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ELEC264: Signals And Systems
Topic 1: LTI Systems
Aishy Amer
Concordia University
Electrical and Computer Engineering
Figures and examples in these course slides are taken from the following sources:
• A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997
• M.J. Roberts, Signals and Systems, McGraw Hill, 2004
• J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003
Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
2
Signals and Systems
A signal is any physical phenomenon which conveys information (e.g., human voice)
Systems respond to signals and produce new signals (e.g., Aircraft, human body)
Excitation signals are applied at system inputs
Response signals are produced at system outputs
3
Signals & Systems
Representation, transformation and
manipulation of signals and
information they contain
Modifying and analyzing information
with computers
4
What is a signal ?
A flow of information
Mathematically, x(t)
a function of independent variables
such as time (e.g. speech signal),
position (e.g. image)
t : a common convention is to refer to
the independent variable as time,
although may in fact not
5
Example signals
Speech: 1-Dimension signal as a function of time s(t)
Grey-scale image: 2-Dimension signal as a function of
space I(x,y)
Video: 3 x 3-Dimension signal as a function of space and
time {R(x,y,t), G(x,y,t), B(x,y,t)}
7
Continuous to discrete?
Computers and other digital devices are restricted to discrete time
Take samples of the continuous signal => discrete
Analog to digital conversion (ADC)
]5.42.34.25.0[][
)( 4
nx
etx
t
9
Key History of Signals &Systems
Prior to 1950’s: analog signal processing using
electronic circuits or mechanical devices
1950’s: computer simulation before analog
implementation, thus cheap to try out
1965: Fast Fourier Transforms (FFTs) by
Cooley and Tukey – make real time digital
signal processing (DSP) possible
1980’s: IC technology boosting digital signal
processing (DSP)
10
Typical system components
Physical signals Analog signals Digital signals
Transducers
e.g. microphones
Analog-to-digital
converters
Digital-to-Analog
converters
Output devices
11
Applications of Signals &Systems
Speech processing
Enhancement – noise filtering
Coding, synthesis and recognition
Image processing
Enhancement, coding, pattern recognition (e.g. OCR)
Multimedia processing
Media transmission, digital TV, video conferencing
Communications
Biomedical engineering
Navigation, radar, GPS
Control, robotics, machine vision
12
Pros and cons of DT Signals
&System
Pros Easy to duplicate
Stable and robust: not varying with temperature,
storage without deterioration
Flexibility and upgrade: use a general computer or
microprocessor
Cons Limitations of ADC and DAC
High power consumption and complexity of a DSP
implementation: unsuitable for simple, low-power
applications
Limited to signals with relatively low bandwidths
13
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
14
Types of signals
A variable (or multiple variables) that changes in time
Speech or audio signal: Amplitude that varies in time
Temperature readings at different hours of a day
Stock price changes over days
…
More generally, a signal may vary in time, 1D, and/or in space, 2-D
A picture: the color varies in a 2-D space
A video sequence: the color varies in 2-D space and in time
15
Types of signals
The independent variable may be either continuous or discrete
Continuous-time signals: The time varies continuously
Discrete-time signals are defined at discrete times
• Represented as sequences of numbers
• Discrete ~ Countable
The signal amplitude may be either continuous or discrete
Analog signals: both time and amplitude are continuous
Digital signals: both are discrete
Signal processing systems classification follows the same lines
16
Types of signals
Signals are functions x(t) = et/4
to manipulate them we apply
calculus on Continuous-Time (CT) signals
algebra on Discrete-Time (DT) signals
CT Signal x(t) is a continuous-value function
DT signal x[n] is a sequence of real or complex numbers
x[n] = [0.5 2.4 3.2 4.5]• x[0] =0.5, x[1]=2.4,…
17
Basic Signals
Sinusoidal
Exponential
Unit Impulse
Unit Step
An arbitrary signal can be expressed as a sum of manysinusoidal signals with different frequencies, amplitudesand phases
00
01)(
t
ttu
1)( dtt
)cos()( 0ttx
tjetx 0)(
00 2, f
00
01)(
t
tt
18
Periodic CT Signals
A CT signal is periodic if there is a positive value
for which
The fundamental period of is the smallest positive
value of for which the equation above holds
Examples:
is periodic with fundamental period
30),3
cos()( tttx
19
Sinusoidal Signals
Sinusoidal signals: important because they can be used to synthesize any signal Phase shift: how much the sinusoidal signal is shifted away from t=0
Music notes are essentially sinusoids at different frequencies
f0 = 1000Hz
f0 = 2000Hz
20
Constant Signal
Case:
•Let the fundamental frequency be zero,
i.e.,constant signal (d.c) has zero
rate of oscillation
•x(t) is periodic with period T for any positive
value of T
Thus fundamental period is undefined
0o
fT
10
f=0
22
Frequency content in signals
A constant : only zero frequency (DC) component
A sinusoid : Contain only a single frequency component
Slowly varying : contain low frequency only
Fast varying : contain very high frequency
Sharp transition : contain from low to high frequency
Music: :
contain both slowly varying and fast varying components
23
What is frequency of an
arbitrary signal?
