Frequency-domain data

Fig. 6.4 Graphical display of frequency domain data for a first-order system.

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

Most of the early vibration analysis was carried out using analog equipment, which necessitated the use of time-domain data. The reason for this is that it was difficult to convert time-domain data to frequency-domain data. Therefore, frequency-domain capability was not available until microprocessor-based analyzers incorporated a straightforward method (i.e.. Fast Fourier Transform, FFT) of transforming the time-domain spectmm into its frequency components. 

Time-domain data are presented with amplitude as the vertical axis and elapsed time as the horizontal axis. Time-domain profiles are the sum of all vibration components (i.e., frequencies, impacts, and other transients) that are present in the machine-train and its installed system. Time traces include all frequency components, but the individual components are more difficult to isolate than with frequency-domain data. 

Most predictive-maintenance programs rely almost exclusively on frequency-domain vibration data. The microprocessor-based analyzers gather time-domain data and automatically convert it using Fast Fourier Transform (FFT) to frequency-domain data. A frequency-domain signature shows the machine s individual frequency components, or peaks. 

While frequency-domain data analysis is much easier to learn than time-domain data analysis, it does not provide the ability to isolate and identify all incipient problems within the machine or its installed system. Because of this. 

Repeated twisting of the spindle s tube or the solid shaft used in jackshafts results in a reduction in the flexible drive s stiffness. When this occurs, the drive loses some of its ability to absorb torsional transients. As a result, damage may result to the driven unit. Unfortunately, the limits of single-channel, frequency-domain data acquisition prevents accurate measurement of this failure mode. Most of the abnormal vibration that results from fatigue occurs in the relatively brief time interval associated with startup, when radical speed changes occur, or during shutdown of the machine-train. As a result, this type of data acquisition and analysis cannot adequately capture these... 

Obviously this is a little difficult to interpret, although with experience you can train yourself to extract all the frequencies by eye... (only kidding ) The FID is a time domain display but what we really need is a frequency domain display (with peaks rather than cosines). To bring about this magic, we make use of the work of Jean Baptiste Fourier (1768-1830) who was able to relate time-domain to frequency-domain data. These days, there are superfast algorithms to do this and it all happens in the background. It is worth knowing a little about this relationship as we will see later when we discuss some of the tricks that can be used to extract more information from the spectrum. 

Recently, a method used for the analysis of frequency-domain data has been proposed for the analysis of time-domain images. The AB-plot or phasor plot provides a useful graphical representation of lifetime data that can be used for the segmentation of the images prior to data fitting [47, 48], With this method, data fitting may be avoided in many instances. 

Analysis This pull-down menu is only available for frequency domain data (spectra) and allows a few simple analytical tasks to be performed such as peak picking, calibration, integration or simple spectral analysis. 

Frequency-domain measurements of fluorescence energy transfer are used to determine the end-to-end distance distribution of donor-acceptor D-A) pairs linked by flexible alkyl chains. The length of the linker is varied from 11 to 2B atoms, and two different D-A pairs are used. In each case the D-A distributions are recovered from global analysis of measurements with different values for the FSrster distance, which are obtained by collislonal quenching of the donors. In all cases essentially the same distance distribution Is recovered from the frequency-domain data for each value of tha Ffirster distance. The experimentally recovered distance distributions are compared with those calculated from the RIS model. The experimentally recovered distance distributions for the largest chain molecules are In agreement with the predictions of the RIS model. However, the experimental and RIS distributions are distinct for the shorter D-A pairs. 

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.

TIME and FREQUENCY DOMAIN DATA in MAGNETIC RESONANCE SPECTROSCOPY... 

Fourier transform voltammetry — Analysis of any AC or transient response using (fast) Fourier transformation (FFT) and inverse (fast) Fourier transformation (IFFT) to convert time domain data to the frequency domain data and then (often) back to time domain data but separated into DC and individual frequency components [i-ii]. See also - Fourier transformation, AC voltamme-... 

The signal from staircase voltammograms can be further analyzed by - Fourier transformation and analysis of the frequency domain data [iv]. 

When the frequency-domain data are plotted out, each peak in the spectrum will reach a minimum (flat line) when... 

The 13C spectrum of 2-chlorobutane, first encountered in Figure 7.1, consists of signals at 8 60.4 (CH), 33.3 (CH2), 24.8 (CHj), and 11.0 (CH3). In our experiment, we will first collect the i3C DEPT spectrum (using the pulse sequence in Figure 12.15), with the variable y pulse width set to zero degrees. The process is then repeated 18 more times, with ,. incremented by 10° each time. Finally, the data are subjected to Fourier transformation to give a frequency-domain data set with the F2 axis corresponding to 13C chemical shift and the F axis to y. 

Obviously, the time-domain data acquired by means of the OHD-OKE measurement agrees completely with the frequency-domain data obtained by means of the depolarized LS measurements for a wide frequency range from 0.1 cm to 250 cm . ... 

A working understanding of statistics is essential for the analysis of experiments conducted in the frequency domain, such as impedamce spectroscopy. The objective of this chapter is to provide an overview of concepts and definitions used in statistics at a level sufficient to understand the interpretation of frequency domain data. 

In the absence of instmment-induced correlations, stochastic errors in the frequency-domain are normally distributed. The appearance of a normal distribution of frequency-domain stochastic errors can be regarded to be a consequence of the Central Limit Theorem applied to the methodology used to measure the complex impedance. ° This result validates an essential assumption routinely used during regression analysis of impedance (and other frequency-domain) data. 

While there many examples of multivariate analysis involving NMR time domain data,24-36 Yvi II be assumed in this review that the data matrix of interest is (or is derived from) the frequency domain data /y(otherwise specified. In most situations the NMR spectrum of the /th sample (different mixture, time, etc.) can be described as the linear combination of comp pure component spectrum, each with a concentration... 

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