Digital Signal Processing Reference
In-Depth Information
relation was used for the chosen bandwidth 6667 Hz with the aim of achieving
a spectral resolution ω = 0.02 ppm. This would delineate isoleucine and
valine, i.e., the two most closely lying metabolites. The closest integer in the
form 2
m
for the FID length required in the FFT is thus N = 2
15
= 32768
= 32K (K = 1024). The FFT spectrum in Ref. [343] was computed by
zerofilling the two encoded FIDs each to 64K. It has been verified [27, 29]
that 32K signal points augmented by 32K zeros is the first FID length, which
simultaneously yields the converged absorption spectra and resolves all the
metabolites in the Fourier absorption spectra.
As discussed elsewhere, the FPT resolution is not predetermined by 2π/T
such that a shorter FID length could su
ce and in the present problem, we
used a total signal length N = 1024. Using (3.15) via 2C
k
/C
ref
we extracted
the input peak amplitudes from the tabulated data from Ref. [343]. The
reference concentration C
ref
was taken to be the largest concentration (6536
M/L) from Ref. [343]. This was the median lactate concentration in the
malignant ovarian samples. The phases φ
k
(1≤k≤12) from d
k
were all set
to zero, so that every amplitude d
k
becomes real, d
k
=|d
k
|. The linewidths
in Ref. [343] were estimated to be approximately 1 Hz. We allowed the line
widths to have small variations within the interval{8.21, 8.32}×10
−4
ppm
(labeled as Im(ν
k
) in the Tables).
The diagonal FPT
(−)
was used to analyze the FIDs. The expansion coef
}of the polynomials P
−
K
and Q
−
K
were computed by solving
the systems of linear equations from chapter 4 (section 4.10) by treating the
product in G
N
(z
−1
)∗Q
K
(z
−1
) = P
K
(z
−1
), as a convolution [10, 11]. In
order to extract the peak parameters, we solve the characteristic equation
Q
−
K
(z
−1
) = 0.
ficients{p
−
r
,q
−
s
This leads to K unique roots z
−
k
(1≤k≤K), so that the
sought ω
−
k
is deduced via ω
−
k
= (i/τ) ln (z
−
k
).
Peak area is proportional to the concentration of the metabolite, relative
to the reference concentration, which here is 6536 M/L, the median lac
tate concentration in the ovarian cancer fluid samples. This was the largest
concentration from Ref. [343], as mentioned. The metabolite concentrations
{C
−
k
}are thus computed via C
−
k
= 3268|d
−
k
|M/L ww.
9.3.1
Pade versus Fourier for MRS data derived from benign
ovarian cyst fluid
Table 9.2
shows the spectral parameters reconstructed by the FPT for the data
derived from benign ovarian cysts, at three signal lengths, N/32 = 32,N/16 =
64 and N/8 = 128. At N/32 = 32 signal points (upper panel), the results are
shown for spectral parameters of the reconstructed nine peaks out of twelve
that should be present. Isoleucine and threonine were undetected at N/32,
and only one peak was identified in the region between 3.07 ppm and 5.22 ppm.
The spectral parameters and computed concentration were fully correct only
for glucose at 5.220345 ppm. This shows that 32 FID points are insu
cient
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