recursive_filters/quantization_of_variables.ipynb

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    "# Realization of Recursive Filters\n",
    "\n",
    "*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 [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantization of Variables and Operations\n",
    "\n",
    "As for [non-recursive filters](../nonrecursive_filters/quantization_effects.ipynb#Quantization-Effects), the practical realization of recursive filters may suffer from the quantization of variables and algebraic operations. The effects of [coefficient quantization](quantization_of_coefficients.ipynb) were already discussed. This section takes a look at the quantization of variables.  We limit the investigations to the recursive part of a second-order section (SOS), since any recursive filter of order $N \\geq 2$ can be [decomposed into SOSs](cascaded_structures.ipynb).\n",
    "\n",
    "The computation of the output signal $y[k] = \\mathcal{H}\\{ x[k] \\}$ by a difference equation involves a number of multiplications and additions. As discussed already for [non-recursive filters](../nonrecursive_filters/quantization_effects.ipynb#Quantization-of-Signals-and-Operations), multiplying two numbers in a binary representation (e.g. [two's complement](https://en.wikipedia.org/wiki/Two's_complement) or [floating point](https://en.wikipedia.org/wiki/Floating_point)) requires requantization of the result to keep the word length constant. The addition of two numbers may fall outside the maximum/minimum values of the representation and may suffer from clipping.\n",
    "\n",
    "The resulting round-off and clipping errors depend on the number and sequence of algebraic operations. These depend on the structure used for implementation of the SOSs. For ease of illustration we limit our discussion to the [direct form I and II](direct_forms.ipynb). Similar insights can be achieved in a similar manner for other structures."
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    "### Analysis of Round-Off Errors\n",
    "\n",
    "Round-off errors are a consequence of reducing the word length after a multiplication. In order to investigate the influence of these errors on a recursive filter, the statistical model for [round-off errors in multipliers](../nonrecursive_filters/quantization_effects.ipynb#Model-for-round-off-errors-in-multipliers) as introduced for non-recursive filters is used. We furthermore neglect clipping. \n",
    "\n",
    "The difference equation for the recursive part of a SOS realized in direct form I or II is given as\n",
    "\n",
    "\\begin{equation}\n",
    "y[k] =  x[k] - a_1 \\, y[k-1] - a_2 \\, y[k-2]\n",
    "\\end{equation}\n",
    "\n",
    "where $a_0 = 1$, $a_1$ and $a_2$ denote the coefficients of the recursive part. Introducing the requantization after the multipliers into the difference equation yields the output signal $y_Q[k]$\n",
    "\n",
    "\\begin{equation}\n",
    "y_Q[k] = x[k] - \\mathcal{Q} \\{ a_{1} \\, y[k-1] \\} - \\mathcal{Q} \\{ a_{2} \\, y[k-2] \\}\n",
    "\\end{equation}\n",
    "\n",
