On Coded Mary Frequency Keying

1
On Coded M-ary Frequency Keying

• A.J. Han Vinck

2
Content

• Motivation
• Impulsive noise, broadcast, background
• Definition
• Convolutional permutation code
• Random access code

3
Motivation

• We want to use constant envelope modulation
  M-FSK
• Error correcting codes for
• Impulsive noise (broad band)
• Permanent disturbances (narrow band)
• Back ground noise

4
idea
Transmit messages as ? sequences (code words)
of length M where all M symbols are
different ? minimum distance ( of
differences) D
Example M 3 D 2 Code 123 312 231
132 321 213
f
time
5
Communication structure example
( 3,2,1 )
encoder
modulator
message
t
M-FSK or PPM
f
Time and frequency diversity !
t

• gt T
• gt T
• gt T

3 Energy detectors
f
6
2 Code words M 5 at D 3
C1 1,2,4,3,5 C2 1,2,3,5,4
frequency
time
7
narrow band noise
C1 1,2,4,3,5
frequency
Narrow band
time
8
broad band noise (impulse)
broad band
C1 1,2,4,3,5
frequency
time
9
background noise
C1 1,2,4,3,- C2 1,2,-,5,4 Both code words 4
agreements!
frequency
insert delete
time
10
Performance for energy/threshold detector

• Error events agree with codewords in ? 1
  position
• ? D error events are needed
• to create an additional codeword or detection
  error

11
Summary code constructions
It can be shown that Simple code constructions
exist for D 2 (all permutations) D
3 ( all even permutations) D M ( all
cyclic shifts ) D M-1, where M is prime
12
Non-coherent detection (FFT)
Envelope detector
y1
filter matched to f1
1
Quantize gt Th 1 lt Th 0
Envelope detector
y2
filter matched to f2
0
X
???
???
Envelope detector
yM
0
filter matched to fM
sample
1 0 0 0
transmit
? ? ? ?
Detect Presence of code sequence
0 0 1 0
? ? ? ?
0 1 0 0
? ? ? ?
0 0 0 1
? ? ? ?
13
Decision matrix structure
hard soft softer
softest
1 0 0 1
1 0 0 1
1 3 4 1
10 3 2 11
0 0 1 0
1 1 1 0
3 1 1 0
3 9 9 2
0 1 0 0
0 1 0 0
2 2 3 4
4 6 1 0
0 0 0 0
0 0 1 1
4 4 2 2
1 1 7 4
select above ranking
calculate largest threshold T
likelihoods
like adding received energy
0 0 0 0
Needs full channel knowledge
T.6E1/2
0 0 0 0
1 1 1 1
0 0 0 0
complexity
14
problem

• Complexity of decoding (exhaustive search)
• Low efficiency

15
Coding gain

• Coding gain (soft) for AWGN only

Pe 10-1 10-2
10-3
M 4, D 4 3 dB !
uncoded
coded
0 3 6
dB
16
Permutation convolutional codes
0
213
mapping
1
Dfree ( permutation conv. code) 5 3 8
IDEA convert binary output to permutation
codewords keep ( or increase) distance if
possible!
17
Conversion binary to M-ary
Distance tables

• conv. code output permutation code word
• 00 01 10 11 231 213 132 123
• 00 0 1 1 2 231 0 2 2
  3
• 01 1 0 2 1 213 2 0 3
  2
• 10 1 2 0 1 132 2 3 0
  2
• 11 2 1 1 0 123 3 2 2
  0

1!
18
For R 1/2

• for constraint length 2 convolutional code free
  distance 5
• After mapping Permutation free distance 8
• ( shortest path has length 3)
• More distance increased mappings exist
• Ex D 2 !
• 4 bits (16 symbols) ? M 6
• 5 bits (32 symbols) ? M 7
• 6 bits (64 symbols) ? M 8

19
Dfree2
Dfree8
Dfree8
20
Code partitioning A lt n M users D n-1
n
n
n
???
M
???
???
???
User 1
User 2
User M-1
D n-1 C M (M-1)
differences n-1 ? agreements (
interference ) 1 ? Pe 0! ? lt n active users
can never create a non participating code
word! Normalized sum rate same as single user!
21
Simple code using permutation codes

• M prime calculations modulo M
• M-1 Users 0 lt Ilt M
• Message 0 ? m lt M
• n ? M
• encode message m for user I gt o as
• CI(m) ( m , I ) 1 1 ? ? ? 1
• 0 1 ? ? ? n-1
• signature
• CI(m) is a permutation code word for a code with
  D n-1

22
example

• M 5, n 3 user 2
• Decoding for user I check for I m

( 3, 2 ) 1 1 1 ( 3, 0, 2 )
0 1 2
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Error probability for n ? A ? M active users
For permutation codes
For random codes
Note R(TDMA)
24
Performance
M 128, n 4
A 24
10-1
random
A 16
Pe
A 75
A 8
M 256, n 8
10-3
A 50
10-5
.1 .2 .3
25
conclusions

