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Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Discrete Convolution
CONTENTS
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Fundamentals
Properties
Discrete convolution
Applications
DFT and inverse DFT
FFT and inverse FFT
Differential calculus
FUNDAMENTALS
( f * g )[ n]  m f [m]g[n  m]  m f [n  m]g[m]
( f * g N )[ n]  mN 10


k  

f [m  kN] g N [n  m]
( f * g N )[ n]  mN 10 f [m]g N [n  m]
 nm0 f [m]g[n  m]  mN 1n1 f [m]g[ N  n  m]
 mN 10 f [m]g[( n  m) mod N ]  ( f *N g )[ n]
COMPLEX ROOTS OF ONE
e2ik / n , k  0,1,...,n 1
n 1
eiu  cos u  i sin u
n0 , 1n ,...,nn1, (Zn ,)
82
i
83
80  88
-1
84
85
main
81
1
87
-i 86
COMPLEX ROOTS OF ONE. PROPERTIES
dk
n, k  0, d  0, dn
 nk
(nk n / 2 )2  (nk )2
n  1, k  0, k mod n  0,
n1
 j 0 (nk ) j  0
DFT
yk  A(nk )  nj10 a jnkj
y  DFTn (a)
INVERSE DFT
 y0  
 y  
 1  
 y2  


y
 3  
   

 
 yn1  
1
1
1
1

1
1
n
n2
n3
n2
n4
n6
n3
n6
n9





nn1
n2( n1)
n3( n1)


1

1
Vn1 : ( j, k )  nkj / n
1 n1
k 0 yknkj
n
a * b  DFT2n1( DFT2n (a)  DFT2n (b))
a  DFTn1 ( y), a j 

  a0 


nn1   a1 
n2( n1)   a2 


3( n 1)
n
  a3 
  



( n 1)( n 1)
 an1 
n
1
CONVOLUTION APPLICATIONS
Convolution and related operations are found in many applications of engineering and mathematics.
In electrical engineering, the convolution of one function (the input signal) with a second function (the
impulse response) gives the output of a linear time-invariant system (LTI). At any given moment, the
output is an accumulated effect of all the prior values of the input function, with the most recent values
typically having the most influence (expressed as a multiplicative factor). The impulse response function
provides that factor as a function of the elapsed time since each input value occurred.
In digital signal processing and image processing applications, the entire input function is often
available for computing every sample of the output function. In that case, the constraint that each
output is the effect of only prior inputs can be relaxed.
Convolution amplifies or attenuates each frequency component of the input independently of the
other components.
In statistics, as noted above, a weighted moving average is a convolution.
In probability theory, the probability distribution of the sum of two independent random variables is the
convolution of their individual distributions.
In optics, many kinds of "blur" are described by convolutions. A shadow (e.g., the shadow on the table
when you hold your hand between the table and a light source) is the convolution of the shape of the
light source that is casting the shadow and the object whose shadow is being cast. An out-of-focus
photograph is the convolution of the sharp image with the shape of the iris diaphragm. The photographic
term for this is bokeh.
CONVOLUTION APPLICATIONS
Similarly, in digital image processing, convolutional filtering plays an important role in many important
algorithms in edge detection and related processes.
In linear acoustics, an echo is the convolution of the original sound with a function representing the
various objects that are reflecting it.
In artificial reverberation (digital signal processing, pro audio), convolution is used to map the impulse
response of a real room on a digital audio signal (see previous and next point for additional information).
In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta
pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse.
In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a
convolution-superposition algorithm.
In physics, wherever there is a linear system with a "superposition principle", a convolution operation
makes an appearance.
In kernel density estimation, a distribution is estimated from sample points by convolution with a kernel,
such as an isotropic Gaussian. (Diggle 1995).
In computational fluid dynamics, the large eddy simulation (LES) turbulence model uses the convolution
operation to lower the range of length scales necessary in computation thereby reducing computational
cost.
Thank you for your attention
DISCRETE MATHEMATICS
Czesław Smutnicki