    "where $\\mathcal{Q} \\{ \\cdot \\}$ denotes the requantizer. Requantization is a non-linear process which results in a requantization error. If the value to be requantized is much larger that the quantization step $Q$, the average statistical properties of this error can be modeled as additive uncorrelated white noise. Introducing the error into above difference equation gives\n",
    "\n",
    "\\begin{equation}\n",
    "y_Q[k] = x[k] - a_1 \\, y[k-1] - e_1[k] - a_2 \\, y[k-2] - e_2[k]\n",
    "\\end{equation}\n",
    "\n",
    "where the two white noise sources $e_1[k]$ and $e_2[k]$ are assumed to be uncorrelated to each other. This difference equation can be split into a set of two difference equations\n",
    "\n",
    "\\begin{align}\n",
    "y_Q[k] &= y[k] + e[k] \\\\\n",
    "y[k] &=  x[k] - a_1 \\, y[k-1] - a_2 \\, y[k-2] \\\\\n",
    "e[k] &= - e_1[k] - e_2[k] - a_1 \\, e[k-1] - a_2 \\, e[k-2]\n",
    "\\end{align}\n",
    "\n",
    "The first difference equation computes the desired output signal $y[k]$ as a result of the input signal $x[k]$. The second one the additive error $e[k]$ due to requantization as a result of the requantization error $- (e_1[k] + e_2[k])$ injected into the recursive filter.\n",
    "The power spectral density (PSD) $\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ of the error $e[k]$ is then given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = | H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})|^2 \\cdot (\\Phi_{e_1 e_1}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + \\Phi_{e_2 e_2}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}))\n",
    "\\end{equation}\n",
    "\n",
    "According to the model for the requantization errors, their PSDs are given as $\\Phi_{e_1 e_1}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\Phi_{e_2 e_2}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{Q^2}{12}$. Introducing this together with the transfer function of the SOS yields\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\left| \\frac{1}{1 + a_1 \\, \\mathrm{e}^{\\,-\\mathrm{j}\\,\\Omega} + a_2 \\, \\mathrm{e}^{\\,-\\mathrm{j}\\,2\\,\\Omega}} \\right|^2 \\cdot \\frac{Q^2}{6}\n",
    "\\end{equation}"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Round-off error of a SOS\n",
    "\n",
    "The following example evaluates the error $e[k] = y_Q[k] - y[k]$ for a SOS which only consists of a recursive part. The desired system response $y[k]$ is computed numerically by floating point operations with double precision, $y_Q[k]$ is computed by applying a uniform midtread quantizer after the multiplications. The system is excited by uniformly distributed white noise. Besides the PSD $\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$, the signal-to-noise ratio (SNR) $10 \\cdot \\log_{10} \\left( \\frac{\\sigma_y^2}{\\sigma_e^2} \\right)$ in dB of the filter is evaluated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SNR due to requantization: 45.442600 dB\n"
     ]
    },
    {
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.signal as sig\n",
    "\n",
    "\n",
    "N = 8192  # length of signals\n",
    "w = 8  # wordlength for requantization of multiplications\n",
    "\n",
    "\n",
    "def uniform_midtread_quantizer(x):\n",
    "    \"\"\"Uniform mid-tread quantizer w/o limiter.\"\"\"\n",
    "    return Q * np.floor(x / Q + 1 / 2)\n",
    "\n",
    "\n",
    "def no_quantizer(x):\n",
    "    \"\"\"Dummy quantizer.\"\"\"\n",
