# Signal and Systems by akkapolkinbuangam

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Outlines
2

   Signals
   Systems
   For examples

CHAPTER 1
SIGNALS AND SYSTEMS
Lecture 1

SIGNAL                                                    Signal Classification
3                                                         4

Signals are functions of independent variables               Type of Independent Variable
that carry information. For example:                        Time is often the independent variable.
 Electrical signals ---voltages and currents in a
Example: the electrical activity of the heart
circuit                                                     recorded with chest electrodes – the
Acoustic signals ---audio or speech signals
 A       ti i    l       di          h i     l               electrocardiogram (ECG or EKG).
(analog or digital)
 Video signals ---intensity variations in an image
(e.g. a CAT scan)
 Biological signals ---sequence of bases in a
gene

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Independent Variable
Dimensionality
5                                                       6

   The variables can also be spatial                      An independent variable can be 1-D (t in the EKG)
or 2-D (x, y in an image).

   For this course: Focus on 1-D for mathematical
simplicity but the results can be extended to 2-D or
even higher dimensions. Also, we will use a generic
In this example, the signal is the intensity as a           time t for the independent variable, whether it is time
function of the spatial variables x and y.                or space.

CT SIGNALS
7                                                       8

Continuous-Time (CT) signals:
x(t), t — continuous values

Discrete Time
Discrete-Time (DT) signals:
x[n], n— integer values only
   Most of the signals in the physical world are CT
signals, since the time scale is infinitesimally fine,
so are the spatial scales.
   E.g. voltage & current, pressure,
temperature, velocity, etc.

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DT SIGNALS                                                                            Many human-made DT Signals
9                                                                                     10

   x[n], n — integer, time varies discretely                                            Ex.#1 Weekly Dow‐                      Ex.#2 digital
Jones industrial                        image
average

   Examples of DT signals in nature:
Courtesy of Jason Oppenheim.
 DNA base  sequence                                                                                                             Used with permission.

 Population of the nth generation of certain species                                Why DT? — Can be processed by modern digital
computers and digital signal processors (DSPs).

CONVERSIONS BETWEEN SIGNAL
TYPES
Signals Energy and power
11                                                                                    12
x(t)
Continuous-Value            Total energy of continuous-time signal x(t) over
t     Continuous-Time
Signal
the time interval t1≤ t ≤ t2
(k-1)t      kt     (k+1)t   (k+2)t                                                                     2
xt  dt
Sampling                                                                                                        t2

x[n]
Continuous-Value
n     Discrete Time
Discrete-Time                                   t1
k-1             k    k+1       k+2
Signal                      Total energy of discrete-time signal x[n] over the
Quantizing               x[n]
Discrete-Value               time interval t1≤ t ≤ t2
n     Discrete-Time
Signal                                          n2

 xn
k-1            k    k+1       k+2                                                                        2
Encoding             x(t)
111         001      111       011            Discrete-Value                                 n  n1
t Continuous-Time
Signal
(k-1)t      kt     (k+1)t   (k+2)t

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13                                                                    14

   Total energy of continuous-time signal x(t) over                     The time-average power over an infinite time
an infinite time interval                                             interval
                                                  1
E  lim  xt  dt                     xt  dt                                         xt 
T       2                        2                                            T        2
P  lim                      dt
T   T                                                              T    2T   T

   Total energy of discrete-time signal x[n] over an
infinite time interval                                               Discrete-time signal x[n]
N                                                                               N
1
 xn               xn                                                       N xn
2                    2                                                       2
E  lim                                                                 P  lim
N 
n N               n  
N    2 N  1 n

TRANSFORMATIONS OF THE
AMPLITUDE SCALING
INDEPENDENT VARIABLE
15                                                                    16

   Amplitude scaling                                                          x(t)                                       x[n]
 x(t)  A·x(t)
 x[n]  A·x[n]
t                                   n
   Time shift (Time delay or time advance)
   x(t)  x(t-t0)
(t)    (t t
   x[n]  x[n-n0]
½x(t)                                      (5/4)x[n]
   Time reversal
   x(t)  x(-t)
   x[n]  x[-n]                                                                             t                                   n
   Time scaling
   x(t)  x(At)

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TIME SHIFT                                                                               Time shifting or delay
17                                                                                       18

x[n]                                         Time-shifting occurs in many real physical systems.
x(t)
   Examples:
 Listening to someone talking 2m away. Received
signal will be delayed, but the delay won’t be
t             0                            n                    ti    bl
noticeable.
x(t+t0)                 x[n-n0]                                        Satellite communication systems (delay can be
noticeable if ground stations are not directly below the
satellite)
-t0              t                0                         n
n0                                    Transmitted signal Ag(t)
t0 ,n0 > 0                                                        Received signal Bg(t-to), with B<A, due to attenuation.

