CORES Group Meeting 5/2009

1. CORES Group Meeting 5/2009 Jared Dulmage Dr. Danijela Cabric

2. Outline • Background • Goals • Literature background • References • Solutions • Analysis Details • Current effort

3. Background • Project funded by California Department of Transportation (CalTrans) and California Partners for Advanced Transit and Highways • Intelligent Transportation Systems (ITS) aims to improve traffic flow and auto safety by giving drivers and planners real-time information on the local and regional traffic environment • Warn of approaching emergency vehicles • Warn drivers of sudden breaking ahead • Notify drivers of impending construction zones • Allow traffic managers to monitor real-time traffic conditions • Dedicated Short-Range Communications (DSRC) covers a variety of wireless technologies that are targeted at enabling Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) communications • IEEE 802.11p is a developing standard for WiFi-like wireless for V2V and V2I • Packet structure and MAC very similar to IEEE 802.11a

4. Goals • Measure the impact of system design decisions on system adequacy for applications • Safety messages require high reliability (connectivity) and low latency (or high minimum/average throughput?) • How can we optimize certain parameters (e.g. packet length, modulation/coding, bandwidth, MAC parameters) to improve performance metrics? • Questions for physical layer analysis • How does physical layer design impact PER? • Pilot structure, bandwidth, mod/coding, channel estimation/tracking accuracy, etc. • Can we optimize or guide the physical layer design from the desired system performance parameters?

5. Literature Background • PER analysis incorporates several issues: • MM – Mis-matched decoding • ML decision metric (minimum Euclidean distance) assumes perfect CSI; estimated CSI used in its place • CF – Correlated fading (Rayleigh or Rician) • Received symbols are NOT independent • CM – Coded modulation • Whole codeword (symbol sequence in packet) is the observation • FBL – Finite block length, imperfect interleaving, discrete constellations • Information theoretic arguments do not apply • APE – Arbitrary pilot schemes and channel estimation algorithms • Cannot rely on simplifications due to specific parameters or algorithms

6. Literature Background • Summary of problem characteristics covered in a selection of the literature

7. References • Gideon Kaplan and Shlomo Shamai. “Achievable Performance over the Correlated Rician Channel.” IEEE Trans. On Comm., vol. 42, no. 11, Nov. 1994. • P. Piantanida, G. Matz, and P. Duhamel. “Estimation-induced outage capacity of Ricean Channels.” SIGPROC Advances in Wireless Comms, p. 1-5, July 2006. • S. Sadough, P. Piantanida, and P. Duhamel. “Achievable Outage Rates with Improved Decoding of BICM Multiband OFDM under Channel Estimation Errors.” Asilomar, 2006. [online: www.arXiv.org.] • Muriel Medard. “The Effect upon Channel Capacity in Wireless Communications of Perfect and Imperfect Knowledge of the Channel.” IEEE Trans. Info. Theory, vol. 46, no. 3, May 2000. • Sanjiv Nando and Kiran Rege. “Frame Error Rates for Convolutional Codes on Fading Channels and the Concept of Effective Eb/No.” IEEE Trans. On Vehicular Tech., vol. 47, no. 4, Nov. 1998. • M.P. Fitz, J. Grimm, S. Siwamogsatham. “A New View of the Performance Analysis Techniques in Correlated Rayleigh Fading.” IEEE Trans. Info. Theory, vol. 48, is. 4, p. 950-956, Apr 2002. • J. Jootar, J. Zeidler, J.G. Proakis. “Performance of Convolutional Codes with Finite-Depth Interleaving and Noisy Channel Estimates.” IEEE Trans. Comm., vol. 54, No. 10, p. 1775-1786, Oct 2006. • A. Dogandzic. “Chernoff Bounds on Pairwise Error Probabilities of Space-Time Codes.” IEEE Trans. Info. Theory, Vol. 49, No. 5, p. 1327-1336, May 2003. • S. Shamai, I Sason. “Variations on the Gallager Bounds, Connections, and Applications.” IEEE Trans. Info. Theory, vol. 48, no. 12, p. 3029-3051, Dec 2002. • J-C. Guey, M. P. Fitz, M. R. Bell, W-Y. Kuo. “Signal Design for Transmitter Diversity Wireless Communications Systems Over Rayleigh Fading Channels.” IEEE Trans. On Comm., vol. 47, no. 4, p. 527-537, Apr 1999.

