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

B4d a nd t herefore t he a nswer given by t he c

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Unformatted text preview: y t he c alculator ( tan- l (j3) == - 71.6°) is c orrect (see Fig. B.4d). 1 - j3 == v'loe- j7 1. 6 ° i i • 1m 1m o C omputer E xample C B.1 E xpress t he following numbers in polar form: ( a) 2 + j 3 ( b) - 2 + j l MATLAB f unction cart2pol(a,b) c an b e used t o convert t he complex number a + j b t o its polar form. Re ..... 3 e-i3 • = -3 -31t !!. 2 (a) Re ..... [ Zangle.ln.J'ad,Zmag]=cart2pol(2,3) Z angle_inJad = 0 .9828 Zmag = 3.6056 Z angle_in_deg=Zangle.ln..rad * ( 180 / pi) Z angle_in_deg-56.31 T herefore z == 2 + j 3 == 3.6056ej56.31 ° (d) (c) i i 1m 1m -41t 41t Re ..... (b) [ Zangle.ln.J'ad , Zmag] = eart2pol( - 2,1) Z angle_inJad = 2 .6779 Zmag = 2.2361 Z angle_in_deg=Zangle.ln..rad*(180/pi) Z angle_in_deg=153.4349 Therefore z = - 2 + j l == 2.2361ej153.4349' F ig. B .5 F rom polar t o C artesian form. o Note t hat M ATLAB a utomatically takes care of t he q uadrant in which t he complex number lies. 0 • E xample B .2 R epresent t he following numbers in t he complex plane a nd express t hem in Cartesian form: ( a) 2e j ,,/3 ( b) 4 e- j3 ,,/4 ( e) 2 e j ,,/2 ( d) 3 e- j3 " ( e) 2e j b ( f) 2 e- j4". ( a) 2e j ,,/3 == 2 (cos ~ + j sin ~) == 1 + j V3 ( f) (e) (see Fig. B.5a) ( b) 4 e- j3 ,,/4 == 4 (cos ¥ - jsin ¥) = - 2V2 - j2V2 (see Fig. B.5b) ( e) 2e M2 == 2 (cos ~ + j sin ~) == 2(0 + j 1) == j 2 (see Fig. B.5c) jh ( d) 3 e= 3(cos 371' - jsin 371') == 3 (-1+jO) == - 3 (see Fig. B.5d) j4 ( e) 2 e " == 2 (cos 471' + j sin 471') == 2(1 + jO) = 2 (see Fig. B.5e) ( f) 2 e- j4" = 2(cos 471' - j sin 471') = 2(1 - jO) = 2 (see Fig. B.5f) • C omputer E xample C B.2 R epresent 4 e- j ¥ in C artesian form. MATLAB function poI2cart(8, r ) converts t he complex number r ejD t o C artesian form. [ Zreal,Zimag]=poI2eart(- 3*pi/ 4 ,4) Z real--2.8284 Zimag=-2.8284 Therefore 4 e- j ¥ = - 2.8284 - j2.8284 0 Arithmetical Operations, Powers, and Roots of Complex Numbers To perform addition a nd s ubtraction, complex numbers should be expressed in Cartesian form. Thus, if 12 B ackground Zl B.1 13 Complex Numbers Zl = 3 + j 4 = 5 ei53 .1' ~ a nd ( 3+j4)(2-j3) j3) = (2 + j 3)(2 - 1 8-j1 1 8-j1_18 = 22 + 32 = - 1-3- - .1 13 - J 13 D ivision: P olar F orm t hen Zl + Z2 = (3 + j 4) + (2 + j 3) = 5 + j 7 Zl I f Z l a nd Z 2 a re g iven in p olar form, we would need t o convert t hem i nto C artesian form for t he p urpose of adding (or s ubtracting). M ultiplication a nd division, however, can b e c arried o ut i n either C artesian o r polar form, although t he l atter proves t o b e m uch m ore convenient. T his is because if Zl a nd Z 2 a re expressed in p olar form as a nd Z2 j53 1 5e . ' = _ 5_ ej (53.1'-56.3') = _ 5_e-j3.2° V13e j56 .3' V13 V13 • I t is c lear from t his e xample t hat m ultiplication a nd division are easier t o accomplish in p olar form t han i n C artesian form. • E xample B .4 For Zl = 2e j ,,/4 and Z2 = 8e j ,,/3, find ( a) 2Z1 - Z2 ( b) 1.. (c) ~ ( d) ijZ2 Zl Z2 ( a) Since subtraction cannot be performed dire...
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