XIII. Walsh Transform (Hadamard Transform)

1
XIII. Walsh Transform (Hadamard Transform)
? 13-A Ideas of Walsh Transforms

• ? 8-point Walsh transform
• ? Advantages of the Walsh transform
• (1) Real
• (2) No multiplication is required
• (3) Some properties are similar to those of
  the DFT

2

• ? Forward and inverse Walsh transforms are
  similar.
• forward
  , inverse
•  
• ? Alternative names of the Walsh transform
• Hadamard transform, Walsh-Hadamard
  transform
•  
• ? Orthogonal Property
  if m0 ? m1
• ? Zero-Crossing Property
• ? Even / Odd Property
•  
• Useful for spectrum analysis
• Sometimes also useful for implementing the
  convolution

3

• Walsh transform ? DFT, DCT ??????
• , DFTm, n exp(?j2? m
  n/N),

4

• References for Walsh Transforms
• 1 K. G. Beanchamp, Walsh Functions and Their
  Applications, Academic Press, New York,
  1975.
• 2 B. I. Golubov, A. Efimov, and V. Skvortsov,
  Walsh Series and Transforms Theory and
  Applications, Kluwer Academic Publishers, Boston,
  1991.
• 3 H. F. Harmuth, Applications of Walsh
  functions in communications, IEEE
  Spectrum, vol. 6, no. 11, pp. 82-91, Nov. 1969.
  
• 4 H. F. Harmuth, Transmission of Information by
  Orthogonal Functions, Springer-Verlag, New
  York, 1972.

5
? 13-B Generate the Walsh Transform
2-point Walsh transform
4-point Walsh transform
How do we obtain the 2k1-point Walsh transform
from the 2k-point Walsh transform ?
Step 1
Step 2 ?? sign changes ? rows ???????
6
?? ?? row ? sign change ?,???????
0, 1, 2, 3, .., 2k-1 ? ?? row ? sign
change ?,??????? 0, 3, 4, 7, .., 2k1-1, 1,
2, 5, 6, .., 2k1-2,
? row ?index ?0 ?? ? ? n ? row (n is even
and n lt N/2) ? sign change ? 2n
(n is odd and n lt N/2) ? sign
change ? 2n 1
(n is even and n ? N/2) ? sign change ? 2n1-N
(n is odd and n ?
N/2) ? sign change ? 2n-N
??? sign change ???? ? row ????
7
sign changes 0 3 1 2
sign changes 0 3 4 7 1 2 5 6
8
Constraint for the number of points of the
Walsh transform N must be a power of 2 (2, 4,
8, 16, 32, ..) Although in Matlab it is
possible to define the 12 ?2k point or the 20
?2k point Walsh transform, the inverse transform
require the floating-point operation.
9
? 13-C Alternative Forms of the Walsh Transform
????

• ? Sequency ordering (i.e., Walsh ordering) .
  using for signal processing
• ? Dyadic ordering (i.e., Paley ordering) ...
  using for control
• ? Natural ordering (i.e., Hadamard ordering)
  using for mathematics

Sequency ordering Dyadic ordering Natural ordering Wm, n
(Gray Code) (Bit Reversal)
row 0 row 0 row 0 1, 1, 1, 1, 1, 1, 1, 1
row 1 row 1 row 4 1, 1, 1, 1, ?1, ?1, ?1, ?1
row 2 row 3 row 6 1, 1, ?1, ?1, ?1, ?1, 1, 1
row 3 row 2 row 2 1, 1, ?1, ?1, 1, 1, ?1, ?1
row 4 row 6 row 3 1, ?1, ?1, 1, 1, ?1, ?1, 1
row 5 row 7 row 7 1, ?1, ?1, 1, ?1, 1, 1, ?1
row 6 row 5 row 5 1, ?1, 1, ?1, ?1, 1, ?1, 1
row 7 row 4 row 1 1, ?1, 1, ?1, 1, ?1, 1, ?1
10
? Dyadic ordering Walsh transform
? Natural ordering Walsh transform
11

