# The Discrete Fourier Transform: Definition as an inner product

## Contents

## 1 Sampled discrete time signals

Consider an -length finite duration discrete time signal, , specified at integer sample time indices .

The signal comprises samples, and therefore runs from . In general, may be complex valued, i.e., , though in most audio-related applications it will be purely real.

## 2 The Discrete Fourier Transform: Definition as an inner product

The DFT, , of the signal is obtained by evaluating the inner product of with an orthogonal set of complex exponentials

(1)

where integer , and is the imaginary number ().

Using the definition on the Mathematical tools page, we can evaluate via

(2)

for sample time index , frequency bin index , and number of time samples included in the transformation (which is also equal to the number of frequency bins). Note that:

- , , and are all integers.
- A conjugation operation acts on , within the summation of equation 2 (as expected).
- There is no explicit definition of `frequency’ when the DFT is defined in this way — i.e., it is agnostic to sample rate (see below for more on this).
- For real valued signals , only half of the spectrum is unique — see here[LINK] for more on this.

The values are sometimes referred to as *complex amplitudes*, since in general they are complex valued for a real valued signal — i.e., each value has a magnitude and phase, or equivalently, a real and imaginary part. Analogous with earlier examples, each value of tells us something about `how much’ of the given complex exponential is present within .