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EE102 – SYSTEMS & SIGNALS
Fall Quarter, 2001.
Instructor: Fernando Paganini.
Signal: Function that describes the
evolution of a variable with time.
• Examples:
– Voltage across an electrical component.
+
V(t)
_
– Position of a moving object.
x(t)
Sound = pressure of the air outside your ear
p
• Information lies in the time evolution.
t
• The signal can be converted to and from other domains:
Electrical (in a stereo), electro-chemical (in your brain).
• What matters is the mathematical structure.
Signal examples (cont):
• Population of a species over time (decades)
• Daily value of the Nasdaq
Time can be continuous (a real number) or discrete
(an integer). This course focuses on continuous time.
System: component that establishes
a relationship between signals
• Example: circuit
Is
R3
V1
R1
Vs
R2
Relationship between voltages and currents.
Io
Systems: examples
• Car: Relationship between signals:
– Throttle/brake position
– Motor speed
– Fuel concentration in chamber
– Vehicle speed.
– …
• Ecosystem: relates populations, …
• The economy: relates GDP, inflation, interest
rates, stock prices,…
• The universe…
Math needed to study signals & systems?
Example 1: static system
Vs
I
R1  R2
Vo
R1
Vs
R2
I
Vs R2
V0 
R1  R2
• Not much math there…
• Time does not enter in a fundamental way.
Example 2: dynamical system
Vo
R
Vs
C
I
dV0
IC
dt
Vs  V0  R I
• Switch closes at t=0.
• For t >= 0, we have the differential equation (ODE)
dV0
Vs  V0  R C
dt
t



RC
Solution: V0 Vs  1  e



Time is essential here.
Dynamic, differential equation
models appear in many systems
• Mechanical system, e.g. the mass-spring
2
system
d x
m
dt
2
k x 0
• Chemical reactions
• Population dynamics
• Economic models
The issue of complexity
• Consider modeling the dynamic behavior of
– An IC with millions of transistors
– A biological organism
• “Reductionist” method: zoom in a component,
write a differential equation model, combine
them into an overall model.
• Difficulty: solving those ODE’s is impossible;
even numerical simulation is prohibitive.
• Even harder: design the differential equation
(e.g., the circuit) so that it has a desired solution.
The “black box” concept
x
y
• Idea: describe a portion of a system by a inputoutput (cause-effect) relationship.
• Derive a mathematical model of this relationship.
This can involve ODEs, or other methods we will
study. Make reasonable approximations.
• Interconnect these boxes to describe a more
complex system.
Definition: Input-Output System
y
x
• The input function x(t) belongs to a space X, and
can be freely manipulated from outside.
• The output function y(t) varies in a space Y, and is
uniquely determined by the input function.
• The relationship between input and output is
described by a transformation T between X and Y.
Notation:
y ( t )  T  x (t ) 
or
y ( )  T  x( )
Example: RC circuit as an input-output system
+
+
R
x(t)
y(t)
C
_
dy
x  y  RC
dt
_
• We assume here that time starts at t=0, y(0) = 0
• To represent the mapping from x to y explicitly, we
must solve the differential equation
dy
  y   x,
dt
1
Here  
RC
y  0   0,
Solution of
dy
  y   x,
dt
y  0   0,
• First, solve homogeneous equation
Solution:
y (t )  Ce
 t
dy
 y 0
dt
• Next, look for a solution of the non-homogeneous
equation. One method: “Variation of constants”, try
a solution of the form
 t
y (t )  C (t )e
dy dC  t
 t

e  C (t )   e 
dt dt
dy
dC  t

 y 
e
dt
dt
Solution of
dy
  y   x,
dt
dy
dC  t
 y 
e  x
dt
dt
t
y  0   0,
dC
t

  e x (t )
dt
C (t )  C (0)    e x( )d

0
t
 y (t )  C (t )e t  e tC (0)    e  ( t  ) x( )d
0
t
Using initial condition y(0)=0,
y (t )    e
0
Input-output representation, y = T[x]
 ( t  )
x( )d