The BICM Capacity of Coherent Continuous-Phase Frequency Shift Keying

Transcription

1 The BICM Capacity of Coherent Continuous-Phase Frequency Shift Keying Rohit Iyer Seshadri 1 Shi Cheng 1 Matthew C. Valenti 1 1 Lane Department of Computer Science and Electrical Engineering West Virginia University October 2, 2007 Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 1 / 21Uni

2 Outline 1 Why Use CPFSK? 2 Coherent Detection 3 Designing a Coded CPFSK System under Bandwidth Constraints 4 Conclusion Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 2 / 21Uni

3 Why Use CPFSK? Bandwidth Efficiency Orthogonal versus Nonorthogonal FSK x(t) = 2Es e j 2πat Ts, a = 0, 1,, M 1 T s Orthogonal FSK Known to achieve Gaussian capacity when the number of tones M goes to infinity. Poor bandwidth efficiency because adjacent frequency tones are at least 1/T S apart. Nonorthogonal CPFSK Saves bandwidth by using modulation index h < 1. Adjacent frequency tones are h/t S apart. Continuous-phase constraint resulting from the accumulated phase φ controls the spectrum. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 3 / 21Uni

4 Why Use CPFSK? Bandwidth Efficiency Orthogonal versus Nonorthogonal FSK x(t) = ( 2Es e j2πh T s φ k + at Ts ), a = 0, 1,, M 1 Orthogonal FSK Known to achieve Gaussian capacity when the number of tones M goes to infinity. Poor bandwidth efficiency because adjacent frequency tones are at least 1/T S apart. Nonorthogonal CPFSK Saves bandwidth by using modulation index h < 1. Adjacent frequency tones are h/t S apart. Continuous-phase constraint resulting from the accumulated phase φ controls the spectrum. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 3 / 21Uni

5 Why Use CPFSK? Bandwidth Efficiency Double-Sided, Normalized 99% CPFSK Bandwidth q=64 3 q=32 q=16 Bandwidth B (Hz/bps) q=8 q=2 q=4 99% Power Bandwidth B = 2B 99 T b. B increases as h and/or M increase h (modulation index) Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 4 / 21Uni

6 Coherent Detection Soft-Output Coherent Detector Noisy CPFSK Signal r(t) = 2Es T s e j2πϕ(t) + n(t) ϕ(t) can be written as ϕ(t) = aht + φ k + T }{{} s }{{}}{{} θ Accumulated phase Channel phase rotation Phase due to a φ k = k 1 i=0 a i mod p h. h = m h /p h. n(t) is AWGN. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 5 / 21Uni

7 Coherent Detection Soft-Output Coherent Detector Trellis Representation for CPFSK a MM x Delay φ k CPE CPFSK can be decomposed in to a CPE and MM. CPE is a rate 1/2 recursive systematic convolutional code over mod p h. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 6 / 21Uni

8 Coherent Detection Soft-Output Coherent Detector Trellis Representation for CPFSK 1 0 Tb 2Tb 3Tb 4Tb Shown is the trellis for MSK (binary CPFSK, h = 1/2). Trellis has p h states, with M branches emerging from each state. Current state is defined as S k = {φ k }. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 6 / 21Uni

9 Coherent Detection Generating Soft-Outputs Soft-Output Coherent Detector Soft-outputs (LLRs) z are obtained using the BCJR algorithm: z k = log P [b k = 1 r] P [b k = 0 r] = log α k(s )γ S(1) k+1 (s, s)β k+1 (s) S (0) α k(s )γ k+1 (s, s)β k+1 (s) α, β are calculated recursively during forward and backward sweeps through the trellis respectively. The branch probabilities are γ k+1 (s, s) = P [ r(t) (S k S k+1 ) = (s s) ] = exp (R{ρ s s}/n 0 ) ρ s s = (k+1)t kt r(t)x (s s) dt x (s s) is the CPFSK signal corresponding to the state transition (S k S k+1 ) = (s s). Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 7 / 21Uni

10 System Model Bit-Interleaved Coded Modulation Encoder b Bit-Interleaver Mapper + Channel Detector Deinterleaver Modulator z Decoder Concatenate a rate r binary code and M-ary modulator using a bit-interleaver. Simplifies design since code alphabet and modulation alphabet need not match. Off-the-shelf capacity-approaching codes are predominantly binary. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 8 / 21Uni

11 System Model Bit-Interleaved Coded Modulation Channel 1 Encoder b Channel 2 z Decoder Channel log 2M Concatenate a rate r binary code and M-ary modulator using a bit-interleaver. Simplifies design since code alphabet and modulation alphabet need not match. Off-the-shelf capacity-approaching codes are predominantly binary. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 8 / 21Uni

12 The Optimization Problem Problem Statement What is the most energy efficient (optimum) combination of M, h, constellation mapping and r for a desired spectral efficiency of η = r/b bps/ Hz? Pick the set (M, h, mapping, and r) that minimizes the E b /N 0 required to signal at some arbitrarily low error rate. It is not clear the coding gain due to a lower r is sufficient to overcome the performance loss caused by scaling M and h. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 9 / 21Uni

