1. • page 1
To: Dr. Jim Gillilan
From: Rob Arnold
Re: 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 are
better handled by evaluating a complex contour integral, usually resulting in much less pain. An example
follows:
Suppose f(z) is analytic inside and on a simple closed contour C except for finitely many isolated singular
points z1, z2, . . .zn interior to C.
Then by the residue theorem,
f z dz j
C
( ) =∫ 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 δT
n
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 define
the output signal of an ideal sampler as follows:
X s x nT nTs
n
*( ) ( )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 output
value 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 K
which has Laplace transform
X s x
s
Ts
s
x T
Ts
s
Ts
s
x T
Ts
s
Ts
s
( ) ( )
exp( )
( )
exp( ) exp( )
( )
exp( ) exp( )
= −
−


 +
−
−
−


 +
−
−
−


+0
1 2
2
2 3
K
[ ]=
− −


 + − + − + − +
1
0 2 2 3 3
exp( )
( ) ( )exp( ) ( )exp( ) ( )exp( )
Ts
s
x x T Ts x T Ts x T Ts K
= −






− −



=
∞
∑x nT nTs
Ts
sn
( )exp( )
exp( )
0
1
The first factor depends on the input signal and the sample period T. The second factor is independent of
the input signal. Note that the first term corresponds to X*(s). The second factor then expresses the effect
of the zero-order hold.
How can we evaluate X*(s)? One method involves taking the Laplace transform by evaluating the
convolution integral:
X s
j
X s dTc j
c j
*( ) ( ) ( )= −
− ∞
+ ∞
∫
1
2π
λ λ λ∆ ,where ∆T s( ) is the Laplace Transform of δT t( )

2. • page 2
∆T s Ts Ts Ts
Ts
( ) exp( ) exp( ) exp( )
exp( )
= + − + − + − + =
− −
1 2 3
1
1
K
∆T s( ) has poles at values of s satisfying exp( )− =Ts 1, or s j
n
T
jn n=





 = = ± ±
2
0 1 20
π
ω , , , ,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 real
constant c.
Then, for the contour shown above, X s
j
X s dT*( ) ( ) ( )= −∫
1
2π
λ λ λ
γ
∆
which can be evaluated by the “residue theorem.” (We picked a contour to ensure that the integrand is
analytic on and around contour γ.) This gives
X s residues of X
T spoles of X
*( ) _ _ ( )
exp( ( ))_ _ ( )
=
− − −





∑ λ
λλ
1
1
which can be evaluated
somewhat less painfully than the infinite series form of X*(s). In practice, you can use the above result
without resorting to the actual setup of any convolution integral or consideration of pole locations and
contours 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 device
which models the analog-to-digital converter, increasingly used in control systems applications as well as in
communications, digital audio (i.e. your CD player or digital VCR), and data acquisition. It turns out that
the residue theorem is an extremely useful result, producing results in these problems not readily obtainable
by other means.”
References:
Charles L. Phillips & H. Troy Nagle, Digital Control System Analysis and Design. Englewood Cliffs, New
Jersey: Prentice Hall, 1995.
B. P. Lathi, Linear Systems and Signals. Carmichael, California: Berkeley-Cambridge Press, 1992.
Math 407 Class Notes.