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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online