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Morten Hjorth-Jensen
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
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'layers-used-to-build-cnns'),
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('Transforming images', 2, None, 'transforming-images'),
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('CNNs in brief', 2, None, 'cnns-in-brief'),
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('Key Idea', 2, None, 'key-idea'),
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('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
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('Convolution Examples: Polynomial multiplication',
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2,
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@@ -358,37 +359,38 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week42-bs057.html#layers-used-to-build-cnns" style="font-size: 80%;">Layers used to build CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs058.html#transforming-images" style="font-size: 80%;">Transforming images</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs059.html#cnns-in-brief" style="font-size: 80%;">CNNs in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs060.html#mathematics-of-cnns" style="font-size: 80%;">Mathematics of CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs061.html#convolution-examples-polynomial-multiplication" style="font-size: 80%;">Convolution Examples: Polynomial multiplication</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs062.html#efficient-polynomial-multiplication" style="font-size: 80%;">Efficient Polynomial Multiplication</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs063.html#a-more-efficient-way-of-coding-the-above-convolution" style="font-size: 80%;">A more efficient way of coding the above Convolution</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs064.html#convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms" style="font-size: 80%;">Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs065.html#principle-of-superposition" style="font-size: 80%;">Principle of Superposition</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs066.html#simple-code-example" style="font-size: 80%;">Simple Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs067.html#wrapping-up-fourier-transforms" style="font-size: 80%;">Wrapping up Fourier transforms</a></li>
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<!-- navigation toc: --> <li><a href="#finding-the-coefficients" style="font-size: 80%;">Finding the Coefficients</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs069.html#final-words-on-fourier-transforms" style="font-size: 80%;">Final words on Fourier Transforms</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs070.html#two-dimensional-objects" style="font-size: 80%;">Two-dimensional Objects</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs071.html#cross-correlation" style="font-size: 80%;">Cross-Correlation</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs072.html#more-on-dimensionalities" style="font-size: 80%;">More on Dimensionalities</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs073.html#further-dimensionality-remarks" style="font-size: 80%;">Further Dimensionality Remarks</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs074.html#cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;">CNNs in more detail, building convolutional neural networks in Tensorflow and Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs075.html#setting-it-up" style="font-size: 80%;">Setting it up</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs076.html#the-mnist-dataset-again" style="font-size: 80%;">The MNIST dataset again</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs077.html#strong-correlations" style="font-size: 80%;">Strong correlations</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs078.html#layers-of-a-cnn" style="font-size: 80%;">Layers of a CNN</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs079.html#systematic-reduction" style="font-size: 80%;">Systematic reduction</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs080.html#prerequisites-collect-and-pre-process-data" style="font-size: 80%;">Prerequisites: Collect and pre-process data</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs081.html#importing-keras-and-tensorflow" style="font-size: 80%;">Importing Keras and Tensorflow</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs082.html#running-with-keras" style="font-size: 80%;">Running with Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs083.html#final-part" style="font-size: 80%;">Final part</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs084.html#final-visualization" style="font-size: 80%;">Final visualization</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs085.html#the-cifar01-data-set" style="font-size: 80%;">The CIFAR01 data set</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs086.html#verifying-the-data-set" style="font-size: 80%;">Verifying the data set</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs087.html#set-up-the-model" style="font-size: 80%;">Set up the model</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs088.html#add-dense-layers-on-top" style="font-size: 80%;">Add Dense layers on top</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs089.html#compile-and-train-the-model" style="font-size: 80%;">Compile and train the model</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs090.html#finally-evaluate-the-model" style="font-size: 80%;">Finally, evaluate the model</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs060.html#key-idea" style="font-size: 80%;">Key Idea</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs061.html#mathematics-of-cnns" style="font-size: 80%;">Mathematics of CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs062.html#convolution-examples-polynomial-multiplication" style="font-size: 80%;">Convolution Examples: Polynomial multiplication</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs063.html#efficient-polynomial-multiplication" style="font-size: 80%;">Efficient Polynomial Multiplication</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs064.html#a-more-efficient-way-of-coding-the-above-convolution" style="font-size: 80%;">A more efficient way of coding the above Convolution</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs065.html#convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms" style="font-size: 80%;">Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs066.html#principle-of-superposition" style="font-size: 80%;">Principle of Superposition</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs067.html#simple-code-example" style="font-size: 80%;">Simple Code Example</a></li>
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<!-- navigation toc: --> <li><a href="#wrapping-up-fourier-transforms" style="font-size: 80%;">Wrapping up Fourier transforms</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs069.html#finding-the-coefficients" style="font-size: 80%;">Finding the Coefficients</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs070.html#final-words-on-fourier-transforms" style="font-size: 80%;">Final words on Fourier Transforms</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs071.html#two-dimensional-objects" style="font-size: 80%;">Two-dimensional Objects</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs072.html#cross-correlation" style="font-size: 80%;">Cross-Correlation</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs073.html#more-on-dimensionalities" style="font-size: 80%;">More on Dimensionalities</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs074.html#further-dimensionality-remarks" style="font-size: 80%;">Further Dimensionality Remarks</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs075.html#cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;">CNNs in more detail, building convolutional neural networks in Tensorflow and Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs076.html#setting-it-up" style="font-size: 80%;">Setting it up</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs077.html#the-mnist-dataset-again" style="font-size: 80%;">The MNIST dataset again</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs078.html#strong-correlations" style="font-size: 80%;">Strong correlations</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs079.html#layers-of-a-cnn" style="font-size: 80%;">Layers of a CNN</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs080.html#systematic-reduction" style="font-size: 80%;">Systematic reduction</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs081.html#prerequisites-collect-and-pre-process-data" style="font-size: 80%;">Prerequisites: Collect and pre-process data</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs082.html#importing-keras-and-tensorflow" style="font-size: 80%;">Importing Keras and Tensorflow</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs083.html#running-with-keras" style="font-size: 80%;">Running with Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs084.html#final-part" style="font-size: 80%;">Final part</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs085.html#final-visualization" style="font-size: 80%;">Final visualization</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs086.html#the-cifar01-data-set" style="font-size: 80%;">The CIFAR01 data set</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs087.html#verifying-the-data-set" style="font-size: 80%;">Verifying the data set</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs088.html#set-up-the-model" style="font-size: 80%;">Set up the model</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs089.html#add-dense-layers-on-top" style="font-size: 80%;">Add Dense layers on top</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs090.html#compile-and-train-the-model" style="font-size: 80%;">Compile and train the model</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs091.html#finally-evaluate-the-model" style="font-size: 80%;">Finally, evaluate the model</a></li>
