professor a g constantinides 1 general transforms let be orthogonal, period n define so that
DESCRIPTION
Professor A G Constantinides 3 General Transforms Thus To support circular convolution 1) and realTRANSCRIPT
1 Professor A G Constantinides
General TransformsGeneral Transforms
• Let be orthogonal, period N
• Define
• So that
)(nk
1
0
*
0=
)().(N
njk jk
jkNnn
1
0)()(1)(
N
kk nkH
Nnh
1
0
* )().()(N
nk nnhkH
2 Professor A G Constantinides
General TransformsGeneral Transforms
• Determine conditions to be satisfied by so that
• Let• Then
)(nk )()()().( nhnxkXkH
)().()( kXkHkY
1
0)().().(1)(
N
kk nkHkX
Nny
1
0
1
0
* )()()().(1 N
kk
N
pk nkXpph
N
3 Professor A G Constantinides
General TransformsGeneral Transforms
• Thus• To support circular convolution
• 1) and real
1
0
1
0
* )().()(1).()(N
p
N
kkk pnkX
Nphny
)()().( * pnpn kkk
)()0().( 0 * nnp kkk
)0(1)0(*kk
)()().0( *k pp kk
4 Professor A G Constantinides
General TransformsGeneral Transforms
• 2)• 3) Since fundamental period is N
• 4)
pn )0()().( *kkk nn
)()( nnN kk 1)0()( kk N Np )()()().( nNnNn kkkk 1p )1()1().( * nn kkk
)1()1().( * nn kkk )1()1().( nn kkk )()1().1( nn kkk nkk n )1()(
1)1( Nk
5 Professor A G Constantinides
Number Theoretic TransformsNumber Theoretic Transforms
• Thus in a complex field are the N roots of unity and
• In an integer field we can write
• and use Fermat's theorem • where is prime and is a primitive root• Euler's totient function can be used to
generalise as
)1(kk
Nj
k e.2
)1(
NNk mod 1)1(
PaP mod 11
P a
Na N mod 1
6 Professor A G Constantinides
Number Theoretic TransformsNumber Theoretic TransformsFermat's Theorem: Consider
• Reduce mod P to produce • Since we have
• or• and since there are no other unknown
factors
aPaaa )1(,...,3,2,
1,...,3,2,1 P1),( Pa
)1....(4.3.2.1)1.....(3.2. PaPaaa
PPPaP mod )!1()!1.(1
PaP mod 11
7 Professor A G Constantinides
Number Theoretic TransformsNumber Theoretic Transforms
• Alternatively (perhaps simpler) • For not multiples of P• expanded in
bionomial form produces multiples of P• except for the terms • Thus
,...,,,P)...(
PPP ,...,,
.mod ...)...( PPPPP
8 Professor A G Constantinides
Number Theoretic TransformsNumber Theoretic Transforms
• Now, if the total number of bracketed terms is for this argument less than P say a, then for one has
• ie
• and
1...
PaP mod 1...11 Pa mod
.mod 11 PaP
nkk an )(
9 Professor A G Constantinides
Number Theoretic TransformsNumber Theoretic Transforms
• For example for P=7 the quantity a, known as the primitive root, will be one of the following {2,3,4,5,6}
• Thus for a=2 we have
• We note further that 7mod 117*96426
.7mod 1172*4
10 Professor A G Constantinides
Number Theoretic TransformsNumber Theoretic Transforms
• Thus we have • And hence
• Thus only real numbers are involved in the computation . Moreover, the kernel is a power of 2
7mod 24 1
1
0
* )().()(N
nk nnhkH
7mod4).()(17
0
n
nknhkH