cores group meeting 5/2009
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CORES Group Meeting 5/2009. Jared Dulmage Dr. Danijela Cabric. Outline. Background Goals Literature background References Solutions Analysis Details Current effort. Background. - PowerPoint PPT PresentationTRANSCRIPT
U n W i R e D L a bUCLA Wireless Research and Development
CORES Group Meeting 5/2009
Jared Dulmage
Dr. Danijela Cabric
Slide 2 U n W i R e D L a bUCLA Wireless Research and Development
Outline
• Background• Goals• Literature background• References• Solutions• Analysis Details• Current effort
Slide 3 U n W i R e D L a bUCLA Wireless Research and Development
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
Slide 4 U n W i R e D L a bUCLA Wireless Research and Development
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?
Slide 5 U n W i R e D L a bUCLA Wireless Research and Development
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
Slide 6 U n W i R e D L a bUCLA Wireless Research and Development
Literature Background
• Summary of problem characteristics covered in a selection of the literature
Reference MM CF CM FBL APE
1 X X X
2 X
3 X X X
4 X X X X
5 X X X
6 X X X
7 X X
8 X X
10 X X X X
Slide 7 U n W i R e D L a bUCLA Wireless Research and Development
References
1) Gideon Kaplan and Shlomo Shamai. “Achievable Performance over the Correlated Rician Channel.” IEEE Trans. On Comm., vol. 42, no. 11, Nov. 1994.
2) P. Piantanida, G. Matz, and P. Duhamel. “Estimation-induced outage capacity of Ricean Channels.” SIGPROC Advances in Wireless Comms, p. 1-5, July 2006.
3) 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.]
4) 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.
5) 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.
6) 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.
7) 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.
8) 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.
9) 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.
10) 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.
Slide 8 U n W i R e D L a bUCLA Wireless Research and Development
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
Slide 9 U n W i R e D L a bUCLA Wireless Research and Development
Analysis Summary
1. 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)
2. Determine the PEP conditioned on the pair of packets considered, the channel covariance, and packet structure (wideband/narrowband, pilot pattern, modulation order, etc.)
3. Determine the distribution of PEP over all pairs of packets
4. Find the expected (average) PEP by using 2 and 3
Slide 10 U n W i R e D L a bUCLA Wireless Research and Development
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
TkiT
n
j
k
i
n
hh
x
x
x
11
th
11
,
carrier-sub j on symbol
ICI
ICI
,
packet observed
hhhh
X
X0
0X
X
nXhy
time→ freq→
time→ freq→
Slide 11 U n W i R e D L a bUCLA Wireless Research and Development
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)
period symboldelay, tap ,multipaths channel
ationautocorrel time/freqR,R
0,R,R
,R0,R
,
th
2
1
0
s
TkjL
s
i
n
n0
TL
eiTki
iki
kii
s
T
TT
TT
Ch
Δtime→ Δfreq→
Slide 12 U n W i R e D L a bUCLA Wireless Research and Development
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
oeeo
cccc
eo
cccc
cc
oe
eo
for PEP)p(
)p(||
1
,
CCep
Slide 13 U n W i R e D L a bUCLA Wireless Research and Development
PEP Notation
• Split observation, channel, symbols into data (length N) and pilot portions (length P)
• Block decompose channel covariance matrix
p
d
p
d
p
d
h
hh
X
XX
y
yy
nXhy
,,
channels data/pilot of xcov
syms (pilot)data at channel of cov
E
dp
d(p)
ppd
dpdh
C
C
CC
CChhC H
Slide 14 U n W i R e D L a bUCLA Wireless Research and Development
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)
hhhAhh
nhXhXyv
p
ooo
ˆ~,ˆ
~ˆ
Slide 15 U n W i R e D L a bUCLA Wireless Research and Development
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
oeeo
o
eoee
XX,XXX
hXv,vv
nhXhXhXyv
~ˆ~~~
ˆˆ
Slide 16 U n W i R e D L a bUCLA Wireless Research and Development
Packet Error Rate (PER) computation
• Pair-wise error probability can be written as a quadratic form of complex Gaussian random vectors (QF-CGRV)
AA
CvvC
CC
CCCQ
v
vz
CCQzz
vvcc
hHxxx
1e
1e
1e
1o
1eo
eoH
eoeo
oft determinan
fE
,~
lnp
pp
x
Slide 17 U n W i R e D L a bUCLA Wireless Research and Development
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|)
NRRC
zC
QCΛ
Hh
z
z
ofmatrix covariance
..1,0
1
nk
c
k
n
kjj jk
kk
0
pp
xec
x
k
kxk
Λ
Heo Qzzcc
Slide 18 U n W i R e D L a bUCLA Wireless Research and Development
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 x vary 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
Slide 19 U n W i R e D L a bUCLA Wireless Research and Development
Observations
μ1 μ2 μ3
σ1
σ2 σ3
Slide 20 U n W i R e D L a bUCLA Wireless Research and Development
Observations
μ1 μ2 μ3
σ1 σ2 σ3
Slide 21 U n W i R e D L a bUCLA Wireless Research and Development
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
0,pdf22 2
2
xex
x x
Slide 22 U n W i R e D L a bUCLA Wireless Research and Development
Potential Simplifications
• Assume =±1 and c=±.5 (upper bound of union bound) for all co & ce
)pmf( ing approximat pdf continuousp
valuesame withscount termN,||
Npmf
p2
1pmf
2
1
||
1
2
1
||
1
)p(||
1
,,
,
xx
xxx
x
dxxeex
ece
p
X
Xx
x
x
xx
e
C
CC
C
CC
C
oe
eo
oe
eo
oe
eo
cccc
cccc
cccc
eo cc
Slide 23 U n W i R e D L a bUCLA Wireless Research and Development
Potential Simplifications
• Substitute the equation for threshold x
e
o
C
C
zz
dxzzp
dxxpep
Z
Xx
e
,E2
1
2
12
1
Slide 24 U n W i R e D L a bUCLA Wireless Research and Development
Potential Simplifications
• Co is 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 ≈ Cd for good channel estimate
– Θ = constant matrix
• Θ ≈ 0 for good channel estimate
H
Ho
H
oe
Hooo
eo
XΨX
XΘXXΨXE
ECC
IΦXXC
CC
~~
~2
~~
2
n
z
EC 1o
11E2
1E
2
1zpe
Slide 25 U n W i R e D L a bUCLA Wireless Research and Development
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~?