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</ul>
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</li>
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@@ -404,73 +406,37 @@ MathJax.Hub.Config({
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<a name="part0068"></a>
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<!-- !split -->
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<h2 id="finding-the-coefficients" class="anchor">Finding the Coefficients </h2>
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<h2 id="wrapping-up-fourier-transforms" class="anchor">Wrapping up Fourier transforms </h2>
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<p>
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Because the forces have been applied for a long time, any non-zero
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damping eliminates the homogenous parts of the solution, so one need
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only consider the particular solution for each \( n \).
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<p>
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The problem is considered solved if one can find expressions for the
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coefficients \( f_n \) and \( g_n \), even though the solutions are expressed
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as an infinite sum. The coefficients can be extracted from the
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function \( F(t) \) by
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$$
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\begin{eqnarray}
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\tag{24}
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f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\
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\nonumber
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g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau).
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\end{eqnarray}
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$$
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<p>
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To check the consistency of these expressions and to verify
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Eq. <a href="#mjx-eqn-24">(24)</a>, one can insert the expansion of \( F(t) \) in
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Eq. <a href="._week42-bs067.html#mjx-eqn-23">(23)</a> into the expression for the coefficients in
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Eq. <a href="#mjx-eqn-24">(24)</a> and see whether
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$$
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\begin{eqnarray}
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f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{
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\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t)
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\right\}\cos(n\omega t).
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\end{eqnarray}
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$$
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<p>
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Immediately, one can throw away all the terms with \( g_m \) because they
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convolute an even and an odd function. The term with \( f_0/2 \)
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disappears because \( \cos(n\omega t) \) is equally positive and negative
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over the interval and will integrate to zero. For all the terms
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\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition
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formulas to see that \( \cos(m\omega t)\cos(n\omega
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t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate
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to zero unless \( m=n \). In that case the \( m=n \) term gives
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We can write down the answer for
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\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By
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writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv
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2\pi/\tau \),
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$$
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\begin{equation}
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\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2},
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\tag{25}
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\tag{23}
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F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t).
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\end{equation}
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$$
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<p>
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and
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The solutions for \( x(t) \) then come from replacing \( \omega \) with
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\( n\omega \) for each term in the particular solution,
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$$
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\begin{eqnarray}
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f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\
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x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\
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\nonumber
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&=&f_n~\checkmark.
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\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\
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\nonumber
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\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\
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\nonumber
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\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right).
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\end{eqnarray}
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$$
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<p>
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The same method can be used to check for the consistency of \( g_n \).
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<p>
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<p>
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<!-- navigation buttons at the bottom of the page -->
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@@ -497,7 +463,7 @@ The same method can be used to check for the consistency of \( g_n \).
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<li><a href="._week42-bs076.html">77</a></li>
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<li><a href="._week42-bs077.html">78</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week42-bs090.html">91</a></li>
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<li><a href="._week42-bs091.html">92</a></li>
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<li><a href="._week42-bs069.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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