Digital Signal Processing Reference
In-Depth Information
3.2
where all the parameters are found to have converged in the interval
[N/8,N/4] = [128, 256]. Before full convergence, at the lowest partial signal
is not detected in the FPT
(±)
. However, full convergence of the entire set
of unknowns in both versions of the FPT is reached at N
P
= 260 on panel
(iii) and (vi). Moreover, it is seen on panel (v) that the FPT
(−)
converged
even at N
P
= 220. Convergence of the FPT
(±)
is verified to be maintained at
N
P
> 260 as also implied by Table 3.2.
3.1.4 Graphic presentation of the input data
Figure 3.1
shows the input spectral parameters in the complex planes of
frequency and harmonic variables. These are Argand plots{Re(ν), Im(ν)}
{Re(z
k
), Im(z
k
)}and{Re(z
−
k
), Im(z
−
k
)}displayed on panels (i), (ii) and (v),
respectively. Also shown in panels (iii) and (vi) are the real, Re(c
n
), and imag
inary, Im(c
n
), parts of the FID from (3.1) as a function of time. In these latter
graphs for{Re(c
n
),n}and{Im(c
n
),n}, integers n (0≤N−1) appear on the
abscissa to count time expressed in the units of the sampling rate τ which is
itself given in seconds (s). As usual, the continuous time variable t is digitized
according to t→t
n
≡τn (0≤n≤N−1), so that the total duration T of
the FID is given by T = Nτ. Panel (iv) displays the absolute values|d
k
|of
the amplitudes d
k
as a function of chemical shifts, Re(ν) where|d
k
|= d
k
due
to the special choice of the phases Arg(d
k
)≡φ
k
= 0 (1≤k≤K).
It is seen on panel (i) in Fig. 3.1 that the majority of the input complex fre
quencies have relatively small imaginary parts Im(ν
k
), i.e., they are clustered
in the proximity of the real axis, Re(ν
k
). The physical significance of such a
distribution is that the resonances associated with these smaller imaginary
frequencies Im(ν
k
) have larger lifetimes that are characteristic of more sta
ble transients from the inner part [1.959,4.271] ppm of the whole considered
spectral band [0.985,4.680] ppm in panel (i). Thus, such frequencies lying
close to the real axis in panel (i) will generate narrow spectral lineshapes
for the metabolites n
k
= 5−24 (Gaba - PCho) from the middle part of
remote from the real axis. These larger imaginary frequencies Im(ν
k
) corre
spond to shorter lifetimes that are typical of more unstable transients from
the outer parts [0.985,1.689] ppm and 4.680 ppm on the right and left wings,
respectively, of the entire investigated window [0.985,4.680] ppm in panel (i).
Being located deep in the complex plane, these latter frequencies will produce
broader spectral lineshapes for the metabolites n
k
= 1−4 (mobile lipids) and
n
k
= 25 (water) from the top and bottom parts, respectively, of Table 3.1.
The distribution of the input amplitudes for all the metabolites from Table
3.1 is depicted on panel (iv) in Fig. 3.1 for the interval [0.985,4.680] ppm
of the chemical shifts, Re(ν). In particular, water is seen as having quite a
sizable amplitude relative to a number of lowabundant metabolites.
This
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