FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS in .NET

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FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
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E. Time Reversal:
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Thus, time reversal of x ( t ) produces a like reversal of the frequency axis for X ( o ) . Equation (5.53) is readily obtained by setting a = - 1 in Eq. (5.52).
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F. Duality (or Symmetry):
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The duality property of the Fourier transform has significant implications. This property allows us to obtain both of these dual Fourier transform pairs from one evaluation of Eq. (5.31) (Probs. 5.20 and 5.22).
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Differentiation in the Time Domain:
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Equation (5.55) shows that the effect of differentiation in the time domain is the multiplication of X(w) by jw in the frequency domain (Prob. 5.28).
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Differentiation in the Frequency Domain:
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(-P)x(t)
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dX , (4 o
Equation (5.56) is the dual property of Eq. (5.55).
I. Integration in the Time Domain:
Since integration is the inverse of differentiation, Eq. (5.57) shows that the frequencydomain operation corresponding to time-domain integration is multiplication by l/jw, but an additional term is needed to account for a possible dc component in the integrator output. Hence, unless X(0) = 0, a dc component is produced by the integrator (Prob. 5.33).
J. Convolution:
CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
Equation (5.58) is referred to as the time convolution theorem, and it states that convolution in the time domain becomes multiplication in the frequency domain (Prob. 5.31). As in the case of the Laplace transform, this convolution property plays an important role in the study of continuous-time LTI systems (Sec. 5.5) and also forms the basis for our discussion of filtering (Sec. 5.6). K. Multiplication:
The multiplication property (5.59) is the dual property of Eq. (5.58) and is often referred to as the frequency convolution theorem. Thus, multiplication in the time domain becomes convolution in the frequency domain (Prob. 5.35). L. Additional Properties: If x ( t ) is real, let
where x,( t ) and xo(t) are the even and odd components of x( t 1, respectively. Let
Then
Equation ( 5 . 6 1 ~ ) the necessary and sufficient condition for x( t ) to be real (Prob. 5.39). is Equations (5.61b) and ( 5 . 6 1 ~ ) show that the Fourier transform of an even signal is a real function of o and that the Fourier transform of an odd signal is a pure imaginary function of w .
M. Parseval's Relations:
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
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Equation (5.64) is called Parseual's identity (or Parseual's theorem) for the Fourier transform. Note that the quantity on the left-hand side of Eq. (5.64) is the normalized energy content E of x(t) [Eq. (1.14)]. Parseval's identity says that this energy content E can be computed by integrating Ix(w)12 over all frequencies w . For this reason Ix(w)l2 is often referred to as the energy-density spectrum of x(t), and Eq. (5.64) is also known as the energy theorem. Table 5-1 contains a summary of the properties of the Fourier transform presented in this section. Some common signals and their Fourier transforms are given in Table 5-2.
Table 5-1. Property
Properties of the Fourier Transform
Signal
Fourier transform
Linearity Time shifting Frequency shifting Time scaling Time reversal Duality Time differentiation Frequency differentiation Integration Convolution Multiplication Real signal Even component Odd component Parseval's relations
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
Table 5-2. Common Fourier Transforms Pairs
sin at
THE FREQUENCY RESPONSE OF CONTINUOUS-TIME LTI SYSTEMS
Frequency Response:
In Sec. 2.2 we showed that the output y ( t ) of a continuous-time LTI system equals the convolution of the input x ( t ) with the impulse response h(t 1; that is,
Applying the convolution property (5.58),we obtain
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
[CHAP. 5
where Y(w), X(o), and H(w) are the Fourier transforms of y(f), d t ) , and h(t), respectively. From Eq. (5.66) we have
The function H ( o ) is called the frequency response of the system. Relationships represented by Eqs. (5.65) and (5.66) are depicted in Fig. 5-3. Let H(w) = I H(w)I ei@~(O) (5.68)
Then IH(o)lis called the magnitude response of the system, and 0 , ( 0 ) the phase response of the system.
X(w)
Y(w)=X(w)H(w)
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