are time-varying systems laplace-transformable?

University of Windsor Electrical & Computer Enginee ring Department

Are Time-Varying Systems Laplace-Transformable?

Shervin ErfaniResearch Centre for Integrated MicroelectronicsElectrical and Computer Engineering Department

University of WindsorWindsor, Ontario N9B 3P4, Canada


Outline of the Presentation

• Objectives• From LTV Elements to LTV Systems• Observations on LTV Systems

– Bivariate Time and Bifrequency Characterization– Generalized -Delay System Representation

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– Generalized -Delay System Representation– Circularly Symmetric Functions

• Two-Dimensional Laplace Transform (2DLT)– Laplace –Carson Transform– Transformation of Nonanticipative Systems– Transformation of Synchronized Systems

• Conclusions• References


Objectives � There has been an increasing interest in realizatio n and

implementation of linear time-varying (LTV) and ada ptive systems.

� The rather limited use of LTV networks in analog si gnal processingand communication fields is due largely to lack of powerful meansfor their analysis, synthesis and performance evalu ation.

� The theory of LTV systems has been widely based on the time - domain

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� The theory of LTV systems has been widely based on the time - domain approach.

� The main objective of this project is to provide a unified treatment for the analysis and synthesis of LTV systems.

� The approach to be described consists of extension of Laplace transform techniques commonly used for linear time- invariant (LTI) systems.


Fundamental Input-Output Representation

• The classical theory of variable systems is based o n the solutions of linear ordinary differential equat ions with varying coefficients. – The varying coefficients are functions of an indepe ndent

variable, conveniently called the time .– The time is assumed to be real for physical systems.

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– The time is assumed to be real for physical systems.

∑∑==

=m

k

kk

n

i

ii txtbtyta

0

)(

0

)( )()()()(


Observations on LTV Dynamic systemsObservation 1 – In general, time variables of the signal and system do not have to be synchronized; i.e., the (time) variables of the signal and system are independentof each other.

)(),()(),( txDKtyDL ττ =Observation 2 – At any instant of “t” there is a response, which is a specified function of “τ”.Observation 3 – At any fixed “τ” there is a response, which is a specified function of “t”.Observation 4 – The system response is a function of variations of observation

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Observation 4 – The system response is a function of variations of observationparameter “t” and application parameter “τ”.Observation 5 – A zero-input, SISO LTV system described by:

0)(),( =⋅yDL τis a linear system that its natural frequencies are varying with “τ”. In other words, solutions of this equation are exponential functions of time with varying natural frequencies, as given by:

∑=

−=⋅n

i

ti

iecy0

)()( τα

where )(τα i is a function of variable coefficients of the fundamental equation of the system under consideration.


Linear Time Varying Elements• A single-input single-output (SISO) dynamic system element of

finite order characterized by its input-output rela tionship is said to be linear if the following holds for each t ≥0:

)()()( txthty =

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� Where h(t) is the system function defines the response at time t, denotes the slope of the y-x curve in a rectangular coordinates system.

h(t1)

h(t2)h(t3)

x(t)

y(t)


Linear Time Varying Operator• A SISO dynamic system operation is shown symbolical ly by:

)}({)( txty ο=• The system operator is linear if and only if the fo llowing

relation holds:

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)()()}({)}({)}()({212121

tttttt yyxxxx βαβοαοβαο +=+=+

• The system input can be any function including an impulse or a delta function:

)()();( τδτδ

−= tthty


Observing an Impulse Response Function (1)

• Observation 6 – The following symbolic identity holds:

)()()()( thtth −=− τδττδ

• Observation 7 - The product is different fro m zero at the point t=τ.

)()( τδ −tth

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at the point t=τ.

• Observation 8 - The impulse response of the system has a circular symmetric property with respect to its arg uments tand τ.

);();( tt yy ττδδ

=


Observing an Impulse Response Function (2)

• Observation 9 – In the (t,τ)-plane , due to the circular symmetryand because delta function is an even function, we can define a bivariate response function as:

|)(|),(),();( τδτττ −= ttxthty

• Observation 10 - The ordinary output response at the point t=τ

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• Observation 10 - The ordinary output response at the point t=τis:

or, equivalently:

)()(|)(|),(),()(0

τττδτττ xhdtttxthy =−= ∫+∞

−

)()(|)(|),(),()(0

txthdttxthty =−= ∫+∞

−

ττδττ

Question – Can a system function be equal to an input functio n?Is the system output considered to be still linear?


