The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 70

coe cients{q s

}are accurate. Explicit computations of the polynomial coef

ficients{p r }and{q s }are simplified by the occurrence that the two systems

of equations (2.176) and (2.177) are decoupled. The same remark holds true
for{p − r }and{q − s

}from the two systems of equations (2.178) and (2.179).
Hence, in the FPT (+)

or FPT (−) , only the system (2.176) or (2.178) for the

coe cients{q s }or{q − s

}needs to be solved, respectively. In other words, if

the sets{q ± s }are obtained, it is not necessary to solve the linear equations

from (2.177) and (2.179) for the remaining coe cients{p r }or{p − r }. In fact,

for the known sets{q ± s

}, the equations (2.177) and (2.179) themselves are
the analytical results for{p r

}and{p − r

}, respectively. In practice, regarding
the systems of linear equations (2.176) and (2.178) for the coe cients{q s }

and{q − s }, the standard MATLAB algorithms can be used with the optional

implementation of the conventional SVD for a refinement of the obtained
solutions.
By construction, the GreenPade spectra G ± K (z ±1 ) are meromorphic func
tions and, as such, the only singularities these functions could possibly have
are their poles. For this reason, the poles of these functions are the same
as the corresponding zeros of the denominator polynomials Q ± K (z ±1 ). This
enables the FPT (±) to determine all the fundamental complex frequencies

{ω ± k }(1≤k≤K), counted with their possible multiplicities, by solving the

pertinent characteristic equations

Q ± K (z ±1 ) = 0. (2.180)

These equations represent the wellknown secular equations for the roots in
terms of the harmonic variables z ± k

≡e ±iω ± k

z ± k Im(ω ± k ) > 0.

τ

(2.181)

Physical, genuine roots have Im(ω ± k ) > 0, meaning that all the K poles{z k

}

and{z − k

}are located inside and outside the unit circle, respectively. The

fundamental frequencies ω ± k are deduced from z ± k

via

ω ± k =∓(i/τ)ln(z ± k ) ln(z ± k ) = ln(|z ± k |) + Arg(z ± k ). (2.182)

In the case of nondegenerate resonances, a fixed fundamental frequency is
associated with only one amplitude. As discussed, these resonances yield
pure Lorentzian lineshapes in the corresponding spectrum. Within the Pade
spectral analysis, such amplitudes are the Cauchy residues of the associated
GreenPade functions. For instance

P ± K−1 (z ± k )

) G ± K (z ±1 )}=

d ± k {(z ±1 −z ±1

k

= lim

z ±1 −→z ±1
k

. (2.183)

(d/dz ±1 )Q ± K (z ±1 )

z ±1 =z ± k

Here, the k th amplitude corresponds only to the k th frequency. Hence non
degeneracy.

The Pade reconstructions for the true phases φ k of the k th