Download General Transforms

General Transforms
• Let k (n) be orthogonal, period N
N 1
N
 k (n). j (n) 
0
n 0
*
k= j
k j
• Define h(n)  1 N1H (k ) (n)
k
N k 0
• So that H (k )  N 1 h(n). * (n)

k
n 0
1
Professor A G Constantinides
General Transforms
• Determine conditions to be satisfied by
k (n) so that H (k ). X (k )  x(n) h(n)
• Let Y (k )  H (k ). X (k )
 1 N 1
• Then
y (n)   X (k ).H (k ).k (n)
N k 0
1 N 1 N 1

*
    h( p).k ( p)  X (k )k (n)
N k 0  p 0

2
Professor A G Constantinides
General Transforms
N 1
1 N 1
*


y
(
n
)

h
(
p
).
X
(
k
)

(
n
).

(
p
)


• Thus
k
k


p 0
 N k 0
• To support circular convolution
*
k (n).k ( p)  k (n  p)
• 1) p  0  k (n).k * (0)  k (n)
*
k (0)  1  k (0) and real
 k (0).k ( p )  k ( p )
*
3
Professor A G Constantinides
General Transforms
n p
• 2)
k (n).k (n)  k (0)
• 3) Since fundamental period is N
k ( N  n)  k (n) k ( N )  k (0)  1
p  N k (n).k ( N )  k (n  N )  k (n)
*
p

1
• 4)
k (n).k (1)  k (n  1)
*
k (n).k (1)  k (n  1) k (n).k (1)  k (n  1)
n
k (n  1).k (1)  k (n) k (n)  k (1)
*
4
 k (1)  1
N
Professor A G Constantinides
Number Theoretic Transforms
• Thus in a complex field k (1) are the N
2
j
.k
roots of unity and
k (1)  e N
• In an integer field we can write
N
k (1)  1 mod N
5
• and use Fermat's theorem a P 1  1 mod P
• where P is prime and a is a primitive root
• Euler's totient function can be used to
generalise as aN  1 mod N
Professor A G Constantinides
Number Theoretic Transforms
Fermat's Theorem: Consider
a,2a,3a,..., ( P  1)a
• Reduce mod P to produce 1,2,3,..., P  1
• Since (a, P)  1 we have
a.2a.3a.....( P  1)a  1.2.3.4....( P  1)
P 1
• or a .( P  1)! ( P  1)! mod P
• and since there are no other unknown
P 1
factors a  1 mod P
6
Professor A G Constantinides
Number Theoretic Transforms
• Alternatively (perhaps simpler)
• For  ,  ,  ,...,
not multiples of P
• (      ...   ) P expanded in
bionomial form produces multiples of P
• except for the terms  P ,  P ,..., P
• Thus
(      ...   ) P   P   P  ...   P mod P.
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Professor A G Constantinides
Number Theoretic Transforms
• Now, if the total number of bracketed terms
is for this argument less than P say a, then
for       ...    1
one has
• ie
8
• and
a P  1  1  ...  1 mod P
a
mod P
a P 1  1 mod P.
k (n)  a nk
Professor A G Constantinides
Number 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
2  64  9 * 7  1  1
6
mod 7
• We note further that
4 * 2  7 1  1
9
mod 7.
Professor A G Constantinides
Number Theoretic Transforms
• Thus we have
• And hence
4  21
mod 7
N 1
H (k )   h(n).k * (n)
n 0
7 1
H (k )   h(n).4nk
n 0
10
mod 7
• Thus only real numbers are involved in the
computation . Moreover, the kernel is a
power of 2
Professor A G Constantinides