The technique of representing signals using bases may be the most widely used and powerful tool in signal processing. In this lesson you will learn a general framework for representing finite-duration sampled signals using basis signals. This knowledge will further your understanding of the discrete Fourier transform and prepare you to understand basis expansion methods […]

# Basis Representations of Signals

## Introduction to Wavelets

Wavelets are a powerful class of bases that have varying frequency and duration. This is in contrast to the varying frequency but fixed duration of Fourier methods. In this lesson you will learn how the unique properties of wavelets can be used to localize signals in both time and frequency. You will learn that the […]

## Multiresolution Analysis and the Scaling Function

Wavelets provide a multiresolution analysis of signals. This means they simultaneously describe the low- and high-resolution features of the signal. In this lesson you will learn how multiresolution analysis is performed using a scaling function that defines the lowest resolution view of the signal. Higher resolution views are obtained from scaled versions of the scaling […]

## Multiresolution Analysis and the Wavelet Decomposition

In this lesson you will learn how the wavelet decomposition for a signal follows naturally from a multiresolution analysis viewpoint. Wavelets represent the details needed to increase the resolution of the representation to the next level. This leads to a relationship between the wavelet and scaling function. You will learn that the discrete wavelet transform […]

## The Discrete Wavelet Transform

The discrete wavelet transform is a numerically efficient algorithm for computing the scaling and wavelet coefficients from samples of the signal. In this lesson you will learn how the discrete wavelet transform obtains the scaling and wavelet coefficients without using the continuous-time scaling and wavelet bases. The availability of the discrete wavelet transform is a […]

## Wavelet Selection

In this lesson you will learn several principles that guide choice of a wavelet family. One goal is to efficiently represent a signal, that is, to minimize the number of nonzero terms in the basis expansion for a signal. You will learn how the number of vanishing moments and the support affect the efficiency of […]