Signals, Instruments, and Systems – W5
Introduction to Signal
Processing – Convolution,
Sampling, Reconstruction
1
Motivation from Week 1 Lecture
Processing
Sensing
Processing
Communication
Mobility
In-situ instrument
Visualization
Storing
Transportation channel
Base station
Highlighted blocks are those mainly leveraging the content of this lecture.
2
Convolution
3
Convolution

 f  g (t )   f ( )  g (t   )d

• For each value of t:
1. Flip (reflect) g
2. Shift g by t
3. Multiply f and g
4. Integrate over 
1) g ( )  g ( )
2) g ( )  g (t   )
3) f ( )  g (t   )

4)
 f ( )  g (t   )d

• Note that the result does not depend on !
4
[Matlab demo “cconvdemo”]
http://users.ece.gatech.edu/mcclella/matlabGUIs/
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Examples
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Examples
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Examples
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Examples
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Examples
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Convolution in Time and
Frequency Domains
h(t )  ( f  g )(t )  hˆ( )  fˆ ( )  gˆ ( )
h(t )  f (t )  g (t )  hˆ( )  ( fˆ  gˆ )( )
Ordinary
frequency
Frequency domain
h(t )  ( f  g )(t )  H ( )  F ( )  G ( )
1
h(t )  f (t )  g (t )  H ( ) 
( F  G )( )
2
Angular
frequency
Time domain
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Convolution Properties
Commutativ ity
f g  g f
Associativity
f  ( g  h)  ( f  g )  h
Distributivity
f  ( g  h)  ( f  g )  ( f  h)
Associativity with scalar multiplica tion
a ( f  g )  (af )  g  f  (ag )
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Discrete Convolution

 f  g (t )   f ( )  g (t   )d


 f  g [n ]   f [m]  g[n  m]
m  
Notes:
• Similar to the continuous version
• The integral becomes an infinite sum
• Matlab, operating on a computer (i.e. a digital device) can only emulate
continuity and therefore use the discrete version with an adjustable
discretization level (in time and amplitude) and finite bounds of the
convolution window
13
[Matlab demo “dconvdemo”]
http://users.ece.gatech.edu/mcclella/matlabGUIs/
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Sampling
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Analog – Digital
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8
Signal Amplitude
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Signal Amplitude
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4
2
0
-2
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4
2
0
-2
0
100
200
300
400
500
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Time
• Analog
– Continuous
– Real world
800
900 1000
-4
0
100
200
300
400
500
600
700
800
900 1000
Time
• Digital
– Discrete
– Digital world
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Train of Dirac Impulses
x(t ) 

  (t  nT )
n  
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Train of Dirac Impulses
x (t ) 

  (t  nT )
n  
1
x ( t )  ak 
T
X ( ) 
T /2

x (t )e  j0t dt 
T / 2
1
T
2 
k 2 






T
T


n  

Time domain
Frequency domain
F()
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Sampling in Time Domain
x(t )
p (t ) 

  (t  nT )
Train of Dirac pulses
n  
x p (t )  x(t ) p (t ) 


n  
n  
 x(t ) (t  nT )   x(nT ) (t  nT )
T = sampling period
f = 1/T = sampling
frequency
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Sampling in Frequency Domain
Time domain
Frequency domain
x p (t )  x (t ) p (t )
Convolution property
1
X p ( ) 
X ( ) * P ( )
2
2 Sampling



s
2
T frequency
(
)
P( ) 



k


S
T
k  
1 
X p ( )   X ( ) *  (  kS )
T k  
1 
  X (  kS )
T k  
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Sampling a Band-Limited Signal
X(ω) → spectrum of signal x(t)
with highest frequency < ωm
2
P( ) 
T

