Linear time-invariant (LTI) systems are the most widely used systems in signal processing. In this lesson you will develop a deeper understanding for the role of the impulse response of LTI systems. You will learn how linearity and time invariance result in the convolution sum for expressing the output of an LTI system in terms Read More

## Graphical Evaluation of Discrete-Time Convolution

Blog post: Convolution of Signals: Why? Convolution expresses the output of a linear time-invariant system in terms of the system's impulse response and the input. In this lesson you will learn a graphical approach to evaluating convolution. Learning how to interpret convolution graphically will develop your intuition for understanding how the impulse response characteristics impact Read More

## Graphical Evaluation of Continuous-Time Convolution

Convolution expresses the output of a linear time-invariant system in terms of the system's impulse response and the input. In this lesson you will learn a graphical approach to evaluating convolution for continuous-time systems. Learning how to interpret convolution graphically will develop your intuition for understanding how the impulse response characteristics impact the system output. Read More

## Difference Equations: Solving System Responses with Stored Energy

Difference equations are one of the few descriptions for linear time-invariant (LTI) systems that can incorporate the effects of stored energy - that is, describe systems which are not at rest when the input is applied. In this lesson you will learn how the output of a LTI system described by a difference equation can be expressed as Read More

## Characteristics of Systems Described by Difference Equations

Difference equations are often used to compute the output of a system from knowledge of the input. They are an important and widely used tool for representing the input-output relationship of linear time-invariant systems. In this lesson you will learn how the characteristics of the system are related to the coefficients in the difference equation. You will Read More

## Differential Equations: Solving System Responses with Stored Energy

Differential equation descriptions for continuous-time linear time-invariant systems are unique in that they allow analysis of the effect of stored energy on the system output. You will learn how to represent the output of such a system as a sum of a steady-state and a transient component. The steady-state component is of the same form as Read More

## Characteristics of Systems Described by Differential Equations

Differential equations are used to represent the effects of continuous-time systems on signals. Signal processing methods are often used to compensate for the effect of such systems. In this lesson you will learn how the parameters of the differential equation describe the characteristics of the system. You will learn how the stability of the system Read More

## Two-Dimensional Signal Processing: Discrete Space

Signals and systems are not limited to one-dimensional functions of the independent variable time. This lesson introduces you to signals that are a function of a two discrete-valued space variables. We often call such signals "images". You will learn about the point-spread function, which is the two-dimensional analog of the impulse response for a one-dimensional system. Read More

## Exercises for Impulse Response and LTI Systems Part II

The benefit you derive from these exercises is proportional to the effort you put into them. Here are my suggestions: Try to work the problems without looking at the solution. Don't give up easily. Feel free to refer back to the lessons covering this material. If/when you become stuck, then look at the solution, but Read More

## Exercises for Graphical Evaluation of Discrete-Time Convolution

The benefit you derive from these exercises is proportional to the effort you put into them. Here are my suggestions: Try to work the problems without looking at the solution. Don't give up easily. Feel free to refer back to the lessons covering this material. If/when you become stuck, then look at the solution, but Read More

## Exercises: Graphical Evaluation of Continuous-Time Convolution

The benefit you derive from these exercises is proportional to the effort you put into them. Here are my suggestions: Try to work the problems without looking at the solution. Don't give up easily. Feel free to refer back to the lessons covering this material. If/when you become stuck, then look at the solution, but Read More