• ? binary code to gray
  code
• When N 2k
• gk bk, gq XOR(bq1, bq) for q k ?1, k
  ?2, ., 1
• ? gray code to binary code
• When N 2k
• bk gk, bq XOR(bq1, gq) for q k ?1, k
  ?2, ., 1
•  

12
? 13-D Properties

• (1) Orthogonal Property
• (2) Zero-Crossing Property
• (3) Even / Odd Property
• (4) Linear Property
• If fn ? Fm, gn ? Gm,
  (? means the Walsh transform)
• then a fn b gn ? a Fm b Gm

13
(5) Addition Property ? Addition modulo
2 (denoted by ?) 0 ? 0 1 ? 1 0, 0
? 1 1 ? 0 1, Example
, therefore
3 ? 7 4
14
(6) Special functions ?n 1 when n 0,
?n 0 when n ? 0 ?n ? 1, 1 ? N??n
(7) Shifting property If fn ? Fm,
then fn ? k ? W(k, m)?Fm (8) Modulation
property If fn ? Fm,
then W(k, n)?fn ? Fm ? k (9) Parsevals
Theorem If fn ? Fm,
If fn ? Fm, gn ? Gm,
15

• (10) Convolution Property
• If fn ? Fm, gn ? Gm,
  then hn fn ? gn ? Fm Gm
•   ? means the logical convolution
• hn fn ? gn
  
• For example, when N 8,
• h3 f0g3 f1g2 f2g1 f3g0
  f4g7 f5g6 f6g5
• f7g4
• h2 f0g2 f1g3 f2g0 f3g1
  f4g6 f5g7 f6g4
• f7g5

16
? 13-E Butterfly Fast Algorithm

• (Method 1) John L. Sharks Algorithm

x0 x1 x2 x3 x4 x5 x6 x7
X0 X7 X3 X4 X1 X6 X2 X5
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
17
(Method 2) Manzs Sequence Algorithm
x0 x4 x2 x6 x1 x5 x3 x7
X0 X1 X2 X3 X4 X5 X6 X7
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
There are other fast implementation algorithm for
the Walsh transform.
18
? 13-F Applications

• Walsh transform ??? spectrum analysis,??????convol
  ution
•  
• Applications of the Walsh transform
•  
• Bandwidth reduction
• High resolution
• Multiplexing
• Information coding
• Feature extraction
• ECG signal (in medical signal processing)
  analysis
• Hadamard spectrometer
• Avoiding quantization error

19

• The Walsh transform is suitable for the function
  that is a combination of Step functions

New Applications CDMA (code division multiple
access)
20
? 13-G Jacket Transform
???? 1 ? ?? 4-point Jacket transform
2k1-point Jacket
P row permutation
Ref M.H. Lee, A new reverse Jacket transform
and its fast algorithm, IEEE Trans. Circuits
Syst.-II , vol 47, pp.39-46, 2000.
21
? 13-H Haar Transform
N 2
N 4
N 8
Ref H. F. Harmuth, Transmission of Information
by Orthogonal Functions,
Springer-Verlag, New York, 1972
22
N 16
23
Hm, n ?? (m 0, 1, , 2k -1, n 0, 1, , 2k
-1)
H0, n 1 for all n If 2h ? m lt 2h1
Hm, n 1 for (m - 2h)2k-h ? n lt (m - 2h
1/2)2k-h
Hm, n -1 for (m - 2h 1/2)2k-h ? n lt (m - 2h
1)2k-h
Hm, n 0 otherwise
???? Walsh transforms ?? Applications localized
spectrum analysis, edge detection
Transforms Running Time terms required for NRMSE lt 10?5
DFT 9.5 sec 43
Walsh Transform 2.2 sec 65
Haar Transform 0.3 sec 128
24
XIV. Number Theoretic Transform (NTT)
? 14-A Definition

