From bef9f0c6eb4d75230896bcc050d531743e801d9f Mon Sep 17 00:00:00 2001
From: Mahdi Olfat <mahdiolfat@gmail.com>
Date: Tue, 5 Mar 2024 10:22:37 -0500
Subject: [PATCH] signal eigendecomposition

---
 examples/process_arma.ipynb              |  2 +-
 examples/signal_eigendecomposition.ipynb | 71 ++++++++++++++++++++++--
 2 files changed, 66 insertions(+), 7 deletions(-)

diff --git a/examples/process_arma.ipynb b/examples/process_arma.ipynb
index 866c4a2..07039bf 100644
--- a/examples/process_arma.ipynb
+++ b/examples/process_arma.ipynb
@@ -179,7 +179,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.0"
+   "version": "3.12.1"
   }
  },
  "nbformat": 4,
diff --git a/examples/signal_eigendecomposition.ipynb b/examples/signal_eigendecomposition.ipynb
index 795753f..d1467a5 100644
--- a/examples/signal_eigendecomposition.ipynb
+++ b/examples/signal_eigendecomposition.ipynb
@@ -2,11 +2,18 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [],
    "source": [
-    "import numpy as np"
+    "import os, sys\n",
+    "dir = os.path.abspath('../ssp')\n",
+    "dir = os.path.dirname(dir)\n",
+    "sys.path.append(dir)\n",
+    "\n",
+    "import numpy as np\n",
+    "\n",
+    "from ssp import spectrum"
    ]
   },
   {
@@ -83,7 +90,7 @@
     "Let $ e_i $ be a column vector:\n",
     "\n",
     "$$\n",
-    "e_i = [1, e^{j \\omega_i}, e^{j \\omega_i}, ..., e^{j (M-1) \\omega_i}] ^T\n",
+    "e_i = [1, e^{j \\omega_i}, e^{j 2 \\omega_i}, ..., e^{j (M-1) \\omega_i}] ^T\n",
     "$$\n",
     "\n",
     "If we layout a new matrix with $ p $ many of $ e_i $ column vectors the result is a $ M $ x $ p $ matrix which we represent as:\n",
@@ -114,15 +121,67 @@
    "source": [
     "# Example of four sinusoids\n",
     "\n",
-    "$ \\omega_k $ are $ 0.2\\pi, 0.3\\pi, 0.8\\pi, 1.2\\pi $ and the variance of white noise variance is $ \\sigma_w^2 = 0.5 $.\n",
+    "$ \\omega_k $ are $ 0.2\\pi, 0.3\\pi, 0.8\\pi, 1.2\\pi $ and the variance of white noise variance is $ \\sigma_w^2 = 0.5 $. Number of samples is M = 64 $\n",
     "\n",
-    "M = 64"
+    "$$\n",
+    "x(n) = \\sum_{k = 1}^{4} e^{j(n \\omega_k + \\phi_k)} + w(n)\n",
+    "$$"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "R = 10\n",
+    "N = 64\n",
+    "nvar = 0.5\n",
+    "wk = [0.2, 0.3, 0.8, 1.2]\n",
+    "\n",
+    "rng1 = np.random.default_rng()\n",
+    "signoise = rng1.normal(0, np.sqrt(nvar), (R, N))\n",
+    "\n",
+    "rng2 = np.random.default_rng()\n",
+    "phasek = rng2.uniform(0, 2 * np.pi, (R, N))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Simulation\n",
+    "\n",
+    "For the Pisarenko method, a 5x5 autocorrelation matrix is estimated from the ten simulated 64-sample realizations.\n",
+    "\n",
+    "The other methods all use the samples as the 64x64 sampled matrix over 10 realizations."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
+  "kernelspec": {
+   "display_name": ".env.py.312",
+   "language": "python",
+   "name": "python3"
+  },
   "language_info": {
-   "name": "python"
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.1"
   }
  },
  "nbformat": 4,