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CHAP. 51 in Visual Studio .NET
CHAP. 51 Reading Denso QR Bar Code In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Create QR Code ISO/IEC18004 In VS .NET Using Barcode creator for VS .NET Control to generate, create QR image in .NET applications. FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
Recognize Denso QR Bar Code In VS .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Painting Bar Code In Visual Studio .NET Using Barcode generator for .NET Control to generate, create bar code image in .NET framework applications. Fig. 534 Ideal sampling.
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Print Barcode In VS .NET Using Barcode drawer for .NET framework Control to generate, create bar code image in .NET framework applications. Painting 2D Barcode In Visual Studio .NET Using Barcode drawer for .NET Control to generate, create Matrix Barcode image in .NET applications. ( h ) From Eq. (5.147) (Prob. 5.25) we have
Linear Barcode Generation In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create Linear Barcode image in .NET applications. Standard 2 Of 5 Generation In .NET Using Barcode drawer for VS .NET Control to generate, create Code 2/5 image in VS .NET applications. Let Then, according to the frequency convolution theorem (5.59), we have
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Make Code 128 Code Set A In Java Using Barcode creation for BIRT Control to generate, create USS Code 128 image in BIRT reports applications. UCC.EAN  128 Maker In None Using Barcode creator for Microsoft Word Control to generate, create EAN128 image in Office Word applications. which shows that X,(w) consists of periodically repeated replicas of X(w) centered about kw, for all k. The Fourier spectrum X,(w) is shown in Fig. 534 f ) for T, < r/w, (or w, > 2wM), and in Fig. 534( j ) for T, > r / w M (or w, < 2wM), where w, = 27~/T,. It is seen that no overlap of the replicas X(o  ko,) occurs in X,(o) for w, r 2wM and that overlap of the spectral replicas is produced for w,$ < 2wM. This effect is known as USS Code 39 Drawer In Java Using Barcode encoder for Java Control to generate, create USS Code 39 image in Java applications. Making Barcode In Java Using Barcode generation for BIRT reports Control to generate, create barcode image in BIRT applications. aliasing.
GS1  12 Printer In Java Using Barcode creator for Eclipse BIRT Control to generate, create UPCA Supplement 2 image in BIRT reports applications. Bar Code Encoder In None Using Barcode drawer for Word Control to generate, create barcode image in Office Word applications. 5.59. Let x ( t ) be a realvalued bandlimited signal specified by Show that x ( t ) can be expressed as sin wM(t  kT,) x(kTs) 41) & C =
where T, = rr/w,.  kT) From Eq. ( 5.183 we have
T,X,(w) X ( o  ko,) Then, under the following two conditions, (1) X(o)=O,IwI>o, and (2) T,= we see from Eq. (5.1185 that
CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
Next, taking the Fourier transform of Eq. (5.182), we have
Substituting Eq. (5.187) into Eq. (5.186), we obtain
Taking the inverse Fourier transform of Eq. (5.1881, we get
C k=
sin w M (t  kT,) x(kT,) W M ( kTs) ~
From Probs. 5.58 and 5.59 we conclude that a bandlimited signal which has no frequency components higher that f M hertz can be recovered completely from a set of samples taken at 2 the rate of f, ( 1 fM) samples per second. This is known as the uniform sampling theorem for lowpass signals. We refer to T, = X / W , = 1 / 2 fM ( o M 27r fM as the Nyquist sampling = interval and f, = 1/T, = 2 fM as the Nyquist sampling rate. 5.60. Consider the system shown in Fig. 5  3 5 ( a ) . The frequency response H ( w ) of the ideal lowpass filter is given by [Fig. 5  3 5 ( b ) ] Show that if w , = 0J2, then for any choice of T,, Fig. 535 FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
From Eq. (5.137) the impulse response h ( t ) of the ideal lowpass filter is given by sin w, t h ( t ) = T5at From Eq. (5.182) we have T5wc sin w,t w,t
By Eq. (2.6) and using Eqs. ( 2 . 7 ) and (1.261, the output y ( t ) is given by
Using Eq. (5.1891, we get
T p , sin w,(t  k c ) ~ ( t=) If w, =wJ2, k =  cc
x(kT,)77 w,(t  kT,) then T,w,/a
1 and we have
Setting t
= mT, ( m = integer) and using the fact that w,T, 2 ~we get , sin ~ X (kTS) ( m ) k
Y ( ~ T=) Since
77(m  k ) we have
which shows that without any restriction on x ( t ) , y(mT5)= x(mT,) for any integer value of m . Note from the sampling theorem (Probs. 5.58 and 5.59) that if w, = 2 a / T 5 is greater than twice the highest frequency present in x ( t ) and w , = w J 2 , then y ( t ) = x ( t ) . If this condition on the bandwidth of x ( t ) is not satisfied, then y ( t ) z x ( t ) . However, if w, = 0 , / 2 , then y(mT,) = x(mT5) for any integer value of m . CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
Supplementary Problems
Consider a rectified sine wave signal
x(t) defined by x ( l ) = IAsin.srt( Sketch x ( t ) and find its fundamental period. (6) Find the complex exponential Fourier series of ( c ) Find the trigonometric Fourier series of x ( t 1. x(!). X(t) is sketched in Fig. 536 and T, (c) x ( t ) = 2A 4 A m 1 cos   C  k 2 r t .sr 4k21 Fig. 536 Find the trigonometric Fourier series of a periodic signal
x(t) =t2, x(t) defined by
a < t < .rr
+ 2a) = x ( t ) Am. x ( t ) = 5.63. (ilk cos  kt
x(t) Using the result from Prob. 5.10, find the trigonometric Fourier series of the signal in Fig. 537. A A " 1 2a A ~ S . ~ ( t=   )  sin ko,! wo= 2 .rr & = I k To

