1. A Capacity-Based Search for Energy and Bandwidth Efficient Bit-Interleaved Coded Noncoherent GFSK
Rohit Iyer Seshadri and Matthew C. Valenti
Lane Dept. of Computer Science and Electrical Engineering
West Virginia University
iyerr,

2. “ Which is the optimal combination of channel coding rate and
Problem
“ Which is the optimal combination of channel coding rate and
continuous phase modulation (CPM) parameters for a given bandwidth efficiency and decoder complexity?”
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3. Continuous Phase Modulation
CPM is a nonlinear modulation scheme with memory
Modulation induces controlled inter symbol interference (ISI)
Well suited for bandwidth constrained systems
Phase continuity results in small spectral side lobes
Reduced adjacent channel interference
Constant envelope makes it suitable for systems with nonlinear amplifiers
CPM is characterized by the following modulation parameters
Modulation order M
Type and width of the pulse shape
Modulation index h
Different combination of these parameters result in different spectral characteristics and signal bandwidths
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4. Challenges
CPM includes an almost infinite variations on the modulated signal
Full response, partial response, GFSK, 1-REC, 2-REC, 2-RC etc..
CPM is nonlinear
Problem of finding realistic performance bounds for coded CPM systems is non-trivial
When dealing with CPM systems with bandwidth constraints, lowering the code rate does not necessarily improve the error rate
System complexity and hence the detector complexity must be kept feasible
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5. Uncoded CPM System u: data bits
a: message stream comprised of data symbols from the set { ±1, ± 3,…, ±(M-1)}
x: modulated CPM waveform
r’: signal at the output of the channel. The filter removes out-of band noise
a: symbol estimates provided by the detector
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6. An Uncoded System with Gaussian Frequency Shift Keying
Gaussian frequency shift keying (GFSK) is a widely used class of CPM
e.g. Bluetooth
Baseband GFSK signal during kT ≤ t ≤ (k+1)T
GFSK phase
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7. GFSK Pulse Shape and Uncoded Power Spectrum
The pulse shape g(t) is the response of a Gaussian filter to rectangular pulse of width T
BgT is the normalized 3 dB bandwidth of the filer
Width of the pulse shape depends on BgT
Wider the pulse, greater is the ISI
Smaller values of BgT result in a more compact power spectrum
Here M =2 and h =0.5
2B99Tb quantifies the bandwidth efficiency
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8. Coded GFSK System Suppose we need 2B99Tb =1.04 while using a rate ½
The value of h needs to be lowered, with BgT
unchanged OR
The value of BgT needs to lowered, with h unchanged
Both can be lowered
It is not immediately clear if the performance loss
caused be lowering h and/or BgT will be overcome
by the coding gain
Channel coding improves energy efficiency at the expense of bandwidth efficiency
For our system, coding must be done without bandwidth expansion, i.e. 2B99Tb should remain unchanged
Find the power spectral density for uncoded GFSK
PSD for GFSK using rate Rc code is now
must meet the required spectral efficiency
This implies the GFSK parameters have to be modified for the coded signal
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9. Proposed Coded GFSK System
Noncoherent detection used to reduce complexity
Detector: Soft-Decision differential phase detector (SDDPD), [Fonseka, 2001].
Produces hard-estimates of the modulated symbols
SO-SDDPD generates bit-wise log-likelihood ratios (LLRs) for the code bits
Bit-wise interleaving between encoder and modulator and bit-wise soft-information passed from detector to decoder (BICM)
Shannon Capacity under modulation and detector design constraints used to drive the search for the “optimum” combination of code rates and GFSK parameters at different spectral efficiencies
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10. System Model (t, a) = (t, a) +
Bit-interleaved codeword b is arranged in a
matrix B, such that
Each column of B is mapped to one of M possible symbols to produce a
The baseband GFSK x is sent through a frequency nonselective Rician channel
Received signal at the output of a frequency nonselective, Rician channel, before filtering
r’(t, a) = c(t) x(t, a) + n’(t)
Received signal after filtering
r(t, a) = c(t) x(t, a) + n(t)
Received signal phase
(t, a) =
(t, a) +
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11. SO-SDDPD
Detector finds the phase difference between successive symbol intervals
We assume that GFSK pulse shape causes adjacent symbol interference
The phase difference space from 0 to 2 is divided into R sub-regions
Detector selects the sub-region Dk in which lies
The sequence of phase regions (D0, DI, …) is sent to a branch metric calculator
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12. SO-SDDPD
