (t)(t) is called is called

thethe“unit impulse response”“unit impulse response”

Denoted by Denoted by h(t):h(t):

Example: human eyeExample: human eye

))(()(

))(()(

tfth

txfty

tt

h(t)h(t)

latencylatency

persistencepersistencepeakpeak

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Generalization: Arbitrary inputGeneralization: Arbitrary inputGiven: Given: unit impulse response unit impulse response h(t),h(t), i.e., i.e.,

Find: Find: system response system response y(t)y(t) to an arbitrary input to an arbitrary input x(t)x(t)

Method:Method: express input x(t) as an infinite sum of weighted impulsesexpress input x(t) as an infinite sum of weighted impulses

compute response to each individual impulsecompute response to each individual impulse

weight and add up all the individual responsesweight and add up all the individual responses

-

)()()( daataxtx

)())(( athatf

)())(( thtf

-

-

)()()( or,

))(()())(()(

daathaxty

daatfaxtxfty

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ConvolutionConvolutionDefinition: Definition: y(t)y(t) is the is the “convolution of”“convolution of” x(t)x(t) and and h(t)h(t) if: if:

Notation: Notation:

Properties:Properties:1.1. commutativity:commutativity:

2.2. associativity:associativity:

3.3. distributivity:distributivity:

4.4. scalability:scalability:

5.5. derivatives:derivatives:

-

)()()( daathaxty

))(()()()( thxthtxty

xhhx )()( 2121 hhxhhx

)()()( 2121 hxhxhhx

)()()( ahxhaxhxa

dt

dhxh

dt

dxhx

dt

d )(

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Convolution: exampleConvolution: example

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http://www.jhu.edu/~signals/convolve/index.htmlhttp://www.jhu.edu/~signals/convolve/index.html

Check it out!Check it out!

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Homework: Due 2/19Homework: Due 2/191.1. The output of a particular system S is the time The output of a particular system S is the time

derivative of its input.derivative of its input.a)a) Prove that system S is linear time-invariant (LTI).Prove that system S is linear time-invariant (LTI).

b)b) What is the unit impulse response of this system?What is the unit impulse response of this system?

2.2. Prove Property 5. Prove Property 5. That is, prove that, for an arbitrary That is, prove that, for an arbitrary LTI system, for a given input waveform LTI system, for a given input waveform x(t)x(t), the time , the time derivative of its output is identical to the output of derivative of its output is identical to the output of that system when subjected to the time derivative of that system when subjected to the time derivative of its inputits input. . In other words, differentiation on the input In other words, differentiation on the input and output sides are equivalent.and output sides are equivalent.