Frequency-Domain Characterization

• A linear time-invariant dynamic system described by a homogeneous n th-order differential equation:

• Using the operator d/dt �� s, this equation can be

0)(0∑

=

=n

i

i

ii tydt

da

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0)(0∑

=

=n

i

ii tysa

• Using the operator d/dt �� s, this equation can be written as:

• Thus, the homogenous solution is a linear combination of exponentials:

∑=

−=n

i

tsi

ieaty0

)( Roots of CharacteristicsOperator Equation


Frequency-Domain Characterization (Cont.)• A linear time-invariant dynamic system with the

applied input x(t):

• The system is turned on at t, the impulse-response at t-τ is obtained :

)()(00

txdt

dbty

dt

da

m

kk

k

k

n

i

i

ii ∑∑==

=

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t-τ is obtained :

• We observe:• The response is a bivariate function of t and• Characteristics roots are a function of τ

∑=

−=n

i

tsi

ieaty0

)(),( ττ Roots of CharacteristicsOperator Equation

)();(00

τδτ −=∑∑==

tdt

dbty

dt

da

m

kk

k

k

n

i

i

ii

τ


• The ordinary unilateral Laplace transform of is obtained as:

• This is a function of the variable application time τ.

Laplace Transform of the Impulse Function

)( τδ −t

ττδτδ 11

0

)()}({ sts edtettL −+∞

−

− =−=− ∫

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• This is a function of the variable application time τ.

• A second transformation yields:

210

2

1)}({ 21

ssdeetL ss

D +==− ∫

+∞

−

−− ττδ ττ


Definition of 2DLT – The ordinary unilateral 2DLT is defined as:

Laplace-Carson Transform

20 10 21212211),(),( dtdteetthssH tsts

∫ ∫+∞ +∞ −−=

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Inverse Transformation - The inverse 2DLT is given by:

{ }

( )2 1

1 1 2 2

2 1

12 1 2 1 2

1 2 1 22

( , ) ( , )

1( , )

2

D

j j s t s t

j j

L H s s h t t

H s s e e ds dsj

σ σ

σ σπ

−

+ ∞ + ∞

− ∞ − ∞

=

= ∫ ∫


Observation 11 – For conformal transformation, it is required that the unit function u(t, τ) transforms into itself:

u (t,τ) is equal to 1 when both t and τ are positive, and is equal to zero when at least one of the arguments is negat ive.

Two-Dimensional Laplace Transform (2DLT)

1),(),( 21 =⇔ ssUtu τ

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Observation 12 - Based on the above observation we modify the 2DLT is given by:

τττ τ dtdeethssssHth sts 21

0 0

2121 ),(),(),( −+∞+∞

−∫ ∫=⇔


� The unit–step function u(t, τ) is defined as:

The Laplace-Carson transform of u(t- τ) is

Two-Dimensional Step Function

)()(),( ττ ututu =

22121

21)(),(ss

sdtdeetussssU sts

+=−= −−

+∞+∞

∫ ∫ ττ τ

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Similarly, the L-C transform of u( τ-t) is

Observation 13 - We can write:

The 2DLT transform of is .

≠=

=−−ττ

ττt

ttutu

0

1)()(

210 0

2121 )(),(ss

dtdeetussssU+

=−= ∫ ∫ ττ

21

1

ss

s

+

2

1

s)()( tutu −− ττ


� The unit–impulse function δ(t, τ) is defined as:

The Laplace-Carson transform of δ(t-τ) is

Two-Dimensional Impulse Function

)()(),( τδδτδ tt =

212121

21)(),(ss

ssdtdeetssss sts

+=−=∆ −−

+∞+∞

∫ ∫ ττδ τ

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Similarly, the L-C transform of δ(t,τ) is

Observation 14 - We can obtain the L-C transform of as:

)()( τδ −tth

210 0

2121 )(),(ss

dtdeetssss+

=−=∆ ∫ ∫ ττδ

21),( sst ⇔τδ

∫+∞

+− +=⇔−0

2121)(

21 )()()()( 21 ssHssdehsstth ss τττδ τ


Impulse-Response System Representation

• The more familiar impulse response, using sifting property of

• The system response for a circularly symmetric syst em function can equivalently be written as:

)(),(),( τδττδ −= tthty

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• The more familiar impulse response, using sifting property of the delta function will be:

)()(),()( ττδτττ

τδ hdttthy

t

t

=−= ∫+

−

=

=

� The limits of integration can be extended to infinit y.


� Define the instant at which the input is applied to the system as the origin for time “ t.”

