ECET 307 Analog Networks Signal Processing - Ch 7 - IPFW.edu

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11/15/2006

Ch 7 System Consideration
-

Paul Lin

1

ECET 307


Analog Networks Signal Processing



Ch 7 System Considerations

1 of 3


Fall 2006




http://www.etcs.ipfw.edu/~lin

11/15/2006

Ch 7 System Consideration
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Paul Lin

2

Ch 7: System Considerations


Definition of System


Subsystems or components


System Theory


System Engineering

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Type of Systems


Continuous Systems


Discrete Systems


discrete


Digital Systems


Linear Systems


Non
-
linear Systems


Dynamic Systems


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Ch 7 System Consideration
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Paul Lin

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Signals


Continuous signals x(t)


Discrete signals


x[n] = x(nT)


x(nT)
-

x(t) is sampled by a
Sample
-
Hold circuit at nT clock
pulse


T


sampling interval or sampling
time


Complex signals

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Ch 7 System Consideration
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Paul Lin

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Signals


Complex signals


e
jωt
= cosωt + j sinωt


)
sin
(cos
cos
sin
)
sin
cos
(
)
(
t
j
t
A
j
t
A
j
t
A
t
jA
t
A
dt
d
e
A
j
dt
de
A
Ae
dt
d
t
j
t
j
t
j






















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Ch 7 System Consideration
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Paul Lin

6

Definition of Linear System


Definition of Linear System


A system is said to be linear with
respect to an excitation x(t) and
a response y(t) if the following
two properties are satisfied:


Property 1 (amplitude linearity)
Property 2 (superposition principle)

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Ch 7 System Consideration
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Paul Lin

7

Definition of Linear System


Property 1 (amplitude linearity).
If an excitation x(t) produces a
response y(t), them an
excitation Kx(t) should produce a
response Ky(t) for any value of
K, where K is a constant


11/15/2006

Ch 7 System Consideration
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Paul Lin

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Definition of Linear System


Example: Property 1 (amplitude
linearity).


Multiplication


y(t) = Kx(t)


y(t) = 2 amperes, x(t) = 10 volts


y(t) = 5 amperes, x(t) = 25 volts


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Ch 7 System Consideration
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Paul Lin

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Definition of Linear System


Property 2 (superposition
principle).


If an excitation x1(t) produces a
response y1(t), and an excitation
x2(t) produces a response y2(t),
then an excitation x1(t) + x2(t)
should produce a response y1(t)
+ y2(t) for arbitrary waveforms
x1(t) and x2(t)


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Ch 7 System Consideration
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Paul Lin

10

Definition of Linear System


Addition


x(t) = x1(t) + x2(t) = 10u(t) +
5 cos(t)


x1(t) = 10 u(t)


x2(t) = 5 cos(t)


y(t) = y1(t) + y2(t) = 20 t + 10
sin(t)


y1(t) = 20t


y2(t) = 10 sin(t)

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Ch 7 System Consideration
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Operations of Linear Systems


Continuous Systems


Laplace Transforms


Convolution integral (time)


Correlation integral (time)


Discrete Systems


Z transform


Convolution sum (time)


Correlation sum (time)

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Example: A Circuit System


X(s) =
L

[x(t)]

-

System Input


Y(s) =
L

[y(t)]

-

System Output


Y(s) = G(s) X(s)


G(s) = Y(s)/X(s)

-

System’s
Transfer Function

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Example: A Circuit System (cont.)


G(s) is defined by the natural or
physical property (R, L, C, etc) of the
system, and is not depend on the
type of excitation


The poles and zeros of G(s) are due
only to the circuit or system


The order of the system = the order
of denominator polynomial of G(s)


A circuit containing m non
-
redundant
polynomial is of order m

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Example: Unit Impulse Response


Input/Output


x(t) = δ(t)


X(s) =
L

[x(t)] = 1


Y(s) = G(s)X(s) = G(s) x 1 =
G(s)


Time domain impulse response




y(t) =
L
-
1
[Y(s)]

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Ch 7 System Consideration
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Paul Lin

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Example: Unit Impulse Response
(cont.)


The impulse response of the system

-

the inverse transform of the
transfer



g(t) =
L
-
1
[G(s)]


Frequency Domain (multiplication)



Y(s) = G(s) X(s)


Time Domain (Convolution)



y(t) = g(t) * x(t)

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Ch 7 System Consideration
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Paul Lin

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System Classification by
Applications


Electrical Systems


a combination
of electrical components


Electrical Power Systems


Analog Systems


Digital Signal Processing Systems


Control Systems


Computer
-
based Controlled Space
Shuttle System

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Paul Lin

17

System Classification by
Applications
(cont.)


