Fourier transform response theory

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

Using linear response theory and noting (according to the results at the end of Section 5.1.3) that the (complex) electrical conductivity a is the Fourier transform of the current density autocorrelation function, we obtain from Eqn. (5.75) (see the equivalent Eqn. (5.21))... 

The 8 part in (2.53) is responsible for elastic scattering, whereas the second term, which is proportional to the Fourier transform of C(f), leads to the gain and loss spectral lines. When the system undergoes undamped oscillations with frequency A0, this leads to two delta peaks in the structure factor, placed at spectral line. The spectral theory clearly requires knowing an object different from (o-2(/)), the correlation function [Dattaggupta et al., 1989]. 

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. 

However, from the point of view of linear response theory, the definitions (174) or (178) suffer from several drawbacks. Actually, the function X ( , tw) as defined by Eq. (174) is not the Fourier transform of the function X (, x), but a partial Fourier transform computed in the restricted time interval 0 < x < tw. As a consequence, it does not possess the same analyticity properties as the generalized susceptibility x( ) defined by Eq. (179). While the latter, extended to complex values of co, is analytic in the upper complex half-plane (Smoo > 0), the function Xi ( - tw) is analytic in the whole complex plane. As a very simple example, consider the exponentially decreasing response function... 

Persson and co-workers [265 267] consider a rough, rigid surface with a height prohle h x ). where x is a two-dimensional vector in the x-y plane. In reaction to /z(x) and its externally imposed motion, the rubber will experience a (time-dependent) normal deformation 8z(x, f). If one assumes the rubber to be an elastic medium, then it is possible to relate 52(q, ), which is the Fourier transform (F.T.) of 8z(x, f), to the F.T. of the stress a(q, ). Within linear-response theory, one can express this in the rubber-hxed frame (indicated by a prime) via... 

Equation (4.2.11) describes the response to three delta pulses separated by ti =oi — 02 >0, t2 = 02 — 03 > 0, and t3 = 03 > 0. Writing the multi-pulse response as a function of the pulse separations is the custom in multi-dimensional Fourier NMR [Eml ]. Figure 4.2.3 illustrates the two time conventions used for the nonlinear impulse response and in multi-dimensional NMR spectroscopy for n = 3. Fourier transformation of 3 over the pulse separations r, produces the multi-dimensional correlation spectra of pulsed Fourier NMR. Foinier transformation over the time delays <7, produces the nonlinear transfer junctions known from system theory or the nonlinear susceptibilities of optical spectroscopy. The nonlinear susceptibilities and the multi-dimensional impulse-response functions can also be measured with multi-resonance CW excitation, and with stochastic excitation piul]. 

The selectivity of the excitation is characterized by the bandwidth of the magnetization response. The response spectrum is determined by the Fourier transform of the selective pulse only in first order. Generally, the NMR response is nonlinear, and nonlinear system theory can be applied for its analysis (cf. Section 4.2.2). A model suitable for describing the NMR response in many situations applicable to NMR imaging is given by the Bloch equations (cf. Section 2.2.1). They are often relied upon when designing and analysing selective excitation (Frel). 

A time domain function can be expressed as a Fourier series, an infinite series of sines and cosines. However in practise integrals related to the FOURIER series, rather than the series themselves are used to perform the Fourier transformation. Linear response theory shows that in addition to NMR time domain data and frequency domain data, pulse shape and its associated excitation profile are also a FOURIER pair. Although a more detailed study [3.5] has indicated that this is only a first order approximation, this approach can form the basis of an introductory discussion. 

At this point we emphasize that Eqs. (16H33) should not be understood as a formally rigorous derivation (such derivations by various techniques can be found in the literature [114, 116, 117]) but rather the present treatment (which follows Ref. 78) is a plausibility argument. In this spirit, one can also extend the theory to the interacting case, within the framework of mean field theory the inverse collective response function of the interacting system within RPA is always found from that of the noninteracting system by subtracting the Fourier transform of the interaction [118]. In our case we have a collective structure factor... 

We will now investigate the relationship between frequency domain and time domain responses. In most cases considered in this section a transient electromagnetic field is excited by a step function current in the source. Moreover the theory of the transient induction logging described in this monograph will be developed for this type of excitation. For this reason the relationship between frequency response and transient response corresponding to this single type of excitation will be our principal concern. The information we need is obtained through use of the Fourier transform which takes the well... 

