MY-ASSIGNMENTEXPERT™可以为您提供Stanford EE261 Fourier analysis傅里叶分析的代写代考和辅导服务！
这是斯坦福大学 傅里叶分析课程的代写成功案例。
EE261课程简介
The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.
Topics include: The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems.
Prerequisites
e goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.
Topics include:
- The Fourier transform as a tool for solving physical problems.
- Fourier series, the Fourier transform of continuous and discrete signals and its properties.
- The Dirac delta, distributions, and generalized transforms.
- Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems.
- The discrete Fourier transform and the FFT algorithm.
- Multidimensional Fourier transform and use in imaging.
- Further applications to optics, crystallography.
- Emphasis is on relating the theoretical principles to solving practical engineering and science problems.
EE261 Fourier analysis HELP（EXAM HELP， ONLINE TUTOR）
Let $G$ be a finite cyclic group and $H$ a subgroup. For $\chi \in \hat{G}$ we write
$$
\hat{f}(\chi)=\sum_g f(g) \bar{\chi}(g)
$$
We say $\chi \in \hat{G}^H$ if $\chi$ is constant on the cosets of $H$.
Prove the following analogue of the classical Poisson Summation formula:
$$
\frac{1}{|G|} \sum_{\chi \in \hat{G}^H} \hat{f}(\chi)=\frac{1}{|H|} \sum_{h \in H} f(h) .
$$
(The classical version has $G=\mathbb{R}$ and $H=\mathbb{Z}$, which leads to $\hat{G}^H=\left{e^{2 \pi i n x}: n \in\right.$ $\mathbb{Z}}$.
1
Suppose $f \in L^2(\mathbb{R})$ is supported on $\left[-\frac{1}{2}, \frac{1}{2}\right]$ then we know that $f$ can be recovered from the values of $\hat{f}(n)$ for $n \in \mathbb{Z}$ (the characters form an orthonormal basis). Prove the Shannon Sampling Theorem:
$$
\hat{f}(\xi)=\sum_n \hat{f}(n) \frac{\sin [\pi(n-\xi)]}{\pi(n-\xi)}
$$
(which includes proving convergence of this infinite sum).
Remark: The audible spectrum extends only to about $20 \mathrm{kHz}$. Consequently, as heard by a human, one may regard music as a function whose Fourier transform is supported on a finite interval. The above theorem says that to faithfully reproduce music, one need only sample the signal forty thousand times per second. This is what happens in CD recording.
Given $\omega \in \mathbb{R}^d$, show that the following are equivalent:
(a) For $m \in \mathbb{Z}^d, m \cdot \omega=0$ implies $m=0$.
(b) The curve $t \mapsto t \omega+\mathbb{Z}^d$ is dense in $\mathbb{R}^d / \mathbb{Z}^d$.
(c) For any continuous function $f$ on $\mathbb{R}^d / \mathbb{Z}^d$,
$$
\lim {T \rightarrow \infty} \frac{1}{2 T} \int{-T}^T f\left(t \omega+\mathbb{Z}^d\right) d t=\int_0^1 \cdots \int_0^1 f\left(x+\mathbb{Z}^d\right) d x .
$$
Hint: prove $(\mathrm{a}) \Leftrightarrow(\mathrm{c})$ and then $(\mathrm{c}) \Rightarrow(\mathrm{b}) \Rightarrow(\mathrm{a})$.
Let $d \mu$ be a finite complex measure on $\mathbb{R}$.
(a) Show that
$$
\lim {L \rightarrow \infty} \frac{1}{2 L} \int{-L}^L|\hat{\mu}(\xi)|^2 d \xi=\sum_{x \in \mathbb{R}}|\mu({x})|^2
$$
(finiteness of the measure implies that only countably many terms in the sum are non-zero).
(b) Suppose that $d \mu$ is purely atomic, that is, $d \mu$ is a (countable) linear combination of delta measures. Show that $\hat{\mu}$ is almost periodic.
A function on $f$ on $\mathbb{R}$ is said to be almost periodic if for any $\epsilon>0$, there exists $L>0$ so that any interval of length $L$ contains an $\epsilon$-almost period:
$$
\forall a \in \mathbb{R} \quad \exists p \in[a, a+L] \quad \text { such that } \quad \sup _x|f(x)-f(x+p)|<\epsilon .
$$
Hint: For part (b) begin by considering the case $\hat{\mu}(\xi)=e^{i \xi}+e^{2 \pi i \xi}$.
MY-ASSIGNMENTEXPERT™可以为您提供STANFORD EE261 FOURIER ANALYSIS傅里叶分析的代写代考和辅导服务！