Example : Time shifting                                                                  TIME REVERSAL
19                                                                                       20

x(t)                             x[n]
s(t)                                         s(t)

1                                         1

01                   t                    01                  t
0                 t                  0               n

x(t) = s(t-2)
x[-n]
x(t) = s(t+1)                                     x(-t)
1                                         1

0 1 2 3              t                -1 0 1              t
Shifted to the right or delayed           Shifted to the left or advanced in time                                            t
by -1 sec.
0                                      0                 n
by 2 sec.

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TIME SCALING                                                                      Example
21                                                                                22

x(t)
Examples:                        Given signal x(t)                             x(t)
1
Playing an audio
-1
t          tape at a faster or                                                                                     t
1
x(2t)                                 slower speed.                                                            0              1       2

(a)   Sketch x(t+1)
-½
t
½
(b)   Sketch x(-t+1)
x(t/2)
(c)   Sketch x(3t/2)
(d)   Sketch x(3t/2+1)
t
-2                         2

Example (2)                                                                       Example (3)
23                                                                                24

x(t)                                                                             x(t)
1                                                                                1

t                                                                                           t
1           2                                                                        1          2
x(t+1)                                                                          x(3t/2)
1
1
   (a)                                                                              (c)
t                                                                                   t
-1      0        1           2                                                                 2/3 4/3
0
x(-t+1)
1                                                                                    x(3t/2+1)
1

t
   (b)                   -1      0             1                                    (d)                  -2/3   0 2/3
t

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TYPES OF SIGNALS                                                     Signal Properties
25                                                                   26

C i
Continuous                        Discrete
Di                           Periodic signals
(analog)
(     g)                         (digital)
( g )                        Even and odd signals
   Exponential and sinusoidal signals
   Step and pulse signals
N
Non-                      Non-
N
Periodic                        Periodic
periodic
p                         periodic
p

Analysis Fourier             Fourier      Discrete   Discrete-Time
Tool     Series              Transform    Fourier    Fourier
Series     Transform &
Z-Transform

PERIODIC SIGNALS                                                     PERIODIC SIGNALS (2)
27                                                                   28

   An important class of signals is the class of                    Periodic signals                x(t)
periodic signals. A periodic signal is a                         CT: x(t) = x(t + T)
continuous time signal x(t), that has the property
 x(t)   = x(t+T)                                                                      -2T   -T    0      T   2T       t

 x[n]   = x[n+N]
Ex. 60-Hz power line, computer clock, etc.
DT: x[n] = x[n + N]           x(n)

n

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PERIODIC SIGNALS (3)                                         EXAMPLE
29                                                           30

2
   Given
cost  if t  0
x t   
   cos(t+2π) = cos(t)                                                            sin t  if t  0
   sin(t+2π) = sin(t)
   Is signal periodic?
Are both periodic with period 2π

EXAMPLE (2)                                                  EVEN AND ODD SIGNALS
31                                                           32

   An even signal is identical to its time reversed
signal, i.e. it can be reflected in the origin and is
x(t)
equal to the original:
x(-t) = x(t)
[ ] [ ]
x[-n] = x[n]                0
t
   Signal x(t) is not periodic because x(t) does not           An odd signal is identical to its negated, time
recur at the time origin.                                    reversed signal, i.e. it is equal to the negative
reflected signal                          x(t)
x(-t) = -x(t)
x[-n] = -x[n]                0            t

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EVEN AND ODD SIGNALS (2)                                               EXAMPLE (2)
33                                                                     34

   Any signals can be expressed as a sum of Even                         DT unit-step function. Even signal                               1

,n  0
      ,n  0
x even n    2
and Odd signals. That is                                                                                                                1     ,n  0
1   ,n  0                              1       2
x n                                                         ,n  0
0   ,n  0
½

dd(t),
x(t) = xeven(t) + xodd(t)                                              1
-3 -2 -1
3 2 1      0 1       2 3            n