13 The Optimization Problem Problem Statement 10 0 BER 10-1 M =2, h =1/8, 1REC, r =1/ Uncoded MSK E /N in db b 0 η = 1/1.2 bps/hz. Uncoded MSK. M = 2, h = 1/8, rate 1/2 cdma 2000 turbo code. Coded system performs worse than the uncoded system. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and ElectricalOctober Engineering 2, 2007 West Virginia 9 / 21Uni

14 Optimization Cost Function The Optimization Problem Cost function is obtained from the capacity, under constraints of the choice of modulation, channel and detector. Information theoretic threshold on E b /N 0 required for reliable signaling. Advantages of a capacity-based optimization: Fundamental performance limit for a coded system. Practical indicator of system performance. Quantifies the trade off between CPM parameters and code rate. Requires significantly less simulation time to compute relative to BER simulations using capacity-approaching codes. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 10 / 21Uni

15 Capacity Calculation Capacity C = Eib [( ; z)] b C = Eib [( ; z )] Encoder z Decoder C log 2 M C log 2 M = i = 1 C i Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 11 / 21Uni

16 Capacity Calculation Capacity i(b; z) is the mutual information random variable variable i(b; z) = log 2 + log P [b z] Capacity of the i th parallel channel is the average mutual information C i = E[i(b i ; z i )] = log 2 + E[log P [b i z i ]] Since the capacities of the parallel channels add, C = log 2 M i=1 The capacity in bits per channel use is log 2 + E[log P [b i z i ]] C = log 2 M 1 log 2 M E[log P [b i z i ]] log(2) i=0 Expectation is most conveniently evaluated using Monte-Carlo simulations. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 12 / 21Uni

17 Capacity BICM Capacity of Coherent Binary CPFSK in AWGN h =3/4 25 Capacity (bits per channel use) h =1/2 h =1/5 h =1/10 minimum Eb/N0 in db h =1/10 h =1/ h =1/2 h =3/ Es/N0 in db code rate r (a) capacity versus E s/n 0 (b) minimum E b /N 0 versus code rate Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 13 / 21Uni

18 Capacity Capacity as a Practical Performance Indicator N =762 BER MSK N =1146 N =4602 N =12282 BER simulations using a rate 1/2 cdma 2000 turbo code E b /N 0 in db Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 14 / 21Uni

19 Optimization under Bandwidth Constraints Identify the Minimum Code Rates for some η 1 minimum code rate r' CPFSK 8 CPFSK 4 CPFSK 2 CPFSK h (c) r versus h for different M at η = 1 bps/hz Minimum code rate for some (M, h) is r = ηb. Range of allowable code rates is r [r, 1]. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 15 / 21Uni

20 Optimization under Bandwidth Constraints Generate E b /N 0 versus r Capacity Curves h =1/10 Permissible region 12 minimum Eb/N0 in db Non-permissible region h =1/7 h =1/5 h =1/4 h =3/ code rate r Generate E b /N 0 versus r curves. Identify h that minimizes E b /N 0 over r [r, 1]. (d) E b /N 0 versus r for 2 CPFSK at h = {1/10, 1/7, 1/5, 1/4, 3/10}. Also shown is minimum allowable r for η = 1 bps/hz. h = 1/4 yields the lowest information theoretic E b /N 0. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 16 / 21Uni

21 Optimization under Bandwidth Constraints Identify M, h with the minimum E b /N 0 for some η minimum Eb/N0 in db CPFSK 2 CPFSK 4 CPFSK 8 CPFSK h (e) E b /N 0 as a function of h at different M and natural mapping for η = 1 bps/hz. Pick M, h with the lowest information theoretic E b /N 0. Repeat the search for all considered mapping. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 17 / 21Uni

22 Optimization under Bandwidth Constraints Minimum Required E b /N 0 as a function of h at different η minimum Eb/N0 in db Natural Gray 2-CPFSK 4-CPFSK 8-CPFSK 16-CPFSK spectral efficiency in bps/hz Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 18 / 21Uni

23 Optimum h at different η Optimization under Bandwidth Constraints Natural Gray 2-CPFSK 4-CPFSK 8-CPFSK 16-CPFSK 0.6 h spectral efficiency in bps/hz Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 19 / 21Uni

24 Optimum r as a function of η Optimization under Bandwidth Constraints optimum code rate Natural Gray 2-CPFSK 4-CPFSK 8-CPFSK 16-CPFSK spectral efficiency in bps/hz Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 20 / 21Uni

25 Conclusion Conclusion This paper outlines a methodology for finding the optimum combination of coded CPFSK parameters for a BICM framework. Cost function driving the optimization is the information theoretic limit on E b /N 0 derived from the constrained capacity. Capacity is a practical indicator of system performance due to the availability of off-the-shelf capacity-approaching codes. At smaller spectral efficiencies, it is better to use larger M and h. At higher values of η, there is little benefit of using M > 4. Gray labelling is preferable at high spectral efficiencies, whereas natural labelling is better at lower spectral efficiencies. Iyer Seshadri et al. The ( Lane BICMDepartment Capacity of of Coherent Computer CPFSK Science and Electrical October Engineering 2, 2007 West Virginia 21 / 21Uni