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

He showed t hat every e quation of the n th order has

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Unformatted text preview: complex number. He showed t hat every e quation of the n th order has exactly n solutions (roots), no more and no less. Gauss was also one of the first t o give a coherent account of complex numbers and t o i nterpret t hem as points in a complex plane. I t is he who introduced t he t erm c omplex n umbers a nd paved the way for general and systematic use of complex numbers. T he n umber s ystem was once again broadened or generalized t o include imaginary numbers. Ordinary (or real) numbers became a special case of generalized (or complex) numbers. T he u tility o f complex numbers can b e u nderstood readily by an analogy with two neighboring countries X a nd Y , as illustrated in Fig. B .l. I f we w ant t o travel from City a t o C ity b ( both in Country X ), t he s hortest route is t hrough Country Y, although t he j ourney begins and ends in Country X . We may, if we desire, perform this journey by a n a lternate route t hat lies exclusively in X , b ut this alternate route is longer. In m athematics we have a similar situation with real numbers (Country X ) a nd complex n umbers ( Country Y ). All real-world problems must s tart w ith real numbers, and all the final results must also be in real numbers. B ut t he derivation o f results is c onsiderably simplified by using complex numbers as an intermediary. I t is also possible t o solve all real-world problems by an alternate method, using real numbers exclusively, b ut such procedure would increase the work needlessly. 3 B.1-2 Algebra o f Complex Numbers A complex n umber (a, b) or a + j b c an be represented graphically by a point whose Cartesian coordinates are (a, b) in a complex plane (Fig. B.2). Let us denote this complex n umber by z so t hat F ig. B .1 Use o f complex numbers can reduce t he work. z = a + jb (B.1) T he numbers a a nd b ( the abscissa and the ordinate) of z are the r eal p art and the i maginary p art, respectively, of z. T hey are also expressed as Re z = a 1m z = b Note t hat in this plane all real numbers lie on t he horizontal axis, and all imaginary numbers lie on the vertical axis. Complex numbers may also be expressed in terms of polar coordinates. I f (r, 0) are the polar coordinates of a point z = a + j b (see Fig. B.2), then a =rcosO b =rsinO and z = a + j b = r cos 0 + j r sin 0 = r(cosO+jsinO) i Imaginary b Real """ ................. - b .................... : :,. z· F ig. B .2 R epresentation of a number in t he complex plane. (B.2) 6 Background B.1 T he E uler f ormula s tates t hat e i 1m j8 7 Complex N umbers = cos () + j sin () j re J6 T o prove t he E uler formula, we e xpand e j8 , cos (), a nd sin () using a Maclaurin series e j8 1t (.())3 (.())4 (.())5 (.())6 ( .())2 = 1 + j () + _ J_ + _ J_ + _ J_ + _ J_ + _ J_ + " . 2! 3! 4! 5! 6] ()2 = 1 + j () ()2 ()3 2T - j 3! + 4! + j 5! - ()4 ()6 ()4 ()5 Re ..... 05 -j (b) ()s F ig. B .3 Understanding some useful identities in terms of re j8 . ()7 Hence, it follows t hat ej8 = cos 0 + j sin () (B.3) Usin...
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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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