The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 61

The rhs of (2.149) is the denominator polynomial of the generating fraction
from (2.133). Then substituting the lhs of (2.149) into (2.133) and writing
the resulting polynomial quotient via its partial fractions, we have

≡ d 1

d 2
z 2 −u ++

d K
z K −u
(2.150)

a 0 + a 1 u + a 2 u 2 ++ a K−1 u K−1
b 0 (z 1 −u)(z 2 −u)(z K −u) z 1 −u +

where{d k }are some constants to be determined, and

1
u k .

z k = (2.151)

If each partial fraction on the rhs of (2.150) is expanded into its own binomial
series, we will have

9
=

+

d 1
z 1 −u =

d 1 u 1 (1 + u 1 u + u 1 u 2 + u 1 u 3 ++ u 1 u m +)

d 2

−u = d 2 u 2 (1 + u 2 u + u 2 u 2 + u 2 u 3 ++ u 2 u m +)
d 3

z 2

∞

c n u n .

=

d 3 u 3 (1 + u 3 u + u 3 u 2 + u 3 u 3 ++ u 3 u m +)
.

−u =

;

z 3

n=0

d K
z K −u = d K u K (1+u K u+u 2 K u 2 +u 3 K u 3 ++u K u m +)

(2.152)

Here, when we sum up these K equations, we shall obtain the generating
fraction on the lhs of (2.152), as per (2.149) and (2.150). Consequently, the
sum of K series from the rhs (2.152) ought to be equal to the Taylor expansion
c 0 + c 1 u + c 2 u 2 ++ c m u m +, as per (2.133) and this is symbolically
indicated via

∞

n=0 c n u n after the curly bracket. The superscript + associated
with the curly bracket in (2.152) is written to point at the said summation of
the K equations. This procedure leads to

K

d k

z k −u = A K−1 (u)

(2.153)

B K (u)

k=1

K ∞ ∞

d k u k

(u k u) r c n u n . = (2.154)

k=1 r=0 n=0

After equating the coe cients of the like power of u in (2.154) and setting

u k d k = d k

(2.155)