Signals and Systems
Fall 2003
Lecture #2
9 September 2003
1) Some examples of systems
2) System properties and
examples
a) Causality
b) Linearity
c) Time invariance
SYSTEM EXAMPLES
x(t) y(t)CT System
DT System
x[n]
y[n]
Ex,#1 RLC circuit
Force Balance:
Observation,Very different physical systems may be modeled
mathematically in very similar ways.
Ex,#2 Mechanical system
Ex,#3 Thermal system
Cooling Fin in Steady State
Ex,#3 (Continued)
Observations
Independent variable can be something other than
time,such as space.
Such systems may,more naturally,have boundary
conditions,rather than,initial” conditions.
Ex,#4 Financial system
Observation,Even if the independent variable is time,there
are interesting and important systems which have boundary
conditions,
Fluctuations in the price of zero-coupon bonds
t = 0 Time of purchase at price y
0
t = T Time of maturity at value y
T
y(t) = Values of bond at time t
x(t) = Influence of external factors on fluctuations in bond price
A rudimentary,edge” detector
This system detects changes in signal slope
Ex,#5
0 1 2 3
Observations
1) A very rich class of systems (but by no means all systems of
interest to us) are described by differential and difference
equations.
2) Such an equation,by itself,does not completely describe the
input-output behavior of a system,we need auxiliary
conditions (initial conditions,boundary conditions).
3) In some cases the system of interest has time as the natural
independent variable and is causal,However,that is not
always the case,
4) Very different physical systems may have very similar
mathematical descriptions.
SYSTEM PROPERTIES
(Causality,Linearity,Time-invariance,etc.)
Important practical/physical implications
They provide us with insight and structure that we
can exploit both to analyze and understand systems
more deeply,
WHY?
CAUSALITY
A system is causal if the output does not anticipate future
values of the input,i.e.,if the output at any time depends
only on values of the input up to that time,
Alreal-time physical systems are causal,because time
only moves forward,Effect occurs after cause,(Imagine
if you own a noncausal system whose output depends on
tomorrow’s stock price.)
Causality does not apply to spatially varying signals,(We
can move both left and right,up and down.)
Causality does not apply to systems processing recorded
signals,e.g,taped sports games vs,live broadcast.
Mathematically (in CT),A system x(t) → y(t) is causal if
CAUSALITY (continued)
when x
1
(t) → y
1
(t) x
2
(t) → y
2
(t)
and x
1
(t) = x
2
(t) for all t ≤ t
o
Then y
1
(t) = y
2
(t) for all t ≤ t
o
CAUSAL OR NONCAUSAL
TIME-INVARIANCE (TI)
Mathematically (in DT),A system x[n] → y[n] is TI if for
any input x[n] and any time shift n
0
,
Informally,a system is time-invariant (TI) if its behavior does not
depend on what time it is.
Similarly for a CT time-invariant system,
If x[n] → y[n]
then x[n - n
0
] → y[n - n
0
],
If x(t) → y(t)
then x(t - t
o
) → y(t - t
o
),
TIME-INVARIANT OR TIME-VARYING?
TI
Time-varying (NOT time-invariant)
NOW WE CAN DEDUCE SOMETHING!
These are the
same input!
Fact,If the input to a TI System is periodic,then the output is
periodic with the same period,
“Proof”,Suppose x(t + T) = x(t)
and x(t) → y(t)
Then by TI
x(t + T) → y(t + T),
↑↑
So these must be
the same output,
i.e.,y(t) = y(t + T),
LINEAR AND NONLINEAR SYSTEMS
Many systems are nonlinear,For example,many circuit
elements (e.g.,diodes),dynamics of aircraft,econometric
models,…
However,in 6.003 we focus exclusively on linear systems.
Why?
Linear models represent accurate representations of
behavior of many systems (e.g.,linear resistors,
capacitors,other examples given previously,…)
Can often linearize models to examine,small signal”
perturbations around,operating points”
Linear systems are analytically tractable,providing basis
for important tools and considerable insight
A (CT) system is linear if it has the superposition property:
If x
1
(t) → y
1
(t) and x
2
(t) → y
2
(t)
then ax
1
(t) + bx
2
(t) → ay
1
(t) + by
2
(t)
LINEARITY
y[n] = x
2
[n] Nonlinear,TI,Causal
y(t) = x(2t)Linear,notTI,Noncausal
Can you find systems with other combinations?
- e.g,Linear,TI,Noncausal
Linear,not TI,Causal
PROPERTIES OF LINEAR SYSTEMS
Superposition
If
Then
For linear systems,zero input → zero output
"Proof" 0 = 0? x[n]→ 0?y[n]= 0
Properties of Linear Systems (Continued)
a) Suppose system is causal,Show that (*) holds,
b) Suppose (*) holds,Show that the system is causal,
A linear system is causal if and only if it satisfies the
condition of initial rest:
“Proof”
LINEAR TIME-INVARIANT (LTI) SYSTEMS
Focus of most of this course
- Practical importance (Eg,#1-3 earlier this lecture
are all LTI systems.)
