This series of lessons builds on several of the concepts introduced in the Foundations series. It concerns time-domain descriptions for the characteristics of linear, time-invariant (LTI) systems. You will learn the origins and properties of convolution for describing LTI systems in terms of the impulse response and a procedure for evaluating convolution. You will also […]

This series of lessons reviews the basics of Fourier transforms and series.  You will learn the details of how to represent signals in the frequency domain and the properties of Fourier representations. You will also gain understanding how to use Fourier methods to analyze interactions between signals and systems.  This series is designed to efficiently teach you […]

All signals in the physical world, e.g., light, sound, seismic waves, and so on, have continuous independent variables.  These signals must be sampled to convert them to a sequence of numerical values prior to computer-based signal processing. The Sampling and Reconstruction series of 15 lessons introduces you to the requirements on sampling in order to […]

The discrete Fourier transform or DFT is the frequency domain workhorse of signal processing. It is the only Fourier representation that can be evaluated with a computer. In this series of 13 lessons you will learn how the DFT is related to the discrete-time Fourier transform, and how the DFT can be used to approximate […]

The -transform is an important signal-processing tool for analyzing the interaction between signals and systems. A significant advantage of the -transform over the discrete-time Fourier transform is that the -transform exists for many signals that do not have a discrete-time Fourier transform. Thus, it is a more general analysis tool. In this series of 13 […]

This series of four lessons sets the context for the subsequent two series on designing frequency selective filters. You will learn the different types of frequency selective filters and the difference between infinite and finite impulse response (IIR and FIR) filters. You will also learn how group delay characterizes the phase distortion introduced by a […]

Infinite impulse response or IIR filters are designed using well-established designs for continuous-time filters. In this series of eight lessons you will learn the characteristics of the four widely used types of IIR filters and the principles of converting a continuous-time prototype filter to a discrete-time filter that satisfies your design specifications. In practice the […]

Finite impulse response or FIR filters have the advantage of always being stable, even for very high orders, and can be designed to introduce no phase distortion. In this series of six lessons you will learn about the conditions for an FIR filter to introduce no phase distortion and three different methods for FIR filter […]

The ability to deal with uncertainty in the characteristics of signals is a very important part of advanced signal processing methods. This series of seven lessons introduces you to tools from probability for describing signals that are modeled as having random characteristics. You will learn about auto- and cross-correlation for describing random signals in the […]

Representing signals as a weighted sum (or integral) of certain basis signals is a powerful signal-processing tool. It is the very essence of Fourier transforms. In this series of seven lessons you will learn the general form of basis representations. You will also learn about wavelets as an alternative basis expansion to the sinusoids of […]

This series of eight lessons addresses the signal-processing problem of estimating properties of a random signal from measurements. Five of the eight lessons concern estimation of frequency domain characteristics such as the power spectrum and coherence. You will learn about the periodogram and why averaging is necessary to obtain acceptable estimates of the power spectral […]

This series of six lessons introduces you to the principles of signal estimation and signal detection or hypothesis testing. You will the maximum likelihood criterion for estimation and how to classify different types of hypothesis tests and the metrics used to characterize the performance of detectors such as the probability of correct detection and the […]

This set of lessons covers a wide ranging set of signal-processing methods for minimum mean-squared error filtering and other applications of least squares problems occurring in estimation and imaging applications. This includes adaptive filtering methods such as the least-mean-square (LMS) algorithm.