Where                                                                                                                         +
-3 -2 -1       0 1 2 3        n   =                                          1
 2   ,n  0
Odd signal                            
x odd n    0    ,n  0
xeven(t) = [x(t) + x(-t)]/2                                                                                                             1

,n  0
½             2
xodd(t) = [x(t) - x(-t)]/2
-3 -2 -1     0 1       2 3            n
-½

EXPONENTIAL & SINUSOIDAL
Example
SIGNALS
35                                                                     36

Compute the even part of the function                                     Exponential and sinusoidal signals are
x[n] =   ancos[πfn]                                characteristic of real-world signals and also from
a basis (a building block) for other signals.
   A generic complex exponential signal is of the
xn   x n  a n cosfn   a  n cos fn             form:
xe n                    
2                       2
a n cosfn   a n cosfn                                       x(t) = Ceat

2


a n  a n cos fn                                  where C and a are, in general, complex numbers.
2

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COMPLEX EXPONENTIAL
REAL EXPONENTIAL SIGNALS
SIGNALS
37                                                                         38

   C and a are real                                                          Consider when a is purely imaginary:

x(t) = Ceat                                        x(t) = Ceat                   xt   e j0t
Exponential growth
E      ti l     th                     E      ti l d
Exponential decay               An important property of this signal is that it is
x(t)                                x(t)                      periodic
j 0 t  j 0  t  T 
e            e
a<0                                                   The fundamental period T0 is                  x(t)
a>0                                                                            2
C
T0 
C                                                                                      2                                 0
T0 
t                               t                                     0                                      t

SINUSOIDAL SIGNAL
39                                                                         40

 By   Euler’s relationship, this can be expressed as:
x(t) = Acos(0t+)
e j0t  cos0t   j sin 0t                                                        x(t)
2
Acos
A         T 

e j 0 t , e  j 0       same fundamental period
t

T : fundamental time period
 : fundamental frequency

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Example                                                                  Cont.
41                                                                       42

   x(t) = 2cos(2t +/3)                                                   By Euler's relation
 x(t) = Acos(0t+ )
e j0t  cos0t   j sin 0t 
A = 2,      0 = 2               = /3
Acos = 2cos(/3) =1                                         Sinusoidal signal can be written in terms of
periodic complex exponential.
x(t)                                                      A j j  0 t       A  j  j t
   Period                            2        2                              A cos0t       e e               e e
2 2                                                                                   2                  2
T        2                     1
   2
A                  A  j 0t  
 e j 0t   
t
e
-2
2                  2

Cont.                                                                    Cont.
43                                                                       44

   If we decrease the
A               A                                     magnitude of ω0, we
A cos0t     e j 0t    e  j 0t                        slow down the rate of
2               2                                     oscillation and
therefore increase
th f       i

A cos0t     e Ae j 0t                                the period. Exactly
the opposite effects

A sin  t     mAe                        
j 0t                      occur if we increase
0                                                        the magnitude of ω0.
2
T0 
0

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Cont.                                                                   Cont.
46
45

Fundamental                  t                                             Total energy and average power over one
1               1
x(t) = cos(t)                period         T0         T0
frequency                                                                                         2
e           dt   1  dt  T0
-10    -5     0       5   10                                                             j0 t
-1                                                    E period 
t                                                                      0                       0
1              ()       (2 )
x(t) = cos(2t)                                  1
-10     -5   0        5   10                                        Pperiod    E period
-1                                                                T
t
1                                         Finite average power
x(t) = cos(3t)
T
1               j 0 t 2
e
-10    -5   0        5   10
0T0 = 2πk                          P  lim                                dt  1
-1             k = 0, 1,  2,…                           T  2T
T

Cont.                                                                   Cont.
47                                                                      48

A necessary condition for a complex exponential                            Harmonically related set of complex
to ejωt be periodic with period T0 is                                     exponentials is a set of periodic exponentials
with fundamental frequencies that are all
e joT0  1                                                  multiples of a signal positive frequency ω0.
With implies that
k t   e jk t
0
k  0,1,2,......
ωT0 = 2k , k = 0, ±1, ±2,...
   The fundamental period
We can see that ω must be multiple of ω0. We                                              2   T
define the set of Harmonically related complex                                                0
exponentials                                                                             k0   k

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Example
49                                                                                           50

   Plot signal x(t) =             2ej2t                                                                                    |x(t)| = 2                x(t) = 2t
R e[x(t)]
2
x(t) = 2ej2t = rejt = 2cos(2t) + j2sin(2t)                                                2    2                                               2

