Boxcar functions

The shaip cutoff at the limits -i and t, as illustrated by the boxcar function, often occurs in the frequency domain. In this case the boxcar acts as a low-pass filter in applications in electronics. All frequencies below [l] areunaltgEed, in this ideal case all higher ones are suppressed. 

Fourier transforms boxcar function 274 Cauchy function 276 convolution 272-273 Dirac delta function 277-279 Gaussian function 275-276 Lorentzian function 276-277 shah function 277-279 triangle function 275 fraction, rational algebraic 47 foil width at half maximum (FWHM) 55, 303... 

Figure 2.9 The periodic boxcar function with period 2X.

Changing x to — x in the first integral and recognizing that cos is an even function shows that b is zero for all n. The Fourier series expansion of the boxcar function is therefore... 

Figure 2.10 Partial sums of Fourier components of the boxcar function up to p = 9. Convergence to the boxcar functions is achieved rapidly although ears appear next to discontinuities (Gibbs effect).

Addition of the first components up to p = 9 is shown in Figure 2.10. Reconstruction of the boxcar function is rapid although ears persist next to the edge, a feature characteristic of discontinuities and known as the Gibbs effect , o... 

Such a sine expansion is generally made possible with the condition of zero concentration at x=0 by using an odd function for the initial concentration distribution. A simple example is for initial uniform concentration C0 between 0 and X for which we can assume a fictitious concentration — C0 between — X and 0. Using the results of Chapter 2, the Fourier expansion of the boxcar function which is C0 between 0 and X, and 0 at x=0 and x = X is... 

The function that has such a Fourier expansion while satisfying C(0,f)=C(0,. Y)=0 is the boxcar function. We found in Section 2.6.2 that the a coefficients of this function are 0 for even values and 4/nn for odd values of n. The general solution is therefore... 

A(8) is called the boxcar function. This limit on the retardation leads to a limit on the resolution of 1/2L, so if L - 100 cm, the highest resolution attainable is 0.005 cm-1. By the convolution theorem, the product of two functions in one space is the same as the convolution of the Fourier transforms of the two functions in the reciprocal space. The effect of multiplying by this boxcar function is to convolve each point in the reciprocal wavenumber space with a sine function [sinc(x) = sin(x)/x Figure 4], An undesirable feature of the sine function as a lineshape is the large amplitude oscillation (the first minimum is -22% of the maximum). This ringing can make it difficult to get information about nearby peaks and leads to anomalous values for intensities. This ringing can be removed by the process known as apodization. 

Figure 4. Apodization functions and their Fourier transforms. The top left function is the boxcar function and its FT is the sine function. Note the large amplitude of the secondary minimum and the narrow full width at half maximum, Ao. The bottom pair of figures show the Hamming function and its FT. The secondary oscillations are smaller but the width has grown.

In particular, if Act (r) has the constant value Act, within the cell, then the coefficient in the boxcar function expansion is equal to this constant. 

We can select the basis functions in expansion (9.181) in the form of the boxcar functions multiplied by the values of the vector E (r ) at some internal point of the cell r G ... 

Now we can use also expansion (9.179) of the anomalous conductivity distribution over the boxcar functions. Substituting (9.179) in the last formula, we finally find... 

According to the convolution theorem of Fourier analysis, the Fourier transform of a product of two functions is given by the convolution (here indicated by the symbol ) of their individual Fourier transforms. Hence, the effect of multiplying 1(8) by the boxcar function D(8) is to yield a spectrum that is the convolution of the Fourier transform of 1(8) measured with an infinitely long retardation and the Fourier transform of D(8). The Fourier transform of 1(8) is the true spectrum 5(v), while the Fourier transform of D(8), /(v), is given by... 

Fig. 2.4 Instrumental Line Shape/LA(v) top), which is the Fourier transform of a boxcar function of unit amplitude extending from +A to —A. Fourier transform of an interferogram generated by a monochromatic line at vi = 2/A bottom)...

The spectral results of the simulation are shown in Fig. 5.7 (left) for the central pixel of the gaussian source (blue), the point source (green) and the central pixel of the elliptical source (red). It can be observed that the emission and absorption line positions are detected but present a sine-shape this is due to the boxcar function... 

The FT can be applied to a number of simple functions as presented in pictorial form in Figure 1. The FT of a Gaussian is another Gaussian, the decaying exponential (double-sided) gives a Lorentzian, and the boxcar function gives a sinc(= sin(x)/5t) function. 

Figure 1 The Fourier transforms of Gaussian, double-sided exponential, and boxcar functions.

FD giving the Fourier spectrum bl. This spectrum is then multiplied by an apodization function (a boxcar function is shown in b2). Finally, the Fourier transform of the truncated Fourier spectrum is computed, yielding the smoothed spectrum. 

From Section 2.3 we know that when a cosine wave interferogram is unweighted, the shape of the spectral line is the convolution of the true spectrum and a sine function [i.e., the transform of the boxcar truncation function, 0(8)]. If instead of using the boxcar function, we used a simple triangular weighting function of the form... 

The interferogram as described mathematically is continuous and infinite requiring the Fourier integral to be evaluated over the limits of oo. Since the mirrors cannot move over distances of oo, the actual mirror movement is equivalent to multiplying the infinite interferogram by a boxcar function that has a value at all points between the optical displacement distance, L, and a value of zero everywhere else. 

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