From c1eba09774e0e2e5dc590a345cd1d5ba845971c7 Mon Sep 17 00:00:00 2001
From: Unlambder <31901933+Unlambder@users.noreply.github.com>
Date: Mon, 2 Jul 2018 06:05:31 +0200
Subject: [PATCH] Fixed discrete fourier formula

added a missing "i" in some formulas
---
 chapters/FFT/cooley_tukey.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/chapters/FFT/cooley_tukey.md b/chapters/FFT/cooley_tukey.md
index 080c774da..f88e2ac39 100644
--- a/chapters/FFT/cooley_tukey.md
+++ b/chapters/FFT/cooley_tukey.md
@@ -58,11 +58,11 @@ Truth be told, I didn't understand it fully until I discretized real and frequen
 
 In principle, the Discrete Fourier Transform (DFT) is simply the Fourier transform with summations instead of integrals:
 
-$$X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-2 \pi k n / N}$$
+$$X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-2 \pi i k n / N}$$
 
 and
 
-$$x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{2 \pi k n / N}$$
+$$x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{2 \pi i k n / N}$$
 
 Where $$X_n$$ and $$x_n$$ are sequences of $$N$$ numbers in frequency and real space, respectively.
 In principle, this is no easier to understand than the previous case!
@@ -138,12 +138,12 @@ Butterfly Diagrams show where each element in the array goes before, during, and
 As mentioned, the FFT must perform a DFT.
 This means that even though we need to be careful about how we add elements together, we are still ultimately performing the following operation:
 
-$$X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-2 \pi k n / N}$$
+$$X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-2 \pi i k n / N}$$
 
 However, after shuffling the initial array (by bit reversing or recursive subdivision), we perform the matrix multiplication of the $$e^{-2 \pi k n / N}$$ terms in pieces.
 Basically, we split the array into a series of omega values:
 
-$$\omega_N^k = e^{-2 \pi k / N}$$
+$$\omega_N^k = e^{-2 \pi i k / N}$$
 
 And at each step, we use the appropriate term.
 For example, imagine we need to perform an FFT of an array of only 2 elements.