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

O therwise t he p rocedure remains t he s ame t he p

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Unformatted text preview: e B .ll Expand F (x) into partial fractions i f which agrees w ith t he p revious result. F (x) = 3 x 2 + 9x x2 A Mixture o f t he Heaviside "Cover-Up" and Short Cuts Here m = n = 2 with bn = b2 = 3. +X - 20 6 = 3 x 2 + 9x - 20 (x - 2)(x + 3) Therefore, I n t he above example, after determining t he coefficients ao = 2 a nd k = 1 by t he Heaviside m ethod a s before, we have 4 x 3 + 1 6x 2 + 23x + 13 2 al a2 1 - -;---:-.,-;;,-;----,-- = - - - + - - - + - - + - (x+l)3(x+2) ( x+l)3 ( x+l)2 x +l x +2 T here a re only t wo u nknown coefficients, a l a nd a2. I f we m ultiply b oth sides of t he above e quation b y x a nd t hen let x - > 0 0, we c an eliminate a 1 . T his yields in which kl = 3x 2 + 9x - I 20 = 12 + 18 (2 z =2 and 2 k2 = 3x + 9 x - 20 ~ I 20 = + 3) 5 = 27 - z =-3 ~ =2 27 - 20 ( -3-2) = - 20 = 4 -5 Therefore, F (x) Therefore, ( x - 2)(x 4 x 3 + 1 6x 2 + 2 3x + 13 2 al 3 1 - -,.--.....,...,,-;----- = - - - + - - - + - - + - (x + 1 )3(x + 2) ( x+l)3 ( x+l)2 x +l x +2 T here is now o nly one unknown a I, which c an b e r eadily found by s etting x e qual t o a ny convenient value, say x = O. T his yields ¥ = 2 + al + 3 + ~ =} Improper F (x) with m 8 .5-6 = 3 + _ 2_ + _ 4_ x- 2 x +3 • Modified Partial Fractions F () = x 5x2 + 2 0x + 18 ( x + 2 )(x + 3)2 --...,.-,.----,-= Dividing b oth sides by x yields =n A g eneral m ethod of handling an i mproper function is indicated in t he beginning of t his s ection. However, for a special case where the n umerator a nd d enominator p olynomials of F (x) a re of t he s ame degree (m = n ), t he p rocedure is t he s ame as t hat for a p roper function. We c an show t hat for 20 + 3) O ften we r equire p artial f ractions of t he form (x~f.)r r ather t han ( x_\.jr. T his c an b e achieved by e xpanding F (x)/x i nto p artial fractions. Consider, for example, al = 1 which agrees w ith o ur e arlier answer. B .5-5 = 3 x 2 + 9x - F (x) 5x 2 --= x x (x + 2 0x + 18 + 2 )(x + 3)2 E xpansion of t he r ight-hand side into p artial f ractions as u sual yields F (x) = 5 x + 2 0x + 18 = a l + ~ + _ a_3_ + _ a_4_ x x (x + 2 )(x + 3)2 X X+2 ( x + 3) ( x + 3)2 Using t he p rocedure discussed earlier, we find a l = 1, a2 = 1, a3 = - 2, a nd a4 = 1. T herefore, 2 F (x) = .!.+_1_ _ _ _+_1_ 2 x x x+2 x+3 ( x + 3)2 Now multiplying b oth s ides by x yields t he coefficients k l' k 2,"" k n a re c omputed a s if F (x) were proper. Thus, x 2x x F (x) = 1 + - -2 - --3 + - ( 3)2 x+ x+ x+ T his expresses F (x) as t he s um o f p artial f ractions having t he form (x~f.)r. For q uadratic o r r epeated factors, t he a ppropriate p rocedures discussed in Secs. B.5-2 or B.5-3 s hould b e u sed as if F (x) were proper. I n o ther words, when m = n , 34 Background B .6 Vectors and Matrices An entity specified by n numbers in a certain order (ordered n-tuple) is a n n-dimensional v ector. Thus, an ordered n-tuple ( Xl, X 2, . .. , X n) represents an n-dimensional vector x. Vectors may be represented as a row ( row v ector): X =[XI X2 X nl o r as...
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