The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 58

this goal, we shall develop the generating fraction in a power series expansion.
Then, by using the method of undetermined coe cients, we can write

a 0 + a 1 z ++ a K−1 z K−1
b K + b K−1 z ++ b 0 z K

= c 0 + c 1 z ++ c K z K +. (2.133)

The constants c 0 ,c 1 ,...,c K ,..., and so on from the rhs of (2.133) are the
undetermined coe cients. In order to find their values when the a k 's and b k 's
are known, we multiply both sides of (2.133) by b K +b K−1 u++b 0 u K . Then
passing all the ensuing terms to the same side and ordering them regarding
the power of z will yield the result

9
=
; = 0.

b K c 0 + b K c 1 z + b K c 2 z 2 + b K c 3 z 3 +

−a 0 +b K−1 c 0 z+b K−1 c 1 z 2 +b K−1 c 2 z 3

+

−a 1 z +b K−2 c 0 z 2 +b K−2 c 1 z 3

+

(2.134)

−a 2 z 2 +b K−3 c 0 z 3

+

−a 3 z 3 +

−

In (2.134), every vertical band of coe cients of the same power of z should
be zero and this yields the following equations [138]
c 0 = a 0
b K
c 1 =− 1

b K [b K−1 c 0 ] + a 1 b K

c 2 =− 1

b K [b K−1 c 1 + b K−2 c 0 ] + a 2 b K

c 3 =− 1

b K [b K−1 c 2 + b K−2 c 1 + b K−3 c 0 ] + a 3 b K

.

(2.135)

c K−1 =− 1

b K [b K−1 c K−2 + b K−2 c K−3 + b K−3 c K−4 ++ b 1 c 0 ] + a K−1 b K

c K =− 1

b K [b K−1 c K−1 + b K−2 c K−2 + b K−3 c K−3 ++ b 0 c 0 ]
.

c m =− 1
b K

[b K−1 c m−1 + b K−2 c m−2 + b K−3 c m−3 ++ b 0 c m−K ] (2.136)

.

c m+K =− 1

b K [b K−1 c m+K−1 + b K−2 c m+K−2 + b K−3 c m+K−3 ++ b 0 c m ].
It is seen that the isolated terms a 0 /b K ,a 1 /b K ,a 2 /b K ,..., given by all the
scaled coe cients{a k /b K }(k = 0, 1, 2,...) of the numerator of the generating