Apodization

In spectroscopy applications, g(t) and h(t) from Eq2.18 have specific meanings, in which g(t) is usually a signal or data stream (e.g., Figure 12A), while h(t) is a response function, typically a curve which falls to zero in both directions from its maximum (e.g., Figure 12B). Normally, the response function is broader than the original signal, and thus, the effect of convolution is to smooth the raw signal g(t) in time according to the formula provided by the response function h(t).

Spectra acquired from an instrument have a definite peak shape which associated with the instrument itself. As is mentioned above, in FT-ICR, the natural peak shape is a convolution of both sinc and Lorentzian functions (Figure 8), and the component from the sinc function contains undesirable sidebands at both sides of a peak, which cannot be avoided during spectral acquisition.83 These wiggles contain no useful information but can interfere with the identification of adjacent peaks of low intensity. Therefore, during data processing, the time-domain transient is often multiplied by a window function prior to FT (Figure 12) to minimize the sideband intensities and smooth the line shape. Such procedure is called “apodization”, meaning "removing the feet", which is also an application of convolution theorem.71 An optimal window function can smooth the sidebands but also inevitably degrades the S/N and resolving power of the spectrum.102

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Figure 2.12 Transient, window function and its corresponding peak shape by simulation. A: original transient of a crude oil sample; B to E: full hanning window, apodized transient, and its corresponding peak shape in the absorption and magnitude-mode; F to I: half hanning window, apodized transient, and its peak shape in the absorption and magnitude-mode.Reprinted from Qi et al.,89 with permission from American Chemical Society, copyright 2012.

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The commonly used window functions in FT-ICR are listed in Table 2.1.71 Theoretical treatments have been applied to FT-ICR transients in order to find the most advantageous window function.103-104 Unfortunately, most window functions produce similar results in a given set of conditions, and the effect of apodization is largely dependent on the ratio of T/τ; however, it has been found

that the magnitude- and absorption-mode spectra do better with different types of apodization.105

Name Full (Magnitude-mode) Half (Absorption-mode)

Rectangular 1 1

Cosine-bell Cos(πn/N-π/2) Cos(πn/2N)

Hamming 0.54-0.46cos(2πn/N) 0.54+0.46cos(πn/N)

Hann 0.5-0.5cos(2πn/N) 0.5+0.5cos(πn/N)

Gaussian exp(-1/2((n-N/2)/kN/2)2) exp(-1/2((n/2)/kN/2)2) Blackman 0.42-0.5cos(2πn /N)+0.08cos(4πn/N) 0.42+0.5cos(πn/N)+0.08cos(2πn/N)

Triangle 1-|1-2n/N| 1-n/N

Table 2.1 Most commonly used apodization functions in FT-ICR, with the mathematical expressions for both full and half window. N is the total number of data points and n is the index of each data point 0≤n≤N.

The magnitude-mode plot is the standard spectrum produced by all commercial FT-ICR instruments (absorption-mode will be discussed in next section). A transient is a signal of the image current damping in the time-domain. By multiplying the transient with a window function as in Figure 2.12, the overall peak shape changes, which therefore affects the resolution, relative intensity, and S/N of the peaks after FT. In Figure 2.13, a crude oil spectrum was used for demonstration; the Fourier transformed spectra (plotted in same vertical scale)

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show varying spectral intensity after apodization, because the window function changes the signal intensity at both the beginning and the end of the transient. By expanding the spectrum in the low m/z region (Figure 2.13J-L), it is clear to see that, after apodization, the sidebands of the peak are largely suppressed and the line shape becomes much smoother, while at the same time, the peak width is broadened. Such an effect is important for complex spectrum (e.g., from proteomics or petroleomics), where peak intensities vary over 1000x throughout these spectra. By using apodization, the low intensity peaks will suffer much less perturbation from adjacent intense peaks. When the peaks become dense in the high m/z region, apodization (especially a full window) suppresses the signal at the beginning of the transient, broadens the peak width, and consequently makes the doublet and triplet peaks unresolved (Figure 2.13M-O).

Apodization is a standard step during FT-ICR data processing, because it smooths the peak shape and facilitates the assignment. However, the benefit varies with the spectral conditions and also varies for different peaks in the same spectrum. Furthermore, the improved peak shape is generated at the cost of spectral resolution, and problems like frequency shifts, from image or space charge will often be “hidden”, as the peak shape is “smoothed out”. Thus, it is advisable to keep the raw spectrum and use apodization carefully.

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Figure 2.13 (A-C): Transient (left) and m/z spectra in magnitude- (middle) and absorption-mode (right) without apodization. (D-F): Transient and spectra with a half Hanning window apodization. (G-I): Transient and spectra with a full Hanning window apodization. (J-O): Close-up of two narrow m/z windows in both magnitude (black) and absorption-mode (grey). (J-L): m/z

414.308−414.335 with no apodization, half Hanning, and full Hanning apodization. (M-O): m/z 751.55−751.75 with no apodization, half Hanning, and

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full Hanning apodization. Reprinted from Qi, et al.,105 with permission from Springer, copyright 2013.