1                       1                            1
t                           t                             t
r=2 2,                                                                                                                    1                            1
-2
Re [x(t)] = 2cos(2t)
Im
Im [x(t)] = 2sin(2t)
2        Very Difficult too plot
mag[x(t)] = r = 2                                                                                                                1
 x(t) = tan-1[sin(2t)/cos(2t)] = 2t                                                                                                                 Re

-2

COMPLEX EXPONENTIAL
Example
SIGNALS
51                                                                                           52

   Plot the magnitude of the signal                                                            Consider complex exponential signal:
xt   e            j 2t
e     j 3t

   Solution                                                                                                  x(t) = Ceat
xt   e   j 2.5t
e   0.5t
e   j 0.5t
  2e   j 2.5t
cos0.5t         Where C and a are
Wh        d

C = |C|ej
xt   2 cos0.5t                                                                                     a = r + jω0

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53                                                                 54

xt   Ce at  C e j e r  j0 t  C e rt e j 0t  
Ce at  C e rt cos 0 t     j C e rt sin 0t   
For r = 0, the real and imaginary parts of complex
exponential are sinusoidal, For r > 0, they
correspond to sinusoidal signals multiplied by a
growing exponential, and for r < 0 they
correspond to sinusoidal signals multiplied by a
decaying exponential.

Discrete-Time Complex
Exponential and Sinusoidal Signals
55                                                                 56

Complex exponential signal or sequence                                If C and α are real
|| > 1,  > 0
x[n] = Cn
x[n] =   Cn
Where C and α are complex numbers
x[n] = Ceβn                                                                     || < 1,  > 0

Where
 = eβ

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57                                                 58

|| > 1,  < 0        If β is purely imaginary

xn  e j0 n
Euler s
By Euler's relation

|| < 1,  < 0
e j0 n  cos 0 n  j sin 0 n
and
A j  j  0 n A  j   j 0 n
A cos0 n         e e         e e
2             2

59                                                 60

   Example of x [n]=Acos(0n+)

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COMPLEX EXPONENTIAL
SIGNALS
61                                                                       62

   General Complex Exponential Signals
C n  C e n cos0 n     j C e n sin 0 n   
x[n] = Cn

Where C and  are
C = |C|ej

   e j    0

Periodicity Properties of Discrete-
Time Complex exponentials
63                                                                       64

   Two properties of continuous-time counterpart e
j 0 t
   Two properties of discrete-time counterpart               e j0 n
 The   larger the magnitude of ω0 the higher is the                        First property
rate of oscillation in the signal                                             The signal with frequency ω0 is identical to the signals
j 0 t                                                                       with frequencies ω0 ± 2π, ω0 ± 4π and so on. Therefore we
 e        is periodic for any value of ω0                                       need only consider a frequency interval of length 2π .

e j 0  2 n  e j 2n e j0 n  e j0 n
j0 n
    e       dose not have a continually increasing rate of
oscillation as ω0 is increased in magnitude. We increase
the rate of oscillation when ω0 increase from 0 to π and
We decrease the rate of oscillation when ω0 increase
from π to 2π.

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65                                                                66

   Second property                                                  There must be an integer of m
 Concerns the periodicity of the discrete-time
complex exponential                                                   0N = 2m
 Signal    e j0 n to be periodic with period N > 0
2 0
or           
j 0 n        j 0  n  N                                 N   m
e            e
Or          e j 0 N  1

0N must be a multiple of 2

Comparison of the signals ejωot and ejωon
67                                                                68

   Comparison of the signals ejωot and ejωon

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DISCRETE –TIME UNIT IMPLUSE
70

   The discrete unit impulse(or unit sample) signal
69
is defined:
0 n  0
x[ n ]   [ n ]  
The Unit Impulse and                                                             1 n  0
Unit Step Functions

DISCRETE –TIME STEP                                  DISCRETE – TIME UNIT
SIGNALS                                              IMPLUSE AND STEP SIGNALS
71                                                    72