Sinusoidal signals have a distinct (unique) frequency
An arbitrary signal x(t) does not have a unique frequency
x(t) can be decomposed into many sinusoidal signals
with different frequencies, each with different magnitude
and phase
)32
3cos(7)
4cos()( tttx
24
Deterministic vs. random
signals
A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table
Future values of the signal can be calculated from past values with complete confidence
A random signal cannot be described by a mathematical formula
has a lot of uncertainty about its behavior
Future values of a random signal cannot be accurately predicted
Future values can usually only be guessed based on the averages of sets of signals
atetx )(
26
Types of Signals:Continuous-Time (CT) vs. Discrete-time (DT)
Analog vs. digital
Periodic vs. non-periodic
28
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
29
Sinusoidal & Exponential
•Play a central role in signals &
systems
•Serve as building block for many
other signals
30
Periodic Complex Exponential
& Sinusoidal Signal
•Sinusoidal signal
A CT sinusoidal signal
1. has one unique frequency
Two signals with different
frequencies are never
identical
2. is always periodic for any
tjte oo
tj o sincos
angle Phase :
Period lFundementa :
2 Frequency lFundementa /1
2 frequency Angular :
at t signal theof value: x(t)
instant time: t
Signal : x
Amplitude Signal :A
)cos()(
0
000
000
0
T
TTf
f
tAtx
• Complex exponential signal:
They can be written in terms of sinusoidal
signals
0
Src: Wikipedia
31
Complex numbers
sin2
cos2
sincos
:relation '
. , |z|r
z, of phaseor angle theis and
z, of magnitude theis 0r where
-:formPolar
numbers, real are b and a and 1
,
:forr rectangulaor
jee
ee
je
sEuler
z
rez
jwhere
jbaz
Cartesian
jj
jj
j
j
32
tjjtjj
ooo ee
Aee
AtA
22)cos(
:lexponentiacomplex
periodic of in termswritten
becan period, lfundamenta
with signal, sinusoidal The
Periodic Complex Exponential
& Sinusoidal Signal
33
ondcycles/secor Hertzin frequency lfundamenta theof
condradians/sein freq.angular lfundamenta of2 o
period lfundamenta the
with
})(
{.)sin(
})(
Re{.)cos(
:followsIt
oT
tojeIMAtoAor
tojeAtoA
Periodic Complex Exponential & Sinusoidal Signal
34
Periodic Complex Exponential
A necessary condition for a complex
exponential tje to be periodic with period To is 1oTj
e
,...2,1,0,2T i.e. .2 of
multiple a is that implieswhich
o kk
To
00 .)( TjtjTtj
eee
35
Harmonically Complex
Exponential Signals
0o
k
,.,frequency positive single
a of multiples arethat sfrequencie
lfundamenta with lsexponentia
periodic ofset :)( 0
kei
Aetx
k
tjk
k
angei
To
oo
o
k .. of multipleinteger an bemust then
,2
define weIf
K
TT 0
36
Harmonically Complex
Exponential Signals
||
periods lfundamenta and |k| sfrequencie
lfundamenta with periodic is )( 0, k
constant. a is )( 0,kFor
2,...1,0,k ,)(
o
k
TT
tx
tx
etx
o
k
k
tjk
ko
37
Harmonically Complex
Exponential Signals
.Tlength of interval any time
during periods lfundamenta its of |k|exactly
throughgoesit as well,as T periodwith
periodic still is )( harmonickth The
o
o
txk
38
General Complex Exponential
Signals
.|||C|
)(Then
.ja and |C|C
:formcartesian in a
and formpolar in expressed is C If
numbers.
