1 appendix 02 linear systems - time-invariant systems linear system linear system f(t) g(t)
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Appendix 02Appendix 02Linear Systems - Time-invariant systemsLinear Systems - Time-invariant systemsAppendix 02Appendix 02Linear Systems - Time-invariant systemsLinear Systems - Time-invariant systems
LinearSystem
LinearSystemf(t)f(t) g(t)g(t)
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Linear SystemLinear SystemLinear SystemLinear System )()( tfLtg
A linear system is a system that has the following two properties:
Homogeneity:
Scaling:
)()()()()()( 212121 tgtgtfLtfLtftfL
)()()( tgtfLtfL
The two properties together are referred to as superposition.
)()()()( 2121 tfLtfLtftfL
33
A time-invariant system is a system that has the propertythat the the shape of the response (output) of this systemdoes not depend on the time at which the input was applied.
)()( TtgTtfL
If the input f is delayed by some interval T,the output g will be delayed by the same amount.
Time-invariant SystemTime-invariant SystemTime-invariant SystemTime-invariant System )()( tfLtg
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Linear time-invariant systems have a very interesting (and useful) response when the input is a harmonic.If the input to a linear time-invariant system is a harmonicof a certain frequency , then the output is also a harmonicof the same frequency that has been scaled and delayed:
)( Ttjtj eeL
Harmonic Input FunctionHarmonic Input FunctionHarmonic Input FunctionHarmonic Input Function )()()()(
)(
tfHtfLtg
etf tj
55
The response of a shift-invariant linear system to a harmonic inputis simply that input multiplied by a frequency-dependent complex number (the transferfunction H()).A harmonic input always produces a harmonic output at the same frequency in a shift-invariant linear system.
Transfer Function H(Transfer Function H())Transfer Function H(Transfer Function H())
)(),(),(
),(),(
)()()()(
),()(
)()()(
),()()(
)()(
)(
)(
1122
1
)(122
11
)(12
1
HHTtHtH
eTtHeeTtH
TtgTtfLtfLtg
etHetgeeLeeeL
eLTtfLtfLtg
etHeLtfLtg
eTtftf
etf
tjjTTtj
tjTjTjtjTjtjTj
Ttj
tjtj
Ttj
tj
)()()()(
)(
tfHtfLtg
etf tj
)()()()(
)(
tfHtfLtg
etf tj
66
Transfer FunctionTransfer FunctionConvolutionConvolutionTransfer FunctionTransfer FunctionConvolutionConvolution
)()()()(
)(
tfHtfLtg
etf tj
)()()(
)()(
j
Tj
eHH
eHH
nConvolutio )()(*)(
ansformFourier tr )()()(
)()()()(
)(
dfthfhtg
FHG
tfHtfLtg
etf tj
)()(
)()()()()()(
)(
tLth
dfthtfHtfLtg
etf tj
h(t)h(t)f(t)f(t) g(t)g(t)
H()H()F()F() G()G()
77
ConvolutionConvolutionConvolutionConvolution
dfthtg
thth
TTthth
dfTTthdTfTthtg
dTfthTtg
dfthtg
TtTt
)()()(
)(),(
),(),(
)(),()(),()(
invarianceshift )(),()(
)(),()(
dfthfhtg )()(*)(
h(t)h(t)f(t)f(t) g(t)g(t)
)()(
)()()()(
tLth
dfthtfLtg
88
Impulse Response [1/4]Impulse Response [1/4]Impulse Response [1/4]Impulse Response [1/4]
