Consider the signal in .NET

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EXAMPLE 3.1. Consider the signal
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x ( t ) =e-O1u(t) Then by Eq. (3.3) the Laplace transform of x(t) is
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because lim, , e-("'")' ,
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only if Re(s + a ) > 0 or Reb) > -a.
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Thus, the ROC for this example is specified in Eq. (3.9) as Re(s) > -a and is displayed in the complex plane as shown in Fig. 3-1 by the shaded area to the right of the line Re(s) = -a. In Laplace transform applications, the complex plane is commonly referred to as the s-plane. The horizontal and vertical axes are sometimes referred to as the a-axis and the jw-axis, respectively.
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EXAMPLE 3.2.
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Consider the signal
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~ ( t = -e-"u( - t ) ) Its Laplace transform X(s) is given by (Prob. 3.1)
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Thus, the ROC for this example is specified in Eq. (3.11) as Re(s) < -a and is displayed in the complex plane as shown in Fig. 3-2 by the shaded area to the left of the line Re(s) = -a. Comparing Eqs. (3.9) and (3.11), we see that the algebraic expressions for X(s) for these two different signals are identical except for the ROCs. Therefore, in order for the Laplace transform to be unique for each signal x(t), the ROC must be specified as par1 of the transform.
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
[CHAP. 3
s-plane
Fig. 3-1 ROC for Example 3.1.
C. Poles and Zeros of X ( s 1:
Usually, X(s) will be a rational function in s, that is,
The coefficients a, and b, are real constants, and m and n are positive integers. The X(s) is called a proper rational function if n > m, and an improper rational function if n I m. The roots of the numerator polynomial, z,, are called the zeros of X(s) because X(s) = 0 for those values of s. Similarly, the roots of the denominator polynomial, p,, are called the poles of X(s) because X(s) is infinite for those values of s. Therefore, the poles of X(s) lie outside the ROC since X(s) does not converge at the poles, by definition. The zeros, on the other hand, may lie inside or outside the ROC. Except for a scale factor ao/bo, X(s) can be completely specified by its zeros and poles. Thus, a very compact representation of X(s) in the s-plane is to show the locations of poles and zeros in addition to the ROC. Traditionally, an " x " is used to indicate each pole location and an " 0 " is used to indicate each zero. This is illustrated in Fig. 3-3 for X(s) given by
Note that X(s) has one zero at s = - 2 and two poles at s factor 2.
- 1 and s
- 3 with scale
D. Properties of the ROC:
As we saw in Examples 3.1 and 3.2, the ROC of X(s) depends on the nature of d r ) . The properties of the ROC are summarized below. We assume that X(s) is a rational function of s.
CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
Fig. 3-2 ROC for Example 3.2.
Fig. 3-3 s-plane representation of
X ( s ) = (2s
+ 4)/(s2 + 4s + 3).
Property 1: The ROC does not contain any poles. Property 2: If
x(t)
is a
finiteduration
signal, that is,
x(r) =0
except in a finite interval r , 5 r 2 r ,
< m), then the ROC is the entire s-plane except possibly s = 0 or s = E. Property 3: If x ( t ) is a right-sided signal, that is, x ( r ) = 0 for r < r , < m, then the ROC is of the form
< I , and
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
[CHAP. 3
where a,,, equals the maximum real part of any of the poles of X(s). Thus, the ROC is a half-plane to the right of the vertical line Reb) = a,,, in the s-plane and thus to the right of all of the poles of Xb).
Property 4: If x(t) is a left-sided signal, that is, x(t) = O for t > t, >
-=, then the ROC is of the
form
where a,,, equals the minimum real part of any of the poles of X(s). Thus, the ROC is a half-plane to the left of the vertical line Re(s) =amin the s-plane and thus to the left in of all of the poles of X(s).
Property 5:
If x(t) is a two-sided signal, that is, x(t) is an infinite-duration signal that is neither right-sided nor left-sided, then the ROC is of the form
where a, and a, are the real parts of the two poles of X(s). Thus, the ROC is a vertical strip in the s-plane between the vertical lines Re(s) = a, and Re(s) = a,. Note that Property 1 follows immediately from the definition of poles; that is, infinite at a pole. For verification of the other properties see Probs. 3.2 to 3.7.
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