the capacity of noncoherent continuous-phase frequency...

The Capacity of Noncoherent Continuous-PhaseFrequency Shift Keying

Shi Cheng1 Rohit Iyer Seshadri1 Matthew C. Valenti1 Don Torrieri2

1Lane Department of Computer Science and Electrical EngineeringWest Virginia University

2US Army Research Lab

March 14, 2007

Shi Cheng et al. ( Lane Department of Computer Science and Electrical Engineering West Virginia University, US Army Research Lab)The Capacity of Noncoherent CPFSK March 14, 2007 1 / 20

Outline

1 Motivation

2 Noncoherent CPFSK

3 Capacity Analysis under Bandwidth Constraint

4 Application

5 Conclusion


Motivation

Bandwidth Efficiency

xi (t) =1√TS

ej π(2k−(M−1))t

TS , k = 0, 1, · · · ,M − 1

Orthogonal FSK

Known to achieve Gaussian capacity when the number of tones M goesto infinityAdjacent frequency tones are at least 1/TS apart.

Nonorthogonal FSK

Nonorthogonal FSK saves the bandwidth by using modulation indexh < 1. Adjacent frequency tones are h/TS apart.Full response CPM with rectangular pulse shape, also called CPFSKPartial response CPM has more compact bandwidth, but leads tocomplex signal processing


Motivation

Bandwidth Efficiency

xi (t) =1√TS

ej“

hπ(2k−(M−1))tTS

+φ”, k = 0, 1, · · · ,M − 1

Orthogonal FSK

Known to achieve Gaussian capacity when the number of tones M goesto infinityAdjacent frequency tones are at least 1/TS apart.

Nonorthogonal FSK

Nonorthogonal FSK saves the bandwidth by using modulation indexh < 1. Adjacent frequency tones are h/TS apart.Full response CPM with rectangular pulse shape, also called CPFSKPartial response CPM has more compact bandwidth, but leads tocomplex signal processing


Motivation

Bandwidth of CPFSK

h

Ban

dwid

th (

Hz/

bps)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

M=2M=4

M=8

M=16M=32

M=64

99% Power Bandwidth


Motivation

CPFSK Detection

Coherent Detection

Decoding through trellish needs to be rationalPhase synchronization

Noncoherent Detection

Symbol by symbol noncoherent detection


Noncoherent CPFSK

Noncoherent CPFSK Discrete Time Model

y = ae jθ√Esx + n

n is colored noise, with E (nnH) = N0K

x is chosen from columns of K = [k0, k1, · · · , kM−1]

p(y|x = kν) ∝ I0(2a

√Es

N0|yν |

)


Noncoherent CPFSK

A Binary Example: Minimum Shift Keying (MSK)

MSK: M = 2, h = 1/2

The normalized correlation matrix for n is

K =

[1 −0.6366j

0.6366j 1

]The modulator selects the columns of K, depending on the M-aryinput d .

d = 0 d = 1

x =

[1

−0.6366j

]x =

[0.6366j

1

]


Noncoherent CPFSK

Noncoherent CPFSK Detector Diagram

∫ ⋅dt

cos(2πf0t)

sin(2πf0t)

∫ ⋅dt

log I0( )

∫ ⋅dt

cos(2πfM-1t)

sin(2πfM-1t)

∫ ⋅dt

log I0( )


Capacity Analysis under Bandwidth Constraint

Capacity Calculation

Instantaneous capacity

i(x; y) = log M − log

∑x′∈S p(y|x′)p(y|x)

.

Capacity

I (x; y) = log M − Ex

[log

∑x′∈S p(y|x′)p(y|x)

].

Monte Carlo simulation



Binary Noncoherent CPFSK Capacities in AWGN

−10 −5 0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Es/No in dB

Mut

ual I

nfor

mat

ion

h=0.2

h=1

h=0.4

0.6

h=0.8dashed line

(a) channel capacity versus ES/N0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

10

15

20

25

Min

imum

Eb/

No

in d

B

Code rate r

h=0.2

h=0.4

h=0.6

h=0.8

h=1

(b) minimum Eb/N0 versus coding rate



Binary Noncoherent CPFSK Capacities in AWGN underBandwidth Constraint

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

10

15

20

25

h

Min

imum

Eb/

No

in d

B

η = 0

η=1/3

η = 1/2

η = 1

η (bits/s/Hz) is the Bandwdith Efficiency



Noncoherent CPFSK Capacities in AWGN Channel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

h

Min

imum

Eb/

No

in d

Bη = 1/2 (solid lines)η = 0 (dashed lines)

M=2

4

8

16

32

64



Noncoherent CPFSK Capacities in AWGN Channel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

η in bps/Hz

Min

imum

Eb/

No

in d

B

M=64

M=2

M=4

M=8

M=16

M=32



Noncoherent CPFSK Capacities in Rayleigh Channel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

h

Min

imum

Eb/

No

in d

B

η = 1/2 (solid lines)

η = 0 (dashed lines)M=2

4

8

16

32



Noncoherent CPFSK Capacities in Rayleigh Channel

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

22

η in bps/Hz

Min

imum

Eb/

No

in d

B

M=64

M=2

M=4

M=8M=16

M=32


Application Coded Noncoherent CPFSK in Frequency Hopping Networks

FH Network Scenario

d=1

Interference Nodes(d<4)

Multiple access interference

Tradeoff on the bandwidth of each sub-channel and the possibility ofinterference

Equal received average SNR, Rayleigh fading, Interference nodessubject to log normal shadowing σ = 8dB



FH Network Scenario

Total bandwidth W = 2000/Tu

h M Coding rate Number of channels

1 4 2048/6144 315

1 8 2048/6144 244

0.46 4 2048/3456 1000

0.32 8 2048/3840 1000

All users use the same modulation and coding strategy

Channel estimator using EM algorithm, the same as the one oforthogonal FSK



Noncoherent CPFSK FH Network against MAI

Users

Eb/N

o (d

B)

0 10 20 30 40 506

8

10

12

14

16

18

20

22

24

8CPFSK h = 14CPFSK h = 18CPFSK h = 0.324CPFSK h = 0.46

SNR collected atBER = 10−4

UMTS turbocode

32 hops


Conclusion

Conclusion

This paper outlines a methodology for finding the coded modulation(CM) capacity of CPFSK with noncoherent detection.

For a specific number of tones M and spectral efficiency η, theminimum required Eb/N0 can be optimized over the modulation indexh and coding rate r .

For a fixed spectral efficiency, it is better to use a higher order M andsmaller h. However, when the spectral efficiency is sufficiently high,the benefit is small by using M > 8.


Conclusion

Thank you


the capacity of noncoherent continuous-phase frequency...

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