8. Solutions • Simulation campaign • Cycled through wide range of system parameters and computed the metrics of interest (PER, average latency) • Gives some insight into specific channel scenarios and parameter ranges • Analytical PER • Continuing effort with several advantages • Allows arbitrary specification of parameters • Rapid generation of results for range of parameters • Clear objective function for optimization • May provide deep insight into the general relationships between parameters and performance • May prove extremely accurate over a variety of parameter settings

9. Analysis Summary • Upper bound PER as the sum of the probability of declaring an error packet when a different packet transmitted • union bound (UB) of pair-wise error probabilities (PEP) • Determine the PEP conditioned on the pair of packets considered, the channel covariance, and packet structure (wideband/narrowband, pilot pattern, modulation order, etc.) • Determine the distribution of PEP over all pairs of packets • Find the expected (average) PEP by using 2 and 3

10. System Model • Linear, frequency domain model for OFDM • X = M-ary symbols, h = channel, n = noise • Packet has n OFDM symbols with k sub-carriers • Bytes/packet B = n×k×M/8 • E.g. k=48 (IEEE 802.11), M=4 (QPSK), n=10, B=240 bytes time→ freq→ time→ freq→

11. System Model • OFDM channel time/frequency covariance matrix Ch • i = time index, k = frequency index • Covariance matrix is symmetric Toeplitz, block Toeplitz • E.g. slow, frequency-flat, Rayleigh fading • ICI=0 (in Xi on previous slide) • R(i,k) = R(i) (constant for all k) Δtime→ Δfreq→

12. Union Bound • Union bound (UB) is upper bound on packet error rate (PER) • Sum of pair-wise error probabilities (PEP) • Variables to UB-PER • co – transmitted packet (codeword) of length B bytes • ce – error packet (codeword) of length B bytes • There are many terms in pe -- O(216×B) • For 100 byte packet, there are 21600 > 10480 > googol > atoms in the universe terms • For coded modulation, restrictions on ce given co but still many terms

13. PEP Notation • Split observation, channel, symbols into data (length N) and pilot portions (length P) • Block decompose channel covariance matrix

14. PEP computation • Assume xo=m(co;M) be the transmitted M-ary symbol vector corresponding to the codeword bytes co • Define vector vo as the difference between the observation and the channel mis-matched (estimated) corrupted symbol xo • Linear estimator (A) assumed unbiased (h~ has 0-mean) • vo~ N(0,Co) : 0-mean Complex Gaussian Random Vector (CGRV) • NOTE: assume data portion unless explicitly subscripted (e.g. h=hd)

15. Packet Error Rate (PER) computation • Define ve as error vector from xo to alternative estimated received vector xe • ve~ N(e,Ce) : Complex Gaussian Random Vector (CGRV) • e = 0 for Rayleigh fading • e ≠ 0 for Rician fading

16. Packet Error Rate (PER) computation • Pair-wise error probability can be written as a quadratic form of complex Gaussian random vectors (QF-CGRV)

17. Pair-wise Error Probability (PEP) • When pair-wise error probability (PEP) argument expressed as QF-CGRV, a closed form expression has been derived [6,10] • Pilot pattern, linear estimation matrix, packet pair co and ce all incorporated into R • Result depends on eigenvalues of CzQ and threshold x=ln(|Co|/|Ce|)

