The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 62

we arrive at

9
=
; .

c 0 = d 1 + d 2 + d 3 ++ d K
c 1 = d 1 u 1 + d 2 u 2 + d 3 u 3 ++ d K u K
c 2 = d 1 u 1 + d 2 u 2 + d 3 u 3 ++ d K u 2 K
.
c n = d 1 u 1 + d 2 u 2 + d 3 u 3 ++ d K u K
.

(2.156)

Hence, the obtained general term of the form

K

c n = d 1 u 1 + d 2 u 2 + d 3 u 3 ++ d K u K = d k u k

(2.157)

k=1

coincides with the complete integral (2.138) of the K th order difference equa
tion (2.137). This sets the prescription for the usage of the first K results
c 0 ,c 1 ,c 2 ,...,c K−1 deduced from the generating fraction. As mentioned, these
results c 0 ,c 1 ,c 2 ,...,c K−1 contain the coe cients a 0 ,a 1 ,a 2 ,...,a K−1 of the
numerator polynomial of the generating fraction. In order to find all the terms
c 0 ,c 1 ,c 2 ,..., of the rhs of (2.154) to the recurring order K, where c n is the
general member of the Taylor series

∞

n=0 c n u n , it is su cient to have only

the K starting values{c r }(0≤r≤K−1) that should be given in advance.

By means of these K initial c n 's, one could continue the sequence forward or
backward using the difference equation (2.137). In this way, one can either
predict the new c n 's beyond the originally given{c r

}(0≤r≤K−1) or
retrieve all the preceding c n 's by recurring with the descending value of the
su ces. In other words, if the K terms are given c s+1 ,c s+2 ,...,c s+K , one
could prolong/extrapolate this latter sequence by means of the relationship

9
=
; . (2.158)

c s+K+1 =− 1
b K

(b K−1 c s+K + b K−2 c s+K−1 ++ b 0 c s+1

c s+K+2 =− 1

b K (b K−1 c s+K+1 + b K−2 c s+K ++ b 0 c s+2
.

Thus the new c n 's are predicted using the previous ones, so in general

K−1

c s+K+m =− 1
b K

(predicting the new c n ′ s). (2.159) b r c r+s+m

r=0

Alternatively, the calculation could also be carried out by recurring towards