   The discrete unit step signal is defined:           Note that the unit
impulse is the first
difference (derivative) of
0 n  0                  the step signal
x[n]  u[ n]  
1 n  0                   [n]  u[n]  u[n  1]
n<0
   Similarly, the unit step is
the running sum
(integral) of the unit
impulse.
n
un       m
n>0

m  

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73                                                           74

n                                               Unit impulse sequence can be used to sample
un       m                                            the value of a signal at n = 0, since  [n] is
m                                               nonzero(and equal to 1) only for n = 0
n<0
   Change m to k = n – m
x[n] [n] = x[0] [n]
0
un     n  k 
k                                               A unit impulse  [n-n0] at n = n0

   n  k                            n>0

k 0
x[n] [n-n0] = x[n0] [n-n0]

CT UNIT STEP FUNCTIONS
75                                                           76

   The continuous unit step signal is defined:
t                              du t 
x(t )  u (t )    ( )d           t  
                              dt
0 t  0
x(t )  u (t )  
1 t  0                                               u(t)                             (t)
1                               1/

0               t                   0             t

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The CT Unit Step and CT Unit
CT UNIT IMPLUSE FUNCTIONS
Impulse
77                                                                                                       78

    (t) is a short pulse, of duration  and with unit                                                    As  approaches zero, x(t) approaches a CT
area for any value of . As   0, (t) becomes
narrower and higher, maintaining its unit area.                                                         unit step and x(t) approaches a CT unit impulse
x(t)                                x'(t)
 t   lim   t                                                                                                1
 0                                                                                                                             1/
 (t)                              k (t)
1                              k                                                              -/2 /2                t        -/2 0 /2                t
   The CT unit step is the integral of the CT unit
0               t                  0                      t                      impulse and the CT unit impulse is the
    Note: It is not the shape of the function that is                                                       generalized derivative of the CT unit step
important in the limit, but it’s the area. The impulse                                                                 t
u(t )    ( )d                                           du t 
has an area of one.

 t  
dt

Relationship between u(t) and (t)                                                                       SAMPLING PROPERTY
79                                                                                                       80

Interval of integration                                                   Interval of integration           The continuous unit impulse signal is defined:
()                           (t-)                                                                                                                        (t)
                 
0            t 
                                                                          t                    2
t           0                                  t          0                                                          1               
             t 
                2                                           t
-/2         /2
Interval of integration
  t                   Interval of integration           Let x(t) be finite and continuous at t = 0.
()                                                        (t-)                                  (t)                                           (t)
1/                                           1/
                                                                                              x(t)                 x(0)
0        t                                        0      t            
t                                                 
u t      d                             u t     t   d                               - /2
t                 - /2
t
/2                                              /2
                                                0

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SAMPLING PROPERTY                                                                                    Properties of the CT Impulse
81                                                                                                   82

The area under the product of the two functions is                                                      The sampling (also called sifting) property

 xt  t  t dt  xt 
 2
1
 xt dt  A                                                                                       0         0

  2                                                                        The sampling property “extracts” the value of a
As the width of x(t) approaches zero x(t)  x(0)
zero,                                                              function at
a point.
2
1                    1                                                           The scaling property
lim A  x0 limdt  x0  lim   x0 
 0                                                                                                             1
 0
 2
 0 
 at  t0    t  t0 
a
The CT unit impulse is implicitly defined by                                                         This property makes the impulse different from
                                                        ordinary function
x0    t xt dt


Properties of the CT Impulse
Example
(continued)
83                                                                                                   84

   The equivalence property                                                                            Consider the discontinuous signal x(t)
xt A   t  t0   A  xt0  t  t0 
x(t)A(t-t0)
A(t-t0)              x(t)
A/                                            Ax(t0)/

x(t0)
t0                  t                                  t0             t
t0- /2        t0+ /2                                 t0- /2        t0+ /2               Calculate and graph the derivative of this
signal

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85

86     SYSTEM
Continuous‐Time and
Discrete‐Time Systems

SYSTEMS                                                     SYSTEM (2)
87                                                         88

For the most part, our view of systems will be from            Eg. Voltage buffer
an input-output perspective:
A system responds to applied input signals, and
its response is described in terms of one or
more output signals

x(t)      CT system          y(t)
x(t)         System   y(t)
x[n]      DT system          y[n]

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EXAMPLES OF SYSTEMS (2)                                      EXAMPLES OF SYSTEMS (3)
89                                                          90

 RLC circuit – an electrical system                         By KVL, we have
Fine the relationship of input voltage vs and output                vs t   vR t   vc t   Rit   vc t 
voltage vc in RC circuit                                   The current in this circuit is
dvc t 
it   C
dt
Replace i(t)
dvc t 
vs t   RC             vc t 
dt
or
dvc t  1              1
    vc t      vs t 
dt       RC           RC