    "    return x\n",
    "\n",
    "\n",
    "def sos_df1(x, a, requantize=None):\n",
    "    \"\"\"Realization of a recursive SOS with round-off of multiplications.\"\"\"\n",
    "    y = np.zeros(len(x) + 2)  # initial value appended\n",
    "    for k in range(len(x)):\n",
    "        y[k] = x[k] - requantize(a[1] * y[k - 1]) - requantize(a[2] * y[k - 2])\n",
    "\n",
    "    return y[0:-2]\n",
    "\n",
    "\n",
    "# cofficients of the SOS\n",
    "p = 0.90 * np.array([np.exp(1j * np.pi / 3), np.exp(-1j * np.pi / 3)])\n",
    "a = np.poly(p)\n",
    "# quantization step\n",
    "Q = 1 / (2 ** (w - 1))\n",
    "\n",
    "# compute input signal\n",
    "x = np.random.uniform(low=-1, high=1, size=N)\n",
    "# compute output signals w and w/o requantization\n",
    "yQ = sos_df1(x, a, requantize=uniform_midtread_quantizer)\n",
    "y = sos_df1(x, a, requantize=no_quantizer)\n",
    "# compute requantization error\n",
    "e = yQ - y\n",
    "# Signal-to-noise ratio\n",
    "SNR = 10 * np.log10(np.var(y) / np.var(e))\n",
    "print(\"SNR due to requantization: %f dB\" % SNR)\n",
    "\n",
    "# estimate PSD of requantization error\n",
    "nf, Pxx = sig.welch(e, window=\"hamming\", nperseg=256, noverlap=128)\n",
    "Pxx = 0.5 * Pxx  # due to normalization in scipy.signal\n",
    "Om = 2 * np.pi * nf\n",
    "# compute frequency response of system\n",
    "w, H = sig.freqz([1, 0, 0], a)\n",
    "\n",
    "\n",
    "# plot results\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.plot(Om, Pxx / Q**2 * 12, \"b\", label=r\"$\\hat{\\Phi}_{ee}(e^{j \\Omega})$\")\n",
    "plt.plot(w, np.abs(H) ** 2 * 2, \"g\", label=r\"$|H(e^{j \\Omega})|^2$\")\n",
    "plt.title(\"Estimated PSD and transfer function of requantization noise\")\n",
    "plt.xlabel(r\"$\\Omega$\")\n",
    "plt.ylabel(r\"$Q^2/12$\")\n",
    "plt.axis([0, np.pi, 0, 100])\n",
    "plt.legend()\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Small Limit Cycles\n",
    "\n",
    "Besides the requantization noise, recursive filters may be subject to periodic oscillations present at the output. These undesired oscillations are termed *limit cycles*. Small limit cycles emerge from the additive round-off noise due to requantization after a multiplication. The feedback in a recursive filter leads to a feedback of the requantization noise. This may lead to a periodic output signal with an amplitude range of some quantization steps $Q$, even after the input signal is zero. The presence, amplitude and frequency of small limit cycles depends on the location of poles and the structure of the filter. A detailed treatment of this phenomenon is beyond the scope of this notebook and can be found in the literature."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Small limit cycles of a SOS\n",
    "\n",