complex are a'' and C''both where,)(
)()(
o
tjttjj
at
j
at
oo eeCee
Cetx
e
Cetx
39
General Complex Exponential
Signals
l.exponentia decayingby multipled signals sinusoidal is x(t)
0, If
l.exponentia growingby multipled signals sinusoidal is x(t)
0, If
.sinusoidal are partsimaginary & real the
0, If
)sin(||)cos(||)(
relation, sEuler' Using
teCjteCCetx o
t
o
tat
41
pi=3.142;
t=-10:.1:10;
f=2000;
w=2*pi*f;
sigma=0.1;
x=zeros(size(t));
x=exp((sigma+w*i)*t);
theta=pi/4;
c=1*exp(i*theta);
y=c*x;
subplot(2,1,1);
plot(t,y);
grid;
sigma=-0.1;
x=zeros(size(t));
x=exp((sigma+w*i)*t);
theta=pi/4;
c=1*exp(i*theta);
y=c*x;
subplot(2,1,2);
plot(t,y);
grid;
end;
Matlab Program for
Growing & Decaying Sinusoids
42
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
43
Periodic Signals: Examples
Planet and satellite orbital positions
Phases of the moon
Firing pattern of spark plugs in a car traveling at a constant speed
Blinker lights in automobiles
Angular position of a pendulum in antique clocks
Migration pattern of birds
44
Periodic Signals:
Sinusoidal Signals
Sinusoidal signal:
Unique frequency e.g., 10Hz
An arbitrary x(t):
No unique frequency
x(t) = summation of sine or cosine
functions at different frequencies
47
Sum of CT periodic signals
The period of the sum of CT periodic functions is the
least common multiple of the periods of the individual
functions summed
If the least common multiple is infinite, the sum is aperiodic
49
Aperiodic CT Signals
A function that is not periodic is called
aperiodic
Aperiodic signal
Examples:
)()( nTtxtx
otherwise 0
30)3
cos()(
tttx
50
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
51
Signal Power and Energy
Signal: A function of a time-varying amplitude
Signal: Many different physical entities
No unit for energy/power
Often, a signal is a function of varying amplitude over time
A good measurement of the strength of a signal would be the area under the function
But this area may have a negative part which does not have less strength than a positive signal of the same size
This suggests either squaring the signal or taking its absolute value, then finding the area under that curve
Energy/power: strength of the signal
54
Signal Energy and Power
A signal with finite signal
energy is called an energy
signal
If the signal does not
decay infinite energy
A signal with infinite signal
energy and finite average
signal power is called a
power signal
60
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
61
Transformations of CT
signals
Modifying a signal x(t) through
1. Transformation of the independent
variable t e.g., x(t/2)
2. Combination of signals:
y(t) = x(t) w(t)
y(t) = x(t)/w(t)
y(t)=sin(t)x(t)
y(t)=dx/dt
65
Transformation of Independent
Variable or Modification of
independent variable t
Modifying signals through elementary
transformations
Examples of elementary transformation
time shift, x(t-t0)
time reversal, x(-t)
time scaling, x(0.5t)
69
Time scaling of continuous
signal
x(t)
x(2t)
x(t/2)
t
t
t
Compression a>1
Linearly stretching a<1
70
Transformation of Independent
Variable: Applications
Play music at faster or slower speed
Aircraft control system:
Input correspond to pilot action
Actions are transformed by electrical &
mechanical system of the aircraft to
changes to aircraft trust or position control
surfaces such as the rudder & ailerons
Finally these changes affect the dynamics
& kinematics such as the aircraft velocity
and heading
71
Transformations of CT signals:
Time Shifting Applications
Time-shifting occurs in many real physical systems:
Listening to someone talking 2m away
Received signal will be delayed, but the delay won’t be noticeable
Satellite communication systems (delay can be noticeable if ground stations are not directly below the satellite)
Radar systems:• Transmitted signal Ax(t)
• Received signal Bx(t-to), with B<A, due to attenuation
72
Transformations of CT signals:
Time Scaling Applications
Examples:
Playing an audio tape at a faster or slower speed
Doppler effect: standing by the side of a road
while a fire truck approaches and then passes by
73
Modification of independent
variable (time axes)
)( is what fixed afor
)( is what fixed afor
ofargument theis :
of parameters are :,
of t variableindependen theis :
offunction a :
)()(
tx
tx
xt
x
xt
tx
txty
• Recommended approach:
o Sketch y(t) for a selected set of t until y(t) becomes clear
