JOINT ITERATIVE DETECTION AND DECODING IN THE PRESENCE OF PHASE NOISE AND FREQUENCY OFFSET

Transcription

1 JOINT ITERATIVE DETECTION AND DECODING IN THE PRESENCE OF PHASE NOISE AND FREQUENCY OFFSET Alan Barbieri, Giulio Colavolpe and Giuseppe Caire Università di Parma Institut Eurecom Dipartimento di Ingegneria dell Informazione Mobile Communications Department Parco Area delle Scienze 181/A 2229 Route de Cretes Parma Italy Sophia Antipolis France IEEE International Conference on Communications (ICC 05) This work is funded by the European Space Agency, ESA-ESTEC, Noordwijk, The Netherlands 2

2 Motivations OUTLINE System model Proposed Algorithm Numerical results Conclusions 1

3 MOTIVATIONS (1/2) To find a detection algorithm for linear modulations affected by a strong time-varying phase noise and an uncompensated frequency offset The proposed algorithm should be able to work in iterative decoding schemes (such as those for turbo-codes and LDPCs) It can use soft estimations for code symbol probabilities available in iterative decoding schemes Time varying phase noise is due to the oscillator instabilities in both transmitter and receiver; this is especially true in satellite communications (where very high frequencies are used) The Tikhonov parametrization algorithm [Colavolpe et al., IEEE JSAC, 2005] is a soft-in soft-out low complexity algorithm able to cope with strong zero-mean phase noise not so robust to frequency offset (i.e., the linear component of the phase process) 2

4 MOTIVATIONS (2/2) However, constant frequency offset is usually present due to the mismatch between trasmitter and receiver oscillators average frequencies, or due to the Doppler effect. Classical algorithms for frequency estimation (e.g., the Rife and Boorstyn algorithm), usually require large preambles and a considerable computational complexity in order to make a correct estimation; moreover, they may fail due to the time-varying nature of the phase We would like to obtain a new (possibly low-complexity) algorithm, robust to both phase noise and frequency offset, using a mathematical derivation similar to the one of the Tikhonov algorithm 3

5 SYSTEM MODEL (1/2) A sequence of M-ary code symbols {c k }, obtained from the encoding of a sequence of information bits and a proper mapping on a multilevel constellation C, is transmitted from epoch 0 to epoch K 1 To avoid phase ambiguity problems, pilot symbols must be also inserted in the sequence {c k } The entire sequence is denoted in vector notation as c = (c 0, c 1,..., c K 1 ) 4

6 SYSTEM MODEL (2/2) We will focus on linear modulations on a channel characterized by a time-varying phase θ k and an uncompensated frequency offset ν; the discrete-time model (true in the absence of strong phase variations, which may cause ISI) becomes where (Wiener model) r k = c k e jθ k + w k θ k = θ k 1 + 2πνT + k k i.i.d. Gaussian increments with zero mean and standard deviation σ, ν frequency offset uniformly distributed in [ ν 0, ν 0 ] and w k are the zero-mean white noise samples with variance σ 2. 5

7 PROPOSED ALGORITHM (1/9) The proposed algorithm is based upon a Bayesian approach, in which the unknown parameters are treated as random variables, and uses the FG/SP framework An overall factor graph taking into account both the code constraints and the channel behavior is built Variable nodes representing the channel parameters are explicitly introduced in the FG For the computation of the marginal pmfs P (c k r), the average over channel parameters of the joint conditional distribution of c, θ and ν is performed by the SP algorithm 6

8 PROPOSED ALGORITHM (2/9) Having defined φ = 2πνT, the joint a posteriori distribution of symbols and unknown parameters may be expressed as (for the Wiener model) where p(c, θ, φ r) p(r c, θ)χ(c)p(θ φ) = χ(c)p(θ 0 ) χ(c)p(θ 0 ) f k (c k, θ k ) = exp 1 K 1 Πk=0 p(r k c k, θ k ) K 1 Πk=0 f k(c k, θ k ) 2σ 2 r k c k e jθ k 2 p(θ 0 ) = 1 2π, θ 0 [0, 2π) p(θ k θ k 1, φ) = p (θ k θ k 1 φ) = K 1 K 1 Πk=1 p(θ k θ k 1, φ) Πk=1 p(θ k θ k 1, φ) 1 e (θ k θ k 1 φ)2 2σ 2 2πσ 2 7

9 PROPOSED ALGORITHM (3/9) Code constraints, χ(c) c k 1 c k c k p(r k 1 c k 1, θ k 1 ) p(r k+1 c k+1, θ k+1 ) p(r k c k, θ k )... θ k 1 θ k θ k+1... p(θ k 1 θ k 2, φ) p(θ k θ k 1, φ) φ p(θ k+1 θ k, φ) Corresponding FG. It contains a lot of cycles with girth 4 the SP algorithm does not converge. p(φ) 8