    "The following example illustrates small limit cycles for the system investigated in the previous example. The input signal is uniformly distributed white noise till time-index $k=256$ and zero for the remainder."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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wIDYxMyA2MzUgNTUwIDYzNSA2MTUgMzUyIDYzNSA2MzQgMjc4IDI3OCA1NzkgMjc4IDk3NCA2MzQgNjEyCjYzNSA2MzUgNDExIDUyMSAzOTIgNjM0IDU5MiA4MTggNTkyIDU5MiA1MjUgNjM2IDMzNyA2MzYgODM4IDYwMCA2MzYgNjAwIDMxOAozNTIgNTE4IDEwMDAgNTAwIDUwMCA1MDAgMTM1MCA2MzUgNDAwIDEwNzAgNjAwIDY4NSA2MDAgNjAwIDMxOCAzMTggNTE4IDUxOAo1OTAgNTAwIDEwMDAgNTAwIDEwMDAgNTIxIDQwMCAxMDI4IDYwMCA1MjUgNjExIDMxOCA0MDEgNjM2IDYzNiA2MzYgNjM2IDMzNwo1MDAgNTAwIDEwMDAgNDcxIDYxNyA4MzggMzYxIDEwMDAgNTAwIDUwMCA4MzggNDAxIDQwMSA1MDAgNjM2IDYzNiAzMTggNTAwCjQwMSA0NzEgNjE3IDk2OSA5NjkgOTY5IDUzMSA2ODQgNjg0IDY4NCA2ODQgNjg0IDY4NCA5NzQgNjk4IDYzMiA2MzIgNjMyIDYzMgoyOTUgMjk1IDI5NSAyOTUgNzc1IDc0OCA3ODcgNzg3IDc4NyA3ODcgNzg3IDgzOCA3ODcgNzMyIDczMiA3MzIgNzMyIDYxMSA2MDgKNjMwIDYxMyA2MTMgNjEzIDYxMyA2MTMgNjEzIDk5NSA1NTAgNjE1IDYxNSA2MTUgNjE1IDI3OCAyNzggMjc4IDI3OCA2MTIgNjM0CjYxMiA2MTIgNjEyIDYxMiA2MTIgODM4IDYxMiA2MzQgNjM0IDYzNCA2MzQgNTkyIDYzNSA1OTIgXQplbmRvYmoKMTcgMCBvYmoKPDwgL1EgMTggMCBSIC9rIDE5IDAgUiAveSAyMCAwIFIgPj4KZW5kb2JqCjI1IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjE1ID4+CnN0cmVhbQp4nDVQS64DMQjb5xS+QKVAEgjnmap6m95/+zCoixHW4B/x65g4hpcs+BL4VrxlrNgwU3zHUi2kdmDrQHXhXMUz5AqOXohN7LTgXLa5SaSuIEMiSiPbckOXULQpOiV/b5pbuim3J02abtPKwNTLkLMjiBhKBktQ07VowqLtqfid8oy/OosXfscOK3Tc4dNTEfU9w2c+iEu9xvao2ZlEui/IkNOaSwlNNNnR78gUzjw3pJDOIEFSXIJrZZD96ZejAxIwMbcsQHZXopwlacfSvzN40ucfYVxUjQplbmRzdHJlYW0KZW5kb2JqCjI2IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzA3ID4+CnN0cmVhbQp4nD2SS24DMQxD9z6FLhDA+tme86Qoupjef9snJemKHNkWRWqWukxZUx6QNJOEf+nwcLGd8jtsz2Zm4Fqil4nllOfQFWLuonzZzEZdWSfF6oRmOrfoUTkXBzZNqp+rLKXdLngO1yaeW/YRP7zQoB7UNS4JN3RXo2UpNGOq+3/Se/yMMuBqTF1sUqt7HzxeRFXo6AdHiSJjlxfn40EJ6UrCaFqIlXdFA0Hu8rTKewnu295qyLIHqZjOOylmsOt0Ui5uF4chHsjyqPDlo9hrQs/4sCsl9EjYhjNyJ+5oxubUyOKQ/t6NBEuPrmgh8+CvbtYuYLxTOkViZE5yrGmLVU73UBTTucO9DBD1bEVDKXOR1epfw84La5ZsFnhK+gUeo90mSw5W2duoTu+tPNnQ9x9a13QfCmVuZHN0cmVhbQplbmRvYmoKMjcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA3MyA+PgpzdHJlYW0KeJwztjRQMFCwMFPQNTQ2VDCyNFYwNzNQSDHkAgqBWLlcMLEcMMvMEsQyNDdDYumaGUJlkVgg43K4YAbnwMzL4crgSgMAHokWlQplbmRzdHJlYW0KZW5kb2JqCjI4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggNjkgPj4Kc3RyZWFtCnicM7Y0UDBQsDRX0DU0NlQwNjBRMDczUEgx5IIxc8EssGwOF0wdhGUGYhgZmiCxzIDGgSXhDJAZOXDTcrgyuNIA+qkWRQplbmRzdHJlYW0KZW5kb2JqCjI5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjQ5ID4+CnN0cmVhbQp4nD1QO45EIQzrOYUv8CTyI3AeRqstZu/frgOaKVBMfrYzJNARgUcMMZSv4yWtoK6Bv4tC8W7i64PCIKtDUiDOeg+IdOymNpETOh2cMz9hN2OOwEUxBpzpdKY9ByY5+8IKhHMbZexWSCeJqiKO6jOOKZ4qe594FiztyDZbJ5I95CDhUlKJyaWflMo/bcqUCjpm0QQsErngZBNNOMu7SVKMGZQy6h6mdiJ9rDzIozroZE3OrCOZ2dNP25n4HHC3X9pkTpXHdB7M+Jy0zoM5Fbr344k2B02N2ujs9xNpKi9Sux1anX51EpXdGOcYEpdnfxnfZP/5B/6HWiIKZW5kc3RyZWFtCmVuZG9iagozMCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDEzNiA+PgpzdHJlYW0KeJxNj0EOAzEIA+95hZ9AIEB4z1ZVD9v/X0vYdtMLHsmAbFEGgSWHeIcb4dHbD99FNhVn45xfUiliIZhPcJ8wUxyNKXfyY4+AcZRqLKdoeF5Lzk3DFy13Ey2lrZeTGW+47pf3R5VtkQ1Fzy0LQtdskvkygQd8GJhHdeNppcfd9myv9vwAzmw0SQplbmRzdHJlYW0KZW5kb2JqCjMxIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggOTQgPj4Kc3RyZWFtCnicRY3BEcAgCAT/VEEJCgraTyaTh/b/jRAyfGDnDu6EBQu2eUYfBZUmXhVYB0pj3FCPQL3hci3J3AUPcCd/2tBUnJbTd2mRSVUp3KQSef8OZyaQqHnRY533C2P7IzwKZW5kc3RyZWFtCmVuZG9iagozMiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM0MSA+PgpzdHJlYW0KeJxFUktuRDEI279TcIFI4ZeQ87Squpjef1ubTNXN4AlgbHjLU6ZkyrC5JSMk15RPfSJDrKb8NHIkIqb4SQkFdpWPx2tLrI3skagUn9rx47H0RqbZFVr17tGlzaJRzcrIOcgQoZ4VurJ71A7Z8HpcSLrvlM0hHMv/UIEsZd1yCiVBW9B37BHfDx