Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 180

5.3 Pade canonical spectra
Substituting (5.5) and (5.7) into (5.9) gives the canonical forms of the poly
nomial quotients from the FPT, i.e., P ± K (z ±1 )/Q ± K (z ±1 )

K

(z ±1 −z ± r,P )

P ± K (z ±1 )
Q ± K (z ±1 )

= p ± K
q ± K

r=1

. (5.9)

K

(z ±1 −z ± s,Q )

s=1

The spectra from (5.9) can also be expressed by a single product symbol as

K

(z ±1 −z ± k,P )
(z ±1 −z ± k,Q ) .

P ± K (z ±1 )
Q ± K (z ±1 )

= p ± K
q ± K

(5.10)

k=1

Physically, the degree K of the denominator polynomials in the FPT (±) rep

resents the total number K T of poles, K T ≡K. Otherwise, the number K T

is defined as the sum of the numbers of the genuine (K G ) and spurious (K S )
poles, K T = K G + K S . Genuine poles are, in fact, the signal poles that
represent the true, physical content of the investigated FID. Spurious or ex
traneous poles are the nonphysical ingredient of the input FID and, as such,
ought to be dropped from the final results of the spectral analysis. By defini
tion, a noiseless input FID possesses no spurious information. Nevertheless,
spuriousness can still appear in the spectral analysis of a noiseless FID in all
signal processors. Among the major origins of this computationally generated
noise (without counting the obvious roundoff errors) is underestimation or
overestimation of the otherwise unknown, exact number K G . Generally, spu
rious poles predominantly consist of Froissart doublets via the couples of the
coinciding Froissart zeros and poles

z ± k,P = z ± k,Q k∈K F . (5.11)

Here, the collectionK F represents the set of counting indices k for Froissart
poles, whose number is denoted by K F . There also exist some extraneous
isolated poles (called “ghost” poles) that do not have the matching zeros
to form the corresponding pairs. Moreover, there could be some extraneous
isolated zeros (named “ghost” zeros) that do not have the associated matching
poles. For example, a “ghost” zero in the FPT (+) is the point z = 0 which
is one of the K zeros of P K (z). In numerical computations by means of the
FPT (+) , one would easily detect the trivial zero z = 0 for any order K.
However, in signal processing, this zero should be discarded from the outset,
since the point z = 0 is excluded from the domain of definition of the original