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Unformatted text preview: w York, 1963.
9 . Gibbs, W. J ., N ature, vol. 59, p. 606, April 1899.
1 0. Bacher, M., Annals of Mathematics, (2), vol. 7, 1906.
1 1. Carslaw, H. S., Bulletin, American Mathematical Society, vol. 31, pp. 420-424,
1. Problems F ig. P 3.1-4 3 .1-5 I f x (t) a nd yet) a re o rthogonal, t hen s how t hat t he e nergy o f t he s ignal x (t) + yet) is
identical t o t he e nergy of t he s ignal x (t) - yet) a nd is given by E x + E y . E xplain t his
r esult using vector concepts. I n g eneral, show t hat for orthogonal signals x( t) a nd
yet) a nd for a ny p air o f a rbitrary c onstants C l a nd C2, t he energies o f CIX(t) + C2y(t)
a nd CIX(t) - c2y(t) a re i dentical, given by d Ex + c~Ey. 1 ,(1) X (I) I- 14 (1) 1 3(1) 0 .707 070:1
1- 3 .1-1 Derive E quation (3.6) in a n a lternate way by observing t hat e = ( f-cx) a nd I- 0
- 0.707 F ig. P 3.2-1 o 228 3 .2-1 3 S ignal R epresentation b y O rthogonal S ets 229 P roblems F ind t he c orrelation coefficient cn of signal x (t) a nd each of t he four pulses h it), h (t),
h it), a nd } 4(t) d epicted in Fig. P3.2-1. W hich pair of pulses would you select for a (a) binary communication in order to provide maximum margin against noise along t he
t ransmission p ath?
3 .3-1 L et XI(t) a.nd X2(t) b e two signals orthonormal ( that is, w ith u nit energies) over an
interval f rom t = t l t o t2. Consider a signal f it) where o -21l1t o -11l1t [ IIl1t (b) 21l1t T his s ignal c an b e represented by a two-dimensional vector f( C I, C 2).
( a) D etermine t he v ector representation of t he following six signals in t he twodimensiona.l vector space:
(i) h it) = 2XI(t) - X2(t) = - XI(t) + 2X2(t)
h it) = - X2(t) ( ii) h it)
( iii) + 2X2(t)
= 2XI(t) + X2(t) ( iv) f4(t) = Xl(t) ( v) f s(t)
( vi) f6(t) = 3XI(t) ( b) P oint o ut p airs of mutually orthogonal vectors among these six vectors. Verify
t hat t he p airs o f signals corresponding t o t hese orthogonal vectors are also orthogonal. 3 .4-1 ( a) Sketch t he signal f it) = t 2 for all t a nd find t he t rigonometric Fourier series cp(t)
t o r epresent f it) over t he i nterval ( -1, 1). Sketch cp(t) for all values of t . 3 .4-2 ( a) S ketch t he signal f it) = t for all t a nd find t he t rigonometric Fourier series cp(t)
t o r epresent f it) over t he interval (-11",11"). S ketch cp(t) for all values of t . 3 .4-3 For each o f t he p eriodic signals shown in Fig. P3.4-3, find t he c ompact trigonometric
Fourier series a nd sketch t he a mplitude a nd p hase spectra. I f e ither t he sine or cosine
t erms are a.bsent in t he Fourier series, explain why. 3 .4-4 ( a) F ind t he t rigonometric Fourier series for x (t) shown in Fig. P3.4-4.
( b) T he s ignal x (t) is t he t ime-inverted signal cp(t) in Fig. 3.7b. Thus, x (t) = cp(-t).
Hence, t he F ourier series for x (t) c an be obtained by replacing t w ith - t in t he Fourier
series [Eq. (3.56)] for cp(t). Verify t hat t he F ourier series thus obtained is i dentical to
t hat found in p art ( a).
( c) Show t hat, in general, t...
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