• M-FSK connected with Permutation codes
• As error correcting codes
• for impulsive and narrow band noise
• provides frequency and time diversity
• As multi user codes with known interference
• Extension to larger M gt 4 easy

26
Some references

• A.J. Han Vinck, Coded Modulation for PLC, AEU,
  Vol. 2000, pp. 45-49
• H.C. Ferreira and A.J. Han Vinck, Interference
  cancellation with permutation trellis code, VTC,
  Fall 2000, pp. 4.5.1.3.1.- 7
• J. Haering and A.J. Han Vinck, Performance bounds
  for optimum and suboptimum reception under Class
  A impulsive noise, IEEE Tr. Comm. July 2002.
• O. Hooijen, Ph.D. Thesis, University of Essen,
  1998, Aspects of residential Powerline
  communications.
• A.J. Han Vinck and J. Keuning, On the Capacity of
  the T-user M-frequency Multiple Access Channel,
  IEEE Trans. Inf. Th., Nov. 1996, pp. 2235-238.

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Reed-Muller based codes

• Starting code m 1 C 00,01,10,11
• has minimum distance 2m-1
• 2m1 code words U of length 2m
• NEW code of with 2m2 code words of length 2m1
• (U,U) and (U,U)
• ? distance (U,U), (U,U) 2m
• ? distance (U,U), (V,V) 2 2m-1 2m
• ? distance (U,U), (V,V) 2 2m-1 2m
  (use compl.prop.)

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Construction by example (m3 layers)

• Start Remove 000 and 111 from all RM codes
• Layer 1 m 3 231-2 14 code words
  distance 23-1 4
• 0000,1111 0011,1100 0101,1010
• 2. Layer 2 m 2 221-2 6 code words
  distance 222-1 4
• 0000,1111 0000,1111 0000,1111
  0000,1111 0000,1111 0000,1111
• 0011,0011 0101,0101 0110,0110
  1010,1010 1001,1001 1100,1100
• 3. Layer 3 m 1 211-2 2 code words
  distance 421-1 4
• 0000,1111 0000,1111
• 0011,0011 0011,0011 01234567 10325476
• 0101,0101 1010,1010
• PROPERTY all cloumns are different d 2m-1
  C (2m-1 2)(2m-1 2) ??? 2

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Multi user communication

• Binary transmission
• 1 Transmit code sequence
• 0 transmit common sequence
• For A active users no other code word generated
  for
• A ( agreements ? M-D ) lt M
• minimum distance D

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Multi user communication

• Performance M-ary sequences of length M,
• SUM-RATE
• Example M 5, D 4 C 120
• A lt 5 R ? 4/25 lt 1/5
• Note for TDMA R lt 5LogM/600 log5/120

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Dfree2
Dfree8
Dfree8
32
Dfree8
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Example M 5
0 1 2 0 2 4 0 3 1 0 4 3 1 2 3 1 3 0 1
4 2 1 0 4 2 3 4 2 4 1 2 0 3 2 1 0 3 4 0
3 0 2 3 1 4 3 2 1 4 0 1 4 1 3 4 2 0
4 3 2
4 users ? 2 active D 2 n D 1 R
2log25/15 RTDMA 2log25/20
Codewords for user 4
34
Non-coherent detection Quadrature receiver
using correlators
()2
cos2?fit
?
sin2?fit
()2
35
Code parameters (1)

• It can be shown that
• Q1 when do we achieve equality?
• Q2 if not, what is the upperbound
• References
• -Ian Blake, Permutation codes for discrete
  channels (1975, IT)
• -P. Frankl and M. Deza, On the max. of
  Permutations with
• given Max. Or Min. Distance (19977, Jrnl of
  Comb. Th.)

36
Code parameters (2)

• Simple code constructions D 2, 3, M
• all cases M lt 7 solved
• Interesting cases left ?

D 2 3 4 5 6 7 6 x x x 18
x C 18 is the Klöve (2000) result 7 x x
? ? x x proof that the max code size is
18 8 x x ? ? x x for D 5, M
6. 9 x x ? ? ? X M! 720
possible code words 10 x x ? ? ? ?
37
Code constructions

• D M
• D 2
• D 3
• D M-1
• others

38
Upperbound on cardinality

• Order codewords specified by set of M-D symbols
• Set 1 1, 2, x, x, x 2, 1, x, x, x
• Set 2 x, 1, x, 2, x x, 2, x, 1, x
• Set 3 x. 1, 2, x, x x, 2, 1, x, x
• etc.
• For distance D, set constains ? (M-D)! D
  codewords
• There are different sets.
• Hence

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Simple codes D M

• Cyclic permutation of M integers has D M
• C M! / (M-1)! M
• Example 1 2 3
• 3 1 2
• 2 3 1

