Time and frequency domains

Before discussing the Fourier transform, we will first look in some more detail at the time and frequency domain. As we will see later on, a FT consists of the decomposition of a signal in a series of sines and cosines. We consider first a signal which varies with time according to a sum of two sine functions (Fig. 40.3). Each sine function is characterized by its amplitude A and its period T, which corresponds to the time required to run through one cycle (2ti radials) of the sine function. In this example the frequencies are 1 and 3 Hz. The frequency of a sine function can be expressed in two ways the radial frequency to (radians per second), which is 

The radial frequency co of a periodic function is positive or negative, depending on the direction of the rotation of the unit vector (see Fig. 40.5). co is positive in the counter-clockwise direction and negative in the clockwise direction. From Fig. 40.5a one can see that the amplitudes (A jp) of a sine at a negative frequency, -co, with an amplitude. A, are opposite to the values of a sine function at a positive frequency, co, i.e. = Asin(-cor) = -Asin(co/) = This is a property of an antisymmetric function. A cosine function is a symmetric function because A -Acos(-co/) = Acos(cor) = A. (Fig. 40.5b). Thus, positive as well as negative 

With increasing interest in time-resolved impedance measurements but also with the demand of parallel measurements, fast methods based on time domain approach move more and more into the focus. Although time and frequency domain are well defined, they are often not clearly presented. Especially, when the impedance spectrum changes with time, a joint analysis in terms of time and frequency dependence is often accompanied by uncertainties in wording. 

The transformation between time and frequency domain requires linear and time invariant systems. Practically, linear refers to the relation between current and voltage within the 

Frequency domain in general means the description of any physical quantity, X, as a function of frequency, w = 2Tcf, as independent variable X = f(w). 

Having time, t, as independent variable, yields time domain X = f(t). 

The physical quantities describing passive electrical behavior of material in time and frequency domain are clearly distinguished. The complex, frequency-dependent impedance (impedance spectrum) exists only in the frequency domain, whereas the impulse answer is the respective property in the time domain. A single relaxation process yields a dispersion region (e.g., 3-dispersion) with a characteristic frequency (e.g., wb) in the frequency domain that corresponds to a relaxation strength and relaxation time (time constant) in the time domain. 

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In the work presented here, a slightly different two-parameter transient model has been used. Instead of specifying a center frequency b and the bandwidth parameter a of the amplitude function A(t) = 6 , a simple band pass signal with lower and upper cut off frequencies and fup was employed. This implicitly defined a center frequency / and amplitude function A t). An example of a transient prototype both in the time and frequency domain is found in Figure 1. 

Figure 1 Example of signal prototype in the time and frequency domains.

Standard procedures that are used for testing of construction materials are based on square pulse actions or their various combinations. For example, small cyclic loads are used for forecast of durability and failure of materials. It is possible to apply analytical description of various types of loads as IN actions in time and frequency domains and use them as analytical deterministic models. Noise N(t) action as a rule is represented by stochastic model. 

While the data are collected in the time domain by scaiming a delay line, they are most easily interpreted in the frequency domain. It is straightforward to coimect the time and frequency domains tln-ough a Fourier transform... 

Johnson A E and Myers ABA 1996 A comparison of time- and frequency-domain resonance Raman spectroscopy in triiodide J. Cham. Phys. 104 2497-507... 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

As said before, there are two main applications of Fourier transforms the enhancement of signals and the restoration of the deterministic part of a signal. Signal enhancement is an operation for the reduction of the noise leading to an improved signal-to-noise ratio. By signal restoration deformations of the signal introduced by imperfections in the measurement device are corrected. These two operations can be executed in both domains, the time and frequency domain. 

Fig. 3.5. Connection between characteristic signal functions in time and frequency domain...

While publications on fluorescence lifetime imaging microscopy (FLIM) have been relatively evenly divided between time and frequency domain methods, a majority of the 10 most highly cited papers using FLIM have taken advantage of the frequency domain method [1, 2-9]. Both techniques have confronted similar challenges as they have developed and, as such, common themes may be found in both approaches to FLIM. One of the most important criteria is to retrieve the maximum information out of a FLIM... 

