Relocated images, equation environment, copyright

Author: spors

images/systems-1.png

@@ -6,7 +6,7 @@
    "source": [
     "# Realization of Non-Recursive Filters\n",
     "\n",
-    "*This jupyter/Python notebook is part of a [collection of notebooks](../index.ipynb) in the masters module [Digital Signal Processing](http://www.int.uni-rostock.de/Digitale-Signalverarbeitung.48.0.html), Comunications Engineering, Universität Rostock. Please direct questions and suggestions to <mailto:[email protected]>.*"
+    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [[email protected]](mailto:[email protected]).*"
    ]
   },
   {
@@ -31,11 +31,13 @@
     "\n",
     "An LTI system can be characterized completely by its impulse response $h[k]$ \n",
     "\n",
-    "![Linear time-invariant system](../images/systems-2.png)\n",
+    "![Linear time-invariant system](LTI_system_td.png)\n",
     "\n",
     "The output signal $y[k]$ is given by (linear) convolution of the input signal $x[k]$ with the impulse response $h[k]$\n",
     "\n",
-    "$$ y[k] = x[k] * h[k] = \\sum_{\\kappa = -\\infty}^{\\infty} x[\\kappa] \\; h[k-\\kappa] $$\n",
+    "\\begin{equation}\n",
+    "y[k] = x[k] * h[k] = \\sum_{\\kappa = -\\infty}^{\\infty} x[\\kappa] \\; h[k-\\kappa]\n",
+    "\\end{equation}\n",
     "\n",
     "Two aspects of this representation become evident when inspecting above equation:\n",
     "\n",
@@ -45,11 +47,15 @@
     "\n",
     "The second aspect prohibits a practical realization. In order to be able to realize a non-recursive filter by convolution, the output at time-instant $k$ should only depend on the input signal $x[k]$ up to time-index $k$\n",
     "\n",
-    "$$ y[k] = \\sum_{\\kappa = -\\infty}^{k} x[\\kappa] \\; h[k-\\kappa] $$\n",
+    "\\begin{equation}\n",
+    "y[k] = \\sum_{\\kappa = -\\infty}^{k} x[\\kappa] \\; h[k-\\kappa]\n",
+    "\\end{equation}\n",
     "\n",
     "This is the case when the impulse response is causal, hence when $h[k] = 0$ for $k<0$. However, this still requires knowledge of the input signal for all past time-instants. If we further assume that the input signal is causal, $x[k] = 0$ for $k<0$, we get\n",
     "\n",
-    "$$ y[k] = \\sum_{\\kappa = 0}^{k} x[\\kappa] \\; h[k-\\kappa] $$"
+    "\\begin{equation}\n",
+    "y[k] = \\sum_{\\kappa = 0}^{k} x[\\kappa] \\; h[k-\\kappa]\n",
+    "\\end{equation}"
    ]
   },
   {
@@ -60,11 +66,15 @@
     "\n",
     "Many practical systems have an impulse response of finite length $N$ or can be approximated by an impulse response of finite length\n",
     "\n",
-    "$$ h_N[k] = \\begin{cases} h[k] & \\text{ for } 0 \\leq k < N \\\\ 0 & \\text{ otherwise} \\end{cases} $$\n",
+    "\\begin{equation}\n",
+    "h_N[k] = \\begin{cases} h[k] & \\text{ for } 0 \\leq k < N \\\\ 0 & \\text{ otherwise} \\end{cases}\n",
+    "\\end{equation}\n",
     "\n",
     "Such an impulse response is denoted as [*finite impulse response*](https://en.wikipedia.org/wiki/Finite_impulse_response) (FIR). Introducing $h_N[k]$ into above sum and rearranging terms yields\n",
     "\n",
-    "$$ y[k] = \\sum_{\\kappa = 0}^{k} x[\\kappa] \\; h_N[k-\\kappa] = \\sum_{\\kappa = 0}^{N-1} h_N[\\kappa] \\; x[k-\\kappa]$$\n",
+    "\\begin{equation}\n",
+    "y[k] = \\sum_{\\kappa = 0}^{k} x[\\kappa] \\; h_N[k-\\kappa] = \\sum_{\\kappa = 0}^{N-1} h_N[\\kappa] \\; x[k-\\kappa]\n",
+    "\\end{equation}\n",
     "\n",
     "Hence for a causal input signal $x[k]$ and a FIR the output of the system can be computed by a finite number of operations. \n",
     "\n",
@@ -77,17 +87,7 @@
    "source": [
     "**Copyright**\n",
     "\n",
-    "<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
-    "  <a rel=\"license\"\n",
-    "     href=\"http://creativecommons.org/publicdomain/zero/1.0/\">\n",
-    "    <img src=\"http://i.creativecommons.org/p/zero/1.0/88x31.png\" style=\"border-style: none;\" alt=\"CC0\" />\n",
-    "  </a>\n",
-    "  <br />\n",
-    "  To the extent possible under law,\n",
-    "  <span rel=\"dct:publisher\" resource=\"[_:publisher]\">the person who associated CC0</span>\n",
-    "  with this work has waived all copyright and related or neighboring\n",
-    "  rights to this work.\n",
-    "</p>"
+    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples, 2016*."
    ]
   }
  ],

