Rectangular function Fourier transform

Sine function sinc(x) = sin(x)/x. The sine function is most often encountered as the (frequency-domain) Fourier transform of a (time-domain) rectangular pulse. 

FIGURE 10.7 Impulse response function of the seven-point rectangular smoother window function used in Figure 10.6. Note that the Fourier transform of a step function has the form sin(x)/a . 

Numerically the convolution of a step scan is merely the application of a sliding weighted mean (e.g. like the Savitzky-Golay method). The Fourier transform of the rectangular function has the shape of sin(nv)/(nv) (whereby n is inversely proportional to the width of the rectangle) and unfortunately approaches 0 only very slowly. To make do with a small number of points for a convolution, one must tolerate a compromise and renounce the ideal rectangular shape of the low pass filter (in the frequency domain). 

Fig. 5.3.5 [Cal2] The sine pulse, (a) The amplitude of the rf carrier is modulated by a truncated sine function, (b) The magnitude of the Fourier transform of the pulse is a rectangular function distorted by wiggles in the centre and near the edges.

Another model for the electron density gradient was proposed by Blundell and analyzed by Vonk it consists of a linear density change in the interface [21, 28]. In this model, called the geometric linear model, the smoothing function is of rectangular type (Fig. 19.10) and its Fourier transform is given by... 

Figure 3. A schematic representation of a rectangular truncation function and its Fourier transform, the sine function.

We want to calculate the far field or Fraunhofer region for a square aperture. This aperture can be represented in one dimension as a rectangular or Rect function. Hence the Fourier transform of this will be a one dimensional sine function, where sinc(x) = sin(x)/x. The sine finction is shown in Fig. 1.3 and is one of the fundamental structures that dominates the information generated by computer generated holograms... 

Let us now apply the direct Fourier transform to find the frequency spectrum associated with the rectangular pulse shown in Figure 9.4. For convenience, a unit-area pulse has been selected. It is customary to display the aperiodic function in a symmetric manner (as shown) when possible, as this simplifies the expression for f(co). This, of course, is subject to the limitations of the physical system involved. The pulse is defined in real time by... 

A few typical apodizing functions and their Fourier transforms (instrumental functions) are shown in Figure 4.11, together with the rectangular function and its Fourier transform. 

The rectangular function and its Fourier transform are referred to in Section 4.4.1.1. The rectangular function II (x) on the jc axis is defined in the following way. 

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