MATH 414 Lecture 37

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Exam Discussion

Problem 2

Frequency response: 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 \sqrt{2\pi} \, \hat{h}(\omega) = \frac{1}{5} \, \frac{1-\mathrm{e}^{-5i\,\omega}}{i \,\omega}}

Convolution filter:

${\begin{aligned}\left(f*h\right)(t)&=\int _{-\infty }^{\infty }h(\tau )\,f(t-\tau )\,\mathrm {d} \tau \\&={\frac {1}{5}}\,\int _{0}^{5}f(t-\tau )\,\mathrm {d} \tau \\&={\frac {1}{5}}\,\int _{t-5}^{t}f(\sigma )\,\mathrm {d} \sigma \end{aligned}}$

Where $\sigma =t-\tau$

3 Cases:

1. $t<0$, then $\sigma \in \left[t-5,0\right)$: thus $(f*h)=0$.
2. $t\in \left(0,5\right)$, then $\sigma \in \left[t,t-5\right]$: thus $(f*h)={\frac {1}{5}}\,{\cancel {\int _{t-5}^{0}f(\sigma )\,\mathrm {d} \sigma }}+{\frac {1}{5}}\,\int _{0}^{t}f(\sigma )\,\mathrm {d} \sigma ={\frac {1}{15}}\,\left(1-\mathrm {e} ^{-3t}\right)$
3. $t>5$, 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 \sigma > 0} : thus 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 * h) = \frac{1}{15} \, \left( \mathrm{e}^{-3(t-5)} - \mathrm{e}^{-3t} \right)}

Problem 4

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_2} defined by 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 a^2 = \left( 3, 1, -2, 0, -3, 9, -3 \right)} (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 a_k^2 = 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 k < 0} or 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 k > 7}

Haar Wavelet Decomposition into 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 b^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 a^1} , 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 a^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 f_2 = f_0 + w_0 + w_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 \begin{align} a_k^{j-1} &= \frac{1}{2} \, \left( a_{2k}^j + a_{2k+1}^j \right) \\ b_k^{j-1} &= \frac{1}{2} \, \left( a_{2k}^j - a_{2k+1}^j \right) \end{align} }

First decomposition level 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 a^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 b^1} gives:

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} a^1 &= \left( 2, -1, 3, -\frac{3}{2} \right) \\ b^1 &= \left( 1, -1, -6, -\frac{3}{2} \right) \end{align}}

Corresponding to 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 k = 0,1,2,3} (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 a_k^1 = b_k^1 = 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 k < 0} or 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 k > 3} )

Second decomposition level 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 a^0} , 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 b^0} gives:

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} a^0 &= \left( \frac{1}{2}, \frac{3}{4} \right) \\ b^0 &= \left( \frac{3}{2}, \frac{9}{4} \right) \end{align}}

Corresponding to 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 k = 0,1} (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 a_k^0 = b_k^0 = 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 k < 0} or 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 k > 1} )

MRA Summary

So far, we've assumed that 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 are real. Let's keep that assumption and see what we have so far:

Multi-Resolution analysis:

scaling 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 \phi(x) = \sum_{k\in \mathbb{Z}} p_k \, \phi(2x-k)}

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 = 2 \, \int_{-\infty}^\infty \phi(x) \, \phi(2x-k) \,\mathrm{d}x}

wavelet

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 \psi(x) = \sum_{k \in \mathbb{Z}} \left( -1 \right)^k \, p_{1-k} \, \phi(2x-k)}

decomposition

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 a_k^{j-1} = \frac{1}{2} \,\sum_{m \in \mathbb{Z}} p_{m-2k} \, a_m^j}

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 b_k^{j-1} = \frac{1}{2} \, \sum_{m \in \mathbb{Z}} p_{1 - m + 2k}}

reconstruction

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 a_k^j = \sum_{m \in \mathbb{Z}} p_{k-2m} \, a_m^{j-1} + \sum_{m \in \mathbb{Z}} \left( -1 \right)^k p_{1-k+2m} \, b_m^{j-1}}

are these 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} indexes correct?