- The powerful analysis tools associated
with LTI systems
A basic fact,If we know the response of an LTI
system to some inputs,we actually know the response
to many inputs
Example,DT LTI System
Fall 2003
Lecture #2
9 September 2003
1) Some examples of systems
2) System properties and
examples
a) Causality
b) Linearity
c) Time invariance
SYSTEM EXAMPLES
x(t) y(t)CT System
DT System
x[n]
y[n]
Ex,#1 RLC circuit
Force Balance:
Observation,Very different physical systems may be modeled
mathematically in very similar ways.
Ex,#2 Mechanical system
Ex,#3 Thermal system
Cooling Fin in Steady State
Ex,#3 (Continued)
Observations
Independent variable can be something other than
time,such as space.
Such systems may,more naturally,have boundary
conditions,rather than,initial” conditions.
Ex,#4 Financial system
Observation,Even if the independent variable is time,there
are interesting and important systems which have boundary
conditions,
Fluctuations in the price of zero-coupon bonds
t = 0 Time of purchase at price y
0
t = T Time of maturity at value y
T
y(t) = Values of bond at time t
x(t) = Influence of external factors on fluctuations in bond price
A rudimentary,edge” detector
This system detects changes in signal slope
Ex,#5
0 1 2 3
Observations
1) A very rich class of systems (but by no means all systems of
interest to us) are described by differential and difference
equations.
2) Such an equation,by itself,does not completely describe the
input-output behavior of a system,we need auxiliary
conditions (initial conditions,boundary conditions).
3) In some cases the system of interest has time as the natural
independent variable and is causal,However,that is not
always the case,
4) Very different physical systems may have very similar
mathematical descriptions.
SYSTEM PROPERTIES
(Causality,Linearity,Time-invariance,etc.)
Important practical/physical implications
They provide us with insight and structure that we
can exploit both to analyze and understand systems
more deeply,
WHY?
CAUSALITY
A system is causal if the output does not anticipate future
values of the input,i.e.,if the output at any time depends
only on values of the input up to that time,
Alreal-time physical systems are causal,because time
only moves forward,Effect occurs after cause,(Imagine
if you own a noncausal system whose output depends on
tomorrow’s stock price.)
Causality does not apply to spatially varying signals,(We
can move both left and right,up and down.)
Causality does not apply to systems processing recorded
signals,e.g,taped sports games vs,live broadcast.
Mathematically (in CT),A system x(t) → y(t) is causal if
CAUSALITY (continued)
when x
1
(t) → y
1
(t) x
2
(t) → y
2
(t)
and x
1
(t) = x
2
(t) for all t ≤ t
o
Then y
1
(t) = y
2
(t) for all t ≤ t
o
CAUSAL OR NONCAUSAL
TIME-INVARIANCE (TI)
Mathematically (in DT),A system x[n] → y[n] is TI if for
any input x[n] and any time shift n
0
,
Informally,a system is time-invariant (TI) if its behavior does not
depend on what time it is.
Similarly for a CT time-invariant system,
If x[n] → y[n]
then x[n - n
0
] → y[n - n
0
],
If x(t) → y(t)
then x(t - t
o
) → y(t - t
o
),
TIME-INVARIANT OR TIME-VARYING?
TI
Time-varying (NOT time-invariant)
NOW WE CAN DEDUCE SOMETHING!
These are the
same input!
Fact,If the input to a TI System is periodic,then the output is
periodic with the same period,
“Proof”,Suppose x(t + T) = x(t)
and x(t) → y(t)
Then by TI
x(t + T) → y(t + T),
↑↑
So these must be
the same output,
i.e.,y(t) = y(t + T),
LINEAR AND NONLINEAR SYSTEMS
Many systems are nonlinear,For example,many circuit
elements (e.g.,diodes),dynamics of aircraft,econometric
models,…
However,in 6.003 we focus exclusively on linear systems.
Why?
Linear models represent accurate representations of
behavior of many systems (e.g.,linear resistors,
capacitors,other examples given previously,…)
Can often linearize models to examine,small signal”
perturbations around,operating points”
Linear systems are analytically tractable,providing basis
for important tools and considerable insight
A (CT) system is linear if it has the superposition property:
If x
1
(t) → y
1
(t) and x
2
(t) → y
2
(t)
then ax
1
(t) + bx
2
(t) → ay
1
(t) + by
2
(t)
LINEARITY
y[n] = x
2
[n] Nonlinear,TI,Causal
y(t) = x(2t)Linear,notTI,Noncausal
Can you find systems with other combinations?
- e.g,Linear,TI,Noncausal
Linear,not TI,Causal
PROPERTIES OF LINEAR SYSTEMS
Superposition
If
Then
For linear systems,zero input → zero output
"Proof" 0 = 0? x[n]→ 0?y[n]= 0
Properties of Linear Systems (Continued)
a) Suppose system is causal,Show that (*) holds,
b) Suppose (*) holds,Show that the system is causal,
A linear system is causal if and only if it satisfies the
condition of initial rest:
“Proof”
LINEAR TIME-INVARIANT (LTI) SYSTEMS
Focus of most of this course
- Practical importance (Eg,#1-3 earlier this lecture
are all LTI systems.)
- The powerful analysis tools associated
with LTI systems
A basic fact,If we know the response of an LTI
system to some inputs,we actually know the response
to many inputs
Example,DT LTI System
