Signal-noise separation via Froissart doublets - Signal Processing in MRS with Biomedical Applications - page 181

Maclaurin expansion (4.79), from which the FPT (+) is derived. Obviously,
this latter point is associated precisely to z −1 =∞in the FPT (−) , since in
this version of the FPT, the harmonic expansion variable is z −1 . Thus, the
“ghost” zero z −1 =∞is one of the K zeros of P − K (z −1 ) in the FPT (−) and,
therefore, it should be ignored for the same reason used to eliminate the point
z = 0 from the FPT (+) . Naturally, in numerical computations, the point
z −1 =∞cannot be reached exactly. Nevertheless, one of the zeros from the
reconstructed set{z − k,P }must have a very large real and imaginary part, and

this will become more pronounced by increasing the degree K of P − K (z −1 ) in
the FPT (−) . The two “ghost” zeros, z = 0 and z −1 =∞in the FPT (+) and
FPT (−) , respectively, have been found in our computations, exactly as per
the outlined description. Moreover, the computations from chapters 3 and
6 with noisefree and noisecorrupted FIDs did not detect any “ghost” poles
and, therefore, it follows

K T = K G + K F . (5.12)

Note that although the numerator polynomials P ± K (z ±1 ) are of degree K, the
total number of their zeros will be K−1 instead of K because of the excluded
“ghost” zero in each polynomial. This was already discussed in chapter 3 and
will be addressed again in chapter 6 in more detailed illustrations.

5.4 Signal-noise separation with exclusive reliance upon
resonant frequencies
As explained, the sets of all the poles{z ± k,Q }consist of the two disjoint subsets

of the genuine and Froissart poles

{z ± k,Q ={z ± k,Q ⊕{z ± k,Q

} } }

k∈K F .

(5.13)

k∈K T k∈K G

Here,K

T is the
set of all the values of k. Since there are no common elements in the two subsets
{z ± k,Q

G is the set of the counting indices k for the genuine poles andK

and{z ± k,Q } k∈K F , their sums from (5.13) represent the socalled

direct sums as labeled by the usual symbol⊕for disjoint sets. Therefore, it
is su cient to count the number K F of Froissart doublets to obtain the exact
number K G of the genuine poles through K G = K T −K F as in (5.12). In
practice, as soon as the FPT (±) are found to fully converge, one should make
a straightforward binning of all the reconstructed poles{z ± k,P

} k∈K G

} k∈K T into two

sets{z ± k,Q } k∈K G depending on whether or not z ± k,Q = z ± k,P ,

i.e., whether or not (5.11) is fulfilled. This grouping allows an unambiguous
recovery of the true number K G of the genuine poles associated with the exact

and{z ± k,Q } k∈K F