1. Polynomials - Representation - Convolution 2. Complex Roots of Unity 3. DFT 4. FFT 5. Euler’s formula

1. Convolution and FFT
Chenghao Jin

2. 2018-08-16 2
Polynomials (Representation and Convolution)1
Complex Roots of Unity2
DFT33
FFT44
Euler’s formula5

3. 1. Polynomials

4. Convolution theorem
 Convolution theorem
 Convolution in one domain (e.g., time domain) equals point-
wise multiplication in the other domain (e.g., frequency
domain).
4

5. Polynomials
 Polynomials
 A polynomial in the variable x over the field F is a function
A(x) that can be represented as
 For example:
 Degree: 4
5
tcoefficienisawhere
xaxaxaaxaxA
i
n
n
n
i
i
i
1
1
2
210
1
0
)( 



  
432
43825)( xxxxxA 

6. Representations of polynomials
 Coefficient representation
 Example:
 Point-value representation of a polynomial A(x)
 A set of n point-value pairs where xk is distinct and yk = A(xk) for 0  k  n – 1
 A polynomial has many point-value representations.
 Example:
6




1
0
)(
n
i
i
i xaxA),,,,( 1210  naaaaa 
)4,3,8,2,5(
43825)( 432


a
xxxxxA
)},(,),,(),,{( 111100  nn yxyxyx 
x
y = A(x)
xj
yj
)105,2(),22,1(),5,0(),12,1(),73,2(
105,22,5,12,73:)(
2,1,0,1,2:
43825)( 432




   k
k
xA
x
xxxxxA

7. Polynomial Addition(Coefficient representation)
 Formal representation
 Addition: O(n)
 Example:
7
10where)()()(
then,)(and)(
1
0
1
0
1
0










njforbacxcxBxAxC
xbxBxaxA
jjj
j
n
j
j
j
n
j
j
j
n
j
j
1
111100
1
1
2
210
1
1
2
210
)()()()()()(
)(
)(









n
nn
n
n
n
n
xbaxbabaxBxAxC
xbxbxbbxB
xaxaxaaxA



)4,7,6,4(4764)(
)2,0,4,5(245)(
)6,7,10,9(67109)(
32
3
32



xxxxC
xxxB
xxxxA

8. Polynomial Multiplication (Coefficient representation)
8
220
:
where)()()(
then,)(and)(
0
22
0
1
0
1
0














njfor
bacnconvolutio
bacxcxBxAxC
xbxBxaxA
i
k
kiki
i
n
j
i
i
n
i
i
i
n
i
i

...)()(
)()(
)()()(
2
201102100100
1
1
2
210
1
1
2
210







xbababaxbababa
xbxbxbbxaxaxaa
xBxAxC
n
n
n
n 
65432
3
32
12144420758645)()()(
245)(
67109)(
xxxxxxxBxAxC
xxxB
xxxxA



Multiplication: O(n2)
Example:
Formal representation:

9. Polynomial Addition(Point-value representation)
 Polynomial Addition(Point-value representation): O(n)
 Example
9
)},(,),,(),,{(:)(
)},(,),,(),,{(:)(
)},(,),,(),,{(:)(
10
111111000
111100
111100





nnn
nn
nn
k
zyxzyxzyxxC
zxzxzxxB
yxyxyxxA
nkfordistinctisx



)}59,3(),18,2(),3,1(),2,0{(:
)}37,3(),13,2(),3,1(),1,0{(:
)}22,3(),5,2(),0,1(),1,0{(:
)3,2,1,0(
1:)(
12:)(
23
3
C
B
A
x
xxxB
xxxA
k







10. Polynomial Multiplication(Point-value representation)
 Problem O(n)
 if 𝐴 and 𝐵 are of degree-bound 𝑛, then 𝐶 is of degree-bound 2𝑛.
 Example
10
)}814,3(),65,2(),0,1(),1,0(),2,1(),9,2(),340,3{(:
)}37,3(),13,2(),3,1(),1,0(),2,1(),3,2(),20,3{(:
)}22,3(),5,2(),0,1(),1,0(),1,1(),3,2(),17,3{(:
)3,2,1,0,1,2,3(
1:)(
12:)(
23
3






