The Laplace transform of a sampled signal
x(t)=X n
x(nTs)δ(t−nTs) (9.1)
is given by
X(s)=X n
x(nTs)L[δ(t−nTs)]
=X n
x(nTs)e−nsTs (9.2)
By lettingz=esTs, we can rewrite Equation (9.2) as
Z[x(nTs)]= L[xs(t)] z
=esTs
=X n
x(nTs)z−n (9.3)
which is called the Z-transform of the sampled signal.
Remarks The function X(s) in Equation (9.2) is different from the Laplace transforms we considered before:
n Letting s=j, X()is periodic of period2π/Ts(i.e., X(+2π/Ts)=X()for an integer k). Indeed, X(+2π/Ts)=X
n
x(nTs)e−jn(+2π/Ts)Ts=X
n
x(nTs)e−jn(Ts+2π)=X()
n X(s)may have an infinite number of poles or zeros—complicating the partial fraction expansion when finding its inverse. Fortunately, the presence of the{e−nsTs}terms suggests that the inverse should be done using the time-shift property of the Laplace transform instead, giving Equation (9.1).
nExample 9.1
To see the possibility of an infinite number of poles and zeros in the Laplace transform of a sam- pled signal, consider a pulsex(t)=u(t)−u(t−T0)sampled with a sampling periodTs=T0/N.
Find the Laplace transform of the sampled signal and determine its poles and zeros.
Solution
The sampled signal is
x(nTs)=
1 0≤nTs≤T0 or 0≤n≤N 0 otherwise
9.2 Laplace Transform of Sampled Signals 513
with Laplace transform
X(s)=
N
X
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
=ej2πk kinteger, −∞<k<∞
orsk= −j2πk/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
=ej2πm minteger, −∞<m<∞
orsm= −j2πm/((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
Letting r=eσTs andω=Ts, we have that
z=rejω
which is a complex variable in polar form, with radius0≤r<∞and angleωin radians. The variable r is a damping factor andωis the discrete frequency in radians, so the z-plane corresponds to circles of radius r and angles−π≤ω < π.
n Let us see how the relation z=esTs maps the s-plane into the z-plane. Consider the strip of width 2π/Ts across the s-plane shown in Figure 9.1. The width of this strip is related to the Nyquist condition establishing that the maximum frequency of the analog signals we are considering is
M=s/2=π/Ts where s is the sampling frequency and Ts is the sampling period. If Ts→0, we would be considering the class of signals with maximum frequency approaching∞—that is, all signals.
The relation z=esTs maps the real part of s=σ +j, Re(s)=σ, into the radius r=eσTs≥0, and the analog frequencies−π/Ts≤≤π/Ts into −π ≤ω < π, according to the frequency connection ω=Ts. Thus, the mapping of the jaxis in the s-plane, corresponding toσ =0, gives a circle of radius r=1or the unit circle.
The right-hand s-plane,σ >0, maps into circles with radius r>1, and the left-hand s-plane,σ <0, maps into circles of radius r<1. Points A, B, and C in the strip are mapped into corresponding points in the z-plane as shown in Figure 9.1. So the given strip in the s-plane maps into the whole z-plane—
similarly for other strips of the same width. Thus, the s-plane, as a union of these strips, is mapped onto the same z-plane.
n The mapping z=esTs can be used to illustrate the sampling process. Consider a band-limited signal x(t) with maximum frequencyπ/Ts with a spectrum in the band[−π/Ts π/Ts]. According to the relation z=esTsthe spectrum of x(t)in[−π/Ts π/Ts]is mapped into the unit circle of the z-plane from[−π,π) on the unit circle. Going around the unit circle in the z-plane, the mapped frequency response repeats periodically just like the spectrum of the sampled signal.
z=esTs jΩ
A σ B
ω r
A B
C
C
s-plane z-plane
−jπ Ts
jπ Ts
FIGURE 9.1
Mapping of the Laplace plane into thez-plane. Slabs of width2π/Tsin the left-hands-plane are mapped into the inside of a unit circle in thez-plane. The right-hand side of the slab is mapped outside the unit circle. Thej-axis in thes-plane is mapped into the unit-circle in thez-plane. The wholes-plane as a union of these slabs is mapped onto the samez-plane.