From e25c1f9ad51acd46f1dbb33b08c77ab4ca0191bd Mon Sep 17 00:00:00 2001
From: Erik Strand <erik.strand@cba.mit.edu>
Date: Tue, 30 Apr 2019 18:15:44 -0400
Subject: [PATCH] Add to Fourier transform notes
---
_notes/fourier_transform.md | 98 ++++++++++++++++++++++++++-----------
1 file changed, 69 insertions(+), 29 deletions(-)
diff --git a/_notes/fourier_transform.md b/_notes/fourier_transform.md
index 8f4b452..6a569e2 100644
--- a/_notes/fourier_transform.md
+++ b/_notes/fourier_transform.md
@@ -2,34 +2,52 @@
title: Fourier Transforms
---
-To really make sense of chapter 2 I needed to review the properties of Fourier transforms. These
-notes are based on my prior knowledge and some helpful websites:
-- [Properties of Fourier Transform](http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html)
-- [symmetry.pdf](https://www.cs.unm.edu/~williams/cs530/symmetry.pdf)
+## Definition
-Note: I'm sloppy with the proofs here since all physical functions will have the nice properties
-that make the relevant operations valid, but I don't always call of these properties out when they
-are used.
+For a suitable function $$f : \mathbb{R} \rightarrow \mathbb{C}$$, the Fourier transform and inverse
+Fourier transform are defined to be
+$$
+\begin{align*}
+(\mathcal{F} f)(\xi) &= \int_\mathbb{R} f(x) e^{-2 \pi i x \xi} \mathrm{d} x \\
+(\mathcal{F}^{-1} f)(x) &= \int_\mathbb{R} f(\xi) e^{2 \pi i \xi x} \mathrm{d} \xi
+\end{align*}
+$$
-## Basics
+The Fourier transform of $$f$$ is frequently written as $$\hat{f}(\xi) = (\mathcal{F} f)(\xi)$$.
-For a function $$f : \mathbb{R} \rightarrow \mathbb{C}$$, I use the definitions
+Every function in [$$L^1$$](https://en.wikipedia.org/wiki/Lp_space#Lp_spaces) has a Fourier
+transform and inverse Fourier transform, since
$$
-(\mathcal{F} f)(x) = \int_\mathbb{R} f(x') e^{-2 \pi i x' x} \mathrm{d} x'
+\begin{align*}
+\left \vert \hat{f}(\xi) \right \vert
+&\leq \int_\mathbb{R} \left \vert f(x) e^{-2 \pi i x \xi} \right \vert \mathrm{d} x \\
+&= \int_\mathbb{R} \left \vert f(x) \right \vert \mathrm{d} x
+\end{align*}
$$
-$$
-(\mathcal{F}^{-1} f)(x) = \int_\mathbb{R} f(x') e^{2 \pi i x' x} \mathrm{d} x'
-$$
+Furthermore when $$f$$ is in $$L^1$$, then $$\hat{f}(\xi)$$ is a uniformly continuous function that
+tends to zero as $$|\xi|$$ approaches infinity. However $$\hat{f}$$ need not be in $$L^1$$, and not
+every continuous function that tends to zero is the Fourier transform of a function in $$L^1$$
+(indeed describing $$\mathcal{F}(L^1)$$ is an open problem). As such it can be helpful to restrict
+the definition to the [Schwartz space](https://en.wikipedia.org/wiki/Schwartz_space) over
+$$\mathbb{R}$$, where the Fourier transform is an
+[automorphism](https://en.wikipedia.org/wiki/Automorphism).
+
+On the other hand, we'll also want to talk about the Fourier transforms of functions that aren't
+absolutely integrable, or objects that aren't functions at all (like the [delta
+function](https://en.wikipedia.org/wiki/Dirac_delta_function)). So I will tend to be very liberal
+with my application of the transform.
+
+
+## Basic Properties
-The Fourier Inversion Theorem states that $$\mathcal{F} \mathcal{F}^{-1} = \mathcal{F}^{-1}
-\mathcal{F} = \mathcal{I}$$ (where $$\mathcal{I}$$ is the identity operator). This holds for the space
-of functions whose Fourier transforms exist and for which both the function and the transform are
-absolutely integrable and continuous. All claims I make about functions should be interpreted to
-apply only to functions in this space.
