Mobile Communications Handbook, 3rd Edition

1

The Discrete Fourier Transform

 

Bryan Usevitch

1.1 Introduction

1.2 DFT Theory

Fourier Transform Review

The DFT as a Sampled DTFT

The DFT as a Change of Basis

1.3 Properties of the DFT

Periodicity

Circular Shift

Circular Convolution

Duality

Conjugate Property

Symmetries

Parseval's Theorem

Zero Padding and Linear Convolution

1.4 Fast Fourier Transforms

Bibliography

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)

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

where

$$W_N\triangleq {\text{e}}^{-\text{j}\frac{2\pi }{N}}.$$

Conversely, given the N frequency coefficients X(k) of a DFT, ...