otherwise
Tt
T
trect
t
tntrecttnftfS
02
1 )()(
)()( lim)(0
tLthtttnt
recttnt
t
dfthdthfdtLf
ttttnt
rectLtnft
tttnt
recttnfL
t
tntrecttnfLtfLtg
otherwise
Tt
T
trect
t
tntrecttnftftf
tttS
t
tS
t
)()()()()()(
)(lim)(lim)(lim)(lim)(
02
1 )(lim)(lim)(
0000
00
)()(
)()()()(
tLth
dfthtfLtg
99
Impulse Response [2/4]Impulse Response [2/4]Impulse Response [2/4]Impulse Response [2/4]
dfth
tfLtg
)()(
)()(
)()( tLth
h(t)h(t)f(t)f(t) g(t)g(t)
H()H()F()F() G()G())()()( FHG
)]([)(
)()(
)()()(
)()()()(
thFH
tLth
FHG
dfthtfLtg
1010
Impulse Response [3/4]Impulse Response [3/4]ConvolutionConvolutionImpulse Response [3/4]Impulse Response [3/4]ConvolutionConvolution
dfthfhtg )()(*)( t
g(t)
)]([)(
)()(
)()()(
)()()()(
thFH
tLth
FHG
dfthtfLtg
1111
Impulse Response [4/4]Impulse Response [4/4]ConvolutionConvolutionImpulse Response [4/4]Impulse Response [4/4]ConvolutionConvolution
dfthfhtg )()(*)(
)]([)(
)()(
)()()(
)()()()(
thFH
tLth
FHG
dfthtfLtg
= *
1212
ConvolutionConvolutionRulesRulesConvolutionConvolutionRulesRules
'*'**
*)*()*(*
**)(*
*)()()()(*
gfgfgfdt
d
hgfhgf
hfgfhgf
fgdgtfdtgfgf
1313
Some Useful FunctionsSome Useful FunctionsSome Useful FunctionsSome Useful Functions
A
a/2
B
b
a
t
at
a
1lim)(
0
1414
The Impulse Function [1/2]The Impulse Function [1/2]The Impulse Function [1/2]The Impulse Function [1/2]
)()()()()( tfdtfdtf
)(1
)( ta
at
a
t
at
a
1lim)(
0
a
t
at
a
1lim)(
0
)()()()(*)( tfdtftft
The impulse is the identity function under convolution
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The Impulse Function [2/2]The Impulse Function [2/2]The Impulse Function [2/2]The Impulse Function [2/2]
)(1
)( ta
at
)(1
)(
)(1
)(1
)0(1
)(1
0 )(1
0 )(1
0 1
)(
0 1
)(
)()(
ta
at
dttfta
dttfta
fa
dta
tft
a
adta
tft
a
adta
tft
a
adtaa
tft
adtaa
tft
dttfat
a
t
at
a
1lim)(
0
1616
Step Function [1/3]Step Function [1/3]Step Function [1/3]Step Function [1/3]b
b
1717
Step Function [2/3]Step Function [2/3]Step Function [2/3]Step Function [2/3]b
b
dttfdttftu )()()(
t
tdsstu
t
0
1)()(
)()(
)(' tdt
tdutu
1818
Step Function [3/3]Step Function [3/3]Step Function [3/3]Step Function [3/3]b
)()(
)(' tdt
tdutu
)()('
)()()0()0()()(')(')(
)(')()()()()()()('
0)(lim
0
ttu
dttftfffdttfdttftu
dttftutftutftudttftu
tft
1919
Smoothing a function by convolutionSmoothing a function by convolutionSmoothing a function by convolutionSmoothing a function by convolutionb
2020
bEdge enhancement by convolutionEdge enhancement by convolutionEdge enhancement by convolutionEdge enhancement by convolution
dfetf
tfetftfhtg
etth
t
t
t
)()(2
)(*)(*)(2*)(
)(2)(
22
22
22
2/)(
2/
2/
2121
Discrete 1-Dim Convolution [1/5]Discrete 1-Dim Convolution [1/5]MatrixMatrixDiscrete 1-Dim Convolution [1/5]Discrete 1-Dim Convolution [1/5]MatrixMatrix
dfthdtfhtfthtg )()()()()(*)()(
NNN
N
N
jjji
jjijiii
jj
f
f
f
hhh
hhh
hhh
g
g
g
fHg
fhfhfhg
jfjihjifjhifihig
...
