A Capacity-Based Search for Energy and Bandwidth Efficient Bit-Interleaved Coded Noncoherent GFSK

1. A Capacity-Based Search forEnergy 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, mvalenti @csee.wvu.edu

2. Problem “ Which is the optimal combination of channel coding rate and continuous phase modulation (CPM) parameters for a given bandwidth efficiency and decoder complexity?”

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

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

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

6. An Uncoded System withGaussian 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

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 • Smallervalues of BgT result in a more compact power spectrum • Here M =2 and h =0.5 • 2B99Tb quantifies the bandwidth efficiency

8. Coded GFSK System • Suppose we need 2B99Tb =1.04 while using a rate ½ • code , • The value of h needs to be lowered, with BgT • unchanged • OR • The value of BgT needs to lowered, with h unchanged • OR • 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. 2B99Tbshould 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

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

10. System Model • 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) +

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

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 Zcan are arranged in a vector z, such that

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]

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

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/Nofor C =Rclog2M min{Eb/No} =min{Es/No}/Rclog2M

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

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

18. Combination of Code Rates and GFSK Parameters in Rayleigh Fading

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 = 12.93 dB Information theoretic threshold = 11.99 dB Coding gain =16 dB (at BER =10-5)

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

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