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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
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 2 / 20
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
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 3 / 20
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
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 3 / 20
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
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 4 / 20
Motivation
CPFSK Detection
Coherent Detection
Decoding through trellish needs to be rationalPhase synchronization
Noncoherent Detection
Symbol by symbol noncoherent detection
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 5 / 20
Motivation
CPFSK Detection
Coherent Detection
Decoding through trellish needs to be rationalPhase synchronization
Noncoherent Detection
Symbol by symbol noncoherent detection
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 5 / 20
Motivation
CPFSK Detection
Coherent Detection
Decoding through trellish needs to be rationalPhase synchronization
Noncoherent Detection
Symbol by symbol noncoherent detection
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 5 / 20
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 |
)
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 6 / 20
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
]
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 7 / 20
Noncoherent CPFSK
Noncoherent CPFSK Detector Diagram
∫ ⋅dt
cos(2πf0t)
sin(2πf0t)
∫ ⋅dt
log I0( )
∫ ⋅dt
cos(2πfM-1t)
sin(2πfM-1t)
∫ ⋅dt
log I0( )
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 8 / 20
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
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 9 / 20
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
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 9 / 20
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
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 9 / 20
Capacity Analysis under Bandwidth Constraint
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
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 10 / 20
Capacity Analysis under Bandwidth Constraint
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
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 11 / 20
Capacity Analysis under Bandwidth Constraint
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
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 12 / 20
Capacity Analysis under Bandwidth Constraint
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
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 13 / 20
Capacity Analysis under Bandwidth Constraint
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
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 14 / 20
Capacity Analysis under Bandwidth Constraint
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
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 15 / 20
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
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 16 / 20
Application Coded Noncoherent CPFSK in Frequency Hopping Networks
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
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 17 / 20
Application Coded Noncoherent CPFSK in Frequency Hopping Networks
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
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 18 / 20
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.
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 19 / 20
Conclusion
Thank you
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 20 / 20