Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 190

using the property

b 0

= 0 (5.38)

which originates from p 0 ≡0 according to the derivation from Ref. [5, 34].

Thus, using (5.35) and (5.37) we have

9
=

K

P K (z)
Q K (z)

d k z
z−z k,Q

=

k=1

(5.39)

;

K K

d k e 2iπnτ ν k,Q c n = d k z +n

k,Q =

k=1 k=1

9
=
; .

K

P − K (z)
Q − K (z)

= p − 0
q − 0

d − k z −1
z −1 −z − k,Q

+

k=1

(5.40)

K K

= p − 0
q − 0

k,Q = p − 0

d − k e −2iπnτ ν − k,Q c − n d − k z −n

+ +

q − 0

k=1 k=1

5.9 Model reduction problem via Heaviside or Pade par-
tial fraction spectra
In an alternative derivation, we can show that the model order reduction can
also be accomplished by using the formulae (5.35) and (5.37) for the Heaviside
or Pade partial fractions

P ± K T (z ±1 )
Q ± K T (z ±1 )

K G +K F

d ± k z ±1
z ±1 −z ± k,Q

= b ± 0 +

k=1

K G K G +K F

d ± k z ±1
z ±1 −z ± k,Q

d ± k z ±1
z ±1 −z ± k,Q

= b ± 0 + + (5.41)

k=1 k=K G +1

k∈K F

as well as for the time signals

K T K G +K F

c ± n = b ± 0 δ(n) + d ± k z ±n = b ± 0 δ(n) + d ± k z ±n

k,Q k,Q

k=1 k=K G +1

K G K G +K F

= b ± 0 δ(n) + d ± k z ±n d ± k z ±n c ± n

∴

k,Q +

. (5.42)

k,Q

k=1 k=K G +1

k∈K F