o Steps:
1. Rewrite: y(t) = x(α(t+β/ α))
2. Scale by |α|: x(|α|t)
3. Invert x(|α|t) if α<0
4. Shift to the LEFT by |β/ α| if β/ α>0
5. Shift to the RIGHT by |β/ α| if β/ α<0
79
Transformation of CT
Signals: Examples x(t)
x(t+1), x(t) shifted left by 1sect
t
1
1
0
0
1 2
1 2-1
80
Tables of x(t) & x(t+1) & x(-t+1)
t x(t) x(t+1) x(-t+1)
-2 0 0 0
-1 0 1 0
0 1 1 1
1 1 0 1
2 0 0 0
3 0 0 0
81
Example
x(t+1) is x(t) shifted left by 1
x(-t+1) is x(t+1) flipped about t=0t
t
1
1
0
0
1 2
1 2-1
-1
82
Example: Alternative 1
x(t-1) is x(t) shifted right by 1sec
x(-t+1)=x(-1(t-1))
Flip about axis t=1
t
t
1
1
0
0
1 2
1 2-1
83
Example , Method 2
x(-t), flip about axis t=0
x(-t+1), shift right (because -t) by 1t
t
1
1
0
0
1 2
1 2-1
-1
85
Example
x(3t/2), x(t) compressed by 2/3
t
1
0 1 2-1 2/3 4/3
x((3/2)*(t+2/3)),
x(t) compressed by 2/3
& shifted left by 2/3
t
1
0 1-1 2/3-2/3
89
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
90
Even and Odd Signals
An even signal is identical to its time reversed
Example:
An odd signal has the property
92
Even and Odd Parts of CT
Signals
The even part of a CT function is
The odd part of a CT function is
A function whose even part is zero is odd and a function whose odd part is zero is even
The derivative of an even CT function is odd and the derivative of an odd CT function is even
The integral of an even CT function is an odd CT function, plus a constant, and the integral of an odd CT function is even
97
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
98
Singularity Functions
In engineering, we often deal with the idea of an action occurring at a point
Whether it be a force at a point in space or a signal at a point in time, it becomes worth while to develop some way of quantitatively defining this
This leads us to the idea of a unit impulse
Unit impulse & complex exponential functions are the two most important functions in systems and signals courses
99
Singularity Functions
Many useful signals are not continuous or
differentiable at every point in time.
For instance, describe the operation of switching on
or off a signal at some specified time
Singularity functions are a set of functions that
are related to one another via integrals/derivatives
and can be used to mathematically describe signals
with discontinuities.
Example:
),...(),( tut
101
CT Unit Impulse
The CT unit impulse function is represented as
The area under the CT unit impulse is equal to 1
00
01)(
t
tt
1)( dtt
102
Properties of CT Impulse
Sampling (Shifting) Property:
the value of the function at a point
Scaling Property
)/()(||
1 abtbata
105
Relationship: CT unit step and
unit impulse
The CT unit impulse is the first derivative of the
continuous-time unit step
The area under the CT unit impulse is equal to 1
The CT unit impulse function is represented as
106
Relationship: CT unit step and
unit impulse
Example: consider a mass with zero velocity. Assume
that a force is applied to it to change its velocity from
zero to 1 on a surface with no friction. The acceleration
of the mass will be a unit impulse
It can be easily verified that
107
Relationship: CT unit step and
unit impulse
The CT unit step is not differentiable at t=0
One can use continuous approximation to the unit
step
Corresponding unit impulse
113
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
114
Introduction to Systems
To get the output y(t) Apply the system S{} on input x(t)
y(t) is the response of S{} to x(t)
A system: An integrated whole composed of diverse, interacting, specialized parts
System performs a function not possible with any of the individual parts
Any system has objectives
Systems respond to particular signals by producing other signals or some desired behavior
116
Introduction to Systems:
a Communication System
A communication system has an information
signal plus noise signals
This is an example of a system that consists of
an interconnection of smaller systems
Cellphones are based on such systems
119
Introduction to Systems:
Interconnections of Systems
Systems can be interconnected in series (cascade),
parallel, feedback, or combination
120
Introduction to Systems:
Response of Systems
Systems respond to signals and produce new
signals
Real signals are applied at system inputs and
response signals are produced at system outputs
Example: What is the response of a system to a
unit impulse?
