Linear systems

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Linear systems FilteringBased onThe
Scientist and Engineer's Guide to Digital Signal
Processingby Steven W. Smith(http//www.dspguid
e.com/)
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Topics

• Linear systems
• Convolution
• Filtering

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Linear systems
Superposition
Easyproblem 1
Easyproblem N
Easyproblem 3
Easyproblem 2
..........
Linear system
Complicated problem
Superposition
Linear system
Superposition
Easyproblem 1
Easyproblem N
Easyproblem 3
Easyproblem 2
..........
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Linear systems
Signals and systems
Analog elctronics
Input
Output
Computer programs
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Linear systems
Reasons for understanding a system
Input
Output
System
gt Compensate for non-ideal transmission
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Linear systems
Reasons for understanding a system
System
Input
Output
gt Find system transmitted signal -gt received
signal
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Linear systems
Linearity

• Homogeneity xn -gt yn gt kxn -gt kyn

gt Change in the input signal's amplitude -gt
corresponding change in the output signal's
amplitude
Power dissipation P I2R IV V2/R µ V2 gt
not homogenousgt not linear
Resistor
Ohms law I V/R or J sE
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Linear systems
Linearity

• Homogeneity
• Additivity

x1n -gt y1n x2n -gt y2n gt x1n x2n
-gt y1n y2n
gt The added signals pass through the system
without interacting
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Linear systems
Linearity

• Homogeneity
• Additivity

gt The added signals pass through the system
without interacting
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Linear systems
Linearity

• Homogeneity
• Additivity
• (Shift invariance) xn -gt yn gt xns -gt
  yns

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Linear systems
Static linearity
Static response of a linear systemThe output is
the input multiplied by a constant
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Linear systems
Memoryless systems
Memoryless system output depends only on the
present state of the input, and not on its
history.
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Linear systems
Linearity

• Static linearity
• Memorylessness

gt Linear system
DC non-linear
Hysterisis
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Linear systems
Examples
Linear systems

• Wave propagation
• Electrical and electronic circuits
• Differentiation and integration
• Convolution

Non-linear systems

• Intensity of light transmitted through material
  Lambers law
• Multiplication
• Saturation
• Systems with a threshold

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Linear systems
Superposition
Easyproblem 1
Easyproblem N
Easyproblem 3
Easyproblem 2
..........
Linear system
Complicated problem
Superposition
Linear system
Superposition
Easyproblem 1
Easyproblem N
Easyproblem 3
Easyproblem 2
..........
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Linear systems
Superposition and decomposition
x0n
xNn
x2n
x1n
..........
Decomposition
Synthesis
xn
Decomposition
Synthesis
x0n
xNn
x2n
x1n
..........
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Linear systems
Superposition and decomposition
Synthesis
Decomposition
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Linear systems
Superposition and decomposition
Decomposition
Synthesis
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Linear systems
Common decompositions
Signal decomposition
Impulse decomposition
Fourier decomposition
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Linear systems
Impulse decomposition
Synthesis
Decomposition
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Linear systems
Fourier decomposition
Sine
Cosine
Synthesis
Decomposition
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Topics

• Linear systems
• Convolution
• Filtering

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Convolution
Mathematics
Discrete
yi åj hj xi-j
x input signal h system y output signal
Continuous
y(t) ò ht xt-t d t
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Convolution
Delta function and impulse response
Delta function
Impulse response
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Convolution
Delta function and impulse response
Examples for impulse response
System
Impulse response
Filter
Filter kernel
Image system
Point spread function
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Convolution
Convolution and linear systems
System
Input
Output
Convolution
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Convolution
Examples
Input
Output
System
Low-pass filter
High-pass filter
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Convolution
Examples
Input
Output
System
Inversion
Derivative
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Convolution
Understanding convolution
Viewpoint of convolution
Input
System
Output
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Convolution
Understanding convolution
Decomposition
Synthesis
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Convolution
Understanding convolution
Decomposition
Synthesis
Å
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Convolution
Understanding convolution
Decomposition
Synthesis
Å
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Convolution
Understanding convolution
Input
System
Output
How do you really do it?
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Convolution
Understanding convolution
x1 hn-1
x2 hn-2
x3 hn-3
x4 hn-4
x8 hn-8
x0 hn-0
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Convolution
Understanding convolution
Å
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Convolution
The input side algorithm
x input signal h system y output signal
for i 011 yi 0 endfor
Initialization
for i 08 for j 03 yij yij
xihj endfor endfor
Convolution
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Convolution
The input side algorithm
x input signal h system y output signal
yi åj hj xi-j
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Convolution
Mathematics
Discrete
yi åj hj xi-j
x input signal h system y output signal
Continuous
y(t) ò ht xt-t d t
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Convolution
The output side algorithm
Å
y6 x3 h6-3 x4 h6-4 x5
h6-5 x6 h6-6
y6 x3 h3 x4 h2 x5 h1
x6 ho
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Convolution
The output side algorithm
The convolutionmachine
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Convolution
Incomplete information
0
Zero padding
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Convolution
Incomplete information
Input
System
Output
Sine wave DC
Highpass filter
Filtered signal
gt Beginning and ending useless
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Convolution
Examples for convolution

• Identity xn dn xn
• gt d function is the identity for
  convolution
• Amplification/Attenuation xn k dn k
  xn
• Shift (delay/advance) xn dn s xn
  s

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Convolution
Examples for convolution

• Derivative (First difference) yn xn -
  xn-1
• Integration (Running sum) yn xn
  yn-1

Firstdifference
Runningsum
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Convolution
Filters
Low-pass
Exponential
Sinc
Square pulse
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Convolution
High-pass filters

• d function (identity)
• Low-pass filter
• _______________________________
• High-pass filter

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Convolution
High-pass filters
Low-pass
High-pass
Exponential
Sinc
Square pulse
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Convolution
Central limit theorem

• Gaussian distribution
• Amplitude of thermal noise in an electronic
  circuit
• Cross-sectional intensity of a laser beam
• Pattern of holes around dart boards bullys
  eye
• .....

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Convolution
Central limit theorem
Observed variable Sn random processess
Gaussian distributionfor lim n-gt
m mean s2 variance s standard deviation
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Convolution
Central limit theorem
xn
xn xn xn ..
xn xn
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Topics

• Linear systems
• Convolution
• Filtering

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Filtering
Examples
Pillbox
Gaussian
Edge enhancement
Square
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Filtering
Edge Modification
Shift subtract
Edge detection
d function
Edge enhancement
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Filtering
Separability
xr,c vertr x horzc
gt Exercises