The Fourier series represents periodic, continuous-time signals as a weighted sum of continuous-time sinusoids. It is widely used to analyze and synthesize periodic signals. This lesson shows you how to compute the Fourier series coefficients, or weights, from the signal. It also introduces you to the conditions that must be met for a signal Read More

## Square Wave Fourier Series and the Sinc Function

The square wave Fourier series provides fundamental insight into the nature of Fourier series expansions. You will learn the relationship between the width or duty cycle of a square wave and the concentration of the Fourier series coefficients. You will also be introduced to the sinc function, a function that has widespread significance in Read More

## Fourier Series Properties

Fourier series properties provide insight and simplify working with Fourier series expansions. This lesson introduces you to the most important and commonly used Fourier series properties and gives you examples of their use. These properties are direct consequence of the nature of sinusoids. Similar properties exist for the other Fourier representations, so understanding the properties of Read More

## The Fourier Transform: Linking Time and Frequency Domains

The Fourier transform is the primary tool for analyzing signals and signal-processing systems in the frequency domain, especially when signals are sampled or converted from discrete time to continuous time. In this lesson you will learn the definition of the Fourier transform and how to evaluate the corresponding integrals for several common signals. You will Read More

## Properties of the Fourier Transform

In this lesson you will learn several of the most important Fourier transform properties and how to apply them. Understanding the properties of the Fourier transform will help you develop your insight regarding the relationships between time- and frequency-domain representations of signals. You will gain insight how specific manipulations of a signal in either the Read More

## The Discrete-Time Fourier Transform

The discrete-time Fourier transform (DTFT) is the tool of choice for frequency domain analysis of discrete-time signals and signal-processing systems. In this lesson you will learn the definition of the DTFT and how to evaluate the DTFT of several common signals. You will also build your intuition for the relationships between time and frequency domain Read More

## Discrete-Time Fourier Transform Properties

In this lesson you will learn several of the most important discrete-time Fourier transform properties and how to apply them. Understanding the properties of the discrete-time Fourier transform will help you develop your insight regarding the relationships between time- and frequency-domain representations of discrete-time signals. You will gain insight how specific manipulations of a signal Read More

## Fourier Transforms and Discrete-Time Fourier Transforms for Periodic Signals

Frequency-domain analysis of the interaction between a periodic and non periodic signal is performed using the Fourier or discrete-time Fourier transform. This requires the definition of Fourier and discrete-time Fourier transforms for periodic signals. In this lesson you will learn how impulses are used in the frequency domain to obtain Fourier transforms and discrete-time Read More

## Frequency-Domain Descriptions for Continuous-Time Linear Time-Invariant Systems

The frequency response is an extremely insightful description for linear time invariant (LTI) systems. The output of a LTI system at a given frequency is simply the product of the input at that frequency and the frequency response. In this lesson you will learn the relationship between the frequency response, impulse response, differential equation, Read More

## Frequency-Domain Descriptions for Discrete-Time Linear Time-Invariant Systems

The frequency response is an extremely insightful description for linear time invariant (LTI) systems. The output of a LTI system at a given frequency is simply the product of the input at that frequency and the frequency response. In this lesson you will learn the relationship between the frequency response, impulse response, differential equation, Read More

## Two-Dimensional Signal Processing: Continuous Space

The concepts of systems and Fourier transforms are easily applied to signals with more than one independent variable, or dimension. Images and movies are common multidimensional signals. This lesson shows you how to extend your knowledge of one dimensional continuous-time signals to two dimensional continuous-space signals. You will learn the form for two-dimensional convolution and the two-dimensional Fourier Read More