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?
ZIR: Response to initial conditions only, with zero input x(t) = 0.
For many systems, the ZIR is transient and dies out with time.
Systems are nonlinear when the ZIR is considered. Why?
ZSR: Response to input x(t), with all initial conditions (or states) zero.
Many systems are linear and time-invariant with respect to the ZSR.
We will consider the ZSR almost always in this course: we will be concerned with how a system output is affected by the system input. (Most systems have inputs and outputs.)
ZIR and ZSR are related to (but slightly different than) the "homogeneous" and "particular" solutions to differential equations that you studied in MATH 212.
Discrete-time system with input x[n] and output y[n]:
Consider the following two systems, which are examples of "digital filters":
Digital Filter Demonstration
What initial conditions are needed to compute the output sequence y, y, y, … ?
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].