Let be the phase differences corresponding to any transmitted sequence
A branch metric calculator finds the conditional probabilities
Branch metrics sent to a 4-state MAP decoder whose state transition is from to
The SO-SDDPD estimates the LLR for Bi,k
The bit-wise LLRs in Z can are arranged in a vector z, such that
Mention how it differs from sddpd-vd
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13. Capacity Under Modulation, Channel And Receiver Design Constraints
Channel capacity denotes maximum allowable data rate for reliable communication over noisy channels
In any practical system, the input distribution is constrained by the choice of modulation
Capacity is mutual information between the bit at modulator input and LLR at detector output
Constrained capacity in nats is; [Caire, 1998]
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14. Capacity Under Modulation, Channel And Receiver Design Constraints
Constrained capacity for the proposed system is now
In bits per channel use
Constrained capacity hence influenced by
Modulation parameters (M, h and BgT)
Channel
Detector design
Computed using Monte-Carlo integration
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15. Capacity Under Modulation, Channel And Receiver Design Constraints
Scenario:
BICM capacity under constraint of using the SO-SDDPD
SDDPD specifications:
R=40 uniform sub-regions for 2-GFSK
R=26 uniform sub-regions for 4-GFSK
Channel parameters:
Rayleigh fading
GFSK specifications :
M =2, h =0.7, BgT =0.25
M =4, h =0.21, BgT =0.2
Information theoretic minimum Es/No (min{Es/No }) is found by reading the value of Es/No for C =Rclog2M
min{Eb/No} =min{Es/No}/Rc log2M
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16. Capacity-Based Search for Energy and Bandwidth Efficient GFSK Parameters
The search space is over
M ={2, 4}-GFSK
Rc ={6/7, 5/6, 3/4, 2/3, 1/2, 1/3, 1/4, 1/5}
2B99Tb ={0.4, 0.6, 0.8, 0.9, 1.0, 1.2}
BgT ={0.5, 0.25, 0.2}
At each Rc, find h for each value of BgT and M, that meets a desired 2B99Tb
Find min{Eb/No} for all allowable combinations of M, h, BgT, and Rc at each 2B99Tb
At every 2B99Tb, select the GFSK parameters yielding the lowest min{Eb/No}
As an example, consider a rate-5/6 coded, {2,4}-GFSK, with SO-SPDPD based BICM in Rayleigh fading
At each 2B99Tb , there are 6 combinations of M, h and BgT
The numbers denote h values corresponding to GFSK parameters with the lowest min{Eb/No} at the particular min{Eb/No}
For 2B99Tb =1.2 , M =2, h =0.7, BgT =0.25
has the lowest min{Eb/No} with Rc =5/6
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17. Capacity-Based Search for Energy and Bandwidth Efficient GFSK Parameters
A similar search was conducted for all listed values of Rc
This gives the set of M, h, BgT with the lowest min{Eb/No} at different 2B99Tb
for each of the considered code rates
The search is now narrowed to find the combination of Rc and GFSK parameters that have the lowest min{Eb/No} for a particular bandwidth efficiency
As an example, consider SO-SPDPD based BICM in Rayleigh fading, at 2B99Tb =0.8
For the proposed system,
Rc =3/4 with M =4, h =0.25, BgT =0.5 has the best energy efficiency
at 2B99Tb =0.8
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18. Combination of Code Rates and GFSK Parameters in Rayleigh Fading
2B99Tb
Rate M
BgT h
min{Eb/No} dB
0.4
3/4 4
0.2
0.195
18.15
0.6
2/3
0.21
18.08
0.8
0.5
0.25
12.38
0.9
0.24
11.99
1.0
0.3
11.44
1.2
5/6 2
0.7
11.34
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19. Bit Error Rate Simulations
Scenario:
Bit error rate for SO-SDDPD based coded and uncoded systems
Solid curve: System without coding
Dotted curve: Systems with coding (BICM)
Channel parameters:
Rayleigh fading
GFSK specifications :
Coded: M =4, h =0.315, BgT =0.5, Rc =2/3, 2B99Tb =0.9
Uncoded: M =2, h =0.5, BgT =0.3, 2B99Tb =0.9
Simulated Eb/No required for an arbitrarily low error rate = dB
Information theoretic threshold = dB
Coding gain =16 dB (at BER =10-5)
DM1:108.8
DM3:387.2
DM5:477.9
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20. Conclusions
The Shannon capacity of BICM under modulation, channel and detector constraints is a very practical indicator of system performance
Most CPM systems are too complex to admit closed-form solution
The constrained capacity is evaluated using Monte-Carlo integration
A Soft-output, SDDPD is used for noncoherent detection of GFSK signals
For a select range of code rates, spectral efficiencies and GFSK parameters, the GFSK constrained capacities have been calculated
The constrained capacity is used to identify combination of code rates and GFSK parameters with the best energy efficiency for a desired spectral efficiency
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21. Future Work Extend the search space to include M >4
Different values of BgT
SO-SDDPD designed to account for additional ISI
More CPM formats (RC, REC etc..)
Alternative noncoherent receivers
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