Fourier complex exponential

The Fourier transform H(f) of the impulse response h(t) is called the system function. The system function relates the Fourier transforms of the input and output time functions by means of the extremely simple Eq. (3-298), which states that the action of the filter is to modify that part of the input consisting of a complex exponential at frequency / by multiplying its amplitude (magnitude) by i7(/)j and adding arg [ (/)] to its phase angle (argument). 

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

In the case of the reciprocal sum, two methods have been implemented, smooth particle mesh Ewald (SPME) [65] and fast Fourier Poisson (FFP) [66], SPME is based on the realization that the complex exponential in the structure factors can be approximated by a well behaved function with continuous derivatives. For example, in the case of Hermite charge distributions, the structure factor can be approximated by... 

Let us consider the Fourier representation of f(t) in terms of the complex exponentials introduced above. For an arbitrary periodic function f(t), we can write... 

The term Fourier transform usually refers to the continuous integration of any square-integrable function to re-express the function as a sum of complex exponentials. Due to the different types of functions to be transformed, many variations of this transform exist. Accordingly, Fourier transforms have scientific applications in many areas, including physics, chemical analysis, signal processing, and statistics. The continuous-time Fourier transforms are defined as follows [1-3] ... 

By Fourier transformation, a signal is decomposed into its sine and cosine components [Angl]. In this way, it is analysed in terms of the amplitude and the phase of harmonic waves. Sine and cosine functions are conveniently combined to form a complex exponential, coscot 4- i sinwt = exp icomplex amplitudes of these exponentials constitute the spectrum F((o) of the signal f(t), where co = In IT is the frequency in units of 2n of an oscillation with time period T. The Fourier transformation and its inverse are defined as... 

Fourier analysis of descriptors is performed by the DFT, which is the sum of the descriptor g r) over all distances r multiplied by a complex exponential. With a descriptor consisting of n components expressed in its discrete form g[x] x is the index of a discrete component), the DFT can be written as... 

The complex exponential can be broken down into real and imaginary sinusoidal components. The results of the transform are the Fourier coefficients g[u] (or g[u,v]) in frequency space. Multiplying the coefficients with a sinusoid of frequency yields the constituent sinusoidal components of the original descriptor. 

The Laplace transform is similar to a one-sided Fourier transform, except that it has a real exponential instead of the complex exponential of the Fourier transform. If we consider complex values of the variables, the two transforms become different versions of the same transform, and their properties are related. The integral that is carried out to invert the Laplace transform is carried out in the complex plane, and we do not discuss it. Fortunately, it is often possible to apply Laplace transforms without carrying out such an integral. We will discuss the use of Laplace transforms in solving differential equations in Chapter 8. 

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

Wavelets are a set of basis functions that are alternatives to the complex exponential functions of Fourier transforms which appear naturally in the momentum-space representation of quantum mechanics. Pure Fourier transforms suffer from the infinite scale applicable to sine and cosine functions. A desirable transform would allow for localization (within the bounds of the Heisenberg Uncertainty Principle). A common way to localize is to left-multiply the complex exponential function with a translatable Gaussian window , in order to obtain a better transform. However, it is not suitable when <1) varies rapidly. Therefore, an even better way is to multiply with a normalized translatable and dilatable window, v /yj,(x) = a vl/([x - b]/a), called the analysing function, where b is related to position and 1/a is related to the complex momentum. vl/(x) is the continuous wavelet mother function. The transform itself is now... 

What FT does is to project the signal f t) onto the set of sine and cosine basis functions of infinite duration represented by the complex exponential function (Rioul and Vetterli 1991). The transformation (named analysis) is reversible and the recovering of the original function (named synthesis) is done by summing up all the Fourier components multiplied by their corresponding basis function, that is. 

Recall that sines and cosines can be expressed in terms of complex exponential functions, according to Eqs. (4.51) and (4.52). Accordingly, a Fourier series can be expressed in a more compact form ... 

Converting the cosine/sine form to the complex exponential form allows many manipulations that would be very difficult otherwise (for an example, see Section 2 of Appendix A Convolution and DFT Properties). But, if you re totally uncomfortable with complex numbers and Euler s Identity (or with the identities of IS"" century mathematicians in general), then you can write the DFT in real number terms as a form of the Fourier Series ... 

Using the complex exponential form, we can write a more general form of the Fourier synthesis equation ... 

The interpretation is the same as for the Fourier transform defined in section 10.1.6 For each required frequency value, we multiply the signal by a complex exponential waveform of that frequency, and sum the result over the time period. The result is a complex number which describes the magnitude and phase at that frequency. The process is repeated until the magnitude and phase is found for every fi-equency. 

The quantities Aik) and Bik) are interpreted as the amplitudes of the sine and cosine contributions in the range of angular spatial frequency between k and k + dk, and are referred to as the Fourier cosine and sine transforms. If we consolidate the sine and cosine transforms into a single complex exponential expression, we arrive at the complex form of the Fourier integral. This is the integral in Eq. (26.32), known as the Fourier transform, which for the one-dimensional function fix) is... 

Characteristic function. The characteristic function of continuous distributions is the expected value of the complex exponential function e" (where i = — 1). In other words, it is the Fourier transform of the density function / ... 

Equation 1.12 is a Fourier transform of the function g to the function p, where is a function of the Cartesian coordinates x, y, and z, and time t. The function g may be obtained in a corresponding Fourier transformation of T T consists of an unimportant complex exponential and a contour function that defines the wave packet, moving with the velocity and acceleration of the particle. 

The discrete-time Fourier series (DTPS) for discrete-time waveforms x n) of period N can also be given in three forms however, the complex exponential form is by far the most common. 

Fourier waveform analysis Refers to the concept of decomposing complex waveforms into the sum of simple trigonometric or complex exponential functions. 

Fourier Series with Complex Exponential Basis Functions... 

We derive the Fourier transform relations for one-dimensional functions of a single variable x and the corresponding Fourier space (also referred to as reciprocal space) functions with variable k. We start with the definition of the Fourier expansion of a function f x) with period 2L in terms of complex exponentials ... 

A Fourier series is an infinite series of terms that consist of coefficients times sine and cosine functions. It can represent almost any periodic function. The sine and cosine functions can be replaced by complex exponential functions. 

A Laplace transform is a representation of a function that is similar to a Fourier transform except that it uses real exponential functions instead of complex exponential functions. 

We have incorporated the terms with negative exponents into the same sum with the other terms by allowing the summation index to take on negative as well as positive values. A Fourier series does not have to represent a real function and the coefficients do not have to be real. If we represent a real function, the coefficients and bn will be real and the coefficients c will be complex. The orthogonality of the complex exponentials is a little different from the orthogonality of the sine and cosine functions, in that the complex conjugate of one of the functions must be taken before integration. 

The Fourier series using complex exponential basis functions is... 

This passage from sine and cosine to complex exponentials gives both the amplitude of the wave present in the function and the phase of the wave. The Fourier transform can be considered as the limit of the Fourier series of X(t) as T approaches inhnity. This can be illustrated as follows by rewriting Eq. (8.62) with inbnite T... 

The Fourier series can be written in a more compact form with the use of the complex exponential function exp(icot) = cos(cot)-H i sin (ot), where co is the radian frequency 2nf, and coq = 2nlT. Then... 

In practice, it is more convenient to write the Fourier series in terms of complex exponential functions, nsing Euler s formula... 

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