2ugiuCYbBrLoPtZTLU//qHFlzvffdixy6AFqznvsEOAKinE7QFyBna7jYpaABVuotJwqPyem52omyjVen5HAAzDjBywIglWx2+0d4Aln1d6EWNiv0rQFFZQPzI1XbB3jHJSHAW5gaOvXA8xZlwSzjGAkCKveIYevAl2OYvV66ImvAJdbpkL7zCntrm50KTCHetAA5eZMOtq6Oolu3pPIL2Z0VyRozUizg6IZJa0jmC4tKgHlrjXDex4m0jsblX3+4f4ZwvXPbrF0vshMQKZW5kc3RyZWFtCmVuZG9iagozMyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDcyID4+CnN0cmVhbQp4nDMyt1AwULA0ARKGFiYK5mYGCimGXEC+qYm5Qi4XSAzEygGzDIC0JZyCiGeAmCBtEMUgFkSxmYkZRB2cAZHL4EoDACXbFskKZW5kc3RyZWFtCmVuZG9iagozNCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDQ3ID4+CnN0cmVhbQp4nDMyt1AwULA0ARKGFiYK5mYGCimGXJYQVi4XTCwHzALRlnAKIp7BlQYAuWcNJwplbmRzdHJlYW0KZW5kb2JqCjM1IDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM5Ci9TdWJ0eXBlIC9Gb3JtIC9UeXBlIC9YT2JqZWN0ID4+CnN0cmVhbQp4nOMyNDBTMDY1VcjlMjc2ArNywCwjcyMgCySLYEFkM7jSABXzCnwKZW5kc3RyZWFtCmVuZG9iagozNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE2MyA+PgpzdHJlYW0KeJxFkDsSAyEMQ3tOoSP4IwM+z2YyKTb3b2PYbFLA01ggg7sTgtTagonogoe2Jd0F760EZ2P86TZuNRLkBHWAVqTjaJRSfbnFaZV08Wg2cysLrRMdZg56lKMZoBA6Fd7touRypu7O+UNw9V/1v2LdOZuJgcnKHQjN6lPc+TY7orq6yf6kx9ys134r7FVhaVlLywm3nbtmQAncUznaqz0/Hwo69gplbmRzdHJlYW0KZW5kb2JqCjM3IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjE4ID4+CnN0cmVhbQp4nD1QuY0EMQzLXYUaWMB67alnFotLpv/0SPn2ItEWRVIqNZmSKS91lCVZU946fJbEDnmG5W5kNiUqRS+TsCX30ArxfYnmFPfd1ZazQzSXaDl+CzMqqhsd00s2mnAqE7qg3MMz+g1tdANWhx6xWyDQpGDXtiByxw8YDMGZE4siDEpNBv+uco+fXosbPsPxQxSRkg7mNf9Y/fJzDa9TjyeRbm++4l6cqQ4DERySmrwjXVixLhIRaTVBTc/AWi2Au7de/hu0I7oMQPaJxHGaUo6hv2twpc8v5SdT2AplbmRzdHJlYW0KZW5kb2JqCjM4IDAgb2JqCjw8IC9GaWx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   "source": [
    "# compute input signal\n",
    "x = np.random.uniform(low=-1, high=1, size=256)\n",
    "x = np.concatenate((x, np.zeros(1024)))\n",
    "# compute output signal\n",
    "yQ = sos_df1(x, a, requantize=uniform_midtread_quantizer)\n",
    "\n",
    "# plot results\n",
    "np.seterr(divide=\"ignore\")\n",
    "plt.figure(figsize=(10, 3))\n",
    "plt.plot(20 * np.log10(np.abs(yQ)))\n",
    "plt.title(\"Level of output signal\")\n",
    "plt.xlabel(r\"$k$\")\n",
    "plt.ylabel(r\"$|y_Q[k]|$ in dB\")\n",
    "plt.grid()\n",
    "\n",
    "plt.figure(figsize=(10, 3))\n",
    "k = np.arange(1000, 1050)\n",
    "plt.stem(k, yQ[k] / Q)\n",
    "plt.title(\"Output signal for zero input\")\n",
    "plt.xlabel(r\"$k$\")\n",
    "plt.ylabel(r\"$y_Q[k] / Q$ \")\n",
    "plt.axis([k[0], k[-1], -3, 3])\n",
    "plt.grid()"
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   "source": [
    "**Exercise**\n",
    "\n",
    "* Estimate the period of the small limit cycles. How is it related to the poles of the system?\n",
    "* What amplitude range is spanned?\n",
    "\n",
    "Solution: The period of the small limit cycles can be estimated from the second illustration as $P = 6$. The normalized frequency of a harmonic exponential signal with the same periodicity is given as $\\Omega_0 = \\frac{2 \\pi}{P} = \\frac{\\pi}{3}$. The poles of the system can be extracted from the code of the first example as $z_{\\infty 0,1} = 0.9 \\cdot e^{\\pm j \\frac{\\pi}{3}}$. The periodicity of the small limit cycles is hence linked to the normalized frequency of the poles. The amplitude range spanned by the small limit cycles is $\\pm 2 Q$."
