Excitation waveforms, frequency-domain

In the frequency domain, any periodic excitation, r.(t), can be described by a sum of sinusoidally modulated light waveforms at harmonics of the fundamental frequency of the excitation... 

Fig. 4.52. Time-domain (left) and frequency-domain (right) excitation waveforms (a), (b) rectangular pulses (c) chirp excitation (d), (e) SWIFT excitations, with (e) formed to eject a certain mass range from the cell. Reproduced from Ref. [201] by permission. John Wiley Sons, 1998.

Sinusoidal voltammetry (SV) is an EC detection technique that is very similar to fast-scan cyclic voltammetry, differing only in the use of a large-amplitude sine wave as the excitation waveform and analysis performed in the frequency domain. Selectivity is then improved by using not only the applied potential window but also the frequency spectrum generated [28]. Brazill s group has performed a comparison between both constant potential amperometry and sinusoidal voltammetry [98]. 

FT/ICR experiments have conventionally been carried out with pulsed or frequency-sweep excitation. Because the cyclotron experiment connects mass to frequency, one can construct ("tailor") any desired frequency-domain excitation pattern by computing its inverse Fourier transform for use as a time-domain waveform. Even better results are obtained when phase-modulation and time-domain apodization are used. Applications include dynamic range extension via multiple-ion ejection, high-resolution MS/MS, multiple-ion simultaneous monitoring, and flatter excitation power (for isotope-ratio measurements). 

Figure 2. Time-domain excitation waveforms (left) and corresponding frequency-domain magnitude-mode spectra (right) of four excitation waveforms used in FT/ICR. A time-domain rectangular rf pulse gives a "sine" excitation spectrum in the frequency-domain. A time-domain frequency-sweep gives a complex profile described by Fresnel integrals. Single-scan time-domain noise gives noise in the frequency-domain. Finally, Stored Waveform Inverse Fourier Transform (SWIFT) excitation can provide an optimally flat excitation spectrum (see Figure 3 for details).

The best approach, adapted from an earlier proposal by Tomlinson and Hill (19), is to specify the desired frequency-domain excitation profile in advance, and then syntheize its corresponding time-domain representation directly via inverse Fourier transformation. The result of the Tomlinson and Hill procedure is shown at the bottom of Figure 2, in which a perfectly flat, perfectly selective frequency-domain excitation is produced by the time-domain waveform obtained via inverse Fourier transformation of the desired spectrum. 

The introduction of the SWIFT technique (10,14,21,22) makes possible FT/ICR frequency-domain excitation with the same mass resolution as has already been demonstrated for FT/ICR detect ion, provided only that sufficient computer memory is available to store a sufficiently long time-domain waveform. When ejection must be performed with ultrahigh mass resolution over a wide mass range, a simple solution is to use two successive SWIFT waveforms first, a broad-band low-resolution excitation designed to eject ions except over (say) a 1 amu mass range and then a second SWIFT waveform, heterodyned to put 2 8K data points spanning a mass range of 1-2 amu. 

Stored waveform inverse Fourier transform (SWIFT) pulses [17] have been applied as a means of broadband ejection of matrix ions generated by Cs+ desorption [18]. These pulses are generated by taking the inverse Fourier transform of the desired frequency domain spectrum and applying the stored time domain waveform to the endcap electrodes via an arbitrary waveform generator. The magnitude of the SWIFT pulse determines the degree of excitation for ions of specific secular frequencies. 

For that purpose, the quartz crystal is simultaneously excited at two frequencies. The response at the lower frequency is processed by a feedback loop dedicated to measure and automatically compensate Cq. The response at the higher frequency is processed by a phase-locked loop that continuously maintains and tracks oscillations at/s. The voltage waveform V hl is the sum of the two sinusoidal signals V h. with frequency/h generated by the voltage controlled oscillator VCO, and Vl, with frequency/, lower than/n, generated by the auxihary oscillator OSC. The frequency/r of the signal Vu is taken as the output frequency/out of the whole circuit. In the frequency domain, the ex-... 