EXAMPLES OF SYSTEMS (4)                                      EXAMPLES OF SYSTEMS (5)
91                                                          92

   RLC circuit – an electrical system                         A shock absorber – a mechanical system

Force Balance: f - input, v – output
dv t  v t  1
t
i t   C                v  d                                 dyt 
f t   M Bvt   K  vt 
t
dt      R     L 
dt                 

capacitanceresistance inductance              This equation looks quite familiar

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EXAMPLES OF SYSTEMS (6)                                           EXAMPLES OF SYSTEMS (7)
93                                                                94

   Observation: different systems could be                          A robot car – a close-loop system
described by the same input/output relations

EXAMPLES OF SYSTEMS (8)                                           EXAMPLES OF SYSTEMS (9)
95                                                                96

A thermal system

Observations
 Independent variable can be something other
than time, such as space.
 Such systems may, more naturally, have
t = distance along rod                                  boundary conditions, rather than “initial”
y(t) = Fin temperature as function of position
conditions.
x(t) = Surrounding temperature along the fin

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EXAMPLES OF SYSTEMS (10)                                    EXAMPLES OF SYSTEMS (11)
97                                                         98

   A rudimentary “edge” detector                              Financial System
y[n] = x[n + 1] – 2x[n] + x[n -1]                          Bank account
= {x[n + 1] – x[n]}- {x[n] – x[n -1]}                      y[n] – balance at nth month
= “Second difference”
Second difference                                      x[n] – deposit at nth month
   This system detects changes in signal slope                    r – monthly interest
(a)   x[n] = x[n]     y[n] = 0                             Then
(b)   x[n] = nu[n]    y[n]                                     y[n] = (1+ r)y[n-1] + x[n]

SYSTEM
SYSTEM INTERCONNECTIOINS
INTERCONNECTIOINS(2)
99                                                         100

   An important concept is that of interconnecting            Signal flow (Block) diagram
systems                                                                  System 1         System 2
 To build more complex systems by
interconnecting simpler subsystems
 To modify response of a system
System 1
Parallel
System 2

Feedback              System 1

System 2

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SYSTEM
Example : Circuit system
INTERCONNECTIOINS(3)
101                                                                102

   Feedback interconnection Example
System 1       System 2

System 4
Input
System 3                      Output

Example: A Communication
System
103

   A communication system has an information
104
signal plus noise signals
   This is an example of a system that consists of
an interconnection of smaller systems.                               SYSTEM PROPERTIES
   Cell phones are based on such systems

Noise           Noise   Noise
Noisy
Informatio
n Signal
n Signal

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System properties                                             System properties (cont.)
105                                                           106

WHY?                                                             Systems with and without Memory
 Memoryless   system is the system that its output
   Important practical/physical implications. We                   for each value of the independent variable at a
can make many important predictions of the                      given time is dependent on the input at only that
t   b h i        ith t having to d
system behaviors without h i t do any                                 time.
same ti
mathematical derivations                                     Example of memoryless systems
y[n] = (2x[n]−x2[n])2
   They allow us to develop powerful tools
(transforms, more on this later) for analysis and                 y(t) = Rx(t)
design

System properties (cont.)                                     System properties (cont.)
107                                                           108

   Example of memory systems                                    Invertibility and Inverse Systems

n
A system is said to be invertible if distinct inputs
yn       xk                                 lead to distinct outputs. It the system is
k                                   invertible, then an inverse system exists that,
y
yn  xn  1                                   when cascaded with the original system, yields
t
1                                      an output w[n] equal to the input x[n] to the first
y t          x d
C 

system.

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109                                                                  110

   Example of Inverse system                                           Example of noninvertible discrete-time systems

y[n] = 0  This system produces the zero output
q            y p
sequence for any input
sequence
   The accumulator
y(t) = x2(t)  We cannot determine the sign of
input           from the knowledge of the
output.

System properties (cont.)                                            System properties (cont.)
111                                                                  112

   A system is causal                                                  Mathematically (in CT): A system x(t)  y(t) is
 if the output does not anticipate future values of the             causal if
input, i.e., if the output at any time depends only on
values of the input up to that time.
when x1(t)  y1(t)
()       ()      x2(t)  y2(t)
()      ()
   All real-time physical systems are causal, because
and
time only moves forward, Effect occurs after cause.
(imagine if you own a noncausal system whose output                      x1(t) = x2(t)      for all t < t0
depends on tomorrow’s stock price.)                                 Then
   All memoryless systems are causal, since the output                      y1(t) = y2(t)      for all t < t0
responds only to the current value of the input.