+The [Fourier Inversion Theorem](https://en.wikipedia.org/wiki/Fourier_inversion_theorem) states that
+$$\mathcal{F} \mathcal{F}^{-1} = \mathcal{F}^{-1} \mathcal{F} = \mathcal{I}$$ (where $$\mathcal{I}$$
+is the identity operator). This is strictly true for functions in $$L^1$$ whose transforms are also
+in $$L^1$$, but can also be extended to more general spaces as well.
The Fourier transform is linear:
@@ -38,24 +56,45 @@ $$
$$
If you shift everything in the original basis (usually the time or space domain), you pick up a
-phase shift in the transformed (i.e. frequency) basis. This follows from a simple change of
-variables.
+phase shift in the transformed (i.e. frequency) basis. This follows from a change of variables.
+
+$$
+\begin{align*}
+(\mathcal{F} f(x + x_0))(\xi)
+&= \int_\mathbb{R} f(x + x_0) e^{-2 \pi i x \xi} \mathrm{d} x \\
+&= \int_\mathbb{R} f(x) e^{-2 \pi i (x - x_0) \xi} \mathrm{d} x \\
+&= e^{2 \pi i x_0 \xi} \int_\mathbb{R} f(x) e^{-2 \pi i x \xi} \mathrm{d} x \\
+&= e^{2 \pi i x_0 \xi} \hat{f}(\xi)
+\end{align*}
+$$
+
+The reverse is also true (with a sign difference):
+
+$$
+\begin{align*}
+\mathcal{F}(e^{2 \pi i x \xi_0} f(x))(\xi)
+&= \int_\mathbb{R} e^{2 \pi i x \xi_0} f(x) e^{-2 \pi i x \xi} \mathrm{d} x \\
+&= \int_\mathbb{R} f(x) e^{-2 \pi i x (\xi - \xi_0)} \mathrm{d} x \\
+&= \hat{f}(\xi - \xi_0)
+\end{align*}
+$$
+
+If you expand $$f$$ horizontally, you contract $$\hat{f}$$ both horizontally and vertically.
$$
\begin{align*}
-(\mathcal{F} f(x' + x_0))(x)
-&= \int_\mathbb{R} f(x' + x_0) e^{-2 \pi i x' x} \mathrm{d} x' \\
-&= \int_\mathbb{R} f(x') e^{-2 \pi i (x' - x_0) x} \mathrm{d} x' \\
-&= e^{2 \pi i x_0 x} \int_\mathbb{R} f(x') e^{-2 \pi i x' x} \mathrm{d} x' \\
-&= e^{2 \pi i x_0 x} (\mathcal{F} f(x'))(x)
+\mathcal{F}(f(a x))(\xi)
+&= \int_\mathbb{R} f(a x) e^{-2 \pi i x \xi} \mathrm{d} x \\
+&= \frac{1}{|a|} \int_\mathbb{R} f(x) e^{-2 \pi i x \xi / a} \mathrm{d} x \\
+&= \frac{1}{|a|} \hat{f} \left( \frac{\xi}{a} \right)
\end{align*}
$$
## Fourier Flips
-The Fourier transform has a number of interesting properties related to the flip operator
-$$(\mathcal{R} f)(x) = f(-x)$$. By definition
+The Fourier transform has a number of interesting properties related to the flip (or reversal)
+operator $$(\mathcal{R} f)(x) = f(-x)$$. By definition
$$
(\mathcal{F}^{-1} f)(x) = \int_\mathbb{R} f(x') e^{2 \pi i x' x} \mathrm{d} x'
@@ -73,7 +112,8 @@ $$
\end{align*}
$$
-Thus $$\mathcal{F}^{-1} = \mathcal{F} \mathcal{R} = \mathcal{R} \mathcal{F}$$. This means that
+Thus $$\mathcal{F}^{-1} = \mathcal{F} \mathcal{R} = \mathcal{R} \mathcal{F}$$. (You can also derive
+this using the expansion/contraction formula discussed above). This means that
$$
\mathcal{I}
@@ -175,7 +215,7 @@ when we're dealing with a real function and only care about the magnitude of the
for spectral power analysis).
-## Transforms of Gaussians
+## The Transform of a Gaussian
A Fourier transform that comes up frequently is that of a Gaussian. It can be calculated by
completing a square.
--
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