ELEC 320
Prof. Rich Kozick

Chapter 2 Notes

Systems Modeled by Differential or Difference Equations


Continuous-time system with input x(t) and output y(t):

Many physical systems are accurately modeled by differential equations.

Example: RC circuit . Verify that the model is .

(You will model an RLC circuit for homework.)

Physical principles lead to differential equation models of systems: (see section 2.2 of the text)

This is why differential equations are so important: they are accurate models for the input/output behavior of many real systems.

The general form of a linear differential equation model with constant coefficients is:

where the superscript indicates the number of derivatives of the function.

N is called the order of the system.
(Recall that earlier we referred to the RC circuit as a first-order filter.)

Need N initial conditions to get a complete solution. Why?



Important points and a preview of the rest of the course:


Discrete-Time Systems

Discrete-time system with input x[n] and output y[n]:

Consider the following two systems, which are examples of "digital filters":

  1. y[n] = 0.5 ( x[n] + x[n-1] ) for n = 0, 1, 2,
  2. y[n] = 0.9 y[n-1] + x[n] for n = 0, 1, 2,

Digital Filter Demonstration

What initial conditions are needed to compute the output sequence y[0], y[1], y[2], ?

Difference equations are to discrete-time systems what differential equations are to continuous-time systems. The notes above regarding linearity, time-invariance, ZIR, and ZSR also apply to difference equations.

Discrete-time systems are becoming more common every day due to the trend toward digital processing. However, many physical systems and signals are continuous-time by nature. Thus we need to be comfortable with both continuous- and discrete-time systems and signals.

For systems 1 and 2, compute the output y[n] for the input x[n] shown below.

 

To prepare for convolution in Chapter 3: What is the "impulse response" h[n] of each system when the input x[n] is a "unit impulse" that equals 1 when n = 0 and equals 0 otherwise?

MATLAB program illustration: Computes and plots the output from systems 1 and 2 for the input x[n].