## Discrete-Time Signals and Systems

###### We investigate how to solve a difference equations.  We first look at solving via an iterative/numerical approach, and then learn how to solve for the total solution by finding the zero-input response and zero-state response.  A "classical" approach that finds the total solution as a sum of the natural response and forced response is also examined.

Introduction to Difference Equations

9/20/18

Running Time: 12:13

This 15-video series introduces the concept of a discrete-time difference equation and how to solve difference equations in general.

In this first video, we show several examples of a difference equation and then define two important standard forms of a difference equation; the Delay Operator Form and the Advance Operator Form.  We also show how difference equations can be shifted in time and how to convert a given difference equation into the standard Delay Operator form.

Iterative Solution of A Difference Equation

9/20/18

Running Time: 7:02

This video explains how to solve a difference equation using an iterative approach.  Given initial conditions for the difference equation, subsequent values of the solution can be computed recursively one-at-at-time.  While this approach doesn't yield an analytic equation as the solution, it is often useful to use this approach to solve for a handful of values of the overall solution.  This iterative approach can also be easily implement in Matlab or another program to compute large numbers of values of the solution in a FOR loop.

Difference Equation Operator Notation

9/22/18

Running Time: 6:30

We introduce some new notation called the "advance operator" written as "E" that allows us to compactly write time advances in a difference equation.  For example, instead of writing x[k+3] we can write E^3x[k] where the order of the exponent indicates the number of time advances to perform.  This new notation is convenient as it lets us describe a difference equation compactly as Q[E]y[k] = P[E]x[k] where Q[E] and P[E] are both polynomials in E.

Difference Equation Zero-Input Response Theory

9/22/18

Running Time: 18:35

This video introduces the concept of the zero-input response of a difference equation.  The zero-input response, denoted as y0[k], is the solution of a difference equation assuming zero input.  As such, this solution is only due to the initial conditions of the system.

The basic form of the  zero-input response is derived and we find that the zero-input response must be a linear combination of the characteristic modes of the system, where the characteristic modes are exponential functions.  We discuss the case where the roots of the system characteristic equation are distinct, repeated, and complex.

While this video just develops the general theory, terminology, and form of the zero-input response, the videos that follow work specific examples.

Difference Equation Zero-Input Response Example #1 (Distinct Roots)

9/24/18

Running Time: 6:46

The zero-input response of a discrete-time linear system is a linear combination of the characteristic modes of the system.  The characteristic modes of the system can be found from the roots of the system characteristic equation.  In this video, we find the roots of the characteristic equation and solve for the zero-input response using the system initial conditions for the case when all roots of the system are distinct.

Difference Equation Zero-Input Response Example #2 (Repeated Roots)

9/24/18

Running Time: 6:52

The zero-input response of a discrete-time linear system is a linear combination of the characteristic modes of the system.  The characteristic modes of the system can be found from the roots of the system characteristic equation.  In this video, we find the roots of the characteristic equation and solve for the zero-input response using the system initial conditions for the case when there are repeated roots of the system.

The previous video in this playlist worked an example where the roots were distinct.  The next video in this series work an examples where the roots are complex.

Difference Equation Zero-Input Response Example #3 (Complex Roots)

9/26/18

Running Time: 11:31

The zero-input response of a discrete-time linear system is a linear combination of the characteristic modes of the system.  The characteristic modes of the system can be found from the roots of the system characteristic equation.  In this video, we find the roots of the characteristic equation and solve for the zero-input response using the system initial conditions for the case when the system has complex-conjugate roots.

Difference Equation Impulse Response Solution via Iterative Approach

9/26/18

Running Time: 7:17

The impulse response of a discrete-time LTI system is defined as the output of the system at rest when a unit impulse is applied at time zero.  In this case, we use the notation h[k] to denote the system output and h[k] is called the system impulse response.

In this video, we show how to solve for the impulse response using an iterative/numerical approach when given a difference equation that describes the system.  In the next video, we'll develop an analytic equation that can be used to write an equation for the impulse response for all time.