## 物理代写|傅立叶光学代写Fourier optics代考|Extrapolation of bandlimited functions

The dual orthogonality and completeness properties of the prolate spheroidal functions can be utilized for extrapolation of bandlimited functions that are known over a truncated region. Suppose a function $g(x)$ bandlimited to the frequency band $f_x:(-B, B)$ is known over a finite range $x:(-L, L)$ then one may expand it using the prolate spheroidal function set as:
$$g(x)=\sum_n a_n \phi_n(x),$$
where the coefficients $a_n$ may be determined using the orthogonality of prolate functions over the region $x:(-L, L)$ :
$$a_n=\frac{1}{2 B \lambda_n} \int_{-L}^L d x g(x) \phi_n(x) .$$
Further one may in principle extend the function beyond the range $x:(-L, L)$ by using the same coefficients $a_n$. The possibility of determining the function beyond the range $x:(-L, L)$ is however practically governed by the accuracy of our knowledge of $g(x)$ over the finite range. If for example, the values of $g(x)$ are determined by some experimental measurement, they will contain some noise which translates to noise on the coefficients $a_n$. In Eq. (3.33) above, the evaluation of the coefficient $a_n$ involves division by the corresponding eigenvalue $\lambda_n$. As we have already seen above, the eigenvalues are typically very small when the index $n>4 R T$, and the corresponding prolate functions have most of their energy concentrated beyond the range $x:(-L, L)$. So although such higher order nite range where the function $g(x)$ is known, their inclusion in the series representation involves division of corresponding coefficients $a_n$ by small numbers. Since the coefficients $a_n$ are noisy, such division by small numbers can amplify the noise making the series representation meaningless.

## 物理代写|傅立叶光学代写Fourier optics代考|Operational introduction to Fast Fourier Transform

Fourier transforms will be encountered time and again in this book and it is important for students and practitioners to have a basic operational understanding of numerical routines or functions readily available for implementing Fourier transform on discretely sampled signals and images. This chapter does not by any means discuss the details of the discrete Fourier transform and fast Fourier transform algorithms for which excellent literature is already available. The aim here is to provide sufficient information so that when using standard computational tools or libraries for FFT operation, a user may be able to make sense of the results. A few important aspects regarding the usage of 2D Fourier transform functions for simulating optical imaging systems are also pointed out along the way.

With an introduction to the sampling ideas as discussed in Chapter 3 , we may now represent a signal over length $2 L$ and an effective bandwidth $2 B$ by means of samples $g(0 / 2 B), g(1 / 2 B), \ldots$, $g((N-1) /(2 B))$ with $N \approx 4 B L$. The discrete Fourier transform (DFT) of the signal is typically defined as:
$$G\left(\frac{m}{2 L}\right)=\sum_{n=0}^{N-1} g\left(\frac{n}{2 B}\right) \exp (-i 2 \pi m n / N) .$$
The corresponding inverse discrete Fourier transform may be de-fined as:
$$g\left(\frac{n}{2 B}\right)=\frac{1}{N} \sum_{m=0}^{N-1} G\left(\frac{m}{2 L}\right) \exp (i 2 \pi m n / N) .$$
We notice that the discrete Fourier transform operations above are approximations to the continuous integral version of the Fourier transform when the signal is defined only over $x:(-L, L)$ by means of a discrete set of samples. The space domain samples have a periodicity of $1 /(2 B)$ while the frequency domain periodicity is given by $1 /(2 L)$. The factors $(2 B)$ and $(2 L)$ are usually omitted from most standard definitions but we will retain them here explicitly.

## 物理代写|傅立叶光学代写傅里叶光学代考|带限函数的外推

$$g(x)=\sum_n a_n \phi_n(x),$$
，其中系数$a_n$可以用延长函数在区域$x:(-L, L)$:
$$a_n=\frac{1}{2 B \lambda_n} \int_{-L}^L d x g(x) \phi_n(x) .$$

## 物理代写|傅立叶光学代写傅立叶光学代考|快速傅立叶变换操作介绍

$$G\left(\frac{m}{2 L}\right)=\sum_{n=0}^{N-1} g\left(\frac{n}{2 B}\right) \exp (-i 2 \pi m n / N) .$$

$$g\left(\frac{n}{2 B}\right)=\frac{1}{N} \sum_{m=0}^{N-1} G\left(\frac{m}{2 L}\right) \exp (i 2 \pi m n / N) .$$

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