121
Response of systems
The response of a system to an impulse is
called "impulse response" h(t)
The impulse response h(t) completely
characterize a Linear Time-Invariant (LTI)
system
122
Basic System Properties
Linearity or superposition property
Linear system
possesses the property of superposition
any constant values a and b, the following equation is satisfied
It can be easily verified that for linear systems:
an input which is zero for all time,
results in an output which is zero for all time
124
Basic System Properties:
Examples
Linearity:
linear? )()( Is 2 txty
)()(
2
)()()]()([)(Then
)()()(Let
21
2
2
2
21
2
1
2
21
2
21
21
tyty
xbxabxxa
tybtyatxbtxaty
txbtxatx
TI? )()( Is 2 txty Time Invariance:
x(t) ~ (x(t))2
x(t-t0) ~ (x(t-t0))2 = y(t-t0)
125
Basic System Properties
Memory:
System is memoryless if its output for each value of the independent variable at a given time is dependent only on the input at that same time
Memoryless CT system: the input-output relationship of a resistor
Examples: y(t) = x(t-1); y(t) = x(t/2);
)()( tRitv
126
Basic System Properties
Invertibility:
System is invertible if distinct inputs lead to
distinct outputs
If a system is invertible
• inverse system exists, when cascaded with the
original system, yields an output equal to the input
to the first system.
Example
x(t) MP3 y(t) MP3 x(t) ?
INV
SSx(t) y(t) x(t)
127
Basic System Properties
Causality: A system is causal (or non-anticipative) if the output
at any time depends only on values of the input at the present time and in the past
All memoryless systems are causal
Examples of causal systems
Examples of non-causal systems
Causal signals are zero for all negative t
Anti-causal signals are zero for all positive t
Non-causal signals have non-zero values in both positive and negative t
128
Basic System Properties
Stability: Stable system: small inputs lead to responses that do
not diverge
BIBO stable system: bounded input results in a
bounded output. If |x(t)|<∞ |y(t)|<∞
Stable system is always BIBO stable but a BIBO
stable system is not necessarily stable
Is the accumulator system stable?
Examples:
129
CT Systems: Example
A system is defined by the following relationship: Is this system: BIBO Stable; Casual; Linear; Memoryless; Time-Invariant;
Invertible?
All answers must be justified (i.e. a simple “Yes” or “No” is not sufficient).
The system is Stable:
So, for any bounded input, the output is bounded.
The system is Casual: Output at time t depends on input at time t/2 - which is the past. The system is Casual.
The system is Linear: Consider:
)2/()2/sin()( txtty
1)2/sin(1 t
)()()2/()2/sin()2/()2/sin(
))2/()2/()(2/sin()2/()2/sin()(
)()()(
)2/()2/sin()(
)2/()2/sin()(
2121
2133
213
22
11
tbytaytxtbtxta
tbxtaxttxtty
tbxtaxtxlet
txtty
txtty
130
CT Systems: Example
The system is not Memoryless: Output does not solely depend on current input values (i.e. depends on past input values).
The system is not Time-invariant:
)_(
)2/()2/sin()2/()2/sin()(
)()(
)2/)(()2/)sin(()_(
)2/()2/sin()(
0
011
01
000
tty
ttxttxtty
ttxtxlet
ttxtttty
txtty
131
CT Systems: Example
The system is not Invertible: There
are Different Inputs which lead to the
same Outputs
0)2/()sin()2/()2/sin()(
)2()(
0)2/()0sin()2/()2/sin()(
)()(
)2/()2/sin()(
221
2
111
1
txtxtty
ttx
txtxtty
ttx
txtty
132
Outline Introduction
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
133
Discrete-time signals
Sequences of numbers
Periodic sampling of an
analog signal
integeran is where
]},[{
n
nnxx
period. sampling thecalled is where
),(][
T
nnTxnx a
x[1]
x[2]
x[n]x[-1]
x[0]
ADC
136
Basic DT Signals
Sinusoidal
Exponential
Unit Impulse
Unit Step00
01][
n
nnu
)cos(][ 0nnx
njenx 0][
00 2, f
00
01][
n
nn
137
DT Sinusoidal signals
angle Phase :
Period lFundementa :
Frequency lFundementa : /1
2 frequency Angular :
nat signal theof value:x[n]
index time:n
Signal :x
Amplitude Signal :A
)sin(][
0
00
000
0
N
NF
F
nAnxA DT sinusoidal signal
1. is NOT always periodic
** Periodic only if its frequency
is a rational number
2. Two signals with different
frequencies maybe identical
njnj oo ee)2(
141
Decaying or
Damped Sinusoids
Examples:
oResponse of RLC circuits
oMechanical systems having both
damping & restoring forces e.g.
