Component shape spectrum

We present the absorption component shape spectra and the total shape spectra as reconstructed by the FPT for the normal breast data in Figure 6.10 at three partial signal lengths Np = 1000,1500, and 2000. The top right panel (iv) indicates that at Np = 1000, the absorption total shape spectrum has converged. In contrast, for the component shape spectrum (top left panel (i)), there is only one peak (phosphoethanolamine. A = 5) at 3.22 ppm that has been overestimated, whereas phosphocholine, (A = 4) has not been detected. 

On the middle left panel (ii) of Figure 6.10 at Np = 1500, the component shape spectrum has converged with resolution of both peaks A = 4 and 5. The correct heights are now displayed for these and all the other peaks. Note that the phosphocholine peak is very small and completely underneath the phosphoethanolamine peak. The bottom panels of Figure 6.10 display the... 

The converged absorption component shape spectrum (top panel (i)) and total absorption shape spectrum (lower panel (ii)) for prosfafe cancer between 2.40 and 3.70 ppm at Np = 800 are compared in Figure 6.26. Strikingly, the serrated peaks on the total shape spectrum only suggest the number of underlying resonances. The converged componenf specfrum is essential to visualize the actual number and structure of resonances. For example, from the small polyamine peaks, it would be difficult to know that there are actually two components. 

On the total shape spectrum, the citrate doublets around 2.53 and 2.73 ppm look like broad peaks, and there is only a hint of the doublet structures shown in the component spectra. Although phosphocholine at 3.23 ppm and glyc-erophosphocholine at 3.24 ppm are clearly distinguished in the component shape spectrum, that is not at all true on the total shape spectrum. 

Figure 4.1 Time and frequency domain data in signal processing in the noiseless case using the fast Fourier transform (FFT) and fast Pad6 transform (FPT). Top panel (i) the input FID (to avoid clutter, only the real part of the time signal is shown). Middle panel (ii) absorption total shape spectrum (FFT). Bottom panel (iii) absorption component (lower curves FPT) and total (upper curve FPT) shape spectra. Panels (ii) and (iii) are generated using both the real and imaginary parts of the FID.

Absorption Component Shape Spectra Absorption Total Shape Spectrum... 

Figure 4.8 Absorption component shape spectra (left) and absorption total shape spectra (right) from the FPTf 1 near full convergence for signal lengths Np = 180,220,260. On panel (iv) for Np = 180, the total shape spectrum reached full convergence, despite the fact that on panel (i) for the corresponding component shape spectra, the 11th peak is missing and the 12th peak is overestimated.

In MRS, the encoded data are heavily packed time signals that decay exponentially in an oscillatory manner. These time domain data are not directly interpretable. The corresponding total shape spectrum is obtained by mathematical transformation of fhe FID info its complementary representation in the frequency domain. This fofal shape specfrum provides qualitative information, but not the quantitative one about the actual number of metabolites that underlie each peak or the relative strength of individual components, their abundance, etc. At best, the FFT takes us only to this second step. More information is needed before fhe mefabolifes can be identified and their concentrations reliably determined, and from fhe fofal shape spectrum alone, this can only be guessed. 

FADE COMPONENT SHAPE SPECTRA (Left), TOTAL SHAPE SPECTRA (Right) PARTIAL SIGNAL LENGTHS Np = 1000,1500, 2000 Absorption Component Shape Spectra Absorption Total Shape Spectrum... 

The Pad6-reconstructed absorption component shape spectra for the normal glandular prostate are shown in Figure 6.18 at Np = 54 and Np = 800. The upper panel reveals that at Np = 54, only 12 of fhe 27 resonances were resolved. These correctly resolved peaks were mainly at the two extremes of the spectrum (lactate and alanine at 1.33 and 1.49 ppm and myoinositol... 

The total absorption shape spectra reconstructed by the FPT at Np = 54 (upper panel (i)) and Np = 800 (lower panel (ii)) for the prostate cancer data are shown in Figure 6.25. Similar to what was seen for the component absorption spectrum for the prostate cancer data, lactate and alanine at 1.33 ppm and 1.49 ppm and myoinositol and lactate at 4.07 ppm and... 

The middle trace in Fig. 8 is the line shape observed using the spin echo technique at a pulse separation of t = 0.25 msec (180°-t-90° pulse sequence). Reimer and Duncan (1983) have determined that the FID and echo line shapes of Fig. 8 are different, and the difference spectrum is plotted at the bottom of Fig. 8. Furthermore the T2 as measured by the echo technique is very different for these two components. The spectrum with the longer T2 (—1.4 msec) in the middle of Fig. 8 is best fit with a Lorentzian line shape centered at 175 ppm with respect to a standard reference ( P in 85%H3P03). The broad line at the bottom of Fig. 8 is best fit with a Gaussian line shape centered at 70 ppm. 

Brown and collaborators interpret their spectra as showing that S q, t) in some systems has multiple slow modes, some being -independent while others scale as (43,48). Brown, et a/. s interpretation potentially explains all slow mode behaviors, namely in different circumstances the slow mode is dominated by a -dependent or by a -independent component. A spectrum whose modes have -dependent shapes might in some cases also be described as a mixture of and °-dependent relaxations whose relative amplitudes are not constant. The relationship between the spectral analyses of Brown and Stepanek(29), who interpret their spectra via a regularized Laplace transform method, and the work of Phillies and collaborators(87), who interpret S q,t) as a sum of stretched exponentials whose parameters depend on q and c, has not been completely analyzed. The latter interpretation has the virtue of supplying quantitative parameters for further analysis. 

Quasi-resonant and resonant transition switching power supplies have a much more attractive radiated spectral shape. This is because the transitions are forced to be at a lower frequency by the resonant elements, hence only the low frequency spectral components are exhibited (below 30MHz). The lower rate of change during the transitions are responsible for behavior. The higher frequency spectral components are almost non existent. The near-held radiated spectrum of a quasi-resonant, hyback converter are shown in Figure E-2. The quasi-resonant and soft switching families of converters are much quieter and easier to hlter. 

When an element is present on the surface of a sample in several different oxidation states, the peak characteristic of that element will usually consist of a number of components spaced close together. In such cases, it is desirable to separate the peak into its components so that the various oxidation states can be identified. Curve-fitting techniques can be used to synthesize a spectrum and to determine the number of components under a peak, their positions, and their relative intensities. Each component can be characterized by a number of parameters, including position, shape (Gaussian, Lorentzian, or a combination), height, and width. The various components can be summed up and the synthesized spectrum compared to the experimental spectrum to determine the quality of the fit. Obviously, the synthesized spectrum should closely reproduce the experimental spectrum. Mathematically, the quality of the fit will improve as the number of components in a peak is increased. Therefore, it is important to include in a curve fit only those components whose existence can be supported by additional information. 

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