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

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Nov 24, 2013 (4 years and 7 months ago)

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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

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Ch 7 System Consideration
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Paul Lin

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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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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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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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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

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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

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Ch 7 System Consideration
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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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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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Definition of Linear System

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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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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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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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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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)
-

version of s(t)

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

y(t) = s(t)

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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

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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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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
)
(
)
(
)
(
'

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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

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Paul Lin

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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 ..};

//the x[N] array

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

dx = x[k]

x[k
-
1];

y[k] = dt/T;

….

}

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Paul Lin

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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
(
)
(
{
)
(
)
(

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Paul Lin

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Example: Integral/Summation
(cont.)

C
-
Program Implementation

void main(){

float T = 0.01;

const int N = 10

float x[N];

float y = 0;

//them in the x[N] array

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

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

y = T*y;

….

}

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System
-
Based Problem Solving

Problem Statement

System Analysis

System Requirements

System Design

Modeling and Simulation

Performance and Analysis

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System
-
Based Problem Solving
(cont.)

System Components: Software,
Hardware, Documentation, etc

System Prototyping

System Testing

System Integration

Lesson Learned