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

3 using b3 in b2 yields za t he graphical

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Unformatted text preview: g (B.3) in (B.2) yields z=a T he graphical representation of a number z a nd its conjugate z ' is d epicted in Fig. B.2. Observe t hat z ' is a m irror image of z a bout t he horizontal axis. To find t he c onjugate o f a ny n umber, we n eed o nly t o r eplace j by - j in t hat n umber (which is t he same as changing t he sign of its angle). T he s um of a complex number a nd its conjugate is a r eal number equal to twice t he real p art of t he number: + jb = re z + z ' = (a + j b) + (a - jb) = 2 a = 2 Re z j8 (B.4) Thus, a complex number can be expressed in Cartesian form a + j b or polar form rejO w ith a =rcosO, b =rsin() a nd +b 2 , o= t an- 1 G) T he p roduct of a complex number z a nd its conjugate is a real number t he s quare of t he m agnitude of t he number: 2 z z' = (a + j b)(a - jb) = a 2 + b = (B.6) Izl. I n a complex plane, r e j8 r epresents a point a t a d istance r from t he origin and a t an angle 0 w ith t he horizontal axis, as shown in Fig. B.3a. For example, t he n umber - 1 is a t a u nit d istance from t he origin a nd has a n angle 7r or - 7r (in fact, any o dd multiple of ±7r), as seen from Fig. B.3b. Therefore, n o dd integer a nd (B.7) Also (B.11) T he n umber 1, o n t he o ther hand, is also a t a u nit d istance from t he origin, b ut has a n angle 27r (in fact, ±2n7r for any integral value of n ). Therefore, n integer (B.8) (B.12) T he n umber j is a t u nit d istance from t he origin a nd i ts angle is 7r / 2 (see Fig. B.3b). Therefore, C onjugate o f a C omplex N umber We define z ', t he c onjugate of z = a + j b, as = I zle- jLz (B. lOb) In fact, Lz = 0 z '=a-jb=re- j8 Izl2 Iz12, U nderstanding Some Useful I dentities Observe t hat r is t he distance of t he p oint z from t he origin. For this reason, r, (B.10a) (B.5) r is also called t he m agnitude (or a bsolute v alue) of z a nd is d enoted by Similarly () is c alled t he angle of z a nd is d enoted by Lz. Therefore Izl = Re ..... 1tI2 (a) s in()=()--+---+,,· 3! 5! 7! r =Va 2 1t12 ()6 6! - ". c os()=l--+---+-··· 2! 4! 6! 8! 03 - 1t -\ i 1m Similarly, (B.9a) (B.9b) T hus (B.13a) 8 B ackground B .l Complex Numbers I n fact, = ±j i = 1, 5, 9, 13,· . . i 1m (B.13b) n = 3, 7, 11, 1 5,··· e ±jn1r/2 1m (B.13c) n 9 a nd 2 +j3 These results are summarized in Table B.1. - 2+j I 153.4' T ABLE B .l r e 0 1 ±7r ±n7r 1 ±2n7r 1 ±7r / 2 1 ±n7r / 2 ±27r ± n7r/2 re 2 j8 =1 = -1 e ±jn1r = - 1 e ±j21r = 1 e ±j2n1r = 1 e ±j1T/2 = ± j e ±jn1T/2 = ± j e ±jn1T /2 = ' f j Re - . Re - . -2 (a) e jO (b) i e ±j1r 1m i R e-. -2 Re - . 1m n odd integer n integer n n = 1 ,5,9,13, . .. = 3 ,7,11,15, . .. -3 - 2-j3 (c) (d) F ig. B .4 From Cartesian to polar form. T his discussion shows t he usefulness of t he graphic picture of r e j8 . T his picture is also helpful in several other applications. For example, to determine t he limit of e (a+jw)t as t - + 0 0, we n ote t hat Now t he m agnitude of e jwt is u nity regardless of t he value of w or t because e jwt = r e j8 w ith r = 1. T herefore, e at d etermines t he b ehavior of e (a+jw)t as...
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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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