nonrecursive_filters/LTI_system_td.png

@@ -6,7 +6,7 @@
    "source": [
     "# Realization of Non-Recursive Filters\n",
     "\n",
-    "*This jupyter/Python notebook is part of a [collection of notebooks](../index.ipynb) in the masters module [Digital Signal Processing](http://www.int.uni-rostock.de/Digitale-Signalverarbeitung.48.0.html), Comunications Engineering, Universität Rostock. Please direct questions and suggestions to <mailto:[email protected]>.*"
+    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [[email protected]](mailto:[email protected]).*"
    ]
   },
   {
@@ -28,19 +28,27 @@
     "\n",
     "The output signal $y[k]$ of a non-recursive filter with a finite impulse response (FIR) $h[k]$ of length $N$ is given as\n",
     "\n",
-    "$$ y[k] = h[k] * x[k] = \\sum_{\\kappa = 0}^{N-1} h[\\kappa] \\; x[k - \\kappa] $$\n",
+    "\\begin{equation}\n",
+    "y[k] = h[k] * x[k] = \\sum_{\\kappa = 0}^{N-1} h[\\kappa] \\; x[k - \\kappa]\n",
+    "\\end{equation}\n",
     "\n",
     "where $x[k]$ denotes the input signal. The quantized impulse response $h_Q[k]$ (quantized filter coefficients) is yielded by quantizing the impulse response $h[k]$\n",
     "\n",
-    "$$ h_Q[k] = \\mathcal{Q} \\{ h[k] \\} = h[k] + e[k] $$\n",
+    "\\begin{equation}\n",
+    "h_Q[k] = \\mathcal{Q} \\{ h[k] \\} = h[k] + e[k]\n",
+    "\\end{equation}\n",
     "\n",
     "where $e[k] = h_Q[k] - h[k]$ denotes the [quantization error](../quantization/introduction.ipynb#Model-of-the-Quantization-Process). Introducing $h_Q[k]$ into above equation and rearranging results in\n",
     "\n",
-    "$$ y_Q[k] = \\sum_{\\kappa = 0}^{N-1} h[k] \\; x[k - \\kappa] + \\sum_{\\kappa = 0}^{N-1} e[k] \\; x[k - \\kappa] $$\n",
+    "\\begin{equation}\n",
+    "y_Q[k] = \\sum_{\\kappa = 0}^{N-1} h[k] \\; x[k - \\kappa] + \\sum_{\\kappa = 0}^{N-1} e[k] \\; x[k - \\kappa]\n",
+    "\\end{equation}\n",
     "\n",
     "The input signal $x[k]$ is filtered by the quantization noise $e[k]$ and superimposed to the desired output of the filter. The overall transfer function $H_Q(e^{j \\Omega})$ of the filter with quantized filter coefficients is given as\n",
     "\n",
-    "$$ H_Q(e^{j \\Omega}) = \\sum_{k=0}^{N-1} h[k] \\; e^{-j \\Omega k} + \\sum_{k=0}^{N-1} e[k] \\; e^{-j \\Omega k}  = H(e^{j \\Omega}) + E(e^{j \\Omega}) $$\n",
+    "\\begin{equation}\n",
+    "H_Q(e^{j \\Omega}) = \\sum_{k=0}^{N-1} h[k] \\; e^{-j \\Omega k} + \\sum_{k=0}^{N-1} e[k] \\; e^{-j \\Omega k}  = H(e^{j \\Omega}) + E(e^{j \\Omega})\n",
+    "\\end{equation}\n",
     "\n",
     "Hence, the quantization of filter coefficients results in a linear distortion of the desired frequency response. To some extend this distortion can be incorporated into the design of the filter. However, the magnitude of the quantization error $| E(e^{j \\Omega}) |$ cannot get arbitrarily small for a finite quantization step $Q$. This limits the achievable attenuation of a digital filter with quantized coefficients. It is therefore important to normalize the filter coefficients $h[k]$ before quantization in order to keep the relative power of the quantization noise small."
    ]
@@ -145,11 +153,15 @@
     "\n",
     "As outlined above, multipliers require a requantization of the result in order to keep the wordlength constant. The multiplication of a quantized signal $x_Q[k]$ with a quantized factor $a_Q$ can be written as\n",
     "\n",
-    "$$ y_Q[k] = \\mathcal{Q} \\{ a_Q \\cdot x_Q[k] \\} = a_Q \\cdot x_Q[k] + e[k]$$\n",