• Number Theoretic Transform and Its Inverse
• Note
• (1) M is a prime number , (mod M) ???? M ???
• (2) N is a factor of M-1
• (Note when N ? 1, N must be prime to M)
• (3) N-1 is an integer that satisfies (N-1)N
  mod M 1
• (When N M -1, N-1 M -1)

25
(4) a is a root of unity of order N
When a satisfies the above equations and N M
-1, we call a the primitive root.
26

• Example 1
• M 5 ? 2 ?1 2 (mod 5) ?2 4 (mod 5)
  ?3 3 (mod 5) ?4 1 (mod 5)
• When N 4
•  
• When N 2

27
Example 2
M 7 ? cannot be 2 but can be 3. ? 2 ?1
2 (mod 7) ?2 4 (mod 7) ?3 1 (mod 7) ?
3 ?1 3 (mod 7) ?2 2 (mod 7) ?3 6 (mod
7) ?4 4 (mod 7) ?5 5 (mod 7)
?6 1 (mod 7)
28
Advantages of the NTT
Disadvantages of the NTT
29
???? SCI Papers ????
?????? SCI ??,impact factor.????? SCI ? impact
factor???????? SCI Papers? Impact factor ??????
SCI ?? Science Citation Index
(A) SCI ????ISI Web of Knowledge
??? ISI Web of Knowledge
http//admin-apps.webofknowledge.com/JCR/JCR?RQHO
ME
????You do not have a session?
establish a new session
??
?????????,????????? ISI ???????, ?????
ISI Web of Knowledge
30
(B) ?????????? SCI journal,???? impact factor
??? Search for a specific journal,?? SUBMIT
????????,?? SEARCH
??? Title Word,????????????????
31
?????,???????? SCI ??
??????????? impact factor
Impact Factor (????)
32
(C) ?? impact factor (????) ??? journal
?????,??????????,??? journal ? impact factor ??
????, impact factor ? 1.5 ??? journals,??????????
Nature ? impact factor ? 36.1 Science ? impact
factor ? 31.4 IEEE ?????? impact factors ??? 1 ?
5 ?? IEEE Trans. Image Processing? impact factors
? 3.0 ?? IEEE Trans. Signal Processing? impact
factors ? 2.65 ?? ???????? impact factors ? 0.6
? 1.5 ??
33
(D) ?????????? SCI journals
???
??? ISI Web of Knowledge ??,?? View a group
of journals by ?Subject Category?,?? SUBMIT
34
??,??????? category,?? SUBMIT,????????? SCI
journals
35
?????????? SCI journals
???
??? ISI Web of Knowledge ??,?? Search for
a specific journal,SUBMIT ??
???? Title Word,???????
?? Search ??,????????????????????? journals
36
(E) EI (Engineering Village)
???? www.engineeringvillage.org http//www.engi
neeringvillage.com/search/quick.url
??????????? EI http//tul.blog.ntu.edu.tw/archives
/4627
(F) SSCI (Social Science Citation Index)
?????????
http//www.thomsonscientific.com/cgi-bin/jrnlst/jl
options.cgi?PCJ
37
??????????
????????????????????????????????,??????????,??????
???????????????,???????????,???????????????????,??
????????????,????????,??????????,?????????????????
?????????????????? (1) ?????????(??)
??????????????????????????
????????????,????????,??????????????????????,?????
???????????? (2) ??????????????
??????????????,?????,???????????,???????????,?????
38
(3) ????????,????????????? ??????????????????? (
4) ???????????,????????????? (5)
????????,???? (6) Previous work (?????????)
??????,?????????Introduction ?? Previous work
??????????????? (7) ?????????????
(8)??????????????,?????????
??????????????????????????????(????)
?????????? (9) ?????,?????????????????????????????
?????
39
(10) ??? Conference ???????????????,??????????????
?,????????????
(11)?????,?????????,????????? (12)
??,??????? ???????,?????????????? ??????????