C
B
A
x
xxxB
xxxA
k
)},(,),,(),,{(:)(
)},(,),,(),,{(:)(
)},(,),,(),,{(:)(
121212111000
12121100
12121100



nnn
nn
nn
zyxzyxzyxxC
zxzxzxxB
yxyxyxxA




11. Conversion Between Representations
 Evaluation (Coefficient to point-value): O(n2)
 Computing the value of A(x0) at a given point x0
 We can evaluate a polynomial in O(n) using Horner’s rule(for each)
 For example
 Standard form:
 3 additions and 6 multiplications 7·x,9·x·x,2·x·x·x
 Horner’s form:
 3 additions and 3 multiplications
 Interpolation (point-value to Coefficient):
 Inverse of evaluation
 Determining the coefficient form of a polynomial from a point-value representation
 Note Vandermonde matrix is invertible iff xk are distinct
11
)))((()( 12210    nn xaaxaxaxaxA
32
2975)( xxxxA 
xxxxA ))29(7(5)( 








































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







 1
1
0
1
1
2
11
1
1
2
11
1
0
2
00
1
1
0
matrixeVandermond
)()(1
)()(1
)()(1
...
n
n
nnn
n
n
n
n a
a
a
xxx
xxx
xxx
y
y
y

  





12. Conversion Between Representations
 Interpolation (point-value to Coefficient):
 Vandermonde matrix, denoted by V(x0,x1,…,xn-1), whose determinant is
 Coefficient :
 The LU Decomposition algorithm: O(n3)
 A faster algorithm Lagrange’s formula
12
0)(
10
  nkj
jk xx
yxxxVa n
1
11,0 ),,( 
 









1
0 )(
)(
)(
n
k
kj
jk
kj
j
k
xx
xx
yxA

13. The Road So Far
13

14. 2. Complex Roots of Unity

15. Complex Roots of Unity
 Complex nth root of unity
 A complex number such that ωn = 1
 Principal n th root of unity
 There are exactly n complex nth root of unity
 All other complex nth roots of unity are powers of
 n complex roots of unity are equally spaced around the circle of unit radius
centered at the origin of the complex plane
15
n
i
n e


2

1,...,2,1,0,
2
 nkfore n
ki
1210
,,,, n
nnnn  
n
radiansingiven:
unit,imaginary:i
logarithm,naturaltheofbase:
,sincos


e
whereiei

Euler's formula:

16. Complex roots of unity
 8 Complex 8th Roots of Unity
16

17. Cancellation Lemma
 For any integers n  0, k  0, and b > 0,
 Proof
 For any even integer n > 0,
 Example:
17
k
n
dk
dn  
k
n
knidkdnidk
dn ee  
 )()( /2/2
12
2/
 n
n
4
6
24  
4
4/224
6
2
624/26
24 )(  


 i
i
i
eee

18. Halving Lemma
 If 𝑛>0 is even, then the squares of the 𝑛 complex 𝑛th
roots of unity are the 𝑛/2 complex 𝑛/2 th roots of unity
 Proof
 By the cancellation lemma, we have for any
nonnegative integer 𝑘.
 If we square all of the complex 𝑛th roots of unity, then each
𝑛/2 th root of unity is obtained exactly twice
18
k
n
k
n 2/
2
)(  
222222/
)()( k
n
k
n
n
n
k
n
nk
n
nk
n   

19. Summation lemma
 For any integer 𝑛≥1 and nonzero integer 𝑘 not divisible by 𝑛,
 Proof
 Requiring that 𝑘 not be divisible by 𝑛 ensures that the denominator is not 0,
since only when k is divisible by 𝑛
19
0)(
1
0



n
j
jk
n
0
1
1)1(
1
1)(
1
1)(
)(
1
0












k
n
k
k
n
kn
n
k
n
nk
n
n
j
jn
k






Note:
▪ k is not divisible by n
▪ only when k is divisible by n
1k
n

20. 3. DFT

21. Discrete Fourier Transform
21
DFT
110 ,...,, Nxxx
110 ,...,, NXXX





1
0
2N
i
ik
N
j
ik exX

Inverse DFT
110 ,...,, Nxxx
110 ,...,, NXXX




1
0
2
1 N
i
ik
N
j
ik eX
N
x


22. Discrete Fourier Transform
 We wish to evaluate a polynomial of degree bound n at
 assume that 𝑛 is a power of 2
 assume 𝐴 is given in coefficient form
 We define the results
 The vector is the discrete fourier transform
of
22
0 1 1
, , , n
n n n   




