Fourier domain signal

Figure 10.5. (a) Lorentzian band and its P ourier transform, (b) Same curves as in (a) but with a small amount of noise added the noise is particularly recognizable in the region where the Fourier domain signal is low, that is, at the end of the decay function, (c) Result of truncation of the Fourier domain signal to remove the low-SNR region and its transformation back to the spectral domain the noise on the Lorentzian band has been reduced at the expense of resolution. 

Figure 10.7. Fourier differentiation to produce a second derivative spectrum. A Lorentzian band (a) is transformed to the Fourier domain (b). The Fourier domain signal in (b) is multiplied by a parabolic function (c) to yield (d). Inverse Fourier transformation of (d) leads to the second derivative of the original Lorentzian band e).

Figure 10.9. Difference in Fourier domain signals for a narrow Lorentzian band (a) and a broader Lx)r-entzian band (i>).

The simplest operation that can be undertaken is to remove the decay in its entirety. This is done by multiplying the Fourier domain signal by an exponential function that cancels the decay [i.e., [exp(+7iy x )], as... 

This will result in a cancellation of the decay, and the Fourier domain signal becomes a pure tmncated cosine wave, as shown in Figure 10.10. The Fourier transform of a pure truncated cosine wave is a sine function, as shown in Section 2.3. The sine function has a narrower FWHH than almost any other spectral waveform however, it does have large sidelobes. Of course, these could be removed with apo-dization (Section 2.4), but it is usually easier to change the rate of decay in Eq. 10.7. If the Fourier domain array is multiplied by an exponential function with a different FWHH, y such that y < y, that is. 

Figure 10.10. Full Fourier self-deconvolution of a Lorentizan band. The Lorentzian band (a) undergoes Fourier transformation to yield the Fourier domain signal b which has a decay, exp(—ya ). The signal in (b) is multiplied by the inverse exponential decay (c), expC-t-yx), to produce a truncated cosine wave (d). Upon inverse Fourier transformation of (d) a sine waveform is produced (e). The sine waveform has a narrower FWHH than that if the original Lorentzian band.

The functions used to change the decay functions in FSD increase the noise level faster than do the polynomial functions used for spectral derivatives because exponential functions tend to increase more rapidly at high spatial frequencies than polynomials. As a consequence, the effect on the data at high spatial frequencies in the Fourier domain signal (the region where the SNR is the lowest) is most severe. The general rule is that as the FWHH is narrowed by a factor of 2, the SNR of the spectmm decreases by an order of magnitude. In practice, the... 

Figure 10.11. Fourier self-deconvolution can be applied simpiy to reduce the FWHH of a band and still largely retain the lineshape. A Lorentzian band (a) has a Fourier domain signal (b) with a decay, exp(-y c). The signal in ( ) can be multiplied by an exponential function (c), expC+yj ), where / < y. The product, (d), still retains a decay function, but a weaker decay than in the raiginal Fourier domain signal. The inverse Fourier transformation of (

An alternative approach to obtaining microwave spectroscopy is Fourier transfonn microwave (FTMW) spectroscopy in a molecular beam [10], This may be considered as the microwave analogue of Fourier transfonn NMR spectroscopy. The molecular beam passes into a Fabry-Perot cavity, where it is subjected to a short microwave pulse (of a few milliseconds duration). This creates a macroscopic polarization of the molecules. After the microwave pulse, the time-domain signal due to coherent emission by the polarized molecules is detected and Fourier transfonned to obtain the microwave spectmm. 

Apparently, the time-domain and frequency-domain signals are interlinked with one another, and the shape of the time-domain decaying exponential will determine the shape of the peaks obtained in the frequency domain after Fourier transformation. A decaying exponential will produce a Lorentzian line at zero frequency after Fourier transformation, while an exponentially decaying cosinusoid will yield a Lorentzian line that is offset from zero by an amount equal to the frequency of oscillation of the cosinusoid (Fig. 1.23). 

Fourier transformation of Rf pulses (which are in the time domain) produces frequency-domain components. If the pulse is long, then the Fourier components will appear over a narrow frequency range (Fig. 1.24) but if the pulse is narrow, the Fourier components will be spread over a wide range (Fig. 1.25). The time-domain signals and the corresponding frequency-domain partners constitute Fourier pairs. 

Figure 1.26 Free induction decay and corresponding frequency-domain signals after Fourier transformations, (a) Short-duration FIDs result in broader peaks in the frequency domain, (b) Long-duration FIDs yield sharp signals in the frequency domain.

