## Continuous-Time Signals and Systems

###### We regularly use a variety of basis signals in the class to include the unit impulse function, unit step function, unit ramp function, rectangle function, triangle function, sinc function, and exponential functions.  These videos provide definitions and examples of each function type, as well as providing a detailed discussion on the impulse function and its sifting property.

The Continuous-Time Unit Impulse Function 1/4

1/12/16

Running Time: 6:33

The basic definition of the continuous-time unit impulse function delta(t) is provided in this video.  This signal is zero everywhere except at t = 0, and has the property that when integrated across zero its integral is equal to one.

We see how the impulse function can be thought of as the limit of a rectangular function of unit area as its width goes to zero and its height goes to infinity.

The Continuous-Time Unit Impulse Function 2/4

1/12/16

Running Time: 6:13

We continue examining the continuous-time impulse function in this video by exploring what happens when a continuous-time signal x(t) is multiplied by the impulse function delta(t).  We see that the result of this multiplication is simply a delta function whose height/area has been modified by the value of the signal at the location of the impulse.

The Continuous-Time Unit Impulse Function 3/4

1/12/16

Running Time: 8:29

The previous video showed us (both visually and mathematically) what happens when we multiply a continuous-time signal x(t) by an impulse function delta(t).  We use this knowledge to develop the Sifting Property of the impulse function in this video.

The sifting property of the impulse function says that when integrating the product x(t)*delta(t), the result is simply the value of the signal x(t) evaluated at the temporal location of the impulse function.

The Continuous-Time Unit Impulse Function 4/4

1/12/16

Running Time: 5:51

The previous video developed the sifting property of the continuous-time impulse function delta(t).  In this video we use the sifting property of the impulse function to evaluate and simplify a variety of integrals involving products of continuous-time with impulse functions.

The Continuous-Time Unit Step and Unit Ramp Functions

1/12/16

Running Time: 5:50

Having completed our discussion of the continuous-time impulse function delta(t), we now turn our attention to some other commonly used signals in a signals-and-systems course.

The unit step function is a signal that is zero for t less than 0 that "turns on" to a value of 1 at time t = 1, i.e. u(t) = 1 for t greater than or equal to zero.

The unit ramp function is a signal that is zero for t less than 0 and then linearly increases with a slope of one for t greater than zero, i.e. r(t) = t for t greater than or equal to 0.

Signal Construction Example #1

1/12/16

Running Time: 7:02

We often deal with "simple" signals in a signals-and-systems course that can be easily described as a linear combination of time-shifted unit step and unit ramp functions.

In this video example we're provided a sketch of a "simple" signal x(t) and we develop an equation for x(t) in terms of time-shifted unit step and unit ramp functions.

Signal Construction Example #2

1/12/16

Running Time: 4:33

We often deal with "simple" signals in a signals-and-systems course that can be easily described as a linear combination of time-shifted unit step and unit ramp functions.

Just as in the previous video, we again are provided a sketch of a "simple" signal x(t) and we develop an equation for x(t) in terms of time-shifted unit step and unit ramp functions.

Continuous-Time Signal Plotting Example

1/12/16

Running Time: 4:47

The previous two videos provided plots of continuous-time signals and we developed mathematical equations to represent the sketch.

In this video example, we're given the equation of a continuous-time signal x(t) which consists of a linear combination of time-shifted and amplitude-scaled unit step and unit ramp function.

Given this equation, we sketch each component and then find the final signal plot of summing the components at each point in time.

The Continuous-Time Exponential Signal

1/12/16

Running Time: 7:58

We write the general continuous-time exponential signal as exp(st) where in general, "s" is the complex number s = sigma + j*omega.

Depending on the value for "s", this signal can be a complex-valued signal, a real-valued signal, a decaying/growing signal, or an oscillating signal.

This video defines the exponential signal and explores its properties.

The Continuous-Time Rectangle, Triangle, and Sinc Functions

1/16/16

Running Time: 6:48

We complete our discussion of commonly used continuous-time signal models by examining the rectangle function, triangle function, and the sinc function.  These functions are often used/encountered in continuous-time signals-and-systems courses.