� The nonanticipative condition implies:

Nonanticipative System Function

ττ <≡− tforotutxth )()()(

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� Then, we may define h(.) to be zero for negative values of its argument:

τττ >≡− tfortytutxth ),()()()(


� Let us define:

� The 2DLT is:

� Similarly, we define :

Frequency-Domain Representation of Nonanticipative System Functions

<>−

=τ

τττ

tfor

tforthth

0

)(),(1

21

1

0

211)()(),( 12

sssHdtthedessH tss

+=−= ∫ ∫+∞ +∞

−− τττ

τ

>− tforth ττ )(

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� Adding together, we obtain:

<>−

=tfor

tforthth

τττ

τ0

)(),(2

21

2212

)(),( sssHssH +=

{ }21

21212

)()(),(|)(|

ss

sHsHssHthL D +

+==−τ


Symbolic Bifrequency Input-Output System Representation

h(t2)

Black box representation of circularly symmetric li near systems

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h(t2)

H(s2)y(t1) Y(s1)

x(t1) X(s1) t1 = t2-


The 2DLT of General LTV SystemsConsider a SISO LTV system, initially at rest, described by:

An input x(.) is applied to the system at time ξ =t- τ

∑∑==

=m

kk

k

ki

in

ii dt

txdtb

dt

tydta

00

)()(

)()(

12/2/2009 21

Taking a 2DLT, we obtain:

∑∑==

−=m

kk

k

ki

in

ii dt

txdtb

dt

tydta

00

)()(

),()(

ττ

ττττ τ

τξ

τ dtdeedt

txdtbdtdee

dt

tydta sts

m

kk

k

k

tsts

i

in

ii

1212

0 00 0 0

)()(

),()( −−

=

∞

=

−−

=

∞ ∞

∑∫ ∫∑∫ ∫−=

−

This may demand more initialconditions that the problem requires!


Synchronized System Representation

h(t2)

t1 is the observation time of signal and t 2 is the application time to the system.

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h(t2)

H(s2)y(t1, t2 )x(t1)

t2=0

t2=t1=0

t1


The 2DLT of Synchronized LTV Systems� Consider a SISO LTV system described by:

� Taking Laplace transform with respect to τ , we obtain:

∑∑==

=m

kk

k

ki

in

ii dt

txdb

dt

tyda

00

)()(

)()( ττ

12/2/2009 23

� Taking Laplace transform with respect to τ , we obtain:

ττττ ττ ddt

txdebd

dt

tydea

m

kk

ks

ki

is

n

ii ∑ ∫∑ ∫

=

−+∞

∞−

−

=

+∞

∞−

=00

)()(

)()( 22

∑∑==

=m

kk

k

ki

in

ii dt

txdsB

dt

tydsA

02

02

)()(

)()(


The 2DLT of Synchronized LTV Systems (Cont.)

� Taking a second transform with respect to t:

� If the system is initially at rest then for i =0, 1, 2, . . . , n-1.

dtedt

txdsBdte

dt

tydsA ts

m

kk

k

kts

i

in

ii

11

02

02

)()(

)()( −

=

−+∞

∞−=∑∫∑ =

0)( =

i

i

dt

tyd

12/2/2009 24

dt

∑ ∑

∑∑

=

−

=

−

=

−−

=

−=

=

−−−−=

m

k

k

j

jjkkk

m

k

kkkkk

in

ii

xssXssB

xxsxssXssBsYssA

0

1

0

)(1112

0

)()1(21

1111211

02

)0()()(

)0(...)0()0()()()()(

∑∑ ∑

=

=

−

=

−

−= n

i

ii

m

k

k

j

jjkkk

ssAxssXssBsY

012

0

1

0

)(11121

)()0()()()(� Thus


Real-Variable Function Representationin ( t,τ)-Plane

f(p)

τo

12/2/2009 25

• A system function, which has the simple rotational property of circular symmetry is shown in this figure.

• Figure shows conversion of a one-dimensional profile of a system function of p to a two-dimensional function of a complex variable of t and τ.

• The common views of the independent signal and system functions are the projection over t-axis (pure real) and τ-axis (pure imaginary), respectively.

t

τ

p

o

θ


Circularly Symmetric System Transformation

t1

τ

t + τ = t1

Region over which the impulse-response h(t, τ) of a nonanticipative system is defined is the shaded area.

12/2/2009 26

t1t

0

τττ τ dtdexthsY ts )()(),()( +−

∆∫∫=

Iterated LaplaceTransform


Circularly Symmetric Functions (1)

h(t)

t2

θ

h(t)

t2

θ

12/2/2009 27

t1

L tt1

L t

Rotational property of a circularly symmetric funct ion

)],(),([2

1)()sin,cos(),( 1221 tthtththtthth sym +=≡= θθθ

)],(),([2

1)(),( 1221 ssHssHsHsH sym +=≡φ


Other Observations on LTV Systems

• Mathematically speaking, t and τ represent time variables of an applied signal and the corresponding system.

• Variations of an input signal and an autonomous syste m are independent of each other.

12/2/2009 28

• For circularly symmetrical systems, with no loss of generality, we can rewrite the response as:

)(),(),(ln

txty eth ττ −=


Generalized-Delay System Representation

• If h(t) is a (piecewise) continuous function and bounded by a finite number, its 1 st -order and higher-order derivatives exists.