Robotics System


Communication Systems


Computer Systems


Network Systems


Wireless Sensor Networks


Monitoring


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Examples of Circuit Systems
(cont.)


Filters


x(t) = s(t) + n(t)


y(t) = s(t)


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Examples of Circuit Systems
(cont.)


Equalizers


s(t)
-

transmitted through a channel


x(t)
-

received signal is a distorted
version of s(t)


A system that its response to x(t)
equals s(t)


y(t) = s(t)


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Ch 7 System Consideration
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Paul Lin

20

Examples of Circuit Systems
(cont.)


Feedback


A control plant P


Input function


x(t) = U(t), unit step


Output y(t) = U(t
-

t0), a delay version
of the input x(t)


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Examples of Circuit Systems
(cont.)


DC Power Supply


A full
-
wave rectifier system R


x(t) = Acosωt, input


y(t) = K, constant response to
|Acosωt|


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Modeling (representation) of a
Systems


Differential equations


continuous
system


Difference equations


discrete
system


Signal Flow Graph/Block Diagrams


Transfer function


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Models for a Continuous Systems


Differential Equations, DEs


Transfer Functions, G(s), H(s)


Frequency Response, G(jω), H(jω)


State Differential Equations


Unit Impulse (or Impulse
Response), h(t)


Signal Flow Graph or Block
Diagrams

11/15/2006

Ch 7 System Consideration
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Paul Lin

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Models for a Discrete Systems


Difference Equations, DEs


Transfer Functions, G(z), H(z)


Frequency Response, G(ejθ), H(ejθ)


State Difference Equations


Unit Impulse (or Impulse
Response), h(n)


Signal Flow Graph or Block
Diagrams

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Paul Lin

25

Example: Difference Equations


Express the samples y[n] = y(nT) of
the differential equation


of a signal x(t) in terms of samples
x[n] = x[nT] of x(t).


If T is sufficiently small,

dt
t
dx
t
x
t
y
)
(
)
(
)
(
'


T
T
t
x
t
x
t
x
)
(
)
(
)
(
'



11/15/2006

Ch 7 System Consideration
-

Paul Lin

26

Example: Difference Equations
(cont.)


With t = nT, we can represent y[n]
in terms of first difference equation

]
[
1
]
[
]
1
[
]
[
]
[
]}
1
[
]
[
{
1
]
1
[
]
[
]
[
n
x
T
n
y
n
x
n
x
n
x
n
x
n
x
T
T
n
x
n
x
n
y














11/15/2006

Ch 7 System Consideration
-

Paul Lin

27

Example: Difference Equations
(cont.)


C
-
Program Implementation

void main(){


float T = 0.01;


const int N = 10


float x[N];


float dx;


float y[N] = {0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0 ..};




//read 10 items from ADC and save them in

//the x[N] array


for(k =1; k < 10; k++)



dx = x[k]


x[k
-
1];


y[k] = dt/T;


….


}

11/15/2006

Ch 7 System Consideration
-

Paul Lin

28

Example: Integral/Summation


Express sample y[n] = y(nT) of the
integral



Approximating the integral by a
sum.




t
d
x
t
y
0
)
(
)
(













n
k
nT
k
Tx
n
y
nT
x
T
x
T
x
T
x
T
d
x
nT
y
nT
t
1
0
]
[
]
[
)}
(
...
)
3
(
)
2
(
)
(
{
)
(
)
(


11/15/2006

Ch 7 System Consideration
-

Paul Lin

29

Example: Integral/Summation
(cont.)


C
-
Program Implementation

void main(){


float T = 0.01;


const int N = 10


float x[N];


float y = 0;




//read 10 items from ADC and save

//them in the x[N] array


for(k =0; k < 10; k++)



y = y + x[k]; // y += x[k]


y = T*y;


….


}

11/15/2006

Ch 7 System Consideration
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Paul Lin

30

System
-
Based Problem Solving


Problem Statement


System Analysis


System Requirements


System Design


Modeling and Simulation


Performance and Analysis


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Ch 7 System Consideration
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Paul Lin

31

System
-
Based Problem Solving
(cont.)


System Components: Software,
Hardware, Documentation, etc


System Prototyping


System Testing


System Integration


Lesson Learned