automotive suspension system
x[n] = eσn.cos(ω0n)σ>0 : grows
σ<0 : decays
144
Real Exponential Signal
Real-valued discrete exponentials
are used to describe:
1) Population growth as function of
generation
2) Total return on investment as a
function of day, month or
quarter
148
Complex exponential signals
njjnjj
o
oo
nj
o
o
nj
o
n
oo
o
o
eeA
eeA
nA
njne
nAnx
enx
Aenx
SignallExponentia
22)cos(
sincos:relation sEuler' From
radians of units have and
both then ess,dimensionl asn Taking
)cos(][
:signal sinusoidal torelatedclosely is signal This
][
imaginary)purely ( j & 1Alet ,][
149
General Complex
Exponential Sequence
)sin(||||)cos(||||
||||
||||][
,|| and || If
numbers.complex generalin are andA where
,][
00
)( 0
0
0
nAjnA
eA
eeAAnx
eAAe
Anx
nn
njn
njnjn
jj
n
151
General complex exponential
sequence
By analogy with the continuous-time case,
the quantity is called the frequency of
the complex sinusoid or complex
exponential and is the phase
n is always an integer
differences between discrete-time and
continuous-time
0
)sin(||)cos(||||][
,1||When
00
)( 0 nAjnAeAnxnj
153
Periodicity Properties of DT
Complex Exponentials
There are differences in each of the
above properties for the discrete-time
case of nj oe
o
j
o
j
of any valuefor periodic is e 2)
noscillatio of rate theishigher the, islarger The)1
et counterpar CT of properties Two
o
o
t
t
154
Periodicity Properties of DT
Complex Exponentials
signals DTfor 2 of intervalfrequency
consider only to need we,2 ofy periodicit thisof Because
of aluesdistinct v allfor distinct all are signals the
wherebycase CT thefromdifferent very is This
on so and,4,2 sfrequencieat Similarly
at that as same theis 2 frequency at lexponentia the
:2 frequency with lexponentiacomplex DT heConsider t
o
oo
oo
2)2(
o
njnjnjnj ooo eeee
155
Periodicity Properties of DT
Complex Exponentials
2at signal d.c.or sequenceconstant a i.e. 0 todecrease will
n oscillatio r the thereafte, until increasesn oscillatio the
n)oscillatio no sequence,constant (d.c., 0 from Increasing
magnitudein increased is as
noscillatio of rate increasingy continuall a havenot does signal the
signal, DT ofy periodicit implied thisof Because
o
o
o
o
nj oe
156
Periodicity Properties of DT
Complex Exponentials
in timepoint each at sign changing rapidly, oscillates signal the
,)1()(e ,of multiple odd,for :Note
of multiple odd,3,at are sfrequencieHigh 2.
of multipleeven
,2 ,0at occurs sfrequencie low 1.
Therefore,
nj
o
o
o
nnje
or 20
:2 of rangefrequency lFundamenta
157
524.069269
)4.0cos(69cos :Examples
) same thegive(or same theare
- ;2 sfrequencie All
or 20
:2 of rangefrequency lfundamenta a has
integeran is ;)2(
ππ.kπn.
πnπn.(
e
k
e
kee
o
nj
ook
nj
njnkj
k
oo
Periodicity Properties of DT
Complex Exponentials
158
Periodicity Properties of DT
Complex Exponentials
).....3
as same theappears )2
2 as same theappears 0)1
or 20
:2 of rangefrequency lFundamenta
00
00
159
Periodicity Properties of DT
Complex Exponentials
qq 2 as same theappears 0
:2 of rangefrequency lFundamenta
00
160
Periodicity Properties of DT
Complex Exponentials
1)0cos(][ nnx )8/cos(][ nnx )4/cos(][ nnx
)2/cos(][ nnx)2/3cos(][ nnx
)4/7cos(][ nnx )8/15cos(][ nnx )2cos(][ nnx
)cos(][ nnx
161
Periodicity Properties of DT
Complex Exponentials
otherwise periodicnot is andnumber rational a is
2/ if periodic is e signal that themeans This
,2
y equvalentlor ,2 i.e.
