Fetch assignments week 2
This commit is contained in:
441
Week 2/5 Discrete and Fast Fourier Transforms.ipynb
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441
Week 2/5 Discrete and Fast Fourier Transforms.ipynb
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"source": [
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"# CDS: Numerical Methods Assignments\n",
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"\n",
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"- See lecture notes and documentation on Brightspace for Python and Jupyter basics. If you are stuck, try to google or get in touch via Discord.\n",
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"\n",
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"- Solutions must be submitted via the Jupyter Hub.\n",
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"\n",
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"- Make sure you fill in any place that says `YOUR CODE HERE` or \"YOUR ANSWER HERE\".\n",
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"\n",
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"## Submission\n",
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"\n",
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"1. Name all team members in the the cell below\n",
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"2. make sure everything runs as expected\n",
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"3. **restart the kernel** (in the menubar, select Kernel$\\rightarrow$Restart)\n",
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"4. **run all cells** (in the menubar, select Cell$\\rightarrow$Run All)\n",
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"5. Check all outputs (Out[\\*]) for errors and **resolve them if necessary**\n",
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"6. submit your solutions **in time (before the deadline)**"
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]
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"team_members = \"\""
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}
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},
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"source": [
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"## Discrete and Fast Fourier Transforms (DFT and FFT)\n",
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"\n",
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"In the following we will implement a DFT algorithm and, based on that, a FFT algorithm. Our aim is to experience the drastic improvement of computational time in the FFT case."
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]
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},
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"outputs": [],
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"source": [
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"# Import packages here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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}
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"source": [
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"### Task 1\n",
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"\n",
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"Implement a Python function $\\text{DFT(yk)}$ which returns the Fourier transform defined by\n",
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"\n",
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"\\begin{equation}\n",
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"\\beta_j = \\sum^{N-1}_{k=0} f(x_k) e^{-ij x_k}\n",
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"\\end{equation}\n",
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"\n",
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"with $x_k = \\frac{2\\pi k}{N}$ and $j = 0, 1, ..., N-1$. The $\\text{yk}$ should represent the array corresponding to $y_k = f(x_k)$. Please note that this definition is slightly different to the one we introduced in the lecture. Here we follow the notation of Numpy and Scipy.\n",
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"\n",
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"Hint: try to write the sum as a matrix-vector product and use $\\text{numpy.dot()}$ to evaluate it."
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]
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},
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{
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"grade": true,
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"locked": false,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"def DFT(yk):\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"source": [
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"### Task 2 \n",
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"\n",
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"Make sure your function $\\text{DFT(yk)}$ and Numpy’s FFT function $\\text{numpy.fft.fft(yk)}$ return\n",
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"the same data by plotting $|\\beta_j|$ vs. $j$ for\n",
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"\n",
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"\\begin{equation}\n",
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" y_k = f(x_k) = e^{20i x_k} + e^{40 i x_k}\n",
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"\\end{equation}\n",
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"and\n",
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"\\begin{equation}\n",
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" y_k = f(x_k) = e^{i 5 x_k^2}\n",
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"\\end{equation}\n",
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"\n",
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"using $N = 128$ for both routines."
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]
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},
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"source": [
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"### Task 3\n",
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"\n",
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"Analyze the evaluation-time scaling of your $\\text{DFT(yk)}$ function with the help of the timeit\n",
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"module. Base your code on the following example:\n",
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"\n",
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"```python\n",
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"import timeit\n",
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"\n",
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"tOut = timeit.repeat(stmt=lambda: DFT(yk), number=10, repeat=5)\n",
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"tMean = np.mean(tOut)\n",
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"```\n",
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"This example evaluates $\\text{DFT(yk)}$ 5 × 10 times and stores the resulting 5 evaluation times in tOut. Afterwards we calculate the mean value of these 5 repetitions. \n",
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"Use this example to calculate and plot the evaluation time of your $\\text{DFT(yk)}$ function for $N = 2^2, 2^3, ..., 2^M$. Depending on your implementation you might be able to go up to $M = 10$. Be careful and increase M just step by step!"
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]
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},
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{
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"cell_type": "code",
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"grade": true,
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"grade_id": "cell-0ab81532ab86e322",
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"locked": false,
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"points": 4,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"solution": false,
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"source": [
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"### Task 4\n",
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"\n",
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"A very simple FFT algorithm can be derived by the following separation of the sum from\n",
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"above:\n",
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"\n",
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"\\begin{align}\n",
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" \\beta_j = \\sum^{N-1}_{k=0} f(x_k) e^{-ij \\frac{2\\pi k}{N}} \n",
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" &= \\sum^{N/2 - 1}_{k=0} f(x_{2k}) e^{-ij \\frac{2\\pi 2k}{N}} \n",
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" + \\sum^{N/2 - 1}_{k=0} f(x_{2k+1}) e^{-ij \\frac{2\\pi (2k+1)}{N}}\\\\ \n",
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" &= \\sum^{N/2 - 1}_{k=0} f(x_{2k}) e^{-ij \\frac{2\\pi k}{N/2}}\n",
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" + \\sum^{N/2 - 1}_{k=0} f(x_{2k+1}) e^{-ij \\frac{2\\pi k}{N/2}} e^{-ij \\frac{2\\pi}{N}}\\\\\n",
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" &= \\beta^{\\text{even}}_j + \\beta^{\\text{odd}}_j e^{-ij \\frac{2\\pi}{N}}\n",
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"\\end{align}\n",
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"\n",
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"where $\\beta^{\\text{even}}_j$ is the Fourier transform based on only even $k$ (or $x_k$) and $\\beta^{\\text{odd}}_j$ the Fourier transform based on only odd $k$. In case $N = 2^M$ this even/odd separation can be done again and again in a recursive way. \n",
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"\n",
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"Use the template below to implement a $\\text{FFT(yk)}$ function, making use of your $\\text{DFT(yk)}$ function from above. Make sure that you get the same results as before by comparing the results from $\\text{DFT(yk)}$\n",
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"and $\\text{FFT(yk)}$ for both functions defined in task 2.\n",
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"\n",
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"```python\n",
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"def FFT(yk):\n",
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" \"\"\"Don't forget to write a docstring ...\n",
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" \"\"\"\n",
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" N = # ... get the length of yk\n",
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" \n",
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" assert # ... check if N is a power of 2. Hint: use the % (modulo) operator\n",
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" \n",
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" if(N <= 2):\n",
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" return # ... call DFT with all yk points\n",
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" \n",
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" else:\n",
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" betaEven = # ... call FFT but using just even yk points\n",
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" betaOdd = # ... call FFT but using just odd yk points\n",
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" \n",
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" expTerms = np.exp(-1j * 2.0 * np.pi * np.arange(N) / N)\n",
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" \n",
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" # Remember : beta_j is periodic in j !\n",
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" betaEvenFull = np.concatenate([betaEven, betaEven])\n",
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" betaOddFull = np.concatenate([betaOdd, betaOdd])\n",
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" \n",
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" return betaEvenFull + expTerms * betaOddFull\n",
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"```"
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]
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"outputs": [],
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"source": [
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"def FFT(yk):\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()"
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]
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},
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"outputs": [],
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"source": [
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"# Do your plotting here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"solution": false,
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},
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"source": [
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"### Task 5\n",
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"\n",
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"Analyze the evaluation-time scaling of your $\\text{FFT(yk)}$ function with the help of the timeit module and compare it to the scaling of the $\\text{DFT(yk)}$ function."
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]
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},
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{
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"cell_type": "code",
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"metadata": {
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"grade": true,
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},
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"outputs": [],
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"source": [
|
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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||||
"cell_type": "code",
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