18. Computing the Union Bound • Exact computation of UB is infeasible due to large number of terms • Given channel covariance Ch, linear estimation matrix A, and pilot pattern, the matrices Cz, Q, and threshold x depend only on data symbol matrices Xo and Xe(equivalently data codewords co and ce) • How do eigenvalues of CzQ vary with codewords? • How the threshold xvary with codewords? • Can we bound or approximate the eigenvalues CzQ and the threshold as they vary over the codeword pairs? • Empirical statistics of eigenvalues and threshold over many (though small subset of total) codeword pairs may suggest something • Time-varying, flat Rayleigh channel with varying Doppler • Initial PCSI for first data OFDM symbol, assumed static over remaining symbols

19. Observations μ1 μ2 μ3 σ2 σ3 σ1

20. Observations μ1 μ2 μ3 σ2 σ1 σ3

21. Observations • Test case: • Flat Rayleigh fading, time-varying with some normalized Doppler • PCSI for first received symbol • No channel tracking (i.e. initial estimate used over the whole packet) • CzQ has only 2 significant eigenvalues • Eigenvalues are anti-podal (1 = -2) • Results in coefficient ck = ±.5 for k=1,2 in PEP • Threshold x has a Rayleigh or Poisson shaped distribution • Offset μ depends on normalized Doppler spread • Variance depends on constellation

22. Potential Simplifications • Assume =±1 and c=±.5 (upper bound of union bound) for all co & ce

23. Potential Simplifications • Substitute the equation for threshold x

24. Potential Simplifications • Cois full rank (N) • Ф = constant PSD matrix • Ф ≈ 0 for good channel estimate • σn2 = noise variance • E is rank deficient (min(P,D)) • D = # non-0 in xo-xe • Max of D non-trivial factors in product • Ψ = constant PSD matrix • Ψ≈ Cdfor good channel estimate • Θ = constant matrix • Θ ≈ 0 for good channel estimate

25. Current Effort • For a fixed Xo, (Co-1E) is a random variable by factors X~ • X~ has a multinomial distribution (generalization of the binomial distribution) • Binomial distribution is equivalent to Poisson for large number of trials • Can we find a distribution for (Co-1E) based on that of X~?

26. Useful Matrix Perturbation Theory • C.-K. Li, R. Mathias. “The Lidskii-Mirsky-Wielandt Theorem – Additive and Multiplicative Versions.” Sept. 1997. • Let A be n x n Hermitian and A’=SHAS then for indices 1i1 …ikn, kn and λj(A)≠0 we have • I.e. the product of k eigenvalues of A’ the product of those of A multiplied by a factor bounded by the product of the k smallest and largest eigenvalues of SHS • Similar result for general (non-Hermitian) matrices and singular values

27. Potential Simplifications • Restate the product of sums in pe as that of the sum of products • For each product sum we know from LMW theorem ωr(X~) f(r)/f’(r) αr(X~) • f(r) ≥ 0 for all r; f(r) = 0 if r > rank(E)  min{# pilots, dH(X~)}

28. Potential Simplifications • Assuming a Rayleigh distributed threshold, there is a closed form approximation • The offset μ is the minimum threshold x over all pairs of packets • Many optimization techniques available to evaluate the minimum • The “variance” σ2 is related to the mean of the distribution E[x]=σ√(/2) Rayleigh pdf

29. What is x? • x is the threshold in the QF-CGRV PEP • Recall: • z = 2n x 1 • Cz= 2n x 2n • Ch = n x n • σn = noise std dev • X = linear channel estimation matrix • A = [I –X] = n x 2n • B = [I I] = n x 2n

30. Progress Summary • Current error rate bound analysis accounts for all variables of concern • Resulting analysis procedure will be used to evaluate PER performance for specific system implementations • Analysis will also elucidate relationships and trends of parameters on resulting performance • Bound is currently computationally impractical for interesting packet lengths (e.g. > 6 bytes = minimum packet size) • Further simplifications or bounds are being explored • Tightness of bound for use as absolute metric is still to be determined • Asymptotic behavior (i.e. PER floors) will still show sensitivity of system to particular parameter choices regardless