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System properties (cont.)                                            System properties (cont.)
113                                                                  114

   Causal or noncausal                                                 Stability system is one in which small inputs
y(t) = x2(t – 1)                                                     lead to responses that do not diverge
y(5) depend on x(4)            Causal
y(t) = x(t + 1)
Stable system
y(5) = x(6), y depends on future    Noncausal
y[n] = x(– n)
y[5] = x[-5] ok
But y[-5] = x[5]      Noncausal
Unstable system
y[n] = (½)n+1x3[n - 1]
y[5] depend on x[4]            Causal

System properties (cont.)
115                                                                  116

TIME-INVARIANCE (TI) :                                               A system is time invariant if a time shift in input
A system is time invariant if the behavior and                       signal result in an identical time shift in the output
characteristics of the system are fixed overtime.
signal.
Informally, a system is time-invariant (TI) if its behavior
does not depend on what time it is.
Example                                                                                          discrete time
If y[n] is the output of a discrete-time time invariant
- RC circuit is time invariant if the resistance and                 system when x[n] is the input, then y[n-n0] is the
capacitance values R and C constant over time: We would              output when x[n-n0] is applied.
expect to get the same results form an experiment with this
circuit today as we would if we ran the identical experiment          Mathematically (in DT): A system x[n]  y[n] is
tomorrow.                                                               TI if for any input x[n] and any time shift n0,
- On the other hand, if the values of R and C are changed
or fluctuate over time: We would expect the results of our                          If             x[n]  y[n]
experiment to depend on the time a which we run it.                                 then        x[n-n0]  y[n-n0].

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System properties (cont.)                                             System properties (cont.)
117                                                                   118

   In continuous time with y(t) the output                              TIME-INVARIANCE: TIME SHIFT WAVEFORM
corresponding to the input x(t), a time - invariant                  Replace argument t by t – t0 where t0 is amount
system will have y(t-t0) as the output when x(t-t0)                   of shift
is the input.
x[n]                 x[n n
x[n-n0]
   Mathematically (in CT): A system x(t)  y(t) is TI
if for any input x(t) and any time shift n0,

If             x(t)  y(t)
then         x(t-t0)  y(t-t0)                                                        n                          n
1 2 3 4                    n0      n0+4

System properties (cont.)                                             Example
119                                                                   120

   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)
pp        (      )  ()
and            x(t) → y(t)
Then by TI
x(t + T) → y(t + T).
y2(t)  y1(t-2)
↑       ↑                                                           System is not time-invariance
These are        So these must be the
the same         same output, i.e., y(t) =
input!           y(t + T).

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Example (2)                                                               Example (3)
121                                                                       122

   Consider the continuous-time system                                           y1(t) = sin|x1(t)|
   Time shift
y(t) = sin|x(t)|                                       y1(t-t0) = sin|x1(t-t0)|
   Let
 Issystem time invariant?                                                    x2(t) = x1(t-t0)  y2(t) = sin|x2(t)| = sin|x1(t-t0)|
 Check system is time invariant
 Determine      whether the time-invariance property holds
for any input and any time shift t0
Because        y2(t) = y1(t-t0),
Then           this system is time invariant

Example (4)                                                               TIME-INVARIANT OR VARYING ?
123                                                                       124

   Consider the discrete-time system                                             y(t) = x2(t+1)
y[n] = nx[n]                                                                                            TI

y1[n] = nx1[n]                                            y[n] = (½)n+1x3[n – 1]
1

time shifting  y1[n-n0] = (n-n0)x1[n-n0]                                                  Time-varying (NOT time-invariant)

x2[n] = x1[n-n0]
y2[n] = nx2[n] = nx1[n-n0]
y2[n]  y1[n-n0]    Time-Varying system

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System properties (cont.)                                     System Linearity
125                                                           126

   LINEAR AND NONLINEAR SYSTEMS                                     The most important property that a system
possesses is linearity
 Many   systems are nonlinear.
   It means allows any system response to be
Ex.Economic system.                                               analysed as the sum of simpler responses
policies, labor,
Input: fiscal and monetary policies labor                         (
(convolution) )
resources, etc                                                 Simplistically, it can be imagined as a line
 Output: GDP, inflation, etc.                                   Specifically, a linear system must satisfy the two
properties:
System behavior is very unpredictable because it
 Additive: the response to x1(t)+x2(t) is y1(t) + y2(t)
is highly nonlinear.                                             Scaling: the response to ax1(t) is ay1(t) where aC
 However,    we focus exclusively on linear systems.              Combined: ax1(t)+bx2(t)  ay1(t) + by2(t)
 Linear   systems can be analyzed accurately.

