Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 191

Here, the summations in the curly brackets from (5.41) and (5.42) are equal to
zero, because of the vanishing Froissart amplitudes{d ± k }(k∈K

F ), according

to (5.32). This implies

P ± K G +K F (z ±1 )
Q ± K G +K F (z ±1 )

K G +K F

d ± k z ±1
z ±1 −z ± k,Q

= b ± 0 +

k=1

K G

= P ± K G (z ±1 )
Q ± K G (z ±1 )

d ± k z ±1
z ±1 −z ± k,Q

= b ± 0 +

k=1

P ± K G +K F (z ±1 )
Q ± K G +K F (z ±1 )

= P ± K G (z ±1 )
Q ± K G (z ±1 )

∴ (5.43)

K G +K F

= b ± 0 δ(n) + d ± k z ±n c ± n

k,Q

k=1

K G

= b ± 0 δ(n) + d ± k z ±n

k,Q

k=1

K G

d ± k e ±2iπnτ ν ± k,Q = b ± 0 δ(n) +

k=1

K G +K F K G

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

k,Q . (5.44)

k,Q

k=1 k=1

The model order reduction in the FPT (+) can be performed by using either
(5.39) or substituting (5.38) into (5.43) and (5.44) with the results

P K G +K F (z)
Q K G +K F (z)

K G +K F K G

= P K G (z)
Q K G (z)

d k z
z−z k,Q

d k z
z−z k,Q

= = (5.45)

k=1 k=1

K G +K F K G K G

d k e 2iπnτ ν k,Q . c n = d k z +n d k z +n

= = (5.46)

k,Q k,Q

k=1 k=1 k=1