2 of multiple a bemust
.1ely equivalentor ,ee
0,N period with periodic be toefor order In
o
j
oo
o
jj)(j
j
o
ooo
o
n
NnNn
n
N
mmN
N
This is also true for the DT sinusoids
162
Fundamental Period & Frequency of
DT complex exponential
)2
m(N
as written is period lfundamenta The
,N
2 isfrequency lfundamenta its
N, period lfundamenta with periodic ise x[n]If
o
j o
m
o
n
163
DT Sinusoidal Signals
12
1
2
,12/2 because periodic
)12/2cos(][
o
o
nnx
31
4
2 ,31/8 because periodic
)31/8cos(][
oo
nnx
number rational2
,6/1 because periodicnot
)6/cos(][
o
o
nnx
169
nt oo jje and e signal theof Comparison
o
jj
of values
distinctfor signalsDistinct
e e oo nt
o of choiceany for Periodic
ofrequency lFundamenta
o
undefined
2:0
:0
period lFundamenta
o
o
2 of multiplesby separated
of for values signals Identical o
m and 0N integers somefor
,2
ifonly Periodic oN
m
mfrequency lFundamenta 0
)2
(:0
:0
period lFundamenta
o
o
o
m
undefined
170
CT vs. DT : Frequency
Consider a frequency
More generally being an integer,
Same for sinusoidal sequences
So, only consider frequencies in an interval of
such as
njnjnjnjAeeAeAenx 000 2)2(
][
)2( 0
20or 00
rr ,)2( 0
)cos(])2cos[(][ 00 nAnrAnx
2
njrnjnjnrjAeeAeAenx 000 2)2(
][
171
CT vs. DT: Frequency
For a CT sinusoidal signal
For the DT sinusoidal signal
rapidly more and more oscillates )( increases, as
),cos()(
0
0
tx
tAtx
slower become nsoscillatio the,2 towards from increases as
rapidly more and more oscillates ][ , towards0 from increases as
),cos(][
0
0
0
nx
nAnx
172
CT vs. DT: periodicity
CT case: a sinusoidal signal and a complex
exponential signal are both periodic
DT case: a periodic sequence is defined as
where the period N is necessarily an integer.
For sinusoid,
integeran is where
/2or 2 that requireswhich
)cos()cos(
00
000
k
kNkN
NnAnA
nNnxnx allfor ,][][
173
CT vs. DT: periodicity
For complex exponential sequence
Complex exponential and sinusoidal sequences
are not necessarily periodic in n with period
depending on the value of , may not be periodic at
all
Consider
kN
eenjNnj
2for only trueiswhich
,
0
)( 00
)/2( 0
0
16 of period ath wi),8/3cos(][
8 of period ath wi),4/cos(][
2
1
Nnnx
Nnnx
Increasing frequency increasing period!
