 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS in .NET
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS QR Code Scanner In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. QRCode Printer In .NET Framework Using Barcode maker for Visual Studio .NET Control to generate, create QRCode image in .NET framework applications. [CHAP. 5
QR Code Recognizer In Visual Studio .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Creating Barcode In VS .NET Using Barcode generation for .NET Control to generate, create bar code image in VS .NET applications. E. Time Reversal: Barcode Decoder In Visual Studio .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET applications. QR Encoder In Visual C#.NET Using Barcode creation for .NET Control to generate, create Quick Response Code image in .NET framework applications. 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). QR Code JIS X 0510 Generator In .NET Using Barcode creation for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Quick Response Code Creator In Visual Basic .NET Using Barcode creator for Visual Studio .NET Control to generate, create QR image in .NET framework applications. F. Duality (or Symmetry): Draw EAN13 In VS .NET Using Barcode encoder for VS .NET Control to generate, create EAN13 image in .NET framework applications. Bar Code Encoder In VS .NET Using Barcode maker for VS .NET Control to generate, create bar code image in .NET applications. 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). Creating Data Matrix 2d Barcode In Visual Studio .NET Using Barcode printer for VS .NET Control to generate, create Data Matrix ECC200 image in .NET framework applications. ISBN  13 Generation In .NET Using Barcode creator for .NET framework Control to generate, create ISBN  13 image in .NET applications. Differentiation in the Time Domain: Creating UPCA Supplement 5 In Java Using Barcode generation for Java Control to generate, create UPCA image in Java applications. Paint Bar Code In Java Using Barcode drawer for Java Control to generate, create barcode image in Java applications. 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). Scanning Code 39 In Visual Studio .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Barcode Printer In .NET Framework Using Barcode generation for ASP.NET Control to generate, create barcode image in ASP.NET applications. Differentiation in the Frequency Domain: Making DataMatrix In Java Using Barcode printer for Java Control to generate, create DataMatrix image in Java applications. Making EAN 128 In VS .NET Using Barcode creator for Reporting Service Control to generate, create EAN128 image in Reporting Service applications. (P)x(t) Generate Linear 1D Barcode In Java Using Barcode generation for Java Control to generate, create 1D image in Java applications. Code 39 Generation In Visual C#.NET Using Barcode printer for .NET framework Control to generate, create ANSI/AIM Code 39 image in .NET framework applications. 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 timedomain 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 continuoustime 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
[CHAP. 5
Equation (5.64) is called Parseual's identity (or Parseual's theorem) for the Fourier transform. Note that the quantity on the lefthand 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 energydensity spectrum of x(t), and Eq. (5.64) is also known as the energy theorem. Table 51 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 52. Table 51. 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 52. Common Fourier Transforms Pairs
sin at
THE FREQUENCY RESPONSE OF CONTINUOUSTIME LTI SYSTEMS
Frequency Response: In Sec. 2.2 we showed that the output y ( t ) of a continuoustime 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. 53. 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)

