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To: Dr. Jim GillilanFrom: Rob ArnoldRe: An application of the “residue theorem”

I asked you once in Applied Complex Variables class, “What do you do with a complex contour integral?”Well, I found out when I got to Digital Control Systems class that some nasty Laplace transform stunts arebetter handled by evaluating a complex contour integral, usually resulting in much less pain. An examplefollows:

Suppose f(z) is analytic inside and on a simple closed contour C except for finitely many isolated singularpoints z1, z2, . . .zn interior to C.

Then by the residue theorem,

f z dz jC

( ) =∫ 2π (sum of residues of f at points z1, z2, . . .zn)

Consider a continuous function, x(t). The action of sampling x(t) at discrete instants of time (t = kT, k =0,1,2,3 . . .) can (ideally) be expressed as the multiplication of x(t) with a train of “impulse functions” (the

Dirac delta function, given byδ ( ) ,t t= ≠0 0 δ ( )t dt =−∞

∞

∫ 1 ). If we symbolize the impulse train by

δ δTn

t t nT( ) ( )= −=

∞

∑0

, then an ideal sampling of x(t) can be represented mathematically by

x t x t tT*( ) ( ) ( )= δ . This can be interpreted as a carrier signal, δT t( ) , modulated by a signal x(t).Functions with discontinuities at t=kT can cause an ambiguity in the above definition. Therefore, we definethe output signal of an ideal sampler as follows:

X s x nT nTsn

*( ) ( )exp( )= −=

∞

∑0

, where x(t) is the signal at the input of the ideal sampler. Typically,

the output of the sampling stage is sent to a zero-order hold, a circuit which maintains the sampler’s outputvalue for the duration of the sample interval. The sample-and-hold output can be expressed as:

x t x u t u t T x T u t T u t T x T u t T u t T( ) ( )[ ( ) ( )] ( )[ ( ) ( )] ( )[ ( ) ( )]= − − + − − − + − − − +0 2 2 2 3 Kwhich has Laplace transform

X s xs

Tss

x TTs

sTs

sx T

Tss

Tss

( ) ( )exp( )

( )exp( ) exp( )

( )exp( ) exp( )

= −−

+−

−−

+−

−−

+0

1 22

2 3K

[ ]=− −

+ − + − + − +1

0 2 2 3 3exp( )

( ) ( )exp( ) ( )exp( ) ( )exp( )Ts

sx x T Ts x T Ts x T Ts K

= −

− −

=

∞

∑ x nT nTsTs

sn( ) exp( )

exp( )0

1

The first factor depends on the input signal and the sample period T. The second factor is independent ofthe input signal. Note that the first term corresponds to X*(s). The second factor then expresses the effectof the zero-order hold.

How can we evaluate X*(s)? One method involves taking the Laplace transform by evaluating theconvolution integral:

X sj

X s dTc j

c j*( ) ( ) ( )= −

− ∞

+ ∞

∫12π

λ λ λ∆ ,where ∆T s( ) is the Laplace Transform of δT t( )

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∆T s Ts Ts TsTs

( ) exp( ) exp( ) exp( )exp( )

= + − + − + − + =− −

1 2 31

1K

∆T s( ) has poles at values of s satisfying exp( )− =Ts 1, or s jn

Tjn n=

= = ± ±

20 1 20

πω , , , ,K

Choose a contour γ such that all the poles of X(λ) lie within γ, and all the poles of ∆T ( )λ lie to the

exterior of γ. This can be done by choosing the real constant c appropriately.

Pole locations for the integrand given above

The λ-plane is translated horizontally from the s-plane by realconstant c.

Then, for the contour shown above, X sj

X s dT*( ) ( ) ( )= −∫1

2πλ λ λ

γ

∆

which can be evaluated by the “residue theorem.” (We picked a contour to ensure that the integrand isanalytic on and around contour γ.) This gives

X s residues of XT spoles of X

*( ) _ _ ( )exp( ( ))_ _ ( )

=− − −

∑ λ

λλ

11

which can be evaluated

somewhat less painfully than the infinite series form of X*(s). In practice, you can use the above resultwithout resorting to the actual setup of any convolution integral or consideration of pole locations andcontours in the λ-plane.

Q: “What do you do with a complex contour integral?”A: “Calculate a closed form for the Laplace transform of the output of an “ideal” sampler, a devicewhich models the analog-to-digital converter, increasingly used in control systems applications as well as incommunications, digital audio (i.e. your CD player or digital VCR), and data acquisition. It turns out thatthe residue theorem is an extremely useful result, producing results in these problems not readily obtainableby other means.”

References:

Charles L. Phillips & H. Troy Nagle, Digital Control System Analysis and Design. Englewood Cliffs, NewJersey: Prentice Hall, 1995.

B. P. Lathi, Linear Systems and Signals. Carmichael, California: Berkeley-Cambridge Press, 1992.

Math 407 Class Notes.