Equation (5.31) defines the Fourier transform of x(r as in .NET framework

Encoding QR Code JIS X 0510 in .NET framework Equation (5.31) defines the Fourier transform of x(r as

Equation (5.31) defines the Fourier transform of x(r as
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The bilateral Laplace transform of x(t), as defined in Eq. (4.31, is given by
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Comparing Eqs. (5.38) and (5.39), we see that the Fourier transform is a special case of the Laplace transform in which s = j o , that is, Setting s = u + jo in Eq. (5.39), we have
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e-("+~")'dt = X ( u + jw)
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1 [ x ( t ) e-"1
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Y ( x ( t ) e-"'1
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FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
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which indicates that the bilateral Laplace transform of x ( t ) can be interpreted as the Fourier transform of x ( t ) e-"'. Since the Laplace transform may be considered a generalization of the Fourier transform in which the frequency is generalized from jw to s = a + j o , the complex variable s is often referred to as the complexfrequency. Note that since the integral in Eq. (5.39) is denoted by X(s), the integral in Eq. (5.38) may be denoted as X ( j w ) . Thus, in the remainder of this book both X ( o ) and X ( j w ) mean the same thing whenever we connect the Fourier transform with the Laplace transform. Because the Fourier transform is the Laplace transform with s =jo, it should not be assumed automatically that the Fourier transform of a signal ~ ( r is the Laplace ) transform with s replaced by j w . If x ( t ) is absolutely integrable, that is, if x ( r ) satisfies condition (5.37), the Fourier transform of x ( t ) can be obtained from the Laplace transform of x ( t ) with s =jw. This is not generally true of signals which are not absolutely integrable. The following examples illustrate the above statements.
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EXAMPLE 5.1. Consider the unit impulse function S( t ) . From Eq. (3.13) the Laplace transform of S ( t ) is
J ( S ( t ) }= 1
all s
By definitions ( 5 . 3 1 ) and ( 1 . 2 0 ) the Fourier transform of 6 ( t ) is
Thus, the Laplace transform and the Fourier transform of S ( t ) are the same.
EXAMPLE 5.2.
Consider the exponential signal
From Eq. ( 3 . 8 ) the Laplace transform of x ( t ) is given by
By definition (5.31) the Fourier transform of x ( t ) is
Thus, comparing Eqs. ( 5 . 4 4 ) and (5.451, we have
X ( w ) =X(s)ls-jcu
Note that x ( t ) is absolutely integrable.
EXAMPLE 5.3. Consider the unit step function u(t ). From Eq. (3.14) the Laplace transform of u ( t ) is
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
The Fourier transform of u(r) is given by (Prob. 5.30) F{u(t)}
= ns(o)
Thus, the Fourier transform of u(t) cannot be obtained from its Laplace transform. Note that the unit step function u(t) is not absolutely integrable.
PROPERTIES OF THE CONTINUOUS-TIME FOURIER TRANSFORM
Basic properties of the Fourier transform are presented in the following. Many of these properties are similar to those of the Laplace transform (see Sec. 3.4).
B. Time Shifting:
Equation ( 5 . 5 0 ) shows that the effect of a shift in the time domain is simply to add a linear term -ot, to the original phase spectrum 8 ( w ) .This is known as a linear phase shift of the Fourier transform X( w ) .
C. Frequency Shifting:
The multiplication of x ( t ) by a complex exponential signal is sometimes called complex modulation. Thus, Eq. (5.51) shows that complex modulation in the time domain corresponds to a shift of X ( w ) in the frequency domain. Note that the frequency-shifting property Eq. (5.51) is the dual of the time-shifting property Eq. ( 5 . 5 0 ) .
eJ"l)'
D. Time Scaling:
where a is a real constant. This property follows directly from the definition of the Fourier transform. Equation ( 5 . 5 2 ) indicates that scaling the time variable t by the factor a causes a n inverse scaling of the frequency variable o by l / a , as well as an amplitude scaling of X ( o / a ) by l / l a ( . Thus, the scaling property (5.52) implies that time compression of a signal ( a > 1) results in its spectral expansion and that time expansion of the signal ( a < 1 ) results in its spectral compression.
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