  (  k )
S
k  
Sampling frequency
ωs > 2 ωm
1
X ( ) * P ( )
2
1 
  X (  kS )
T k  
X p ( ) 
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Sampling a Band-Limited Signal
X(ξ) → spectrum of signal x(t)
with highest frequency < 2 kHz
Sampling frequency:
5 kHz > 2 * 2 kHz
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Sampling a Band-Limited Signal
X(ξ) → spectrum of signal x(t)
with highest frequency < 3 kHz
Sampling frequency:
5 kHz < 2 * 3 kHz
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Original Signal
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f (t )  sin( 2t )  0.4 sin( 2  2t )  0.2 sin( 2  5t )
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Too Few Samples (1Hz)
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→ Data is lost
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Too Many Samples (100 Hz)
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→Redundant data
→Increase of data size
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Minimal Possible Sampling (> 10 Hz)
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Nyquist–Shannon Theorem
• If a function x(t) contains no frequencies higher than
B Hz, it is completely determined by giving its
coordinates at a series of points spaced 1/(2B)
seconds apart.
• Sampling frequency must be at least two times
greater than the maximal signal frequency
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Sampling in Practice
• Sampling frequency two times greater than
maximal frequency is the limit
• If possible, try to use a sampling frequency
10 times greater than the maximal
frequency
• Audio CD, sampling at 44.1 kHz
Maximal hearable frequency: 20 kHz
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Signal Reconstruction
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p (t ) 

  t  nT 
n  
x(t )
x
x p (t )
H ( )
xr (t )
If there is no overlap between the
shifted spectra the signal xr(t) can be
perfectly reconstructed from x(t)
spectrum of original signal
spectrum of sampled signal
sampling frequency
ωs > 2 ωm
filtering
filter cut-off frequency
ωm < ωc < (ωs –ωm)
X r ( )  X p ( ) H ( )
spectrum of reconstructed
signal
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p (t ) 

  t  nT 
n  
x(t )
x
x p (t )
H ( )
xr (t )
If there is overlap between the
shifted spectra the signal xr(t) cannot be
perfectly reconstructed from x(t)
spectrum of original signal
spectrum of sampled signal
sampling frequency
fs < 2 fm
filtering
filter cut-off frequency fc
fm < fc < (fs –fm)
X r ( )  X p ( ) H ( )
spectrum of reconstructed
signal
32
Time Domain Interpretation
of Signal Reconstruction
p (t ) 

  t  nT 
n  
x
x(t )
xr (t )
ωm < ωc < (ωs –ωm)
xr (t )  x p (t ) * h(t )

 
   xnT  t  nT  * h(t )

 n  

with h(t ) 
T sin c t 
t

 xnT ht  nT 
n  

F-1()

 xnT 
n  
T sin c t  nT 
 t  nT 
The lowpass filter interpolates
the samples assuming x(t)
contains no energy at
frequencies > ωc
(ωc =cutoff frequency)
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From Prof. A. S. Willsky, Signals and Systems course
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Signal Reconstruction Methods
1. Zero-order hold
2. First-order hold – linear interpolation
3. Whittaker-Shannon interpolation (bandlimited interpolation):
x (t ) 
 t  nTs 


x
(
nT
)
sinc


s
T
n  
s



 t 
 

x (t )    x ( nTs )   (t  nTs )   sinc 
 n  

 Ts 
•
•
(Alternative equivalent formulation)
Signal has to be band limited
(i.e. Fourier transform for frequencies greater than B equal 0)
The sampling rate must exceed twice the bandwidth, 2B
1
Ts 
2B
or
f s  2B
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From Prof. A. S. Willsky, Signals and Systems course
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Aliasing
37
No Problems in Reconstruction
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Reconstruction Problems
Overlapping
Alias
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Harmonics
Fundamental Frequency
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Harmonics
1
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0
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-0.2
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0
0.1
Time [s]
1Hz
2Hz
3Hz
4Hz
0.2
0.3
0.4
0.5
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La-Tone (440 Hz) sampled at
44.1 kHz (CD standard)
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La-Tone (440 Hz) sampled at
2 kHz without filtering
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La-Tone (440Hz) sampled at
2 kHz filtered at 1 kHz
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Aliasing Audio Examples
Original sound
Aliases 4 kHz
Correct sampling 4 kHz
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Moiré Pattern
46
Aliasing Video Examples
https://www.youtube.com/watch?v=R-IVw8OKjvQ
http://www.youtube.com/watch?v=jHS9JGkEOmA
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Conclusion
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Take Home Messages
When you sample a signal …
1. Make sure you know what the maximum
frequency fmax is or enforce it through a
low-pass filter
2. Make sure you sample at fs > 2 fmax
(Nyquist rule)
49
Additional Literature – Week 5
Books:
• Ronald W. Schafer and James H. McClellan
“DSP First: A Multimedia Approach”, 1998
• A. Oppenheim and A. S. Willsky with S. Hamid,
“Signals and Systems”, Prentice Hall, 1996.
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