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Consider the signal in .NET
EXAMPLE 3.1. Consider the signal Scan Denso QR Bar Code In .NET Framework Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. Make QR Code In .NET Using Barcode maker for Visual Studio .NET Control to generate, create Quick Response Code image in .NET applications. x ( t ) =eO1u(t) Then by Eq. (3.3) the Laplace transform of x(t) is
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Decoding Barcode In VS .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET framework applications. QR Maker In C#.NET Using Barcode generator for .NET framework Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications. because lim, , e("'")' , Creating QR Code 2d Barcode In VS .NET Using Barcode generation for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Encode QR Code JIS X 0510 In VB.NET Using Barcode printer for Visual Studio .NET Control to generate, create QRCode image in Visual Studio .NET applications. only if Re(s + a ) > 0 or Reb) > a.
Matrix Barcode Drawer In .NET Using Barcode drawer for .NET framework Control to generate, create Matrix 2D Barcode image in .NET framework applications. Code 39 Extended Creation In .NET Using Barcode creator for VS .NET Control to generate, create ANSI/AIM Code 39 image in .NET framework applications. 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. 31 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 splane. The horizontal and vertical axes are sometimes referred to as the aaxis and the jwaxis, respectively. Generate UPCA Supplement 2 In VS .NET Using Barcode encoder for .NET framework Control to generate, create UPC Symbol image in .NET applications. Encode ISBN In .NET Using Barcode printer for Visual Studio .NET Control to generate, create International Standard Book Number image in .NET framework applications. EXAMPLE 3.2.
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Data Matrix Maker In ObjectiveC Using Barcode drawer for iPhone Control to generate, create DataMatrix image in iPhone applications. Barcode Encoder In Java Using Barcode generation for Java Control to generate, create barcode image in Java applications. ~ ( t = e"u(  t ) ) Its Laplace transform X(s) is given by (Prob. 3.1) EAN / UCC  13 Scanner In Visual C#.NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Code 39 Full ASCII Encoder In None Using Barcode drawer for Microsoft Excel Control to generate, create Code39 image in Excel applications. 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. 32 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 CONTINUOUSTIME LTI SYSTEMS
[CHAP. 3
splane
Fig. 31 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 splane 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. 33 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 CONTINUOUSTIME LTI SYSTEMS
Fig. 32 ROC for Example 3.2.
Fig. 33 splane 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 splane except possibly s = 0 or s = E. Property 3: If x ( t ) is a rightsided signal, that is, x ( r ) = 0 for r < r , < m, then the ROC is of the form < I , and
LAPLACE TRANSFORM AND CONTINUOUSTIME LTI SYSTEMS
[CHAP. 3
where a,,, equals the maximum real part of any of the poles of X(s). Thus, the ROC is a halfplane to the right of the vertical line Reb) = a,,, in the splane and thus to the right of all of the poles of Xb). Property 4: If x(t) is a leftsided 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 halfplane to the left of the vertical line Re(s) =amin the splane and thus to the left in of all of the poles of X(s). Property 5: If x(t) is a twosided signal, that is, x(t) is an infiniteduration signal that is neither rightsided nor leftsided, 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 splane 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.

