prb7.tex (4920B)
1 \include{preamble.tex} 2 3 \begin{document} 4 \maketitle 5 \tableofcontents 6 7 \section{Sheet 7} 8 \subsection{Dirac Comb} 9 The Dirac train or Dirac comb on defined in the following way 10 \begin{align} 11 \Sha_m[n] = 12 \begin{cases} 13 1\;\;\;\;\;\; n = 0, \pm m, \pm 2m,\dots\\ 14 0\;\;\;\;\;\; \text{else} 15 \end{cases} 16 \end{align} 17 The dirac comb can be represented in a series of discrete dirac delta's 18 \begin{align} 19 \Sha_m[n] = \sum_{l=-N}^N \delta[n - lm], 20 \end{align} 21 where $\delta[s] = 1$ if $s = 0$ else $0$, for $s \in \mathbb{Z}$. 22 The discrete Fourier transform of the Dirac comb in $\mathbb{C}^N$ is 23 \begin{align} 24 \widehat{\Sha_m[n]} 25 &=\frac{1}{N}\sum_{n=0}^{N-1} \Sha_m[n] e^{-2\pi i \frac{k}{N}n}=\\ 26 &=\frac{1}{N}\sum_{n=0}^{N-1} 27 \left( 28 \sum_{l=-N}^N \delta(n-lm) 29 \right) 30 e^{-2\pi i \frac{k}{N}n}, 31 \end{align} 32 where the summation happens exactly $\frac{N}{m}$ times, then 33 \begin{align} 34 &\frac{1}{m}\sum_{l=-N}^N e^{-2\pi i \frac{k}{N}lm}=\\ 35 &= \frac{1}{m} \sum_{l=-N}^N \delta[k - l\cdot \frac{N}{m}]\qquad 36 \text{(Poisson's summation formula)} \\ 37 &= \frac{1}{m}\Sha_{\frac{N}{m}}[k] 38 \end{align} 39 \subsection{Schwartz Space} 40 The Schwartz space $\mathcal{S}(\mathbb{R}^d)$, for $d \in \mathbb{N}$ is 41 defined as 42 \begin{align} 43 &\mathcal{S} := 44 \bigg\{ 45 f\in\mathcal{C}^\infty(\mathbb{R}^d): 46 \forall\alpha,\beta\in\mathbb{N}^d\;\; \lVert f \rVert_{\alpha,\beta} 47 < \infty 48 \bigg\},\\ 49 &\lVert f \rVert_{\alpha, \beta} := 50 \sup_{x\in\mathbb{R}^d}\left|x^\alpha (D^\beta f) (x) \right|. 51 \end{align} 52 Our aim is to show that if $f\in\mathcal{S}(\mathbb{R})$ then $\hat{f} \in 53 \mathcal{S}(\mathbb{R})$. The condition is obviously 54 \begin{align} 55 &\lVert \hat{f} \rVert_{\alpha, \beta} = 56 \sup_{\xi\in\mathbb{R}}\left|\xi^\alpha (D^\beta \hat{f}) (\xi) 57 \right|<\infty, 58 \end{align} 59 for all $\alpha, \beta \in \mathbb{N}$. 60 We can start with what we know about the Fourier transform 61 \begin{align} 62 \xi^\alpha \hat{f}(\xi) &= \mathcal{F}\left(\frac{1}{(2\pi 63 i)^\alpha}(D^{\alpha}f)(x)\right)\\ 64 D^{\beta}\hat{f}(\xi) &= \mathcal{F}\left( 65 (-2\pi i x)^\beta f(x) 66 \right). 