1
0
)(
n
i
i
i xaxA
),,( 110  naaa a
1,...,2,1,0)(
1
0
 


nkforaAy
n
j
kj
nj
k
nk  




1
0
2N
i
ik
N
j
ik exX

),,( 110  nyyy y
),,( 110  naaa a

23. Discrete Fourier Transform
 Key idea
23
unityofrootnprincipleiswherexchoose thk
k 






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
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







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


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
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







 1
3
2
1
0
)1)(1()1(3)1(21
)1(3963
)1(2642
132
1
3
2
1
1
3
2
1
0
1
1
1
1
11111
)(
)(
)(
)(
)1(
n
nn
n
n
n
n
n
n
n
n
nnnn
n
nnnn
n
nnnn
n
n
n
n
n
n a
a
a
a
a
A
A
A
A
A
y
y
y
y
y

































































1
1
0
1
1
2
11
1
1
2
11
1
0
2
00
1
0
)()(1
)()(1
)()(1
...
n
n
nnn
n
n
n
n a
a
a
xxx
xxx
xxx
y
y
y





DFT:

24. Inverse Discrete Fourier Transform
24
















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

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
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
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


















 1
3
2
1
0
1
)1)(1()1(3)1(21
)1(3963
)1(2642
132
1
3
2
1
0
1
1
1
1
11111
1
n
nn
n
n
n
n
n
n
n
n
nnnn
n
nnnn
n
nnnn
n y
y
y
y
y
n
a
a
a
a
a












Inverse DFT:

25. 4. FFT

26. Fast Fourier Transform(FFT)
 Fast Fourier Transform
 An efficient algorithm for computing the DFT
 Takes advantage of the special properties of the complex roots of unity to
compute DFT𝑛(a) in time Θ(𝑛log𝑛).
 Divide and conquer approach
 Divide and conquer approach
 Divide the polynomial A(x) by splitting it into its even and odd powers
26
1
1
3
3
2
210)( 
 n
n xaxaxaxaaxA 
)()()(
)()()(
)()()(
)(
)(
221
1
3
3
2
210
221
1
3
3
2
210
22
12/
1
2
531
12/
2
2
420
xxAxAxaxaxaxaaxA
xxAxAxaxaxaxaaxA
xxAxAxA
xaxaxaaxA
xaxaxaaxA
oddeven
n
n
oddeven
n
n
oddeven
n
nodd
n
neven


















27. Fast Fourier Transform(FFT)
 The problem of evaluating A(x) at reduces to
 Evaluating the degree-bound n/2 polynomials Aeven(x) and Aodd(x) at the points
 Combining the results by A(x) = Aeven(x2) + xAodd(x2)
 Why bother?
 The list does not contain n distinct values,
but n/2 complex n/2 th roots of unity
 Polynomials Aeven and Aodd are recursively evaluated at the n/2 complex
n/2 th roots of unity
 Sub problems have exactly the same form as the original problem, but are
half the size
27
110
,,, n
nnn  
212120
)(,,)(,)( n
nnn  
212120
)(,,)(,)( n
nnn  

28. Fast Fourier Transform(FFT)
28

29. Fast Fourier Transform(FFT)
29)()()()(
)(1)()1()(
)()()()(
)(1)()1()(
)()(
)()()(
)()(
)1()(
)()(
)1()(
)(
3120
3
4
3120
2
4
3120
1
4
3120
0
4
2
31
2
20
22
3210
3
4
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30. Complexity
 If we can recursively evaluate Aeven and Aodd using the same approach, we get
the following recurrence relation for the running time of the algorithm
 Our trick was to evaluate x at ( positive ) and -x ( negative ).
 But inputs to Aeven and Aodd are always of the form x2 ( positive )!
 How can we apply the same trick?
30
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31. 31

32. 5. Euler’s formula

33. Euler’s formula
 Euler's formula
 Modulus
 Some important cases
33
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34. Euler’s formula
 A point in the complex plane
 can be represented by a complex number written in cartesian coordinates
 Euler's formula provides a means of conversion between cartesian
coordinates and polar coordinates.
 The polar form simplifies the mathematics when used in multiplication or
powers of complex numbers
34
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35. Euler’s formula
 Relationship to trigonometry
 Euler's formula provides a powerful connection between analysis and
trigonometry
35
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