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   "source": [
    "### Large Limit Cycles\n",
    "\n",
    "Large limit cycles are periodic oscillations of a recursive filter due to overflows in the multiplications/additions. As for small limit cycles, large limit cycles may be present even after the input signal is zero. Their level is typically in the range of the minimum/maximum value of the requantizer. Large limit cycles should therefore be avoided in a practical implementation. The presence of large limit cycles depends on the scaling of input signal and coefficients, as well as the strategy used to cope for clipping. Amongst others, they can be avoided by proper scaling of the coefficients to prevent overflow. Again, a detailed treatment of this phenomenon is beyond the scope of this notebook and can be found in the literature."
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Large limit cycles of a SOS\n",
    "\n",
    "The following example illustrates large limit cycles for the system investigated in the first example. In order to trigger large limit cycles, the coefficients of the filter have been doubled. The input signal is uniformly distributed white noise till time-index $k=256$ and zero for the remainder."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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   "source": [
    "def uniform_midtread_quantizer(x, xmin=1):\n",
    "    \"\"\"Uniform mid-tread quantizer with limiter.\"\"\"\n",
    "    # limiter\n",
    "    x = np.copy(x)\n",
    "    if x <= -xmin:\n",
    "        x = -1\n",
    "    if x > xmin - Q:\n",
    "        x = 1 - Q\n",
    "    # linear uniform quantization\n",
    "    xQ = Q * np.floor(x / Q + 1 / 2)\n",
    "\n",
    "    return xQ\n",
    "\n",
    "\n",
    "# compute input signal\n",
    "x = np.random.uniform(low=-1, high=1, size=256)\n",
    "x = np.concatenate((x, np.zeros(1024)))\n",
    "# compute output signal\n",
    "yQ = sos_df1(x, 2 * a, requantize=uniform_midtread_quantizer)\n",
    "\n",
    "# plot results\n",
    "plt.figure(figsize=(10, 3))\n",
    "plt.plot(20 * np.log10(np.abs(yQ)))\n",
    "plt.title(\"Level of output signal\")\n",
    "plt.xlabel(r\"$k$\")\n",
    "plt.ylabel(r\"$|y_Q[k]|$ in dB\")\n",
    "plt.grid()\n",
    "\n",
    "plt.figure(figsize=(10, 3))\n",
    "k = np.arange(1000, 1050)\n",
    "plt.stem(k, yQ[k])\n",
    "plt.title(\"Output signal for zero input\")\n",
    "plt.xlabel(r\"$k$\")\n",
    "plt.ylabel(r\"$y_Q[k]$ \")\n",
    "plt.grid()"
   ]
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   "source": [
    "**Exercise**\n",
    "\n",
    "* Determine the period of the large limit cycles. How is it related to the poles of the system?\n",
    "\n",
    "Solution: The period of the large limit cycles can be estimated from the second illustration as $P = 6$. The normalized frequency of a harmonic exponential signal with the same periodicity is given as $\\Omega_0 = \\frac{2 \\pi}{P} = \\frac{\\pi}{3}$. The poles of the system can be extracted from the code of the first example as $z_{\\infty 0,1} = 0.9 \\cdot e^{\\pm j \\frac{\\pi}{3}}$. The periodicity of the large limit cycles is hence linked to the normalized frequency of the poles."
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "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*."
   ]
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