31. Conclusions • QF-CGRV PEP and union bound comprises a solution that accounts for all variables of concern • Without further bounds and/or simplifications, computation is impractical for even moderate packet lengths • Inherent structure in the QF-CGRV formulation and the constituent input variables offer some computational savings still being explored

32. Future Work • How do input variables generally dictate the eigenvalue distribution? • How do input variables dictate the threshold distribution? • In the specific example case, how do the input variables impact the critical parameters μ and σ? • Can we formulate a more general solution irrespective of the details of the threshold distribution?

33. Appendix • Details of the QF-CGRV derivation

34. Brief Summary • Variables to probability of error union bound (UB) • co – transmitted codeword (packet), length n, of M-ary symbols • ce – error codeword, length n, of M-ary symbols • A – linear channel estimation matrix • Ch – channel covariance matrix • Permutation of channel covariance Ch reflects pilot structure • Size n x n of Ch reflects length of codeword (n)

35. Brief Summary • Quadratic forms of Complex Gaussian Random Vectors (QF-CGRV) gives pair-wise error probability (PEP) in closed form when decision metric is a quadratic form [6,10] • Closed form PEP is a function of the eigenvalues of CzQ • Dependence of PEP on codeword pair results in the number of terms in the union bound to be exponential in the codeword length O(Mn) • Must understand how variables affect eigenvalues to reduce computations and make union bound computation both feasible and tight

36. Bounded PER computation • Combine PEP to determine overall prob. of error • Generally, p(co→c) depends on both codewords • Requires many terms: O(Mn), M=constellation size, n=codeword length • What terms dominate the summation?

37. Packet Error Rate (PER) computation • From the QF, there is a fixed relationship to the requisite error cumulative distribution function (CDF) [10] (Guey, Fitz, et. al) • Performance dictated by eigenvalues of CzQ and threshold x • Given equation valid for unique eigenvalues of CzQ (i.e. multiplicity 1) and Rayleigh fading (0 mean) • Extension to Rician fading (non-0 mean) has a related form [7]

38. Computing the Union Bound • PER depends on: • How eigenvalues of CzQ vary with parameters • How the threshold xdepends on parameters • To simplify error probability computation • Find invariants or bounds between pairs of codewords and eigenvalues • Example below • D = codeword matrix = diagonal of symbols • DDH= energy of symbols in the codeword • For equal energy (E) symbols (PSK), k(Co)=Ek(C)  D

39. Observations • Test case: • Flat rayleigh fading • PCSI for first received symbol • Channel assumed static over codeword • No tracking (i.e. first channel estimate used for whole codeword) • CzQ has only 2 significant eigenvalues • Regardless of c, co,Doppler spread, packet length • Eigenvalues are anti-podal (1 = -2) • CzQ is trace free, i.e. tr(CzQ)=k=0  z • This property makes CzQ a Lie Algebra corresponding to the special linear group sl(n;C) of nxn complex matrices

40. Potential Simplifications • Assume =±1 and c=.5 (upper bounds union bound) for all co & ce • Assume x has a Rayleigh distribution with offset μ and parameter σ2

41. Potential Simplifications • Assume =±1 and c=.5 (upper bounds union bound) for all co & ce • Assume x has a Rayleigh distribution with offset μ and parameter σ2

42. Useful Matrix Perturbation Theory • G.W. Stewart, J.-g. Sun. “Matrix Perturbation Theory.” Academic Press, 1990. • Interleave rule: Let B be a rank r = n – k principle submatrix of A (n x n) then for eigenvalues ordered in non-decreasing order λ1≥ λ2≥… ≥ λn • I.e. the ith eigenvalue of B is within a window defined by the ith and and i+kth eigenvalue of A • Smaller submatrices have wider eigenvalue windows