+    "\\begin{equation}\n",
+    "y_Q[k] = \\mathcal{Q} \\{ a_Q \\cdot x_Q[k] \\} = a_Q \\cdot x_Q[k] + e[k]\n",
+    "\\end{equation}\n",
     "\n",
     "where the round-off error $e[k]$ is defined as\n",
     "\n",
-    "$$ e[k] = y_Q[k] - a_Q \\cdot x_Q[k] $$\n",
+    "\\begin{equation}\n",
+    "e[k] = y_Q[k] - a_Q \\cdot x_Q[k]\n",
+    "\\end{equation}\n",
     "\n",
     "This leads to the following model of a multiplier including round-off effects\n",
     "\n",
@@ -169,7 +181,9 @@
     "\n",
     "The variance (power) of the round-off error is [derived from its PDF](../random_signals/important_distributions.ipynb#Uniform-Distribution) as\n",
     "\n",
-    "$$ \\sigma_e^2 = \\frac{Q^2}{12} $$"
+    "\\begin{equation}\n",
+    "\\sigma_e^2 = \\frac{Q^2}{12}\n",
+    "\\end{equation}"
    ]
   },
   {
@@ -180,19 +194,27 @@
     "\n",
     "Using above model of a multiplier and discarding clipping, a straightforward realization of the convolution with quantized signals would be to requantize after every multiplication \n",
     "\n",
-    "$$ y_Q[k] = \\sum_{\\kappa = 0}^{N-1} \\mathcal{Q} \\{ h_Q[\\kappa] \\; x_Q[k - \\kappa] \\} = \\sum_{\\kappa = 0}^{N-1} h_Q[\\kappa] \\; x_Q[k - \\kappa] + e[\\kappa]$$\n",
+    "\\begin{equation}\n",
+    "y_Q[k] = \\sum_{\\kappa = 0}^{N-1} \\mathcal{Q} \\{ h_Q[\\kappa] \\; x_Q[k - \\kappa] \\} = \\sum_{\\kappa = 0}^{N-1} h_Q[\\kappa] \\; x_Q[k - \\kappa] + e[\\kappa]\n",
+    "\\end{equation}\n",
     "\n",
     "The round-off errors for each multiplication are uncorrelated to each other. The overall power of the round-off error is then given as\n",
     "\n",
-    "$$ \\sigma_e^2 = N \\cdot \\frac{Q^2}{12}$$\n",
+    "\\begin{equation}\n",
+    "\\sigma_e^2 = N \\cdot \\frac{Q^2}{12}\n",
+    "\\end{equation}\n",
     "\n",
     "Many digital signal processors allow to perform the multiplications and additions in an internal register with double wordlength. In this case only the result has to be requantized \n",
     "\n",
-    "$$ y_Q[k] = \\mathcal{Q} \\left\\{ \\sum_{\\kappa = 0}^{N-1}  h_Q[\\kappa] \\; x_Q[k - \\kappa] \\right\\}$$\n",
+    "\\begin{equation}\n",
+    "y_Q[k] = \\mathcal{Q} \\left\\{ \\sum_{\\kappa = 0}^{N-1}  h_Q[\\kappa] \\; x_Q[k - \\kappa] \\right\\}\n",
+    "\\end{equation}\n",
     "\n",
     "and the power of the round-off noise in this case is\n",
     "\n",
-    "$$ \\sigma_e^2 =  \\frac{Q^2}{12} $$\n",
+    "\\begin{equation}\n",
+    "\\sigma_e^2 =  \\frac{Q^2}{12}\n",
+    "\\end{equation}\n",
     "\n",
     "It is evident that this realization is favorable due to the lower round-off noise, especially for filters with a large number $N$ of coefficients."
    ]
@@ -306,17 +328,7 @@
    "source": [
     "**Copyright**\n",
     "\n",
-    "<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
-    "  <a rel=\"license\"\n",
-    "     href=\"http://creativecommons.org/publicdomain/zero/1.0/\">\n",
-    "    <img src=\"http://i.creativecommons.org/p/zero/1.0/88x31.png\" style=\"border-style: none;\" alt=\"CC0\" />\n",
-    "  </a>\n",
-    "  <br />\n",
-    "  To the extent possible under law,\n",
-    "  <span rel=\"dct:publisher\" resource=\"[_:publisher]\">the person who associated CC0</span>\n",
-    "  with this work has waived all copyright and related or neighboring\n",
-    "  rights to this work.\n",
-    "</p>"
+    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples, 2016*."
    ]
   }
  ],

