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

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

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Unformatted text preview: + C2 8 or 8 We c an now choose some other value for x , such as x = 2, t o o btain one more relationship to use in determining Cl a nd C2. I n this case, however, a simple method is t o multiply b oth sides of Eq. (B.43) by x a nd t hen let x - -> 0 0. This yields = - 8, and CI C 2)X (B.43) 2 =1+Cl 2 (B.42) so t hat a nd Therefore, Short-Cuts T he values o f Cl and C2 in Eq. (B.40) can also be determined by using shortcuts. After c omputing k l = 2 by the Heaviside method as before, we let x = 0 on both sides of Eq. (B.40) to eliminate Cl. T his gives us 18 _ 2 C2 13 - + 13 Therefore, C2 = -8 To determine Cl, we multiply b oth sides of Eq. (B.40) by x a nd t hen let x ---> 0 0. Remember t hat w hen x ---> 0 0, only the terms of the highest power are significant. Therefore, 4 a nd = k l + Cl = 2 + C! Cl = 2 I n the proced u re discussed here, we let x = 0 t o determine C2 a nd then multiply b oth sides by x a nd let x ...... 0 0 t o determine Cl. However, nothing is sacred about these values (x = 0 or x = 0 0). We use them because they reduce the number of 1 F (x) = x B .5-3 x +2 + x 2 + 2x + 5 R epeated Factors in Q(x) I f a function F (x) h as a repeated factor in its denominator, it has the form F (x) = P (x) (x - >-Jr(x - C"l)(x - (B.44) (x - aj) (2)'" I ts p artial fraction expansion is given by ao F (x) = (x _ >-)r + (x - a r-l al >-V-I + . .. + (x - >-) kl k2 kj + - - + - - + . .. + - x - a1 x - 0!2 x - aj (B.45) T he coefficients k l' k2, . .. , k j corresponding t o t he unrepeated factors in this equation are determined by t he Heaviside method, as before [Eq. (B.37)]. To find the 30 Background coefficients ao, a i, a2, . .. , a r-l, we multiply b oth sides of Eq. (B.45) by (x - >.)". T his gives us (x - >.)"F(x) = ao + a l(x - >.) + a2(x - >.)2 + . .. + a r-l(x - >.)"-1 ( - >.)" (x - >.)" ( - >.)" + k1x---+ k 2x---+"'+ k n - - x- al x- (B.46) x - an a2 I f we let x = >. o n b oth sides of Eq. (B.46), we o btain >.)"F(x)) 1 x=>' = al T hus, a l is o btained by concealing t he factor (x - >.)" in F (x), t aking the derivative of t he remaining expression, a nd t hen l etting x = A. Continuing in this manner, we find aj = ~ ~ [(x - J. dx J >')"F(x))1 (B.47b) x=>' Observe t hat ( x - >')" F (x) is o btained from F (x) by omitting t he factor (x - A)" from its denominator. Therefore, t he coefficient a j is o btained by concealing t he factor (x - >.)" i n F (x), t aking t he j th derivative of t he remaining expression, a nd t hen l etting x = A (while dividing by j !). • 2 (B.47a) Therefore, ao is o btained by concealing t he factor (x - >.)" in F (x) a nd l etting x = >. in t he r emaining expression (the Heaviside "cover up" m ethod). I f we t ake t he derivative ( with respect to x) of b oth sides of Eq. (B.46), t he r ight-hand side is a 1+ t erms containing a factor ( x - >.) in their numerators...
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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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