Figure 10.8.3 Procedure for generating a complex excitation waveform, b) Shows the chosen amplitudes for the various frequencies and a) shows randomized phase angles. In (c) there is a complex plane representation of the arrays in (a) and (b), (d) The time domain representation, which is subjected to digital/analog conversion to produce (e) and in turn, low-pass filtering yields (/). Only a small part of the waveform period is shown in (e) and (/). [From S. C. Creason et al., 7. Electroanal. Chem., 47, 9 (1973), with permission.]...

Many applications of this strategy are based on extensions of the concepts of impedance developed earlier in this chapter (41-43). However, the excitation waveform is usually an impulse in potential (rather than a periodic perturbation), and a transient current is measured. One records both E t) and i t) as observed functions. Then both are subjected to transformations, and comparisons are made in the frequency domain between E s) and i s). Ratios of the form i s)IE s) are transient impedances, which can be interpreted in terms of equivalent circuits in exactly the fashion we have come to understand. The advantages of this approach are (a) that the analysis of data is often simpler in the frequency domain, (b) that the multiplex advantage applies, and (c) the waveform E(f) does not have to be ideal or even precisely predictable. The last point is especially useful in high-frequency regions, where potentiostat response is far from perfect. Laplace domain analyses have been carried out for frequency components above 10 MHz. 

Tvvo vidcl used approaches are used for lifetime measurcnienis. ilie lime-domain approach and the frt i/iu niy-domain approach. In tinte-domain measurements. a pulsed source is employed and the time-depcndcnr decay of fluorescence is measured. In the frequency-domain method, a sinusoidallv modulated source is used to excite the sample. The phase shift and demodulation of the fluorescence emission relative lo the excitation waveform provide the lifetime information. ( onimercial instrumentation is available to implement both techniques. ... 

If several different masses are present, then one must apply an excitation pulse that contains components at all of the cyclotron frequencies. This is done by using a rapid frequency sweep ( chirp ), an impulse excitation, or a tailored waveform. The image currents induced in the receiver plates will contain frequency components from all of the mass-to-charge ratios. The various frequencies and their relative abundances can be extracted mathematically by using a Fourier transform which converts a time-domain signal (the image currents) to a frequency-domain spectrum (the mass spectrum). 

Lifetimes. The theory of frequency-domain lifetime determinations has been described in detail elsewhere (15-19). Briefly, a high-frequency (MHz - GHz) sinusoidally-modulated light source is used to excite the fluorescent sample. The time-dependent mathematical representation of the excitation waveform (Ex(t)) is given by ... 

The accurate determination of the phase change between the excitation and emission waveforms is the central role in the frequency-domain measurements of luminescence. In the case of prompt fluorophores, the method of choice seems to be heterodyne detection - also called cross-correlation - and subsequent lock-in amplification. In the heterodyne detection, the gain of the photomultiplier is modulated by the frequency m Am where m is the frequency of the excitation modulation. As the consequence the PM signal contains a low-frequency... 

Stored waveform inverse Fourier transformation, technique to create excitation waveforms for ions in FT-ICR mass spectrometer or Paul ion traps. An excitation waveform in the time-domain is generated by taking the inverse Fourier transform of an appropriate frequency domain programmed excitation spectrum, in which the resonance frequencies of ions to be excited are included. This procedure may be used for selection of precursor ions in MS/MS experiments. 

The SOS-TWF approach is somewhat similar to the SWIFT-TWF, introduced by Marshall. " Both create a defined time-domain waveform consisting of many frequencies. The SWIFT-TWF, however, is calculated by taking the inverse Fourier transform of a specified excitation or ejection frequency-domain spectrum. This synthesized time-domain spectrum can be applied to the end caps of the QIT for dipolar excitation similar to the SOS TWF discussed above. Guan and Marshall as well as Julian et al. have discussed the use of SWIFT-TWF in the QIT in detail. 

