1. A Capacity-Based Approach for Designing Bit-Interleaved Coded GFSK with Noncoherent Detection
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)
Phase continuity results in small spectral side lobes
Well suited for bandwidth constrained systems
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, REC, 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
u: bit 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, GSM
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
BT is the normalized 3 dB bandwidth of the filer
Width of the pulse shape depends on BT
Wider the pulse, greater is the ISI
Smaller values of BT 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 BT
unchanged OR
The value of BT 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 BT 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
Spectral scaling (not shifting)
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
The availability of capacity-approaching turbo and LDPC codes make the capacity under BICM a very practical indicator of system performance
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10. System Model (t, a) = (t, a) +
Bit-interleaved codeword b is mapped to symbol sequence a, which is modulated to produce x
The baseband GFSK signal x is sent through a frequency nonselective Rician channel
Received signal at the output of the 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 code bits
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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 BT)
Channel
Detector design
Computed using Monte-Carlo integration
The constrained capacity is used to find the minimum Eb/No required for reliable signaling
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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=26 uniform sub-regions for 4-GFSK
Channel specifications:
Rayleigh
GFSK specifications :
M =4, h =0.21, BT =0.2, 2B99Tb =0.6 with Rc =2/3
min{Es/No} if found at C=Rclog2M
min{Eb/No} = min{Es/No} /C
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16. Optimum Combination of Code Rates And GFSK Parameters In An Ergodic Channel
The search space is
M ={2, 4}- GFSK
Rc ={6/7, 5/6, 3/4, 2/3, 1/2, 1/3, 1/4, 1/5}
BT ={0.5, 0.25, 0.25}
2B99Tb ={0.4, 0.6, 0.8, 0.9, 1.0, 1.2}
At a particular Rc
Find h for each value of BT and M that meets a desired 2B99Tb
Find min{Eb/No} for all allowable combinations of M, h, BT at every 2B99Tb
At each 2B99Tb, select GFSK parameters yielding the lowest min{Eb/No}
Select the combination of Rc and GFSK parameters that have the lowest min{Eb/No} at the desired 99% bandwidth
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17. Optimum Combination of Code Rates And GFSK Parameters In An Ergodic Channel
Scenario:
Information theoretic minimum Eb/No at different 2B99Tb with Rc =5/6
SDDPD specifications:
R=40 uniform sub-regions for 2-GFSK
R=26 uniform sub-regions for 4-GFSK
Channel specifications:
Rayleigh
Search specifications:
At each 2B99Tb, there are 6 combinations of M, h and BT
The numbers denote h values corresponding to GFSK parameters with the lowest min{Eb/No} at the particular bandwidth efficiency
At 2B99Tb =1.2, selecting M =2, h =0.7 and
BT =0.25 yields the lowest min{Eb/No}
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18. Optimum Combination of Code Rates And GFSK Parameters In An Ergodic Channel
Scenario:
Best GFSK parameters for various code rates at 2B99Tb =0.9
SDDPD specifications:
R=40 uniform sub-regions for 2-GFSK
R=26 uniform sub-regions for 4-GFSK
Channel specifications:
AWGN, Rayleigh
Search specifications:
The combination of code rates and GFSK parameters with lowest min{Eb/No} can be identified at the particular 2B99Tb
At 2B99Tb =0.9:
M =4, h =0.24, BT =0.5 with Rc =2/3 (Rayleigh)
M =4, h =0.285, BT =0.5 with Rc =3/4 (AWGN)
yield the best energy efficiency
Notice the trade-off between code rate and energy efficiency
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19. Combination of Code Rates And GFSK Parameters
Rayleigh Fading
2B99Tb
Rate M BT 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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20. Combination of Code Rates And GFSK Parameters
Rician Fading (K =6 dB)
2B99Tb
Rate M BT h
min{Eb/No} dB
0.4
3/4 4
0.2
0.195
15.38
0.6
5/6
0.5
0.18
11.67
0.8
0.29
9.09
0.9
0.285
8.87
1.0
2/3
0.3
8.83
1.2
6/7 2
0.25
0.76
8.39
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21. Conclusions
BICM with a soft-output SDDPD is used for noncoherent detection of GFSK signals
The Shannon capacity of BICM under modulation, channel and detector constraints is evaluated using Monte-Carlo integration
The constrained capacity is used to identify combination of code rates and GFSK parameters with the best energy efficiency and outage probability at a desired spectral efficiency
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22. Future Work Extend the search space to include M >4
Different pulse shapes and signal bandwidths
Alternative receivers
A smarter method to comb the search space
Evolutionary algorithm
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23. Performance In Block Fading
In block-fading a is broken into F blocks, which are transmitted over independent channels
Channel coefficient c(t) =c, remains constant for the entire duration of a block
Instantaneous SNR of the bth block is
When code combining is used at the receiver, the instantaneous capacity for the entire code word is
The information outage probability
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24. Optimum Combination of Code Rates And GFSK Parameters In A Block Fading Channel
Scenario:
Information outage probability with code combining in block fading at F =1 and F = 100 for SO-SDDPD based BICM at 2B99Tb =0.9
At F =1, M =4, h =0.285, BT =0.5 with Rc =3/4 has the lowest information outage probability
At F =100, M =4, h =0.24, BT =0.5 with Rc =2/3 has the lowest information outage probability
The capacity based search also helps in identifying the combination of code rates and GFSK parameters with the lowest outage probability in block fading
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