Signal and Systems
Prof. H. Sameti
Chapter #1:
1)
Signals
2)
Systems
3)
Some examples of systems
4)
System properties and examples
a)
b)
c)
Causality
Linearity
Time invariance
Reformatted version of open course notes from MIT opencourseware
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003-signals-and-systems-spring-2010/lecture-notes/
Book Chapter#: Section#
Signals
 Signals are functions of independent variables that carry
information. For example:

Electrical signals
voltages and currents in a circuit

Acoustic signals
audio or speech signals(analog or digital)

Video signals
intensity variations in an image

Biological signals
sequence of bases in a gene
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Book Chapter#: Section#
The Independent Variables
 Can be continuous
 Trajectory of a space shuttle
 Mass density in a cross-section of a brain
 Can be discrete
 DNA base sequence
 Digital image pixels
 Can be 1-D, 2-D, ••• N-D

For this course: Focus on a single (1-D) independent variable which
we call "time”
Continuous-Time (CT) signals: x(t), t
continuous values
Discrete-Time (DT) signals: x[n] , n
integer values only
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Book Chapter#: Section#
CT Signals
 Most of the signals in the physical world are CT signals—
E.g. voltage & current, pressure, temperature, velocity, etc.
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Book Chapter#: Section#
DT Signals
 x[n], n—integer, time varies discretely
 Examples of DT signals in nature:
 DNA base sequence

Population of the nth generation of certain species
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Book Chapter#: Section#
Many human-made DT Signals
Ex.#1 Weekly
Dow-Jones industrial average
Ex.#2 digital image
Why DT? — Can be processed by modern
digital computers and digital signal
processors (DSPs).
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Book Chapter#: Section#
SYSTEMS
 For the most part, our view of systems will be from an
input-output perspective:
 A system responds to applied input signals, and its
response is described in terms of one or more output signals
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Book Chapter#: Section#
EXAMPLES OF SYSTEMS
 An RLC circuit




Dynamics of an aircraft or space vehicle
An algorithm for analyzing financial and economic factors to
predict bond prices
An algorithm for post-flight analysis of a space launch
An edge detection algorithm for medical images
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Book Chapter#: Section#
SYSTEM INTERCONNECTIOINS
 An important concept is that of interconnecting systems
 To build more complex systems by interconnecting simpler subsystems
 To modify response of a system
Cascade
Parallel
Feedback
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Book Chapter#: Section#
SYSTEM EXAMPLES
 Example. #1 RLC circuit
di (t )
Ri (t )  L
 y (t )  x(t )
dt
dy (t )
i (t )  c
dt

d 2 y (t )
dy (t )
LC

RC
 y (t )  x(t )
2
dt
dt
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

Example. #2 Mechanical system
Force Balance:
d 2 y (t )
dy (t )
M

x
(
t
)

Ky
(
t
)

D
dt
dt 2
d 2 y (t )
dy (t )
M

D
 Ky(t )  x(t )
2
dt
dt

Observation: Very different physical systems may be modeled
mathematically in very similar ways.
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Example. #3 Thermal system

 Cooling Fin in Steady State



t = distance along rod
y(t) = Fin temperature as function of position
x(t) = Surrounding temperature along the fin
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Example. #3 Thermal system (Continued)

d 2 y (t )
 K [ y (t )  x(t )]
2
dt
y (T0 )  y0
dy
(T1 )  0
dt
 Observations


Independent variable can be something other than time,
such as space.
Such systems may, more naturally, have boundary conditions,
rather than “initial” conditions.
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
Example. #4 Financial system
Fluctuations in the price of zero-coupon bonds
t = 0
Time of purchase at price y0
t = T Time of maturity at value yt
y(t) = Values of bond at time t
x(t)= Influence of external factors on fluctuations in bond price
d 2 y (t )
dy (t )



f
y
(
t
),
,
x
(
t
),
x
(
t
),...,
x
(
t
),
t


1
2
N
2
dt
dt


y (0)  y0 , y (T )  yT .
Observation: Even if the independent variable is time,
there are interesting and important systems which have
boundary conditions.
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Example. #5

 A Rudimentary “Edge” Detector
y[n]=x[n+1]-2x[n]+x[n-1]
={x[n+1]-x[n]}-{x[n]-x[n-1]}
= “Second difference”
 This system detects changes in signal slope
a)
b)
x[n]=n → y[n]=0
x[n]=nu[n] → y[n]
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Book Chapter#: Section#
Observations
1)
2)
3)
4)
A very rich class of systems (but by no means all systems
of interest to us) are described by differential and
difference equations.
Such an equation, by itself, does not completely describe
the input-output behavior of a system: we need auxiliary
conditions (initial conditions, boundary conditions).
In some cases the system of interest has time as the
natural independent variable and is causal. However, that
is not always the case.
Very different physical systems may have very similar
mathematical descriptions.
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Book Chapter#: Section#
SYSTEM PROPERTIES
(Causality, Linearity, Time-invariance, etc.)
 WHY?