The development of new oximeters is also in progress, with the application of time- and frequency-domain techniques which are, in principle, capable of discriminating between the absorption and scattering contributions coming from human tissue, thus making possible the detection of tissue oxygenation37 39. 

The Fourier transform (FT) relates the function of time to one of frequency—that is, the time and frequency domains. The output of the NMR spectrometer is a sinusoidal wave that decays with time, varies as a function of time and is therefore in the time domain. Its initial intensity is proportional to Mz and therefore to the number of nuclei giving the signal. Its frequency is a measure of the chemical shift and its rate of decay is related to T2. Fourier transformation of the FID gives a function whose intensity varies as a function of frequency and is therefore in the frequency domain. 

These two experiments are fundamentally different in the nature of the applied deformation. In the case of the relaxation experiment a step strain is applied whereas the modulus is measured by an applied oscillating strain. Thus we are transforming between the time and frequency domains. In fact during the derivation of the storage and loss moduli these transforms have already been defined by Equation (4.53). In complex number form this becomes... 

This comparison between time and frequency domain measurements is performed at submegahertz frequencies in order to avoid the issue of deconvolution of time domain signals. At megahertz frequencies time domain measurements encounter an additional limitation, these signals must be deconvoluted to isolate the sensor response from the instrument response. The need for deconvolutions adds extra software and computation time, which limits the versatility of time domain techniques for real-time applications. No deconvolutions are necessary in the frequency domain as shown below. 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

While in the frequency domain all the spectroscopic information regarding vibrational frequencies and relaxation processes is obtained from the positions and widths of the Raman resonances, in the time domain this information is obtained from coherent oscillations and the decay of the time-dependent CARS signal, respectively. In principle, time- and frequency-domain experiments are related to each other by Fourier transform and carry the same information. However, in contrast to the driven motion of molecular vibrations in frequency-multiplexed CARS detection, time-resolved CARS allows recording the Raman free induction decay (RFID) with the decay time T2, i.e., the free evolution of the molecular system is observed. While the non-resonant contribution dephases instantaneously, the resonant contribution of RFID decays within hundreds of femtoseconds in the condensed phase. Time-resolved CARS with femtosecond excitation, therefore, allows the separation of nonresonant and vibrationally resonant signals [151]. 

An example of a combination of time and frequency-domain masking, using a tone burst, is given in Fig. 1.4. 

Clark et al., 1983] Clark, G., Parker, S., and Mitra, S. (1983). A unified approach to time- and frequency-domain realization of FIR adaptive digital filters. IEEE Trans. Acoust. Speech and Sig. Proc., ASSP-31 1073-1083. 

Figure 1.4 Excitation pattern for a short tone burst. The excitation produced by a short tone burst is smeared out in the time and frequency domain.

Analysis. We can divide analysis algorithms into time and frequency domain processes. Certainly, the division between these categories is arbitrary since we can mix them together to solve an audio problem. However, it suffices for our purposes. 

Continuous functions and signals in the time domain are denoted by lower case letters with the argument in parentheses, e.g. x(t). Sampling at constant intervals A t produces a discrete approximation x[n] to the continuous signal, defined at times f = n A t, n = 0,1,2. Square brackets are used for the arguments of discrete functions. The Fourier transform establishes the connection between the time and frequency domains [76] ... 

In the literature the response theory is largely described in the time-dependent form which requires a somewhat complicated technique of time ordering of the operators and Fourier transformations between time and frequency domains. The static responses which are largely needed in the present book appear as a result of subtle limit procedures for the frequencies flowing to zero. Here we have developed the necessary static results within their own realm. 

The matter of sampling and limited representation of frequencies requires a second look at the representation of data in the time and frequency domains, as well as the transformation between those domains. Specifically, we need to consider the Fourier transformation of bandwidth-limited, finite sequences of data so that the S/N enhancement and signal distortion of physically significant data can be explored. We begin with an evaluation of the effect of sampling, and the sampling theorem, on the range of frequencies at our disposal for some set of time-domain data. 

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