Time-frequency plot

Figures 5, 6 and 7 show a selection of 2D frequency-time plots for various temperatures in D-RADP-20, D-RADP-25 and D-RADP-30, respectively.

Figure 12.5 (a) Frequency-time plots obtained by QCM at gold electrodes in 9 1 TEOS/APTES (molar ratio) solutions following potential steps from 0.0 V (versus Ag/AgCI) to... 

Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

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... 

As shown previously, vibrations can be displayed graphically as plots which are referred to as vibration profiles or signatures. These plots are based on measurable parameters (i.e., frequency and amplitude). Note that the terms profile and signature are sometimes used interchangeably by industry. In this chapter, however, profile is used to refer either to time-domain (also may be called time trace or waveform) or frequency-domain plots. The term signature refers to a frequency-domain plot. 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

We can use the LTI Viewer to do all the plots, not only step and impulse responses, but also more general time response and frequency response plots in later chapters. If we know how to execute individual plot statements, it is arguable whether we really need the LTI Viewer. Nonetheless, that would be your personal choice. We will provide here the basic idea and some simple instructions. 

The whole idea of handling dead time applies to other types of frequency-domain plots, but the Bode plot is the easiest to learn from. 

Keep in mind that we are talking about closedloop stability and that we are studying it by making frequency-response plots of the total openlaop system transfer function. We are also considering openlocp stable systems most of the time. We will show how to deal with openloop unstable processes in Sec. 13.4. 

From this plot, we see that as RF increases the upper -3 dB frequency goes down, or as the gain increases the upper -3 dB frequency goes down. Another way to state this is that the upper -3 dB frequency times the circuit gain is approximately constant. We shall show this in the next exercise. 

We would like the top plot to display the Fourier components and the bottom plot to display the waveform versus time. For a Fourier plot the x-axis is frequency. For a time plot the x-axis is time. Presently both plots use the same x-axis. To allow the plots to have different x-axes, select Plot and then Unsynchronize X Axis ... 

Multiple-point fluorescent deteclion has been proposed to enhance detection sensitivity. This method is based on the use of a detector function, such as the Shah function. The time-domain signals were first detected, and they were converted into a frequency-domain plot by Fourier transformation. Therefore, this technique was dubbed Shah convolution Fourier transform detection (SCOFT). As a comparison, the single-detection point time-domain response is commonly known as the electropherogram [698,699,701]. 

In case of resonance the maximum pressure pulsations come up to 7 bar or 4,8 % and exceed the specified values. For the nominal speed of 90 rpm a time plot of pressure is shown in Fig. 8a). The spectrum of pressure pulsations at the inlet of CFM 1 shows no appreciable pressure amplitudes at frequencies above 50 Hz (Fig. 8b). 

If a time plot shows a shift, trend, or cycle, then in addition to examining the process more closely, we should investigate the sampling frequency and sampling techniques. The importance of the effects of sampling in such cases is often overlooked. Several examples follow. In each case, the process itself could be causing the observed pattern in the time plot, but we show how sampling can also be the culprit. 

Figure 4.4 Time plot with increased sampling frequency.

Fig. 4. Examples of experimental traces (frequency change vs. time plot) for five sequential measurements of serum samples. The marked part is zoomed in the inset figure where individual steps of the assay cycle are shown (2 min baseline, 10 min interaction with sample, 5 min buffer zone, 4 min regeneration, and 2 min baseline). PZ sensor with immobilized anti osteoprotegerin Ab, osteoprotegerin in serum was analyzed, for procedure see (10).

For each individual curve, calculate kobs and f using the nonlinear curve fitting of the frequency vs. time plot to Eq. 3 (see Note 6). 

This effect can also be seen in Fig. 8, in which the resulting bed pulse frequency is plotted as a function of the adjusted liquid feed frequency for three different base flow times. 

The primary information in dynamic controllability analysis is similar to the steady-state analysis, namely the gains for manipulated variables and disturbances on the controlled variables, this time plotted against frequency. This can be obtained from a state-space description (matrices A, B, C, D) or by identification. 

The turnover frequency (TOP) is defined as the number of turnovers per enzyme molecule per second at the start of reaction. In order to calculate this number, an active protein content of 10 wt.% of the immobilized preparation is assumed. The TOP is calculated using the formule TOP = ki (initial substrate concentration) / (total enzyme concentration). The initial rate constant (ki) is the slope of the ln(l-conversion) versus time plot. 

RGURE 7<41 (a) Time-domain plot of two siighlly different frequencies of the same amplitude i and I v (b) Time-domain plot of the sum of the Iwo waveforms In (a), (c) Frequency-domain plot of 1 . (d) Frequency-domain plot of v-,. (e) Frequency-domain plot of the waveform in b. ... 

Figure 9. Plot of dispersion vs. (frequency times absorption), in which frequency is measured in units of (1/t), where t is the relaxation time of the Lorentzian line shape of Equations 1. The straight diagonal line is for a single Lorentzian line. The narrow and wide loops correspond to a spectrum consisting of the sum of two equally intense Lorentzians of the same width, separated by 0.6/t and 1.0/t, respectively. The frequency of the center of the absorption peak is taken as zero (i.e., wq = " in Equations 1). [Taken from ref. 12.]...

Electrochemical impedance spectroscopy is extensively employed for the investigation of SAMs because the broad range of frequencies covered by this technique (usually from 10 to 10 Hz) may allow processes with different relaxation times taking place within the electrified interphase to be detected and sorted out. Unfortunately, the various relaxation times often differ by less than 2 orders of magnitude, thus requiring a certain amount of arbitrariness and of physical intuition for their separation. In fact, it is well known that the same impedance spectrum can often be equally well fitted to different equivalent circuits, which are consequently ascribed to different relaxation processes. Impedance spectra are frequently reported on a Y /co versus Y"/co plot, where Y and Y" are the in-phase and quadrature components of the electrochemical admittance and co is the angular frequency. This plot is particularly suitable for representing a series RC network. Thus, a series connection of R and C yields... 

---