Signals and Systems - Electrical Engineering

9.2 Laplace Transform of Sampled Signals 513

with Laplace transform

X(s)=

∑N

n= 0

e−nsTs=

1 −e−(N+^1 )sTs

1 −e−sTs

The poles are theskvalues that make the denominator zero—that is,

e−skTs= 1

=ej^2 πk kinteger, −∞<k<∞

orsk=−j 2 πk/Tsfor any integerk, an infinite number of poles. Similarly, one can show thatX(s)

has an infinite number of zeros by finding the valuessmthat make the numerator zero, or

e−(N+^1 )smTs= 1

=ej^2 πm minteger, −∞<m<∞

orsm=−j 2 πm/((N+ 1 )Ts)for any integerm. Such a behavior can be better understood when we

consider the connection between thes-plane and thez-plane. n

The History of the Z-Transform

The history of the Z-transform goes back to the work of the French mathematician De Moivre, who in 1730 introduced the

characteristic function to represent the probability mass function of a discrete random variable. The characteristic function

is identical to the Z-transform. Also, the Z-transform is a special case of the Laurent’s series, used to represent complex

functions.

In the 1950s the Russian engineer and mathematician Yakov Tsypkin (1919–1997) proposed the discrete Laplace transform,

which he applied to the study of pulsed systems. Then Professor John Ragazzini and his students Eliahu Jury and Lofti

Zadeh at Columbia University developed the Z-transform. Ragazzini (1912–1988) was chairman of the Department of

Electrical Engineering at Columbia University. Three of his students are well recognized in electrical engineering for their

accomplishments: Jury for the Z-transform, nonlinear systems, and the inners stability theory; Zadeh for the Z-transform

and fuzzy set theory; and Rudolf Kalman for the Kalman filtering.

Jury was born in Iraq, and received his doctor of engineering science degree from Columbia University in 1953. He was

professor of electrical engineering at the University of California, Berkeley, and at the University of Miami. Among his

publications, Professor Jury’s “Theory and Application of the Z-transform,” is a seminal work on the theory and application

of the Z-transform.

Remarks

n The relation z=esTsprovides the connection between the s-plane and the z-plane:

z=esTs=e(σ+j)Ts=eσTsejTs