10 PROPOSED ALGORITHM (4/9) Modified FG (obtained by clustering φ in each θ k node). The application of the SP algorithm allows the computation of the marginals P (c k r) MAP symbol detection is performed Code constraints, χ(c) c k 1 c k c k+1... P u (c k ) P d (c k )... p(r k 1 c k 1, θ k 1 ) p(r k+1 c k+1, θ k+1 ) p(r k c k, θ k ) p d (θ k ) µ k 1 µ k µ k+1 p b (µ k ) p b (µ k+1 ) p(θ k 1 θ k 2, φ)p(φ) p f (µ k 1 ) p f (µ k ) p(θ k θ k 1, φ)p(φ) p(θ k+1 θ k, φ)p(φ) 9

11 PROPOSED ALGORITHM (5/9) We defined µ k = (θ k, φ) as the state of our system Some messages are functions of two continuous variable (θ k and φ); three possible approaches to handle this circumstance: Discretization of both θ k and φ extremely high complexity Canonical distributions we represent the pdfs p f,k and p b,k with given canonical pdfs, described by some parameters Hybrid approach: φ is discretized (i.e., it is constrained to assume values in a finite set), whereas the distribution of θ k, for a given value of φ, is a Tikhonov distribution We follow the last approach; applying Bayes rule p f,k (θ k, φ) = p(θ k r0 k 1, φ = φ (l) )P (φ = φ (l) r0 k 1 ) where φ (l), l = 0,..., L 1 are the L discretization levels of φ and p(θ k r k 1 0, φ = φ (l) ) = e Re [ a (l) f,k e jθ k ] 10

12 PROPOSED ALGORITHM (6/9) Since the pdfs p f,k and p b,k, for any given φ, are constrained to be Tikhonov distribution, we finally obtain [ p f,k (θ k, φ) = γ f,ke (l) Re a (l) ] f,k e jθ k where we have denoted γ (l) f,k = P (φ = φ (l) r k 1 0 ) and a (l) f,k are complex coefficents With this hypothesis, the messages p f,k are perfectly known when the coefficents a (l) f,k and γ (l) f,k are known In the following we will derive simple forward and backward recursions for the coefficents a (l) f,k, γ (l) f,k and a (l) b,k, γ b,k, (l) for each l = 0,..., L 1 11

13 PROPOSED ALGORITHM (7/9) Some approximations are needed in order to derive the algorithm each message p d (θ k ) is approximated as a Tikhonov pdf t(u k ; θ k ) where r k αk u k = 2 2σ 2 + β k α k 2 and α k and β k are the first and second order moments of symbol c k respectively Tikhonov-Gaussian convolution approximation: t(z; x)g(x, σ 2 z ; y)dx t( 1 + σ z 2 ; y) (exact for σ = 0, i.e., constant phase) 12

14 PROPOSED ALGORITHM (8/9) Applying the SP algorithm, we finally find these simple recursive expressions a (l) f,k+1 = a(l) f,k +u k 1+σ 2 a(l) f,k +u k ejφ(l) γ (l) f,k+1 = γ (l) f,k exp { a (l) f,k + u k a (l) f,k with a (l) f,0 = 0, γ (l) f,0 = const } l = 0,..., L 1 Similarly, we find a couple of analogous expressions for the backward recursion The soft-information is finally evaluated through P u,k (c k ) e c k 2 2σ 2 l ( γ f,kγ (l) (l) I 0 a (l) f,k + a (l) ( b,k I 0 a (l) f,k b,k + r kc k ) I0 ( a (l) b,k σ 2 ) ) 13

15 PROPOSED ALGORITHM (9/9) The resulting algorithm consists of rather simple expressions Its complexity is related to the number of discretization levels L and the frame size Concluding remarks: The number and position of pilot symbols are extremely relevant (e.g., 1 pilot every M information symbols leads to an unresolvable ambiguity in frequency resolution of 1 M ) The number and position of discretization levels can be anything; moreover, they can be dynamically changed between different iterations in order to reduce complexity and increase convergence speed 14

16 NUMERICAL RESULTS (1/2) BER known phase proposed NSM, L 0 =256 RB, P=64 RB, P=128 RB, P=256 Kay, P= E b /N 0 (3,6)-regular LDPC code with codewords of length iterations QPSK modulation 1 pilot every 20 code symbols DVB/S2 phase noise model Normalized frequency uniformly distributed in ±2% P is the preamble size (unnecessary for the proposed algorithm) L=11 at the first iteration, L=3 from the second ahead 15

17 NUMERICAL RESULTS (2/2) BER known phase Tikhonov, ν 0 T=0 proposed RB, P=64 RB, P=128 RB, P=256 R=4/5 LDPC code with codewords of length mapped over a 32-APSK constellation (as in the DVB/S2 standard) 40 iterations 1 pilot every 40 code symbols DVB/S2 phase noise model E b /N 0 Normalized frequency uniformly distributed in ±1% L=11 at the first iteration, L=3 from the second ahead 16

18 CONCLUSIONS The problem of iterative detection and decoding of linear modulations over a channel affected by a random time-varying phase and a constant unknown frequency offset has been considered A factor graph taking into account both the code constraints and the channel behaviour was built The proposed algorithm may be seen as an extension of the Tikhonov algorithm, which does not work very well in the presence of a large frequency offset With respect to classical solutions, the proposed algorithm does not require long preambles, exhibits a practically optimal performance and an affordable complexity 17

19 That s all Thank you for your attention 18