Signal Processing and Linear Systems-B.P.Lathi copy

T he duality principle has far reaching consequences

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Unformatted text preview: 9 s pectral spreading can be reduced by increasing t he window width. Therefore, we can achieve a given combination of spectral spread (transition bandwidth) a nd leakage characteristics by choosing a suitable t apered window function of a sufficiently longer width T . References 1. Churchill, R.V., a nd J .W. Brown, Fourier Series and Boundary Value Problems, 3d ed., McGraw-Hili, New York, 1978. 2. Bracewell, R.N., Fourier Transform and Its Applications, revised 2nd Ed., McGraw-Hill, New York, 1986. 3. Guillemin, E.A., Theory o f L inear Physical Systems, Wiley, New York, 1963. 4. Lathi, B.P., Modern Digital and Analog Communication Systems, 3rd ed., Oxford University Press, New York, 1998. 5. J . Carson, "Notes o n Theory o f Modulation" Proc. I RE, v ollO, February 1922, pp. 57-64. 6. J . Carson, " The R eduction of Atmospheric D isturbances" Proc. I RE, vol 16, J uly 1928, pp. 966-975. 7. A rmstrong E.H., "A M ethod o f Reducing Disturbances in Radio Signaling by a System o f Frequency M odulation", Proc. I RE, vol. 24, May 1936, pp. 689-740. 8. Hamming, R.W., Digital Filters, 2nd Ed., P rentice-Hall, Englewood Cliffs N.J. 1983. ' 9. Harris, F .J., "On t he Use of Windows for Harmonic Analysis with t he Discrete Fourier T ransform", Proc. I EEE, vol. 66, N o.1, J anuary 1978, pp. 51-83. Problems 4 .1-1 S how t hat if f (t) is a n e ven function o f t, t hen F(w) = 21 00 f (t) cos w tdt a nd i f f (t) is a n o dd f unction o f t, t hen 1 00 F(w) = - 2j f (t) s in wt dt Hence, prove t hat i f f (t) is a real a nd e ven function o ft, t hen F(w) is a real a nd even ~unction of w. I n a ddition, if f (t) is a real a nd o dd f unction o f t , t hen F(w) is a n I maginary a nd o dd f unction o f w. 4 .1-2 S how t hat for a real f (t), Eq. (4.8b) c an b e e xpressed as 11 f (t) = 71' 0 00 IF(w)1 cos [wt + LF(w)] dw T his is t he t rigonometric form o f t he F ourier integral. C ompare t his w ith t he c ompact t rigonometric F ourier series. 4.1-3 A signal f (t) c an b e e xpressed a s t he s um o f e ven a nd o dd c omponents (see Sec. 1.5-2): f (t) = fe(t) + fort) 3 lO 4 C ontinuous-Time S ignal A nalysis: T he F ourier T ransform f (t) (a) 311 IF (ro) I ( b) 0 1 -- P roblems I~ T IF(O'l 0 -<00 CilCilO Fig. P 4.l-4 ICil- <0 0 <% (a) (b) !tI2 Tn ~~, I I 2 (a) L F(Cil) <% Cil'1- - ",2 (b) F ig. P 4.2-4 F ig. P 4.1-5 f it) ;J1 ~ro)1 Cil ..... -(0)0 - --roLo--"""-t":'o-"----!:-"\, ro- 1 (a) 2 0 t_ O 1- 1_ 0 2 ( b) F ig. P 4.1-6 f .lt) F ( ro) 1 . cos M CO . r o- ,- ,- 0 (a) 1_ -1 r o- = { =} Re[F(w)] a nd 1_ 2 4 .2-3 fo(t) { =} F rom definition (4.8b), show t hat t he inverse Fourier transform o f r ect (W~:o) is sinc (1ft) ellO'. 4 .2-4 F(w), show t hat for real f it), f .(t) 0 F ig. P 4.3-2 (b) F ig. P 4.1-7 ( a) I f f (t) 0 F ind t he inverse Fourier transform of F(w) for t he s pectra i llustrated in Figs. P4.2-4a a nd b. Hint: F(w) = IF(w)lejLF(w) T his problem illustrates how different phase s pectra ( both w ith t he s ame a mplitude s pectrum) represent entirely different signals. Apply t he s ymmet...
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