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

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


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


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?


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


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


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.


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


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


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→


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→


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


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


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


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



• 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


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


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


Observations

μ1 μ2 μ3

σ1

σ2 σ3


Observations

μ1 μ2 μ3

σ1 σ2 σ3


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


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



• Substitute the equation for threshold x

e

o

C

C

zz

dxzzp

dxxpep

Z

Xx

e

,E2

1

2

12

1



• 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


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~?


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



• 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



• 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


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


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


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


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?


Appendix

• Details of the QF-CGRV derivation


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


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


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



• 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


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,


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



• 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



• 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


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

Documents

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