MATH 414 Lecture 33

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Begin Exam 3 content

Multiresolution Analysis (MRA)

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dots \subset V_{j-1} \subset V_j \subset V_{j+1} \subset \dots}
2. Density, approximating everything in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2}
3. Separation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcap V_j = \left\{ 0 \right\}}
4. Scaling: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \in V_j} if and only if $f\left(2^{-j}\,x\right)\in V_{0}$
5. Scaling unctions $\phi$, and $\left\{\phi (x-k)\right\}_{k\in \mathbb {Z} }$ is an orthonormal basis for $V_{0}$

Properties

Orthonormal Bases for Subspaces

$\left\{2^{\frac {j}{2}}\,\phi \left(2^{j}\,x-k\right)\right\}_{k\in \mathbb {Z} }$ is an orthonormal basis for $V_{j}$

Orthogonality:

${\begin{aligned}\int 2^{\frac {j}{2}}\,\phi \left(2^{j}\,x-k\right)\,2^{\frac {j}{2}}\,\phi \left(2^{j}\,x-\ell \right)\,\mathrm {d} x&=2^{j}\,\int \phi \left(2^{j}\,x-k\right)\phi \left(2^{j}\,x-k\right)\,\mathrm {d} x\\&=\int \phi (u-k)\,\phi (u-\ell )\,\mathrm {d} x\\&=\delta _{\ell ,k}\end{aligned}}$

$f(x)\in V_{j}$

$f(2^{-j}\,x)\in V_{0}$

$f(2^{-j}\,x)=\sum _{k\in \mathbb {Z} }c_{k}\,\phi (x-k)$

Let $u=2^{-j}\,x$, then $f(u)=\sum _{k\in \mathbb {Z} }c_{k}\,\phi (2^{j}\,x-k)$

Scaling / Two-Scale Relation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) \in V_0 \subset V_j} . Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \sum_{k \in \mathbb{Z}} \frac{p_k}{\sqrt{2}} \, \sqrt{2} \, \phi(2x-k)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{p_k}{\sqrt{2}} = \left\langle \phi, \sqrt{2} \, \phi(2x-k) \right\rangle} by the definition of vector space projection.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left\langle \phi(x), \sqrt{2} \, \phi(2x-k) \right\rangle = \frac{p_k}{\sqrt{2}} \\ &= \int_{-\infty}^\infty \phi(x) \, \sqrt{2} \, \phi(2x-k) \,\mathrm{d}x \\ &= \sqrt{2} \, \int_{-\infty}^{\infty} \phi(x) \, \phi(2x-k) \,\mathrm{d}x \\ p_k = 2 \, \int_{-\infty}^{\infty} \phi(x) \, \phi(2x-k) \,\mathrm{d}x \end{align}}

Therefore

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \sum_{k \in \mathbb{Z}} p_k \, \phi(2x-k)}

Example: Haar MRA

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \phi(2x) + \phi(2x-1) = p_0 \, \phi(2x) + p_1 \, \phi(2x+1)}

Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_k = \begin{cases} 1 & k \in \left\{ 0,1 \right\} \\ 0 & \mbox{otherwise} \end{cases}}

Example: The Shannon MRA

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_j = \left\{ f \in L^2 ~\mid~ \mathrm{supp}(\hat{f}) \subseteq \left[ -2^j \, \pi, 2^j \, \pi \right] \right\}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \mathrm{sinc}(x)}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\omega)} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\omega) = 0} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \not\in \left[ -\pi, \pi \right]} , we have

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} , the Nyquist frequency
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\Omega} , the Nyquist rate
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{nat} = \nu_{freq} = \frac{\Omega}{2\pi}} , (the natural frequency (also called Nyquist frequency)
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{rate} = \frac{\Omega}{\pi}} , (also called Nyquist rate)

I'm confused now...

The sampling theorem states

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \sum_{k \in \mathbb{Z}} f \left( \frac{\pi \, k}{\Omega} \right) \, \mathrm{sinc} \left( \nu_{freq} \, x - k \right)}

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = 2\pi} . then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{nat} = \frac{\Omega}{2\pi} = 1} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{rate} = 2}

Therefore

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \sum_{k \in \mathbb{Z}} \phi \left( \frac{k}{2} \right) \, \mathrm{sinc} \left( 2x - k \right)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_k = \phi \left( \frac{k}{2} \right) = \mathrm{sinc} \left( \frac{k}{2} \right)} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{sinc} \left( 2x-k \right) = \phi(2x-k)} .

What are the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_k} 's?

In the even case, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{2k} = 0} , and in the odd case, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{2k+1} = \frac{2(-1)^k}{\pi(2\ell+1)}}

Properties

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k \in \mathbb{Z}} p_{k - 2\ell} \, p_k = 2\delta_{\ell,0}}
2. In the above case if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell = 0} , we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k\in \mathbb{Z}} p_k^2 = 2}

Proof:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi(x-\ell) &= \sum_{k \in \mathbb{Z}} p_k \, \phi(2(x-\ell) - k) \\ &= \sum_{k \in \mathbb{Z}} p_k \, \phi(2x - (k+2\ell) \\ &= \sum_{k \in \mathbb{Z}} p_{k-2\ell} \phi (2x-k) \\ \left\langle \phi(x), \phi(x-\ell) \right\rangle &= \sum_{k \in \mathbb{Z}} \frac{p_{k-2\ell}}{\sqrt{2}} \cdot \sqrt{2} \, \phi(2x-k) \\ &= \sum_{k \in \mathbb{Z}} \frac{p_{k-2\ell}}{\sqrt{2}} \cdot \frac{p_k}{\sqrt{2}} \\ &= 2\delta_{\ell,0} \end{align}}

The proof of the latter property comes from taking the Fourier series expansion of the sawtooth wave function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(0) = \frac{\pi}{4} - \sum_{k=0}^{\infty} \frac{1}{(2k+1)^2}}