# FourierTransformsJW

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## Fourier Transform

### Introduction

A Fourier series allows a periodic function to be represented as the sum of sine and/or cosine waves. This is very useful, because functions in the time domain can be expressed in the frequency domain. The frequency domain can, at times, be easier to work with. This is where the Fourier transfrom comes in. It allows for a nonperidic function of time to be converted (or transformed) into a function of frequency.

This conversion is analgous to the conversion from cartesian coordinates to polar or spherical coordinates. The location of the point does not change; only the directions for how to get there.

### Fourier Transform Defined

The Fourier Transform of a function $x(t)$ can be defined as:

$\mathcal{F}[x(t)] = \int_{-\infty}^\infty x(t) e^{-j 2 \pi f t} dt = X(f)$

then the inverse Fourier transform of a function $X(f)$ will be:

$\mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t} df = x(t)$

where $x(t)$ is a function of time and $X(f)$ is the Fourier transform of $x(t)$ and is it's Fourier Transform.

### Some Properties of the Fourier Transform

Given: $x(t) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t} df$

##### Differentiation

$\frac{dx}{dt} = \int_{-\infty}^\infty j 2 \pi X(f) e^{j 2 \pi f t} df = \mathcal{F}^{-1}[j 2 \pi f X(f)]$

##### Time Shift

$x(t-t_o) = \int_{-\infty}^\infty j 2 \pi X(f) e^{j 2 \pi f(t-t_o)} df$

$= \int_{-\infty}^\infty e^{-j 2 \pi f t_o} X(f) e^{j 2 \pi f t} df$

$= \mathcal{F}^{-1}[e^{-j 2 \pi f t_o}X(f)]$

##### Frequency Shift

Given: $X(f) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t}df$

$X(f-f_o) = \int_{-\infty}^\infty X(f) e^{j 2 \pi (f-f_o) t}df = \mathcal{F}[e^{j 2 \pi f_o t}x(t)]$

##### Modulation

$\mathcal{F}[cos(2 \pi f_o t)x(t)] = \int_{\infty}^\infty x(t)cos(2 \pi f_o t) e^{-j 2 \pi f t} dt$

$= \int_{\infty}^\infty \frac{e^{j 2 \pi f_o t} + e^{-j 2 \pi f_o t}}{2} x(t) e^{j 2 \pi f t} dt$

$= \frac{1}{2} \int_{\infty}^\infty x(t) e^{-j 2 \pi (f-f_o) t} dt + \frac{1}{2} \int_{\infty}^\infty x(t)e^{-j 2 \pi (f+f_o) t} dt$

$= \frac{1}{2}X(f-f_o) + \frac{1}{2}X(f+f_o)$