전기전자공학 실험지시서 Lecture 00 - math equation

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• section 3.1, _p97

 

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section 3.1, _p97

This is test inline equation, \(x=Fa\).

A compressible signal $\boldsymbol{x} \in \mathbb{R}^n$ may be written as a sparse vector $\boldsymbol{s} \in \mathbb{R}^n$ in a transform basis $\boldsymbol{\Psi} \in \mathbb{R}^{n*n}$:

\[\boldsymbol{x} = \boldsymbol{\Psi} \boldsymbol{s} \tag{3.1}\]

Specifically, the vector $\boldsymbol{s}$ is called $K$-sparse in $\boldsymbol{\Psi}$ if there are exactly $K$ nonzero elements.

• 책에서 서술된 것 처럼 Fourier, Wavelet TF는 $\boldsymbol{\Psi}$의 구체적인 예로 이해하여도 무관하다.

 

With knowledge of the sparse vector $\boldsymbol{s}$ it is possible to reconstruct the signal $\boldsymbol{x}$ from eq.(3.1). Thus, the goal of compressed sensing is to find the sparsest vector $\boldsymbol{s}$ that is consistent with the measurements $\boldsymbol{y}$.

\[\begin{aligned} \boldsymbol{y} & = \boldsymbol{C} \boldsymbol{x} \tag{3.3} \\ & = \boldsymbol{C} \boldsymbol{\Psi} \boldsymbol{s} \\ & = \boldsymbol{\Theta} \boldsymbol{s} \end{aligned}\] \[\lim_{x\to 0}{\frac{e^x-1}{2x}} \overset{\left[\frac{0}{0}\right]}{\underset{\mathrm{H}}{=}} \lim_{x\to 0}{\frac{e^x}{2}}={\frac{1}{2}}\]

Recorving an audio signal from sparse measurements

% Generate signal, DCT of signal
n = 4096;             % points in high resolution signal
t = linspace(0, 1, n);
x = cos(2* 97 * pi * t) + cos(2* 777 * pi * t);

% Randomly sample signal
p = 128;              % num. random samples, p=n/32
perm = round(rand(p, 1) * n);
y = x(perm);          % compressed measurement