ELEC 320

Prof. Rich Kozick

Prof. Rich Kozick

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)

- Electrical circuits: Voltage/current relations for capacitor and inductor, Kirchhoff's laws
- Mechanical systems: Position, velocity, acceleration relations and Newtons law
- Also fluid dynamics, thermal systems, chemical systems, and other applications

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?

- Differential equations provide models for many systems that engineers work with.
- In MATH 212, you studied methods for solving the differential equations.
- In this course, we will develop several other tools for
understanding systems: convolution, frequency
response/Fourier transform, and transfer function/Laplace
transform. These tools provide significant
*insight*for system design. **Example:**In lab, we designed the RC circuit to function as a low-pass filter with cutoff frequency 1/(RC) rad/sec. Is it clear how to design such a filter using only the differential equation model? The tools of*phasors*and*impedance*provided considerable insight for filter design.- Systems defined by differential equations of the form given above are
*linear*and*time-invariant*if all initial conditions are zero. This can be proven with the same reasoning that we applied to the RC circuit in previous class notes. - Zero-input response (ZIR) versus zero-state response (ZSR):

**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":

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

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].