From 3317084ea93883415dc7ab2145bdb1f9885b9570 Mon Sep 17 00:00:00 2001
From: Sascha Spors <Sascha.Spors@uni-rostock.de>
Date: Mon, 9 Dec 2024 13:39:47 +0100
Subject: [PATCH] fixed typo in gradient of cost function

---
 random_signals_LTI_systems/linear_prediction.ipynb | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/random_signals_LTI_systems/linear_prediction.ipynb b/random_signals_LTI_systems/linear_prediction.ipynb
index ea61a89..df117a9 100644
--- a/random_signals_LTI_systems/linear_prediction.ipynb
+++ b/random_signals_LTI_systems/linear_prediction.ipynb
@@ -66,7 +66,7 @@
     "Above equation is referred to as [*cost function*](https://en.wikipedia.org/wiki/Loss_function) $J$ of the optimization problem. We aim at minimizing the cost function, hence minimizing the MSE between the signal $x[k]$ and its prediction $\\hat{x}[k]$. The solution of this [convex optimization](https://en.wikipedia.org/wiki/Convex_optimization) problem is referred to as [minimum mean squared error](https://en.wikipedia.org/wiki/Minimum_mean_square_error) (MMSE) solution. Minimizing the cost function is achieved by calculating its gradient with respect to the filter coefficients [[Haykin](../index.ipynb#Literature)] using results from [matrix calculus](https://en.wikipedia.org/wiki/Matrix_calculus)\n",
     "\n",
     "\\begin{align}\n",
-    "\\nabla_\\mathbf{h} J &= -2 E \\left\\{ x[k-1] (x[k] - \\mathbf{h}^T[k] \\mathbf{x}[k-1]) \\right\\} \\\\\n",
+    "\\nabla_\\mathbf{h} J &= -2 E \\left\\{ \\mathbf{x}[k-1] (x[k] - \\mathbf{h}^T[k] \\mathbf{x}[k-1]) \\right\\} \\\\\n",
     "&= - 2 \\mathbf{r}[k] + 2 \\mathbf{R}[k-1] \\mathbf{h}[k]\n",
     "\\end{align}\n",
     "\n",
@@ -10436,9 +10436,9 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.13"
+   "version": "3.12.6"
   }
  },
  "nbformat": 4,
- "nbformat_minor": 1
+ "nbformat_minor": 4
 }