...
............
...
...
...
*
)()()()()(*)()(
2
1
11
312
21
2
1
)()(
)()()()(
tLth
dfthtfLtg
2222
Discrete 1-Dim Convolution [2/5]Discrete 1-Dim Convolution [2/5]ExampleExampleDiscrete 1-Dim Convolution [2/5]Discrete 1-Dim Convolution [2/5]ExampleExample
)()(
)()()()(
tLth
dfthtfLtg
2323
Discrete 1-Dim Convolution [3/5]Discrete 1-Dim Convolution [3/5]Discrete operationDiscrete operationDiscrete 1-Dim Convolution [3/5]Discrete 1-Dim Convolution [3/5]Discrete operationDiscrete operation
)()(
)()()()(
tLth
dfthtfLtg
2424
Discrete 1-Dim Convolution [4/5]Discrete 1-Dim Convolution [4/5]Graph - Continuous / DiscreteGraph - Continuous / DiscreteDiscrete 1-Dim Convolution [4/5]Discrete 1-Dim Convolution [4/5]Graph - Continuous / DiscreteGraph - Continuous / Discrete
)()(
)()()()(
tLth
dfthtfLtg
2525
Discrete 1-Dim Convolution [5/5]Discrete 1-Dim Convolution [5/5]Wrapping h index arrayWrapping h index arrayDiscrete 1-Dim Convolution [5/5]Discrete 1-Dim Convolution [5/5]Wrapping h index arrayWrapping h index array
)()(
)()()()(
tLth
dfthtfLtg
2626
Two-Dimensional ConvolutionTwo-Dimensional ConvolutionTwo-Dimensional ConvolutionTwo-Dimensional Convolution
dvduvufvyuxhdvduvyuxfvuhfhyxg ),(),(),(),(*),(
2727
Discrete Two-Dimensional Convolution [1/3]Discrete Two-Dimensional Convolution [1/3]Discrete Two-Dimensional Convolution [1/3]Discrete Two-Dimensional Convolution [1/3]
][
...
][
][
][...][][
............
][...][][
][...][][
[
...
[
][
*
),(),(),(*),(),(
2
1
11
312
21
2
1
,,,,,
NNN
N
N
m nnmnjmijijiji
m n
f
f
f
hhh
hhh
hhh
g
g
g
FHG
fhfhg
nmfnjmihjifjihjig
dvduvufvyuxhdvduvyuxfvuhfhyxg ),(),(),(),(*),(
2828
Discrete Two-Dimensional Convolution [2/3]Discrete Two-Dimensional Convolution [2/3]Discrete Two-Dimensional Convolution [2/3]Discrete Two-Dimensional Convolution [2/3]
000
043
021
43
21F
000
022
011
22
11H
8
2
6
8
3
5
2
1
1
0
0
0
0
4
3
0
2
1
110220000
011022000
101202000
000110220
000011022
000101202
220000110
022000011
202000101
fHg
826
835
211
*FHG
2929
Discrete Two-Dimensional Convolution [3/3]Discrete Two-Dimensional Convolution [3/3]Discrete Two-Dimensional Convolution [3/3]Discrete Two-Dimensional Convolution [3/3]
x Cx C
Summer Scaling factor
Kernel matrix
Input image Output image
Arrayof products
Output pixel
3030
Linear System - Fourier TransformLinear System - Fourier TransformLinear System - Fourier TransformLinear System - Fourier Transform
h(t)
H()
h(t)
H()
f(t)f(t) g(t)g(t)
F()F() G()G()
)()()( FHG
)]([)(
)()(
)()()(
)()()()(
thFH
tLth
FHG
dfthtfLtg
Input function
Spectrum of input function
Output function
Spectrum of output function
Impulse respons
Transfer function
)(
)()(
)(
)(
)(
)()(
1
tfF
tgFFth
tfF
tgF
F
GH
3131
EndEnd