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Simple codes D 2

• The code with all M! permutation has D 2
• C M! / (2-1)! M!
• Proof
• All symbols are different
• Codewords differ in at least 2 positions
• x x a x x x b x
• x x b x x x a x

41
Simple codes D M-1

• Construction for prime P example M 3
• - starting sets B 0 1 2
• 2B 0 2 1
• - add constant vector
• B 0 ? 0 1 2 2B 0 ? 0 2 1
• B 1 ? 1 2 0 2B 1 ? 1 0 2
• B 2 ? 2 0 1 2B 2 ? 2 1 0
• In general C M!/(M-2)! M(M-1)

2
3
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Codes for D 3

• The class of M! Permutations can be divided into
• M! / 2 ODD and M! / 2 EVEN permutations
• for the EVEN permutations D 3
• Example
• 1 2 3 4
• 2 1 4 3
• 3 4 1 2
• 1 2 3 4

even of crossings
43
Codes for D 3 (alternative proof)

• Take all M! Permutations
• Remove distance 2 code words
• fix 2 positions and throw away all codewords at
  distance 2 (needs more explanation)

44
Some special optimum cases

• The Mathieu groups
• M11 D 8 C 7920
• M12 D 8 C 95040
• Codes with distance D M-2 exist for
• M 1 pn C pn(1pn)(pn-1)

45
Mapping results

• Distance increasing
• Distance conservative

46
Distance increased mapping results

• K 4 (16 symbols)
• D 1 M 5 use M 4 and extend
• D 2 M 6 start with D 5, C 18
• K 5 (32 symbols)
• D 2 M 7 use M 6 and extend
• K 6 (64 symbols)
• D 2 M 8 use M 7 and extend

47
Distance increased mapping

• Idea use permutation code larger distance
• less code words
• Example M 4 D 3 C 12
• 000 001 010 011 100 101 110 111
• 1234 1342 1423 3241 4132 2314 2431 2143
• 011 2 1 1 0 3 2 2 1
• 3241 3 3 4 0 4 4 3 3

48
Extension to larger M

• Idea
• 00 01 10 11
• 231 213 132 123 D1
• 000 001 010 011 100 101 110 111
• 4231 4213 4132 4123 3241 3214 3142 3124
• exchange 3 and 4 to increase the distance

49
Communication structure
k
n
Convolutional encoder
mapping
M-tuples
encoding structure
R
Y
Non-coherent demodulation
ML-decoding
estimate
decoding structure
50
Table M 4 ( 4! 24 Words )

• 0000 0001 0010 0011 0100 0101 0110 0111
• 1234 1243 1324 1342 1423 1432 2134 2143
• 1000 1001 1010 1011 1100 1101 1110 1111
• 3214 3241 2314 2341 3421 3412 3124 3142
• Distance conservative mapping! (no increase in D)
• Construction look for smallest M sucht that 2k ?
  M!

51
Non-coherent detection performance

• Decoder outputs sequence at minimum distance
• Error if noise generates valid sequence
• Advantage time and frequency diversity
• robusts against Broad- and narrowband noise

1 0 0 0
1 0 1 0
Detect Presence of code sequence
0 0 1 0
0 0 1 0
0 1 0 0
1 1 1 1
0 0 0 1
0 0 1 1
52
Performance minimum distance decoding

• NOTE Sequences have minimum distance D
• Errors agree with sequences in only 1 positions
• gt D-1 errors are needed to create another
  sequence
• If E correct symbols disappear due to background
  noise
• gt D-1-E errors are needed to create decoding
  errors

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Channel capacity M-FSK multi-access (cont)

• Sender 1 1,2, ? ? ?, M
• Sender 2 1,2, ? ? ?, M
• ? ? ? Y 2M -1
• Sender N 1,2, ? ? ?, M
  output set of input frequencies
• Example for M 3 input 1, 2, 3 output
  (1), (2), (3), (1,3), (1,2), (2,3), (1,2,3)
• SUM CAPACITY ? M-1 bits/transm.
• RANDOM ACCESS CAPACITY ?ln2 (M-1) bits/transm
• Note Simple time sharing gives R log2 M
  bits/transm.

54
M-FSK multi-access (cont)

• Capacity obtaining group time sharing!
• User M2 M 3 (2bits/tr) M 4
  (3 bits/tr.)
• I 0 1 0 1 0 1
• I1 0 2 0 2
• I2 0 3
• Output (0),(1) (0,1),(0,2),(1,2)
  (0,1),(0,2),(0,3)
• Y (0) (0), (1,2,3)
• (0,1,2),(0,1,3),(0,2,3)

Group I
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Interference property
For minimum distance D M-1
difference C M(M-1) Maximum
interference M - D 1 agreement CONCLUSIO
N up to M-1 users uniquely detectable always
one unique position left