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

6 alternately we n ote t hat a jb 1 j v3 hence c

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Unformatted text preview: a phase shift of ±7r amounts t o multiplication by - 1. Therefore, f (t) can also be expressed alternatively as 2e j1r / 3 . f (t) We c an a lso p erform t he r everse o peration, e xpressing f (t) = C c os (wot = - 2 cos (wot + 60° ± 180°) i n t erms o f cos wot a nd s in wot u sing t he t rigonometric i dentity = - 2 cos (wot - 120°) C cos (wot = - 2cos (wot + 240°) F or e xample, = - 3, b = 4, a nd from Eqs. (B.23) + e) = C c os ecos wot - C s in es in wot 10 c os (wot - 60°) = 5 cos wot In practice, a n expression with a n angle whose numerical value is less t han 180° is preferred. ( b) In this case, a + e) + 5 )3 s in wot Sinusoids in Terms o f Exponentials: Euler's Formula S inusoids c an b e e xpressed i n t erms o f e xponentials u sing E uler's f ormula [see Eq. (B.3)] C =)(-3)2+4 2 =5 e = t an- 1 (:::;) = - 126.9° 1. c os <p = - ( eJ'P 2 Observe t hat . + e-J'P ) 1 . . s in <p = 2 j (eJ'P - e-J'P ) ( B.24a) (B.24b) I nversion o f t hese e quations y ields Therefore, f (t) = 5 cos (wot - 126.9°) o Computer Example CB.4 B.3 Express f (t) = - 3 cos wot + 4 sin wot as a single sinusoid. Recall t hat a cos w ot+bsin wot = C cos[wot+tan-1(-b/a)]. Hence, t he a mplitude C a nd t he angle e o f t he resulting sinusoid are t he m agnitude and angle of a complex number a - jb. We use t he ' cart2pol' function t o convert it t o t he polar form t o o btain C a nd e. T heta.deg=-126.8699 - 3 cos wot + 4 sin wot = 5 cos (wot - 126.8699°) ( B.25a) = cos <p - j s in <p ( B.25b) 0 Sketching Signals I n t his s ection we discuss t he s ketching o f a few useful signals, s tarting w ith e xponentials. B.3-1 a =-3;b=4; [ theta,C]=cart2pol(a,-b); T heta.deg=(180/pi)*theta; C ,Theta.deg C=5 Therefore e -i'P cos <p +j s in <p e i'P = This result is r eadily verified in t he phasor diagram in Fig. B.8b. Alternately, a - jb = - 3 - j4 = 5e-j126.9°. Hence, C = 5 a nd e = - 126.9°. • Monotonic Exponentials T he s ignal e - at d ecays monotonically, a nd t he s ignal e at g rows m onotonically w ith t ( assuming a > 0) a s d epicted i n Fig. B.9. F or t he s ake o f simplicity, we s hall c onsider a n e xponential e - at s tarting a t t = 0, a s s hown i n F ig. B .lOa. T he s ignal e - at h as a u nit v alue a t t = O. A t t = l /a, t he v alue d rops t o l /e ( about 37% o f i ts i nitial v alue), a s i llustrated i n Fig. B .lOa. T his t ime i nterval o ver w hich t he e xponential r educes b y a f actor e ( that is, d rops t o a bout 37% o f i ts v alue) is k nown a s t he t ime c onstant o f t he e xponential. T herefore, t he t ime 20 B ackground B.3 Sketching Signals 21 4 0.2 t- o t- (a) 0.5 1.5 (b) (a) F ig. B .9 Monotonic exponentials. constant of e - at i s l /a. Observe t hat t he e xponential is reduced t o 37% of its initial value over any t ime interval of d uration l /a. T his can be shown by considering any set of instants t 1 a nd t 2 s eparated by one time constant so t hat t2 - t1 ( b) =~ 4 e- 2t Now t he r atio of e - at , t o e - at , is given by cos (6t - 60") ( c) t- Fig. B .ll Sketching an exponentially varying sinusoid (b) (a) Fig. B .lO (a) Sketching e - at (b) sketching e - 2t ....
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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