174
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
175
Periodic DT Signals
A periodic DT function is one which is invariant to the
transformation , nn+mN , where N is a period of the
function and m is any integer
A DT signal is periodic with period where is a
positive integer if
The fundamental period of is the smallest positive
value of for which the equation holds
Example:
Is periodic with fundamental period
)3
8()4
3
2m()
2m(N
o
m
number rational a is
2/ if periodic is sinusoid DT o
176
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
182
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
183
Transformations of DT
signals
Modifying a signal x[n] through
1. Transformation of the independent
variable t e.g., y[n]=x[2n+1]
2. Combination of signals:
y[n] = cos[wn](x[n]
y[n]=x[n]-x[n-1]
184
Combination of signals
The product & sum of two sequences x[n] and y[n]:
z[n] = x[n]+y[n]
sample-by-sample production and sum
Multiplication of a sequence x[n] by a number
:multiplication of each sample value by
188
Transformation of Independent
Variable or Modification of
independent variable n
Modifying signals through elementary
transformations
Examples of elementary transformation
time shift, x[n-n0)
time reversal, x[-n]
time scaling, x[n/2]
189
Modification of independent
variable (time axes)
)( is what fixed afor
)( is what fixed afor
ofargument theis :
of parameters are :,
of t variableindependen theis :
offunction a :
)()(
integeran bemust :
nx
nx
xn
x
xn
nx
nxny
n
• Recommended approach:
o Sketch y[n] for a selected set of n until y[n] becomes clear
o Steps:
1. Rewrite: y(n) = x(α(n+β/ α))
2. Scale by |α|: x(|α|n)
3. Invert x(|α|n) if α<0
4. Shift to the LEFT by |β/ α| if β/ α>0
5. Shift to the RIGHT by |β/ α| if β/ α<0
199
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
200
Even and Odd DT Signals
An even signal is identical to its time reversed
An odd signal has the property
Example :
202
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
204
DT Unit Impulse
Unit sample sequence
(discrete-time impulse, impulse)
Any sequence can be represented as a sum of
scaled, delayed impulses
More generally
]5[...]2[]3[][ 523 nanananx
k
knkxnx ][][][
,0,1
,0,0][
n
nn
206
Defined as
Related to the impulse by
Conversely,
Relationship: DT unit impulse
and unit step
,0,0
,0,1][
n
nnu
0
][][][][
or
...]2[]1[][][
kk
knknkunu
nnnnu
]1[][][ nunun
208
Exponential
Extremely important in representing and analyzing LTI systems
Defined as
If A and are real numbers, the sequence is real
If and A is positive, the sequence values are positive and decrease with increasing n
If , the sequence values alternate in sign, but again decrease in magnitude with increasing n
If , the sequence values increase with increasing n
10
1||
nAnx ][
01
n
n
n
nx
nx
nx
22][
)5.0(2][
)5.0(2][
209
Combining basic sequences
An exponential sequence that is zero for n<0
0 ,0
,0,][
n
nAnx
n
][][ nuAnx n
210
Outline
Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties
211
DT Systems
To get the output y[n] Apply the system S{} on input x(n)
y[n] is the response of S{} to x[n]
A system: An integrated whole composed of diverse, interacting, specialized parts
System performs a function not possible with any of the individual parts
Any system has objectives
Systems respond to particular signals by producing other signals or some desired behavior
212
DT systems
A transformation or operator that maps input into
output
Examples:
The ideal delay system
A memoryless system
]}[{][ nxTny
T{.}x[n] y[n]
nnnxny d ],[][
nnxny ,])[(][ 2
213
Basic System Properties
Linearity:
A Linear system possesses the property of superposition
any constant values a and b, the following equation is satisfied
It can be easily verified that for linear systems:
an input which is zero for all time,
results in an output which is zero for all time
214
Basic System Properties
A system is linear if and only if
Combined into superposition
constantarbitrary an is where
][]}[{]}[{
and
][][]}[{]}[{]}[][{ 212121
a
naynxaTnaxT
nynynxTnxTnxnxT
additivity property
scaling property
][][]}[{]}[{]}[][{ 212121 naynaynxaTnxaTnbxnaxT
215
Examples
Accumulator system – a linear system
A nonlinear system
)10log()1log()101log(][3
21
10
10][ and 1][Consider
|)][(|log][
ny
nxnx
nxny
][][])[][(][
][][,][][
][][
21213
2211
nbynaykbxkaxny
kxnykxny
kxny
n
k
n
k
n
k
n
k
216
Basic System Properties
o A TI system is a system for which a time shift or delay of the input
sequence causes a corresponding shift in the output sequence
217
Basic System Properties:
Example
Accumulator system
][][][][ 0101 nnynynnxnx
][][
][][][
][][
01
011
0
0
1
0
nnykx
nkxkxny
kxnny
nn
k
n
k
n
k
nn
k
218
Basic System Properties:Causality
The output sequence value at the index
n=n0 depends only on the input sequence
values for n<=n0
Example
Causal for nd>=0
Noncausal for nd<0
nnnxny d ],[][
219
Basic System Properties: Stability
A system is stable in the BIBO sense if
and only if every bounded input sequence
produces a bounded output sequence
Example
stable
nnxny ,])[(][ 2
220
Outline Introduction
Types of signals
CT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
CT Systems & basic properties
DT Signals
Sinusoidal and exponential signals
Periodic and aperiodic signals
Signal energy and power
Transformation of the independent variable
Even and odd signals
Special signals
DT systems & basic properties