design/Frequency Accuracy.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# NCO Core\n",
    "\n",
    "The NCO core has a few stages - first, we have the frequency _divider_.  This is a simple counter that counts up by one each clock.  When this counter reaches the division ratio, it resets to zero, and triggers a pulse to the main counter.\n",
    "\n",
    "The main counter has width $m$ bits, of which the top $n$ are output.  Each time this receives a pulse, it increases not by one, but by a programmable frequency _multiplier_.\n",
    "\n",
    "# Frequency Accuracy\n",
    "\n",
    "Thus the NCO core has two ways to adjust the frequency - a frequency _multiplier_ and a frequency _divider_.  The frequency multiplier gives us linear control over the frequency, where the divider gives us inverse linear control.\n",
    "\n",
    "We can imagine a few schemes here \n",
    "\n",
    "  1. We rarely change the divider and usually control the multiplier\n",
    "  2. We rarely change the multiplier and usually control the divider\n",
    "  3. We change both all the time\n",
    " \n",
    "We can maintain the absolute best pitch accuracy by changing both often (scheme 3), but we can change frequencies fastest with scheme 1, since we can't adjust the divider quickly in a glitch-free manner (what if the divider's count is already past the new desired division ratio).  So, let's try to use scheme 1 - we usually control pitch with the multiplier, but then use the divider for coarse adjustments, say of octaves.  This is easy to calculate using bitwise-shifts, which is why we choose octaves rather than some other interval.\n",
    "\n",
    "## Linear Control Accuracy\n",
    "\n",
    "Let's calculate the accuracy if we only use linear control (multiplier) within an octave.  That is, we fix the frequency divider and only control the frequency through the linear multiplier.\n",
    "\n",
    "Basically, we have to choose an octave's worth of linear control - a number $k$ such that our frequency multiplier will vary between $2^k$ and $2^(k+1) - 1$.  Unfortunately, given a fixed total main counter width $m$, there's a trade-off in how to choose $k$.  \n",
    "\n",
    "### Accuracy/Aliasing trade-off\n",
    "\n",
    "To see this trade-off, let's fix $m$ to 32 and $n$ to 8.  I.e., the main counter is a 32-bit counter with 8 output bits.\n",
    "\n",
    "First consider just setting $k = 1$.  This will let our frequency vary between $2$ and $3$ - i.e., we'll only have 2 different frequencies per octave!  So, we have horrible frequency accuracy.\n",
    "\n",
    "But, if we go the other way, setting $k = m - n = 24$, the counter can vary from $2^{24}$ to $2^{25} - 1$, a range of $2^{24}$ or $16777216$ steps per octave!  That sounds great - Why wouldn't we do this?  The reason comes from the effective sampling rate.\n",
    "\n",
    "The effective sampling rate, or apparent sampling rate, is $\\frac{F_c}{F_{div}}$  Where $F_c$ is the clock frequency, and $F_{div}$ is the frequency divider.  The dividers and multipliers have to line up such that $ F = \\frac{F_c}{F_{div}} F_{mul}$ - i.e., the higher the multiplier, for the same output frequency, we have a lower effective sampling rate, which means more aliasing artifacts!  \n",
    "\n",
    "So, there's a trade-off between frequency accuracy on one side (large numbers of steps per samples), and aliasing artifacts on the other side.\n",
    "\n",
    "## Choosing $k$\n",
    "\n",
    "Given this trade-off, we want to choose $k$ to be the lowest number possible that will give us frequency errors we can tolerate.  Below we show how to calculate the maximum error for $k = 8$, $k = 12$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "error for k = 8: 6.5272934920950565\n",
      "error for k = 12: 0.40870577007929487\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import math\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def max_error_for_n_linear(n):\n",
    "    def closest_frequency(f_desired):\n",
    "        return np.round(f_desired * 2 ** (n - 1)) / 2 ** (n - 1)\n",
    "\n",
    "    cents = np.arange(1200)\n",
    "    freqs = 2 ** (cents / 1200)\n",
    "    closest_freqs = closest_frequency(freqs)\n",
    "    closest_cents = np.log2(closest_freqs) * 1200\n",
    "    cent_diffs = np.abs(closest_cents - cents)\n",
    "    return np.max(cent_diffs)\n",
    "\n",
    "for k in (8, 12):\n",
    "    print(\"error for k = {}: {}\".format(k, max_error_for_n_linear(k)))\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, looks like $k = 12$ is a good choice.  Below there's code to generate a lookup table:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'uint16_t PITCH_LOOKUP_TABLE[4096] = 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      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import math\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "# input format: an 0.m-bit fixed point number representing log2(t)\n",
    "# output format: an 1.(n-1)-bit fixed point number t\n",
    "def cpp_lookup_table(m, n):\n",
    "    size = 2 ** m;\n",
    "    points = np.arange(size) / size\n",
    "    lin_points = 2 ** points\n",
    "    rounded_points = np.round(2 ** (n-1) * lin_points)\n",
    "    rounded_point_strs = (str(int(f)) for f in rounded_points)\n",
    "    bit_size = 2 ** math.ceil(math.log2(n))\n",
    "    table = \"uint{}_t PITCH_LOOKUP_TABLE[{}] = {{{}}};\".format(\n",
    "            bit_size, size, \",\".join(rounded_point_strs))\n",
    "    return table\n",
    "\n",
    "cpp_lookup_table(12, 12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "2**12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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