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, mvalenti @csee.wvu.edu

“ 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?” 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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) + 7/12/2006

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 7/12/2006

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 7/12/2006

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] 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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} 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006

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 7/12/2006