The dominant application of the discrete Fourier transform is for performing spectral analysis – analyzing the frequency content of signals. The discrete Fourier transform is often used to approximate the Fourier transform of a signal. In this lesson you will learn the three types of approximations involved in using the discrete Fourier transform to represent the Fourier transform. You will also learn […]

# The DFT and Applications

## The Effect of Windowing on the Discrete Fourier Transform Approximation to the Fourier Transform

The most significant effect on the fidelity of the discrete Fourier transform approximation to the Fourier transform is due to truncating or "windowing" the duration of the signal. In this lesson you will gain key insight into the loss of resolution or detail and the loss of dynamic range that is introduced by truncation. A thorough understanding […]

## Windows and the Discrete-Time Fourier Transform: Trading Resolution for Dynamic Range

In this lesson you will learn how to use different windows to manage the loss of resolution and limited dynamic range resulting from truncation. The rectangular window leads to relatively good resolution but very poor dynamic range. Windows with less abrupt transitions provide much better dynamic range at the expense of some loss of resolution. […]

## An Example of Approximating the Fourier Transform with the Discrete Fourier Transform

This lesson illustrates the use of the discrete Fourier transform for approximating the Fourier transform by using a signal with a known Fourier transform. You will see how the choices of window length and discrete Fourier transform length affect the quality of the approximation. This example is the culmination of the concepts discussed in the three preceding lessons and […]

## The Short-Time Fourier Transform and the Spectrogram

The DFT is an extremely powerful tool for assessing the spectral characteristics of signals. In this lesson you will learn about a variation of the DFT called the short-time Fourier transform (STFT). The STFT is used for spectral analysis of signals whose frequency characteristics are changing over time. There are many types of signals with […]

## A Matrix Interpretation of the Discrete Fourier Transform

This lesson teaches you how to interpret the discrete Fourier transform as the inner product between a matrix and a vector and introduces a linear algebraic viewpoint. Linear algebra is an extremely powerful and widely used tool in signal processing. You will be introduced to these ideas in this lesson and build on them in future studies.