Important practical/physical implications

They provide us with insight and structure that we can
exploit both to analyze and understand systems more deeply.
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Book Chapter#: Section#
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.

All real-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.
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Book Chapter#: Section#
CAUSALITY (continued)
 Mathematically (in CT):

A system x(t) →y(t) is causal if
when
and
Then
x1(t) →y1(t) x2(t) →y2(t)
x1(t) = x2(t) for all t≤ to
y1(t) = y2(t) for all t≤ to
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Book Chapter#: Section#
CAUSAL OR NONCAUSAL
𝑦(𝑡 = 𝑥 2 (𝑡 − 1

depends on x(4) … causal
𝐸. 𝑔. 𝑦(5
𝑦(𝑡 = 𝑥(𝑡 + 1

y depends on future
𝐸. 𝑔. 𝑦(5 = 𝑥(6 ,
𝑦[𝑛 = 𝑥[−𝑛

ok, but
𝐸. 𝑔. 𝑦[5 = 𝑥[−5
 𝑦[−5 = 𝑥[5 y depends on future
 𝑦[𝑛
=
1 𝑛+1 3
𝑥 [𝑛
2
noncausal
noncausal
− 1] depends on x(4) … causal
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Book Chapter#: Section#
TIME-INVARIANCE (TI)
 Informally, a system is time-invariant (TI) if its behavior
does not depend on what time it is.
Mathematically (in DT): A system x[n] → y[n] is TI if for
any input x[n] and any time shift n0,
If
x[n] →y[n]
then
x[n -n0] →y[n -n0]
 Similarly for a CT time-invariant system,
If
x(t) →y(t)
then
x(t -to) →y(t -to) .

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Book Chapter#: Section#
TIME-INVARIANT OR TIMEVARYING?
 𝑦(𝑡
 𝑦[𝑛
= 𝑥 2 (𝑡 + 1 is TI
=
1 𝑛+1 3
𝑥 [𝑛
2
− 1] is TV (NOT time-invariant)
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Book Chapter#: Section#
NOW WE CAN DEDUCE
SOMETHING!
 Fact: If the input to a TI System is periodic, then the
output is periodic with the same period.
“Proof”: Suppose
and
Then by TI
x(t + T) = x(t)
x(t) →
y(t)
x(t + T) →y(t + T).
↑
↑
These are the
same input!
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So these must be
the same output,
i.e., y(t) = y(t + T).
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Book Chapter#: Section#
LINEAR AND NONLINEAR SYSTEMS
Many systems are nonlinear. For example: many circuit elements
(e.g., diodes), dynamics of aircraft, econometric models,…
However, in this course 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
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Book Chapter#: Section#
LINEARITY
 A (CT) system is linear if it has the superposition
property:
If
x1(t) →y1(t) and x2(t) →y2(t)
then ax1(t) + bx2(t) → ay1(t) + by2(t)
y[n] = x2[n]
Nonlinear, TI, Causal
y(t) = x(2t)
Linear, not TI, Noncausal
Can you find systems with other combinations ?
-e.g. Linear, TI, Noncausal
Linear, not TI, Causal
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Book Chapter#: Section#
EXAMPLES OF LINEAR SYSTEMS
Ex. 1:
= 𝑥 ∗ (𝑡
 Additive, but not scalable, so not linear
 𝑦(𝑡
Ex. 2:
𝑦

𝑡
=
𝑥 2 (𝑡
𝑥(𝑡−1
Scalable, but not additive, so not linear
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Book Chapter#: Section#
PROPERTIES OF LINEAR SYSTEMS
 Superposition
If
Then
𝑥𝑘 [𝑛 → 𝑦𝑘 [𝑛
𝑎 𝑥 [𝑛 → 𝑘𝑎𝑘 𝑦𝑘 [𝑛
𝑘 𝑘 𝑘
 For linear systems, zero input → zero output
"Proof"
0=0⋅x[n]→0⋅y[n]=0
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Book Chapter#: Section#
Properties of Linear Systems
(Continued)
 A linear system is causal if and only if it satisfies the
condition of initial rest:
𝑥(𝑡 = 0 𝑓𝑜𝑟 𝑡 ≤ 0 → 𝑦(𝑡 = 0 𝑓𝑜𝑟 𝑡 ≤ 𝑡0 (∗
“Proof”
a) Suppose system is causal. Show that (*) holds.
b) Suppose (*) holds. Show that the system is causal.
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Book Chapter#: Section#
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
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Book Chapter#: Section#
Example:
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DT LTI System
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