## Discrete-Time Signals and Systems

###### As we begin our study of discrete-time signals and systems we first examine various ways to classify signals.  These different ways of classifying signals include continuous-time vs. discrete-time, even or odd, periodic or non-periodic, etc.  The ten videos below examine these properties and others in greater detail.

Introduction

3/29/17

Running Time: 6:36

This video introduces the 3 key topics that will be discussed in the subsequent videos.  The 3 topics are:

1) Signal Classification (i.e. signal properties)
2) Common Signal Types
3) Basic Signal Operations

We also provide basic definitions for a signal (i.e. a function of one or more variables that contains information) as well as a system (i.e. an entity that manipulates signals to accomplish some function)

Continuous-Time vs. Discrete-Time

3/29/17

Running Time: 7:18

Videos 2 through 10 of Part 1 discuss various signal properties.

In this video, we define a continuous-time signal (i.e. a signal that is defined at all points in time) versus a discrete-time signal (i.e. a signal that is defined only at discrete points in time).

We plot a continuous-time signal and a discrete-time signal to note the differences between the two signal types.  We also define the sampling period T, sampling rate fs, and how to transform a continuous-time signal x(t) into a discrete time signal x[k] by simply replacing t with t = kT.

Even and Odd Signals

12/13/17

Running Time: 14:42

In this video, we define even and odd discrete-time signals.  The mathematical definition of even and odd is provided and then a variety of even and odd signals are sketched to demonstrate their symmetry properties.

Finally, we show that while some signals are neither even or odd, all signals can be decomposed into a summation of even and odd components via the equation x[k] = xE[k] + xO[k].  In this equation, xE[k] is an even signal and xO[k] is an odd signal.

After proving that the equations for these components are true, a specific "graphical decomposition" problem is worked to show how to use these equations.

Conjugate Symmetric Signals

12/14/17

Running Time: 6:38

In this video we examine complex-conjugate symmetric signals.  This type of symmetry is associated with complex-valued signals.

We say that a complex-valued discrete-time signal is conjugate symmetric if the equality x[-k] = x*[k] holds for all time.  In this equation the symbol * denotes complex conjugation.

Complex conjugate signals always have a real component that is even, and an imaginary component that is odd.

Periodic Signals

1/3/18

Running Time: 6:26

In this video we examine periodic discrete-time signals.

A periodic discrete-time signal satisfies the equality x[k] = x[k+N] for all time for some integer N.  We call N the fundamental period of the discrete-time signal.

Several examples of periodic discrete-time signals are plotted and their periods are found.

Simply-Defined and Piecewise-Defined Signals

1/3/18

Running Time: 5:18

In this video we examine simply-defined and piecewise-defined signals.

Simply-defined signals are defined with a single equation that holds for all time.  Piecewise-defined signals have different equations that are valid for different time intervals.

Deterministic and Random Signals

1/4/18

Running Time: 4:15

In this video we define deterministic and random signals.

Deterministic signals have no uncertainty associated with them and as such can be defined with an equation.  Most undergraduate courses in linear systems analyze deterministic signals and systems almost exclusively.

Random signals (also called random process) have uncertainty associated with them and are more often encountered in graduate-level course.  Understanding how to analyze random processes is also very important, but only analysis of deterministic signals and systems is considered in this course.

Energy and Power Signals

1/4/18

Running Time: 7:19

In this video we define energy and power signals.

The energy of a discrete-time signal is denoted by the real-valued scalar quantity E.  A discrete-time signal x[k] is an energy signal if E is non-zero and finite.

The power of a discrete-time signal is denoted by the real-valued scalar quantity P.  A discrete-time signal x[k] is a power signal if P is non-zero and finite.

This video introduces the energy signal and power signal definitions, as well as the equations for computing E and P.  A detailed example problem is presented in the next video.

Energy and Power Computation Example

1/6/18

Running Time: 8:57

The previous video defined energy and power signals.

In this video, we work several examples where we compute energy (E) and power (P) for different discrete-time signals. Computing these quantities from their definition requires manipulating summations.

Input, Output, and Internal Signals

1/6/18

Running Time: 3:34

In this video we define input, output, and internal signals.

Signals input to a system are called "input signals" and the notation x(t) (or x[k] if discrete-time) is typically used for them.

Signals output from a system are called "output signals" and the notation y(t) (or y[k] if discrete-time) is typically used for them.

If the system consists of various subsystems, other signals within the system may be defined and these are called "internal signals".