{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Cross Correlation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general form of the cross correlation with integration:\n",
    "\\begin{equation}\n",
    "{\\displaystyle (f\\star g)(\\tau )\\ = \\int _{-\\infty }^{\\infty }{{f^{\\star}(t)}}g(t+\\tau )\\,dt}\n",
    "\\end{equation}\n",
    "This can be written in discrete form as:\n",
    "\\begin{equation}\n",
    "{\\displaystyle (f\\star g)[n]\\ = \\sum _{m=-\\infty }^{\\infty }{{f^{\\star}[m]}}g[m+n]}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pylab\n",
    "from matplotlib import pyplot as plt\n",
    "pylab.rcParams['savefig.dpi'] = 300\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from gps_helper.prn import PRN\n",
    "from sk_dsp_comm import sigsys as ss\n",
    "from sk_dsp_comm import digitalcom as dc\n",
    "from caf_verilog.quantizer import quantize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Signals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "prn = PRN(10)\n",
    "prn2 = PRN(20)\n",
    "fs = 625e3\n",
    "Ns = fs / 125e3\n",
    "prn_seq = prn.prn_seq()\n",
    "prn_seq2 = prn2.prn_seq()\n",
    "prn_seq,b = ss.nrz_bits2(np.array(prn_seq), Ns)\n",
    "prn_seq2,b2 = ss.nrz_bits2(np.array(prn_seq2), Ns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Px,f = plt.psd(prn_seq, 2**12, Fs=fs)\n",
    "plt.plot(f, 10*np.log10(Px))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Autocorrelation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, lags = dc.xcorr(prn_seq, prn_seq, 100)\n",
    "plt.plot(lags, abs(r)) # r -> abs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Time Shifted Signals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, lags = dc.xcorr(np.roll(prn_seq, 50), prn_seq, 100)\n",
    "plt.plot(lags, abs(r))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, lags = dc.xcorr(np.roll(prn_seq, -50), prn_seq, 100)\n",
    "plt.plot(lags, abs(r))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## No Correlation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r_nc, lags_nc = dc.xcorr(prn_seq, prn_seq2, 100)\n",
    "plt.plot(lags_nc, abs(r_nc))\n",
    "plt.ylim([0, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculation Space Visualization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from caf_verilog.xcorr import size_visualization\n",
    "size_visualization(prn_seq[:10], prn_seq[:10], 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple Cross Correlation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from caf_verilog.xcorr import simple_xcorr\n",
    "r, lags = simple_xcorr(prn_seq, prn_seq, 100)\n",
    "plt.plot(lags, r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time Shifted Signals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, lags = simple_xcorr(prn_seq, np.roll(prn_seq, 50), 100)\n",
    "plt.plot(lags, abs(np.array(r)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, lags = simple_xcorr(prn_seq, np.roll(prn_seq, -50), 100)\n",
    "plt.plot(lags, abs(np.array(r)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### No Correlation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, lags = simple_xcorr(prn_seq, prn_seq2, 100)\n",
    "plt.plot(lags, abs(np.array(r)))\n",
    "plt.ylim([0, 5000])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dot Product Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To ensure the integration time is filled, the secondary or received signal must be twice the length of the reference signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from caf_verilog.sim_helper import sim_shift\n",
    "center = 300\n",
    "corr_length = 250\n",
    "shift = 25\n",
    "ref, rec = sim_shift(prn_seq, center, corr_length, shift=shift, padding=True)\n",
    "ref = np.array(ref)\n",
    "rec = np.array(rec)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f, axarr = plt.subplots(2, sharex=True, gridspec_kw={'hspace': 0})\n",
    "axarr[0].plot(ref.real)\n",
    "axarr[1].plot(rec.real)\n",
    "plt.xlabel(\"Sample Number\")\n",
    "plt.savefig('prn_seq.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from caf_verilog.xcorr import dot_xcorr\n",
    "ref = np.array(ref)\n",
    "rec = np.array(rec)\n",
    "rr = dot_xcorr(ref, rec)\n",
    "rr = np.array(rr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rxy, lags = dc.xcorr(ref, rec, corr_length)\n",
    "plt.plot(lags, abs(rxy))\n",
    "plt.xlabel(\"Center Offset (Samples)\")\n",
    "plt.grid();\n",
    "plt.savefig('xcorr.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.argmax(rxy) - corr_length"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ref, rec = sim_shift(prn_seq, center, corr_length, shift=shift)\n",
    "ref = np.array(ref)\n",
    "rec = np.array(rec)\n",
    "rr = dot_xcorr(ref, rec)\n",
    "rr = np.array(rr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, axs = plt.subplots(2, sharey=True)\n",
    "axs[0].plot(abs(rr))\n",
    "axs[0].grid(True)\n",
    "axs[1].plot(abs(rr))\n",
    "axs[1].set_xlim([80, 120])\n",
    "axs[1].set_xlabel('Inverse Center Offset (Samples)')\n",
    "axs[1].grid(True)\n",
    "fig.savefig('xcorr_250.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(corr_length / 2) - np.argmax(rr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from caf_verilog.xcorr import XCorr\n",
    "xc = XCorr(ref, rec, output_dir='.')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xc.gen_tb()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}