67 \end{align} 68 Combining the two relations above we get 69 \begin{align} 70 \xi^\alpha (D^\beta \hat{f})(\xi) = 71 \mathcal{F}\left(\frac{(-2\pi i x)^\beta}{(2\pi 72 i)^\alpha}x^\beta(D^{\alpha}f)(x)\right)=: \mathcal{F}(g(x))\\ 73 \end{align} 74 If we call this function $g$, then $g\in\mathcal{S}(\mathbb{R})$ and 75 $g\in L^1(\mathbb{R})$. Applying the Riemann-Lebesgue Lemma we get 76 \begin{align} 77 \hat{g}(\xi) = \int_\mathbb{R} g(x) e^{-2\pi i x \xi}\ dx \longrightarrow 0 78 \;\;\; \text{as $|\xi| \rightarrow \infty$ } 79 \end{align} 80 Thereby $\hat{g} \in \mathcal{S}(\mathbb{R})$ and thus $\hat{f} \in 81 \mathcal{S}(\mathbb{R})$. 82 \subsection{Tempered Distributions} 83 Tempered distributions are the elements of 84 \begin{align} 85 \mathcal{S}'(\mathbb{R}^d) := 86 \bigg\{ 87 L: \mathcal{S}(\mathbb{R}^d) \rightarrow \mathbb{C} | \text{$L$ is 88 linear and continuous} 89 \bigg\}. 90 \end{align} 91 Consider $\xi$ as a tempered distribution, buy acting on $\varphi \in 92 \mathcal{S}(\mathbb{R})$ we have 93 \begin{align} 94 \xi(\phi) = \int_\xi \xi \varphi(\xi)\ d\xi. 95 \end{align} 96 The Fourier transform of $\xi$ is 97 \begin{align} 98 \hat{\xi}(\varphi) 99 &=\xi(\hat{\varphi}) 100 = \int_\mathbb{R} \xi \hat{\varphi}(\xi)\ d\xi\\ 101 &= \int_{\mathbb{R}^2}\xi \varphi(x) e^{2\pi i\xi x}\ dxd\xi\\ 102 &= \int_{\mathbb{R}^2}\varphi(x) \xi e^{2\pi i \xi x}\ dxd\xi\\ 103 &=\int_{\mathbb{R}^2}\varphi(x)\frac{i}{2\pi} \frac{\partial}{\partial x} 104 e^{2\pi i \xi x}\ dxd\xi =\\ 105 &=\frac{i}{2\pi}\int_{\mathbb{R}^2}\varphi(x)\delta'(x)\ dx=\\ 106 &=\frac{i}{2\pi} \delta'(\varphi). 107 \end{align} 108 \subsection{Fourier transform of the Dirac Comb} 109 The general case of the Dirac Comb as a distribution is 110 \begin{align} 111 \Sha_T = \sum_{n \in \mathbb{Z}} \delta_{nT}. 112 \end{align} 113 The Fourier transform of the $\Sha_T$ distribution for $\varphi \in 114 \mathcal{S}(\mathbb{R})$ is 115 \begin{align} 116 \widehat{\Sha_T}(\varphi) 117 &= \sum_{n\in\mathbb{Z}} \hat{\delta}_{nT}(\varphi)\\ 118 &= \sum_{n\in\mathbb{Z}} \delta_{n\omega_0}(\varphi)\\ 119 &=\Sha_{\omega_0}(\varphi). 120 \end{align} 121 The Fourier transform, transforms the period of the combs. 122 \subsection{Shannon Sampling} 123 The Fourier transform of $1_{[-\frac{a}{2}, \frac{a}{2}]}(x)$ is 124 \begin{align} 125 \mathcal{F}\left(1_{[-\frac{a}{2}, \frac{a}{2}]}\right)(\xi) 126 &= \int_\mathbb{R} 1_{[-\frac{a}{2}, \frac{a}{2}]} e^{-2\pi i x \xi}\ 127 dx\\ 128 &= \int_{-\frac{a}{2}}^{\frac{a}{2}} e^{-2\pi i x\xi}\ dx\\ 129 &= \frac{-1}{2\pi i \xi} e^{-2\pi i x 130 \xi}\bigg|_{-\frac{a}{2}}^{\frac{a}{2}}\\ 131 &= \frac{1}{\pi \xi} \frac{1}{2i}\left( 132 e^{pi i a \xi} - e^{-\pi i a \xi} 133 \right)\\ 134 &= \frac{\sin(\pi \xi a)}{\pi \xi} 135 \end{align} 136 137 %\printbibliography 138 \end{document}