 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
2d barcode generator vb.net shown in .NET
shown QRCode Decoder In Visual Studio .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. QR Drawer In Visual Studio .NET Using Barcode creator for Visual Studio .NET Control to generate, create Denso QR Bar Code image in VS .NET applications. Fig. 537 QR Code 2d Barcode Recognizer In .NET Framework Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Encoder In .NET Using Barcode generation for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
Scan Bar Code In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in VS .NET applications. Making QR Code In Visual C# Using Barcode printer for .NET Control to generate, create QR Code 2d barcode image in VS .NET applications. [CHAP. 5
QR Code ISO/IEC18004 Maker In .NET Using Barcode generator for ASP.NET Control to generate, create QR Code image in ASP.NET applications. QR Code Generator In VB.NET Using Barcode printer for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET applications. Derive the harmonic form Fourier series representation (5.15) from the trigonometric Fourier series representation (5.8). Code 39 Extended Generator In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create Code 39 image in VS .NET applications. Generate GS1 RSS In VS .NET Using Barcode creation for VS .NET Control to generate, create GS1 DataBar image in .NET applications. Hint: Bar Code Encoder In VS .NET Using Barcode maker for VS .NET Control to generate, create barcode image in Visual Studio .NET applications. Leitcode Generator In Visual Studio .NET Using Barcode creation for VS .NET Control to generate, create Leitcode image in VS .NET applications. Rewrite a , cos k w , t
Code 128 Code Set B Maker In None Using Barcode encoder for Office Excel Control to generate, create Code 128 Code Set B image in Office Excel applications. UCC  12 Reader In Visual C#.NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications. + b, sin k w , t
Code 128 Recognizer In VS .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications. Scanning USS Code 128 In VB.NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET applications. as cos k w , t
Create Barcode In .NET Using Barcode generator for ASP.NET Control to generate, create bar code image in ASP.NET applications. Drawing Code 128 Code Set C In .NET Using Barcode encoder for ASP.NET Control to generate, create Code 128 Code Set C image in ASP.NET applications. ( 4+ b:) Generate Bar Code In Java Using Barcode encoder for BIRT Control to generate, create barcode image in Eclipse BIRT applications. Data Matrix ECC200 Scanner In C# Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. sin k w , t
and use the trigonometric formula cod A  B) = cos A cos B + sin A sin B.
Show that the meansquare value of a real periodic signal x ( r ) is the sum of the meansquare values of its harmonics. Hint: Use Parseval's identity (5.21) for the Fourier series and Eq. (5.168). Show that if then
Hint: Repeat the timedifferentiation property (5.55). Using the differentiation technique, find the Fourier transform of the triangular pulse signal shown in Fig. 538. Ar..Ad[ sin( w d / 2 ) wd/2 Fig. 538 Find the inverse Fourier transform of
Hint: Differentiate Eq. (5.155) N times with respect to ( a ) . CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
Find the inverse Fourier transform of X(w) = Hint: Note that
2  w 2 + j3w
2  w2 + j3w = 2 + ( jw12 + j3w
Am. x ( t ) = (e'  e  2 ' ) u ( t ) + j w ) ( 2 +j w ) and apply the technique of partialfraction expansion.
Verify the frequency differentiation property (5.561, that is, Hint: Use definition (5.31) and proceed in a manner similar to Prob. 5.28. the Fourier transform of each of the following signals: x ( t ) = cos wotu(t) x ( t ) = sin wotu(t) x ( t ) = e  " ' ~ ~ ~ w ~ tau>(O ) , t x ( t ) = e"'sin w,tu(t), a > 0 Use multiplication property (5.59). Find (a) (b) (c) (dl Hint: ( a ) X ( w ) = S(w 2
 w o ) + S(w + w , ) + 2
iw ( jw)' + w i
(dl X(w)= ( a + jo12
Let x ( t ) be a signal with Fourier transform X ( w ) given by
Consider the signal
Find the value of
Hint: Use Parseval's identity (5.64) for the Fourier transform.
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
Let x ( t ) be a real signal with the Fourier transform X ( w ) . The analytical signal x + ( t ) associated with x ( t ) is a complex signal defined by x + ( t )= x ( t ) + j i ( t ) where
. is the Hilbert transform of x(t 1
( a ) Find the Fourier transform X + ( w ) of x + ( t 1. ( 6 ) Find the analytical signal x + ( t ) associated with cos w,t and its Fourier transform X + ( w ) . Consider a continuoustime LTI system with frequency response H(w). Find the Fourier transform S ( w ) of the unit step response s ( t ) of the system. Hint: Am.
Use Eq. (2.12) and the integration property (5.57). S ( w ) = .rrH(O)G(o)+ ( l / j w ) H ( w ) Consider the RC filter shown in Fig. 539. Find the frequency response H ( w ) of this filter and discuss the type of filter. Ans. H(o)= iw , highpass filter ( l / R C ) +jw
Fig. 539 Determine the 99 percent energy containment bandwidth for the signal
Ans.
W,,= 2.3/a radians/second or f , = 0.366/a
hertz
The sampling theorem in the frequency domain states that if a real signal x ( t ) is a durationlimited signal. that is, CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
then its Fourier transform X ( w ) can be uniquely determined from its values X ( n s r / t , ) at a series of equidistant points spaced n / t , apart. In fact, X ( w ) is given by o t ,  n.rr
Verify the above sampling theorem in the frequency domain.
Hint: Expand x ( t ) in a complex Fourier series and proceed in a manner similar to that for Prob. 5.59.
6
Fourier Analysis of DiscreteTime Signals and Systems
INTRODUCTION
In this chapter we present the Fourier analysis in the context of discretetime signals (sequences) and systems. The Fourier analysis plays the same fundamental role in discrete time as in continuous time. As we will see, there are many similarities between the techniques of discretetime Fourier analysis and their continuoustime counterparts, but there are also some important differences. 6.2 DISCRETE FOURIER SERIES
Periodic Sequences: In Chap. 1 we defined a discretetime signal (or sequence) x [ n ] to be periodic if there is a positive integer N for which x [ n +N] = x [ n ] all n
(6.1) The fundamental period No of x [ n ] is the smallest positive integer N for which Eq. (6.1) is satisfied. As we saw in Sec. 1.4, the complex exponential sequence where no= 27r/Nu, is a periodic sequence with fundamental period Nu. As we discussed in Sec. 1.4C, one very important distinction between the discretetime and the continuoustime complex exponential is that the signals el"^' are distinct for distinct values of wO,but the sequences eiR~~", which differ in frequency by a multiple of 2rr, are identical. That is, Let and more generally,