Slide 26 U n W i R e D L a bUCLA Wireless Research and Development
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 1i1 … ikn,
kn 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
)()(
1 111 SSS
A
ASSS HH
k
k
i
k
ii
k
j i
i
ink
j
j
Slide 27 U n W i R e D L a bUCLA Wireless Research and Development
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~)}
1
0
fE2
1 n
ie ip
indices )rank( ofout of sets
)rank(1:,,)T(
10f,f
1
)T(
E
E
EC 1o
n
iiin
r
jn
n jj
Slide 28 U n W i R e D L a bUCLA Wireless Research and Development
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)
22
2
,
2222
2
2
1
pdf2
1pmf
2
1
)p(||
1
edxee
dxxexe
p
xxx
x
x
x
e
oe
eocc
cceo cc
CC
Rayleigh pdf
Slide 29 U n W i R e D L a bUCLA Wireless Research and Development
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
ECBBCC
XXAACC
CC
oH
ze
HHho
eo
2
ln
n
x
0
pp
xec
x
k
kxk
Λ
Heo Qzzcc
Slide 30 U n W i R e D L a bUCLA Wireless Research and Development
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
Slide 31 U n W i R e D L a bUCLA Wireless Research and Development
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
Slide 32 U n W i R e D L a bUCLA Wireless Research and Development
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?
Slide 33 U n W i R e D L a bUCLA Wireless Research and Development
Appendix
• Details of the QF-CGRV derivation
Slide 34 U n W i R e D L a bUCLA Wireless Research and Development
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)
oeeo
o
ccc
ceo
ccc
ceo
cccc
cccc
c
cccco
oe
eo
oe
e
for PEP)p(
of weight hammingd,d:)(
1 of prob. prioria
)p(1
)p(
o
hh
)(ddd (d)oe
dd
pfreefree
C
C
CC CC C
Slide 35 U n W i R e D L a bUCLA Wireless Research and Development
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
matrix nrank ,
~
2
MMN
AX0
AXXR
NRRCC
oo
Hhz
n
hHxxx
1e
1e
1e
1o
1e
oe
o
Heo
CvvC
CC
CCCQ
vv
vz
Qzzcc
xfE
,
pp
x
Slide 36 U n W i R e D L a bUCLA Wireless Research and Development
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?
(uncoded) 1
of weight hammingd
d:)(
priors equalfor of prob. prioria
)p(1
)p(
1o
,
free
h
h
)(ddd (d)oe
d
dd
pfreefree
cc
cc
c
cccc
o
ccc
ceo
ccc
ceo
o
oe
eo
oe
e
C
C
C
C CC C
Slide 37 U n W i R e D L a bUCLA Wireless Research and Development
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]
0:,0:
λλdiag,
0)0F(
0
)F(p
n11
kkkk
n
kjj jk
kk
xk
xk
c
xec
xec
xx
k
k
k
k
ΛΛΛΛ
QCQCΛ
Qzz
zz
Λ
ΛH
Slide 38 U n W i R e D L a bUCLA Wireless Research and Development
Computing the Union Bound
• PER depends on:
– How eigenvalues of CzQ vary with parameters
– How the threshold x depends 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)=Ek(C) D
HH
o
Ho
DDDD
CC
DDCDC
nk
kkk
1
diagonal,
Slide 39 U n W i R e D L a bUCLA Wireless Research and Development
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
Slide 40 U n W i R e D L a bUCLA Wireless Research and Development
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
xPDF)(p
ofmoment , of MGF)(M
!2
1)1(M
2
1)(p
2
1
PMF||
Np,p
2
1
||2
1
||
1
)p(||
1
d
th
0
,,
,
x
xkmxt
k
mdxxe
xx
xex
ece
p
kx
k
kxd
x
mx
xm
xx
e
C
CC
C
CC
C
oe
eo
oe
eo
oe
eo
cccc
cccc
cccc
eo cc
Slide 41 U n W i R e D L a bUCLA Wireless Research and Development
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
12
erf2
12
1
)1MGF(2
1E
2
1
2
1
2
22
2
22
ee
eee
dxex
ep
x
xxe
Slide 42 U n W i R e D L a bUCLA Wireless Research and Development
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
rikiii 1 ABA