Electrochemical systems can also show a nonlinear response—that is, the current response of an electrochemical system can be composed of a response at the excitation frequency and responses at harmonics of the excitation frequency. Therefore, the frequencies superimposed for a multisine experiment have to be chosen very carefully. To prevent faulty results, the frequencies chosen are usually odd harmonics of the lowest frequency in order to eliminate the second harmonic components, which may be caused by a nonlinear response of the system. Nonlinear behavior of the system would cause additional frequencies to appear in the response signal. The response at these additional hequencies would appear in the time-domain signal at places not occupied by the frequencies of the excitation waveform and at places already occupied by the excitation waveform. Hence, not only nonstationary effects but also nonlinearity of the response appear in the FFT spectra as additional scatter. 

Figure 4. Comparison of a theoretical magnitude-mode excitation spectrum (top) with those detected (on one pair of cell transverse plates) during transmission (on the other pair of cell transverse plates) of a frequency-sweep (middle) or SWIFT (bottom) waveform. The time-domain signals were zero-filled once before Fourier transformation to reveal the full shape of the excitation magnitude spectrum. Note the much improved uniformity and selectivity for SWIFT compared to frequency-sweep excitation.

Their selection should be carefully conducted, since they function as pure weighting factors and therefore they can strongly subdue dispersion and dissipation errors. Also, correction functions df co, At) and LA() of (3.71) and (3.74) are selected to lessen grid discrepancies and certify the proper transition from the continuous physical space to the discretized domain. In fact, their arguments have a substantial contribution in the method s wideband profile and hence involve an in-depth examination. More specifically, by considering the excitation frequency content and duration, they subdue oscillatory or spurious modes that corrupt the final waveform envelope. Probing kf, LA() analysis indicates that its argument should opt... 

The high permeability of the carbonate aquifer (easy flow) provides the right conditions for part of the P-wave energy to compress the fluid and excite slow waves. This phenomenon can also be seen in the time domain. A pressure source having a peak source frequency of 300 Hz excites fast and slow P waves that are recorded by a pressure detector at a horizontal distance of 100 m. The waveforms are calculated for a = 1, 1.5, 2, and 3 cm vugs, in an... 

Fig.l. Composition of an ac excitation signal. (A) Odd harmonics of the base frequency are chosen. All components have the same amplitude but the phase shifts are randomized. (B) An inverse Fourier transformation gives the corresponding time domain waveform. 

Deconvolution of response to frequency-sweep excitation, (a) Cosine Fourier transform of linearly increasing frequency sweep time-domain waveform, (b) Magnitude spectrum of excitation, (c) Cosine Fourier transform of time-domain response to excitation, (d) Magnitude spectrum of response, (e) = (c)/(a). (f) Magnitude spectrum... 

One method for temporal precnrsor isolation is stored waveform inverse Fourier transform (SWIFT) [40]. In this method, the desired freqnency domain profile (all frequencies except that of the ion of interest) is inversely Fonrier transformed to a time domain waveform. This waveform is then applied to the excite electrodes in the ICR cell and, thns, the precursor ions are isolated in the cell. An alternative techniqne for in-cell isolation is correlated sweep excitation (COSE) [41], also known as correlated harmonic excitation fields (CHEF) [42]. This method involves application of radiofrequency pulses to the excite electrodes. The technique correlates the duration and frequency of the RF-pulses with those appropriate to the ions to be isolated. Both SWIFT and COSE are capable of isolating single isotopomers in peptide and protein ions [43-45]. 

FTICR-MS instruments operate on the principle of ion cyclotron resonance. As ions have resonant frequencies, these frequencies can be used to isolate the ions prior to further fragmentation or manipulation. For example, a resonant frequency pulse on the excite plates (E+/— in Figure 2.8b) will eject the ions at, or near, that frequency. Furthermore, frequency sweeps - carefully defined to not excite the ion of interest - can be used to eject unwanted ions. However, the most elegant method for ion isolation is that of Stored Waveform Inverse Fourier Transform (SWIFT) [86] in which an ion-exdtation pattern of interest is chosen, inverse Fourier-transformed, and the resulting time domain signal stored in memory. This stored signal is then clocked-out, amplified, and sent to the excite plates when needed. The typical isolation waveform in SWIFT uses a simple excitation box with a notch at the frequencies of the ion of interest, a few kHz. 

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