Equation frequency domain

If X, y and h are functions with Fourier transforms X, Y and H (real problem), we can write equation (9) in the frequency domain ... 

This is the description of NMR chemical exchange in the time domain. Note that this equation and equation (B2.4.11)) are Fourier transfomis of each other. The time-domain and frequency-domain pictures are always related in this way. 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

The process of going from the time domain spectrum f t) to the frequency domain spectrum F v) is known as Fourier transformation. In this case the frequency of the line, say too MFtz, in Figure 3.7(b) is simply the value of v which appears in the equation... 

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

Since many closed-loop systems approximate to second-order systems, a few interesting observations can be made. For the case when the frequency domain specification has limited the value of Mp to 3 dB for a second-order system, then from equation (6.72)... 

The radiation pressure noise decreases rapidly with frequency, so we focus our attention on the worst case, the low frequency domain. Because of the seismic noise which is very large at low frequency, it will be very difficult to detect gw below a few hertz, so we put this as a limit. Equation (46) exhibits interesting features ... 

The frequency-domain spectrum is computed by Fourier transformation of the FIDs. Real and imaginary components v(co) and ifi ct>) of the NMR spectrum are obtained as a result. Magnitude-mode or powermode spectra P o)) can be computed from the real and imaginary parts of the spectrum through application of the following equation ... 

Equations (40.3) and (40.4) are called the Fourier transform pair. Equation (40.3) represents the transform from the frequency domain back to the time domain, and eq. (40.4) is the forward transform from the time domain to the frequency domain. A closer look at eqs. (40.3) and (40.4) reveals that the forward and backward Fourier transforms are equivalent, except for the sign in the exponent. The backward transform is a summation because the frequency domain is discrete for finite measurement times. However, for infinite measurement times this summation becomes an integral. 

Indirect covariance processing has been further extended by the work of Martin and co-workers83 84 and by Kupce and Freeman85 to include the reconstruction of non-symmetric spectra from pairs of spectra, F and G, that share a common frequency domain according to Equation (3) ... 

It is important to note that if a mixture of fluorophores with different fluorescence lifetimes is analyzed, the lifetime computed from the phase is not equivalent to the lifetime computed from the modulation. As a result, the two lifetimes are often referred to as apparent lifetimes and should not be confused with the true lifetime of any particular species in the sample. These equations predict a set of phenomena inherent to the frequency domain measurement. 

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... 

The value of fEdetermines all other variables in the equations above. In turn, fE is determined by the temporal resolution of interest of the system studied. To resolve an average excited state lifetime t, the required data sampling rate, in frequency domain techniques is at least an order ofmagnitude slower than it is in the time domain as stated by the following relation (when Np > 32 and Nw= 1) ... 

You might remember from your physics that this is the differential equation that describes a harmonic oscillator. The solution is a sine wave with a frequency of l/ip. We will discuss these kinds of functions in detail in Part V when we begin our Chinese" lessons covering the frequency domain. 

In Chap. 12 we will show that we can convert from the Laplace domain (Russian) into the frequency domain (Chinese) by merely substituting ia for s in the transfer function of the process. This is similar to the direct substitution method, but keep in mind that these two operations are different. In one we use the transfer function. In the other we use the characteristic equation. 

These large systems of equations can be solved fairly easily by going into the frequency domain. The procedure is ... 

A new value of frequency is specified and the calculations repeated. Table 12.3 gives a FORTRAN program that performs alt these calculations, The initial part of the program solves for all the steadystate compositions and flow rates, given feed composition and feed flow rate and the desired bottoms and distillate compositions, by converging on the correct value of vapor boilup Vg. Next the coeflicients for the linearized equations arc calculated. Then the stepping technique is used to calculate the intermediate g s and the final P(j transfer functions in the frequency domain. 

At this point it might be useful to pull together some of the concepts that you have waded through in the last several chapters. We now know how to look at and think about dynamics in three languages time (English), Laplace (Russian) and frequency (Chinese). For example, a third-order, underdamped system would have the time-domain step responses sketched in Fig. 14.10 for two different values of the real TOOt. In the Laplace domain, the system is represented by a transfer function or by plotting the poles of the transfer function (the roots of the system s characteristic equation) in the s plane, as shown in Fig. 14.10. In the frequency domain, the system could be represented by a Bode plot of... 

Equation (18.4) expresses the sequence of impulses that comes out of an impulse sampler in the time domain. Equation (18.5) gives the sequence in the Laplace domain. Substituting io) for s gives the impulse sequence in the frequency domain. 

Sampled-data control systems can be designed in the frequency domain by using the same techniques that we employed for continuous systems. The Nyquist stability criterion is applied to the appropriate closedloop characteristic equation to find the number of zeros outside the unit circle. 

S.G. Johnson and J.D. Joannopoulos, Block-iterative frequency-domain methods for Maxwell s equations in a planewave basis , Optics Express, 8, 173-190 (2001). 

MODELING OF ONE-DIMENSIONAL NONLINEAR PERIODIC STRUCTURES BY DIRECT INTEGRATION OF MAXWELL S EQUATIONS IN THE FREQUENCY DOMAIN... 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

Here I describe a simple numerical method for solving Maxwell s equation in the frequency domain. As the structure to be analysed is onedimensional, Maxwell s equations turn into a system of two coupled ordinary differential equations that can be solved with standard numerical routines. 

N. Wiener s solution was originally derived in the frequency domain for time-invariant systems with stationary statistics. In what follows, a mtrix solution derived from such approach but developed in the time domain for time-varying systems and non-stationary statistics will be presented (22-23). An expression for the required transformation H in Equation 7 will be obtained. In all that follows, we shall denote with the best estimate of l.e. an estimate such that ... 

In these formulae, Gd is the desymmetrized profile, Gc is the classical (symmetric) line profile c and T are angular frequency and temperature. In all cases, the desymmetrized function Go(ft>) obeys Eq. 5.73 exactly. We note that at low frequencies, h(o < kT, the four expressions are practically equivalent. However, at high frequencies the results of these desymmetrizations differ strikingly. One needs only to compare the magnitude of the factors of Gc of the upper three defining equations, for ft) — +oo, to realize enormous differences among these. Hence, the question arises as to which one (if any) of these procedures approximates the exact quantum profile, G(co). We note that in EgelstafFs procedure the desymmetrization is accomplished in the time domain rather than the frequency domain. The classical correlation function, Cci(t), and spectral function, Gci (co), are related by Fourier transform. 

The reduction schemes used by Tang et al. [20] to define the surrogate fewer state system follows the method proposed by Shore [62]. The scheme has a compact form when we introduce two orthogonal projection operators P and Q and work in the frequency domain instead of the time domain. The time evolution matrix for the n-state system dynamics, U(f). and its Fourier transform, G(w), satisfy the following equations ... 

Let P be the projection operator onto the subspace composed of the states having stronger couplings within which we try to approximate the system dynamics, and let Q be the projection operator onto the remaining states. We are interested in evaluating the matrix elements of G(o>) within the subspace of P states in the frequency domain. Multiplying both sides of the equation for G(g>) in Eq. (7.5) by P + Q = 1, we find... 

The frequency response of a system may be more easily determined by effecting the substitution s = iw in the appropriate transfer function [where i = y/(—1)] (i.e. mapping the function from the s domain onto the frequency domain(17)). Hence substituting into equation 7.19 ... 

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