nonrecursive_filters/fast_convolution.ipynb

@@ -6,7 +6,7 @@
    "source": [
     "# Realization of Non-Recursive Filters\n",
     "\n",
-    "*This jupyter/Python notebook is part of a [collection of notebooks](../index.ipynb) in the masters module [Digital Signal Processing](http://www.int.uni-rostock.de/Digitale-Signalverarbeitung.48.0.html), Comunications Engineering, Universität Rostock. Please direct questions and suggestions to <mailto:[email protected]>.*"
+    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [[email protected]](mailto:[email protected]).*"
    ]
   },
   {
@@ -28,11 +28,15 @@
     "\n",
     "The [overlap-add algorithm](https://en.wikipedia.org/wiki/Overlap%E2%80%93add_method) is based on splitting the signal $x_L[k]$ into non-overlapping segments $x_p[k]$ of length $P$\n",
     "\n",
-    "$$ x_L[k] = \\sum_{p = 0}^{L/P - 1} x_p[k - p \\cdot P] $$\n",
+    "\\begin{equation}\n",
+    "x_L[k] = \\sum_{p = 0}^{L/P - 1} x_p[k - p \\cdot P]\n",
+    "\\end{equation}\n",
     "\n",
     "where the segments $x_p[k]$ are defined as\n",
     "\n",
-    "$$ x_p[k] = \\begin{cases} x_L[k + p \\cdot P] & \\text{ for } k=0,1,\\dots,P-1 \\\\ 0 & \\text{ otherwise} \\end{cases} $$\n",
+    "\\begin{equation}\n",
+    "x_p[k] = \\begin{cases} x_L[k + p \\cdot P] & \\text{ for } k=0,1,\\dots,P-1 \\\\ 0 & \\text{ otherwise} \\end{cases}\n",
+    "\\end{equation}\n",
     "\n",
     "Note that $x_L[k]$ might have to be zero-padded so that its total length is a multiple of the segment length $P$. Introducing the segmentation of $x_L[k]$ into the convolution yields\n",
     "\n",
@@ -215,21 +219,27 @@
     "\n",
     "This motivates to split the input signal $x_L[k]$ into overlapping segments of length $P$ where the $p$-th segment overlaps its preceding $(p-1)$-th segment by $N-1$ samples\n",
     "\n",
-    "$$ x_p[k] = \\begin{cases}\n",
+    "\\begin{equation}\n",
+    "x_p[k] = \\begin{cases}\n",
     "x_L[k + p \\cdot (P-N+1) - (N-1)] & \\text{ for } k=0,1, \\dots, P-1 \\\\\n",
     "0 & \\text{ otherwise}\n",
-    "\\end{cases} $$\n",
+    "\\end{cases}\n",
+    "\\end{equation}\n",
     "\n",
     "The part of the circular convolution $x_p[k] \\circledast h_N[k]$ of one segment $x_p[k]$ with the impulse response $h_N[k]$ that is equal to the linear convolution of both is given as\n",
     "\n",
-    "$$ y_p[k] = \\begin{cases}\n",
+    "\\begin{equation}\n",
+    "y_p[k] = \\begin{cases}\n",
     "x_p[k] \\circledast h_N[k] & \\text{ for } k=N-1, N, \\dots, P-1 \\\\\n",
     "0 & \\text{ otherwise}\n",
-    "\\end{cases} $$\n",
+    "\\end{cases}\n",
+    "\\end{equation}\n",
     "\n",
     "The output $y[k]$ is simply the concatenation of the $y_p[k]$\n",
     "\n",
-    "$$ y[k] = \\sum_{p=0}^{L/P - 1} y_p[k - p \\cdot (P-N+1) + (N-1)] $$\n",
+    "\\begin{equation}\n",
+    "y[k] = \\sum_{p=0}^{L/P - 1} y_p[k - p \\cdot (P-N+1) + (N-1)]\n",
+    "\\end{equation}\n",
     "\n",
     "The overlap-save algorithm is illustrated in the following diagram\n",
     "\n",
@@ -419,17 +429,7 @@
    "source": [
     "**Copyright**\n",
     "\n",
-    "<p xmlns:dct=\"http://purl.org/dc/terms/\">\n",
-    "  <a rel=\"license\"\n",
-    "     href=\"http://creativecommons.org/publicdomain/zero/1.0/\">\n",
-    "    <img src=\"http://i.creativecommons.org/p/zero/1.0/88x31.png\" style=\"border-style: none;\" alt=\"CC0\" />\n",
-    "  </a>\n",
-    "  <br />\n",
-    "  To the extent possible under law,\n",
-    "  <span rel=\"dct:publisher\" resource=\"[_:publisher]\">the person who associated CC0</span>\n",
-    "  with this work has waived all copyright and related or neighboring\n",
-    "  rights to this work.\n",
-    "</p>"
+    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples, 2016*."
    ]
   }
  ],
@@ -449,7 +449,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.5.0"
+   "version": "3.5.1"
   }
  },
  "nbformat": 4,