PROPERTIES OF LINEAR
LINEARITY
SYSTEMS
127                                                           128

A (CT) system is linear if it has the superposition              Superposition
property:                                                           If                    xk[n]  yk [n]
If         x1(t) → y1(t) and x2 (t) → y2 (t)
then
ax1(t) + bx2(t) → ay1(t) + by2(t)
Then         a x n   a y n
k
k   k
k
k   k

 For linear system, zero input → zero output

“Proof”
0 = 0x[n] → 0y[n] = 0

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Example
129                                                                         130

an +         x(t)                y(t)                 an +             Ex#1 Linear                y(t) = 3x(t)         why?
+                                                   +         Prove       ax1(t) + bx2(t)  ay1(t) + by2(t)  linear
xn(t)                                   xn(t)
Find ax1(t) + bx2(t) = ?
a1
x1(t)  y1(t) = 3x1(t)
a1                 Equivalent
x2(t)  y2(t) = 3x2(t)
x1(t)                                  x1(t)                            x3(t) = ax1(t) + bx2(t)

Linear
(a)                                (b)                        y3(t) = 3x3(t) = 3ax1(t) + 3bx2(t) *
   If the system is linear outputs y(t) in (a) and (b)                     Find ay1(t) + by2(t) = ?
are equal.                                                                  ay1(t) + by2(t) = 3ax1(t) + 3bx2(t) *

131                                                                         132

Ex#2: Is this system linear : y(t) = 3x(t)+2,                              Ex#3 : Consider a system whose input x[n] and
Prove         ax1(t) + bx2(t)  ay1(t) + by2(t)  linear                     output y[n] are related by y[n] = e{x[n]}.
Determine whether or not the system is linear.
x1(t)        y1(t) = 3x1(t) + 2
x2(t)        y2(t) = 3x2(t) + 2                                                   x1[n] = r1[n] + js1[n]  y1[n]= e{x1[n]} = r1[n]
[ ]     [ ] j [ ]        [ ]     { [ ]}      [ ]
x3(t) =       ax1(t) + bx2(t)                                                      x2[n] = r2[n] + js2[n]  y2[n]= e{x2[n]} = r2[n]
x3[n] = r3[n] + js3[n]  y3[n]= e{x3[n]} = r3[n]


y3(t) = 3x3(t) + 2 = 3ax1(t) + 3bx2(t) + 2 *
ay1(t)+by2(t) = a[3x1(t) + 2] + b[3x2(t) + 2]                               x3[n] = ax1[n] + bx2[n] = ar1[n] +jas1[n]+br2[n]+ jbs2[n]
not linear
= a3x1(t) + 2a + 3bx2(t) + 2b *
 y3[n]   = ar1[n] + br2[n]              (1)

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133                                                               134

ay1[n] = ar1[n]                                          But If a = j
by2[n] = br2[n]                                          x3[n] = jx1[n] + bx2[n] = jr1[n] - s1[n]+br2[n]+ jbs2[n]
ay1[n] + by2[n] = ar1[n] + br2[n]        (2)                      y3[n] = e{x3[n]} = -s1[n] + br2[n] (*)

(1) = (2)  It should be Linear                                          jy1[n] = jr1[n]
by2[n] = br2[n]
jy1[n] + by2[n] = jr1[n] + br2[n]     (**)
(*)  (**)  This system is not linear

Example DT LTI System
135                                                               136

   Is this system linear : y(t) = 3x2(t)                              x1[n]                               y1[n]
2
1             1                         1
-1 0 1                                     0 1

x2[n]                    2x1[n-1]             x1[n-2]
Not linear                           3
4
2                         2           2                 2
3                                1             1
0 1 2         -1          0 1 2                1 2 3

y2[n]                2y1[n-1]             y1[n-2]
2                     2
1                                         1
3
1 2                   1 2                      2 3
-1

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