Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 85

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Exact quantum-mechanical, Pade-based
recovery of spectral parameters

Using the FPT, we shall examine a synthesized noisefree FID in order to re
construct the input spectral parameters. This FID, reminiscent of time signals
typically encountered in MRS [88], is a sum of powers of K damped complex
exponentials exp (iω k τ) with fundamental frequencies ω k and stationary am
plitudes d k as in the geometric progression (2.193)

K

d k e inω k τ

c n = Im(ω k ) > 0 0≤n≤N−1. (3.1)

k=1

For the illustrations in this chapter, we set K = 25 and N = 1024. Recall that
ω is the angular frequency, ω = 2πν, and ν is the linear frequency. The finite
rank Green function as the system's response function gives the corresponding
exact complexvalued spectrum (2.162).
The physics of the representation (3.1) dictates that all the signal points
c n must have finite absolute values, |c n

K

k=1 d k z k

| <∞, where the
complex numbers z k = e iω k τ represent the signal poles. This requires that all
these latter input fundamental harmonics z k , that are present in (3.1), must
describe the exponentially decaying transients.

| = |

In other words, each signal

pole z k = e iω k τ must be inside the unit circle,|z k |< 1

K

d k (z k ) n z k = e iω k τ |z k |< 1 c n = (3.2)

k=1

where|c n |<∞. The condition,|z k |=|e iτ Re(ω k )−τ Im(ω k ) |< 1 will be fulfilled
by the imposition of the constraint Im(ω k ) > 0 (1≤k≤K), as in (3.1).
Formally, we can represent the same time signal c n in terms of the inverses
of the signal poles z − k rather than the signal poles z k themselves. However,
in this case, as well, we ought to preserve the same sign of the imaginary
fundamental frequencies, Re(ω k ) > 0. This can be done by writing (3.2) as
the following identity

K

d k e iω k τ ≡ K d k (u k ) −n

c n = Re(ω k ) > 0 (3.3)

k=1 k=1

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