Single-quantum coherence is the type of magnedzadon that induces a voltage in a receiver coil (i.e., Rf signal) when oriented in the xy-plane. This signal is observable, since it can be amplified and Fourier-transformed into a frequency-domain signal. Zero- or multiple-quantum coherences do not obey the normal selection rules and do not... 

Another resolution-enhancement procedure used is convolution difference (Campbell et ai, 1973). This suppresses the ridges from the cross-peaks and weakens the peaks on the diagonal. Alternatively, we can use a shaping function that involves production of pseudoechoes. This makes the envelope of the time-domain signal symmetrical about its midpoint, so the dispersionmode contributions in both halves are equal and opposite in sign (Bax et ai, 1979,1981). Fourier transformation of the pseudoecho produces signals... 

Fx axis The axis of a 2D spectrum resulting from the Fourier transformation of the tx domain signal. 

Frequency spectrum A plot of signal amplitude versus frequency, produced by the Fourier transformation of a time-domain signal. 

It may thus be necessary to calculate the Fourier transform of the measured signal to return to the domain of interpretation, here wavelength or wavenumber. In FTIR the signal is measured in the displacement domain 6 and transformed to the wavelength or wavenumber domain by a Fourier transform. Because the wavelength domain is the Fourier transform of the displacement domain, and vice versa, we say that the spectrum is measured in the Fourier domain. 

Instead of considering a spectrum as a signal which is measured as a function of wavelength, in this chapter we consider it measured as a function of time. The frequency scale in the Fourier domain is given in cycles (v) per second (Hz) or radians (to) per second (s" ). These units are related by 1 Hz = 2k s" . 

Fig. 40.17. Convolution in the time domain offlf) with h t) carried out as a multiplication in the Fourier domain, (a) A triangular signal (w, = 3 data points) and its FT. (b) A triangular slit function h(t) (wi/, = 5 data points) and its FT. (c) Multiplication of the FT of (a) with that of (b). (d) The inverse FT of (c).

These four steps are illustrated in Fig. 40.17 where two triangles (array of 32 data points) are convoluted via the Fourier domain. Because one should multiply Fourier coefficients at corresponding frequencies, the signal and the point-spread function should be digitized with the same time interval. Special precautions are needed to avoid numerical errors, of which the discussion is beyond the scope of this text. However, one should know that when J(t) and h(t) are digitized into sampled arrays of the size A and B respectively, both J(t) and h(t) should be extended with zeros to a size of at least A + 5. If (A -i- B) is not a power of two, more zeros should be appended in order to use the fast Fourier transform. 

Ion detection is carried out using image current detection with subsequent Fourier transform of the time-domain signal in the same way as for the Fourier transform ion cyclotron resonance (FTICR) analyzer (see Section 2.2.6). Because frequency can be measured very precisely, high m/z separation can be attained. Here, the axial frequency is measured, since it is independent to the first order on energy and spatial spread of the ions. Since the orbitrap, contrary to the other mass analyzers described, is a recent invention, not many variations of the instrument exist. Apart from Thermo Fischer Scientific s commercial instrument, there is the earlier setup described in References 245 to 247. 

Stonehouse and Keeler developed an intriguing method for the accurate determination of scalar couplings even in multiplets with partially convoluted peaks (one- or two-dimensional). They recognized that the time domain signal is completely resolved and that convolution of the frequency domain spectrum is a consequence of the Fourier transform of the signal decay. The method requires that the multiplet be centred about zero frequency and this was achieved by the following method ... 

The second spectrum is then subtracted from the first. This may be done either via subtraction of the time domain transients or, alternatively, via subtraction of the Fourier transformed spectra, provided that the absolute intensity of the second is referenced to the first according to the relative magnitudes of the time domain signals. 

The way to ensure a clean extraction of an experimental reference signal is thus to zero-fill the experimental free induction decay s it) once before Fourier transformation, zero completely the imaginary part of the resultant spectmm, and zero all but the reference region to wr of the real part [7], Inverse Fourier transformation then gives a symmetric time-domain signal, the first half of which is the required experimental reference signal Sr t) ... 

Blass (1976a) and Blass and Halsey (1981) discuss data acquisition for a continuous scanning spectrometer in detail. The principal concept is that as a system scans a spectral line at some rate, the resulting time-varying signal will have a distribution of frequency components in the Fourier domain. 

Fig. 2. The behavior of the magnetization vector (i) is shown in response to the application of a single 7i/2 r.f. pulse along V, (ii). The decay of the magnetization vector in the x -y plane yields the received time-domain signal, called the FID, shown in (iii). The result of a Fourier transformation of the FID is the spectrum shown in (iv). For a liquid-like sample, the full-width at half-maximum-height of the spectral signal is l/itV) (Section II.A.2).

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