## 1

## The Discrete Fourier Transform

Bryan Usevitch

Zero Padding and Linear Convolution

### 1.1 Introduction

The discrete Fourier transform (DFT) is a method that represents the frequency content of a finite length time sequence, or sequence of samples. Specifically, it takes *N* input samples *x*(*n*) and converts them into *N* frequency coefficients *X*(*k*)

$$X(k)={\displaystyle \sum _{n=0}^{N-1}x}(n){W}_{N}^{kn},\text{\hspace{1em}}k=0,1,\dots ,N-1,\left(1.1\right)$$

where

$${W}_{N}\triangleq {\text{e}}^{-\text{j}\frac{2\pi}{N}}.$$Conversely, given the *N* frequency coefficients *X*(*k*) of a DFT, ...

Get *Mobile Communications Handbook, 3rd Edition* now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.