The discrete Fourier transform is one of the most important computational tools in signal processing. This lesson briefly introduces you to some of the applications for the discrete Fourier transform, its definition, and develops the relationship between the discrete Fourier transform and the discrete-time Fourier transform. An understanding of this relationship is essential to proper use […]

# The DFT and Applications

## Important Discrete Fourier Transform Properties

In this lesson you will learn several of the most important properties of the discrete Fourier transform (DFT) for signal processing applications. Not surprisingly, these properties are similar to those of the Fourier transform and the discrete-time Fourier transform. The convolution-multiplication property is deferred to a separate lesson due to its importance in using the […]

## Fast Fourier Transform (FFT) Algorithm

In this lesson you will learn the principles at the core of the decimation-in-time fast Fourier transform algorithm. The (re)discovery of the fast Fourier transform algorithm by Cooley and Tukey in 1965 was perhaps the most significant event in the history of signal processing. There is evidence that Gauss first developed a fast Fourier transform-type […]

## Introduction to Circular Convolution and Filtering with the Discrete Fourier Transform

The convolution-multiplication property is one of the most insightful and useful properties of the Fourier transform and discrete-time Fourier transform. This lesson introduces the convolution-multiplication property for the DFT. Multiplication of DFT coefficients corresponds to circular convolution of time signals. You will gain an understanding of the difference between linear and circular convolution that is […]

## Circular Convolution Property of the Discrete Fourier Transform

The Circular Convolution Property of the Discrete Fourier Transform lesson takes a detailed look at the convolution-multiplication property for the DFT. You will learn how to derive this important property, how to evaluate circular convolution, and the relationship between linear or ordinary and circular convolution. Understanding these concepts is key for properly using the DFT […]

## Filtering with the Discrete Fourier Transform

The discrete Fourier transform is often used to implement linear time-invariant filters in a computationally efficient manner. This is due to the availability of computationally efficient or fast algorithms for computing the discrete Fourier transform. You will learn about the overlap and add method for computing convolution in this lesson. Overlap and add exploits the efficient computation […]