filters

low-pass decomposition 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_k = \frac{1}{2} \, p_{-k}}

high-pass decomposition 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 h_k = \frac{1}{2} \, \left( -1 \right)^k \, p_{k+1}}

low-pass reconstruction 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 \tilde{\ell}_k = p_k}

high-pass reconstruction 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 \tilde{h}_k = \left( -1 \right)^k \, p_{1-k}}

Properties of 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

The entire scheme of an MRA depends on 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, which, in turn, may very well depend on the scaling function definition

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 must satisfy the following properties:

• 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}}
• 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_{2k} = \sum_{k \in \mathbb{Z}} p_{2k+1} = 1}

If 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 \phi(x) = \sum_{k \in \mathbb{Z}} p_k \, \phi(2x-k)} , 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 \begin{align} \mathcal{F} \left[ \phi(x) \right] = \hat{\phi}(\xi) &= \mathcal{F} \left[ \sum_{k \in \mathbb{Z}} p_k \, \phi(2x-k) \right](\xi) \\ &= \sum_{k \in \mathbb{Z}} p_k \, \mathcal{F} \left[ \phi(2x-k) \right](\xi) \end{align}}

Let's look at that inner Fourier transform:

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} \mathcal{F} \left[ \phi(2x-k) \right](\xi) &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^\infty \phi(2x-k) \, \mathrm{e}^{-i \, \xi \, x} \,\mathrm{d}x \\ &= \frac{1}{2} \, \frac{1}{\sqrt{2\pi}} \phi(u) \, \mathrm{e}^{-i \, \xi \, \left( \frac{u + k}{2} \right)} \,\mathrm{d}u \\ &= \frac{1}{2} \, \mathrm{e}^{-\frac{i \, \xi \, k}{2}} \cdot \underbrace{\frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^\infty \phi(u) \, \mathrm{e}^{-i \, \frac{\xi}{2} \, u \, \,\mathrm{d}u}}_{\hat{\phi} \left( \frac{\xi}{2} \right)} \end{align}}

Plugging this back in to our Fourier transform of 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} gives:

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} \hat{\phi}(\xi) &= \sum_{k \in \mathbb{Z}} p_k \, \mathcal{F} \left[ \phi(2x-k) \right](\xi) &= \sum_{k \in \mathbb{Z}} p_k \, \left( \frac{1}{2} \, \mathrm{e}^{-\frac{i \, \xi \, k}{2}} \, \hat{\phi} \left( \frac{\xi}{2} \right) \right) \end{align}}

Let 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(z) = \frac{1}{2} \, \sum_{k \in \mathbb{Z}} p_k \, z^k} . 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 \begin{align} \hat{\phi}(\xi) &= P \left( \mathrm{e}^{i \, \frac{\xi}{2}} \right) \, \hat{\phi} \left( \frac{\xi}{2} \right) \\ \end{align}}

We can expand 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 \hat{\phi} \left( \frac{\xi}{2} \right)} in the RHS 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 n} times to get

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} \hat{\phi}(\xi) &= \left( \prod_{r=1}^n P \left( \mathrm{e}^{- \frac{i \, xi}{2^r}} \right) \right) \, \hat{\phi} \left( \frac{\xi}{2^n} \right) \end{align}}

If we let 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 n \to \infty} , 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 \hat{\phi} \left( \frac{\xi}{2^n} \right) \to \hat{\phi}(0) = \frac{1}{\sqrt{2\pi}}} . 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 \begin{align} \hat{\phi}(\xi) &= \frac{1}{\sqrt{2\pi}} \, \prod_{r=1}^\infty P \left( \mathrm{e}^{-\frac{i\,\xi}{2^r}} \right) \end{align}}

For the Haar wavelet, 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{\phi}(\xi) = \frac{1}{\sqrt{2\pi}} \, \prod_{r=1}^\infty \left( \frac{1 + \mathrm{e}^{-\frac{i\,\xi}{2^r}}}{2} \right)}

Now we can construct new scaling functions and wavelets by just working 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{\phi}} .