• The system response can equivalently be written as:

12/2/2009 29

• The system response can be written more compactly a s a generalized-delay operator:

)()(),(

txty etg τ−=

)()( )(

)('

txty et

h

dh∫= −τ

ξξξ


� The Hankel transform is compatible with LTV systems described by a general Bessel equation given as:

The Hankel Transform

)()(1 2

2

2

2

2

txtyat

n

dt

d

tdt

dN

=

±

−+

12/2/2009 30

� The Hankel transform pairs are symmetric because it deals with symmetric functions.

� The 2DLT of a circularly symmetric function with the property lim t�∞ h(t) �0 that is a Hankel transform of order zero.

� This property is quite useful in application of Hankel transforms to LTV systems.

tdttdt


� The Mellin transform is compatible with LTV systems characterized by a general Euler-Cauchy equation given as:

� The impulse response of this nonanticipative Euler-Cauchy LTV system is:

The Mellin Transform

)()(

0

txdt

tydta

i

ii

n

ii =∑

=

)()(1

),( ττ −= tut

gth

12/2/2009 31

� where g(.) is the impulse response of a prototype LTI system obtained by changing the time scale t → - ln t

� The 2DLT in this case becomes the following Mellin Transform pairs:

)()(1

),( ττ

τ −= tugt

th

{ } dttththM s 1

0

)()( −+∞

∫= { } dstsHj

sHM sj

j

−∞+

∞−

−∫=

σ

σπ)(

2

1)(1


Conclusions• The 2DLT techniques are applicable to LTV systems.

• This investigation justifies application of 2DLT as an operational calculus for system characterization, es pecially for analog signal processing problems.

• This approach allows, in effect, two-dimensional t ransform techniques to be used for the time -varying systems in the

12/2/2009 32

techniques to be used for the time -varying systems in the same manner that the conventional frequency-domain techniques are used in connection with fixed system s.

• The 2DLT, Mellin transform, and Hankel transform ca n be derived from the two-dimensional Fourier transform.

• The work presented here opens several areas for fur ther investigations in theory of variable systems.


For Further Information1. S. Efani, “Extending Laplace and Fourier transforms and the case of variable systems: A

personal perspective,” Presentation to IEEE Signal Processing Long Island Section, NY, May 15, 2007.

2. S. Erfani and N. Bayan , “On Linear Time-Varying System Characterizations,” Proc. IEEE EIT 2009, Windsor, ON, Jun. 7-9, 2009, pp. 207-210.

3. S. Erfani and N. Bayan, “Laplace, Hankel, and Mellin Transforms of of Linear Time-Varying Systems,” Proc. IEEE 52th MWSCAS, Cancun, Mexico, Aug. 2-9, 2009, pp. 794-799.

4. V. A. Ditkin and A. P. Prudnikuv, Operational Calculus in Two Variables and Its Applications,Pergaman Press, 1962.

5. D. Voelker and G. Doetsch, Die Zweirdimensional Laplace Transformation, Birkhauser

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Verleg, Basel, 1950.

6. A. H. Zemanian, Generalized Integral Transformations, Dover Publications, New York, NY, 1987.

7. I. N. Sneddon, Fourier Transformations, Dover Publications, New York, NY, 1995.8. N. Bayan and S. Erfani, “Frequency-domain realization of linear time-varying systems by

two-dimensional Laplace transformation,” Proc. NEWCAS, Montreal, Que., June 10-14, 2008, pp. 213-216.

9. A. D. Poularikas, (ed.), The Transforms and Applications Handbook, 2nd Ed., CRC Press, Boca Raton, FL, 2000.

10. L. A. Zadeh, “Time-varying networks, I,” Proc. IRE, vol. 49, October 1961, pp. 1488-1503.


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Shervin ErfaniShervin Erfani is a professor and former Electrical and Computer Engineering Department Head at the University of Windsor, Windsor, Ontario, Canada. His experience spans over 30 years with AT&T Bell Labs, Lucent Technologies, University of Puerto Rico, University of Michigan-Dearborn, Stevens Institute of Technology, and Iranian Naval Academy. His expertise expands over many areas from System Theory to Digital Signal Processing to Network Security Management. He has been a consultant to the industry for a number of years.

Dr. Erfani has published more than 70 technical papers, holds three patents, and is the Senior Technical Editor of the Journal of Network and Systems Management and an associate editor for Computers & Electrical Engineering: An International Journal.

He received a combined B.Sc. and M.Sc. degree in Electrical Engineering from the University of Tehran in 1971, and M.Sc. and Ph.D. degrees, also in Electrical Engineering, from Southern Methodist University in 1974 and 1976, respectively. He was a Member of Technical Staff at Bell Labs of Lucent Technologies in Holmdel, New Jersey, from 1985 to 2001.

are time-varying systems laplace-transformable?

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