The general theory of time-dependent response functions has been described in many publications.2,18,19 The response is non-local in time and the Fourier transforms of the general time-dependent functions lead to the definitions of the frequency-dependent response functions which are the quantities most easily related to experimental measurements and potential applications. The notation... 

Having obtained experimental data in the form of an instrumental response as a function of time (counts N i) or, as we will describe, asymmetry -4(t)), the experimenter seeks to extract information on physical properties and processes in the sample by analyzing the data. When there are clear oscillations in the data, from either an applied transverse field or from well-defined internal fields in well-ordered magnetic states, it is often useftil to transform the data into frequency space by Fast Fourier Transform (FFT, see e.g.. Press et al. 1986), or some alternate transform algorithm (see Alves et al. 1994, Rainford and Daniell 1994). This is particularly useful if a theory that is being tested predicts response as a function of frequency. In xSR of magnetic systems, however, many of the data... 

Let us first consider spectroscopy. Linear-response theory, in particular the fluctuation dissipation theorem - which relates the absorption of an incident monochromatic field to the correlation function of (e.g. dipole) fluctuations in equilibrium - has changed our perspective on spectroscopy of dense media. It has moved away from a static Schrodinger picture -phrased in terms of transitions between immutable (but usually incomputable) quantum levels - to a dynamic Heisenberg picture, in which the spectral line shape is related by Fourier transform to a correlation function that describes the decay of fluctuations. Of course, any property that cannot be computed in the Schrodinger picture, cannot be computed in the Heisenberg picture either however, correlation functions, unlike wave-functions, have a clear meaning in the classical limit. This makes it much easier to come up with simple (semi) classical interpretations and approximations. 

At T 0 the sharp lines corresponding to the harmonic modes are broadened by anharmonic effects until, at high temperature, the simple relationship between vibrational density of states and dynamical matrix is lost. In this regime, and especially for large aggregates, MD is the most suitable tool to compute the vibrational spectrum. Standard linear response theory within classical statistical mechanics shows that the spectrum f(co) is given by the Fourier transform of the velocity-velocity autocorrelation function... 

Pulse compression requires a separate matched filter be supported for each waveform used by the radar system. In practice, this filter is implemented via fast convolution as illustrated in Fig. 17.12. From Fourier transform theory, frequency-domain multiplication is equivalent to time-domain convolution. Hence, receiver digitized output data is input to a FFT, multiplied by the Fourier transform of the matched filter response, and then passed through an inverse FFT (I FFT) to output time-domain data. The matched filter reference functions transforms are generally computed off-line and stored in memory to support real-time processing. Fast convolution significantly reduces the number of operations required compared to time-domain direct convolution of returns and the appropriate matched filter function. 

In the dielectric screening method the electron density response due to the motion of the ions around their equilibrium positions is calculated in first order perturbation theory. The potential energy of the crystal for an arbitrary configuration of the ions is expanded to second order in the ionic displacements from equilibrium. The expansion coefficients of the second order term form a matrix. The Fourier transform of this matrix is the dynamical matrix whose eigenvalues yield the phonon frequencies. The dynamical matrix has an ionic and electronic part. The electronic part can be expressed in terms of the electron density response matrix and of the ionic potential. This method has the advantage over the total energy difference m ethod that the phonon frequencies for any arbitrary wave vector can be calculated without additional difficulties. Furthermore in this method the acoustic sum rule is automatically satisfied as a consequence of the way the dynamical matrix is derived. However the dielectric screening method is limited to harmonic phonons. 

The phenomenological theory of the dielectric relaxation behaviour of linear systems is well-established [1-5]. The fundamental relationship joining the frequency-dependent complex permittivity c(cu) measured at frequency / = (ofln and the transient step-response function t) is the Fourier transform relationship... 

The tensile strength and stress of the IPMC are measured in the same manner as those of the IP. The bending stiffness of a fully hydrated IPMC sample is estimated using the free oscillation attenuation method. By bending the sample to the appropriate initial displacement, the free vibration response can be recorded. The natural frequency of the cantilever, is obtained from the fast Fourier transform of the free vibration response curve. The stiffness of the IPMC, Egg, is determined using Eq. 4, which is based on the thin cantilever beam theory of material mechanics ... 

---