Multipath Resolution Effects in Wideband CDMA Transmission

Rodger E. ZiemerElectrical and Computer Engineering Dept.University of Colorado at Colorado SpringsColorado Springs, CO 80933

2

The Challenge

3G wideband: Mixed traffic, some of which demands wide bandwidth Finer resolution of multipath:

Wider spread bandwidth Directive antennas

Statistics/spectra of multipath: Envelope component partially specular - Ricean model? Phase distributions for tracking loops (Tikonov?) Bathtub Doppler power spectrum no longer valid

Fundamental question: Resolve more paths – power decreases per resolved

path When is additional diversity gain provided by finer path

resolution negated by phase/timing errors?

3

A Related Challenge

Where does bandwidth come from to do this finer resolution?cdma2000 hedges on this by having an RTT option that allows noncontiguous chunks of bandwidth to be used (multicarrier spread spectrum, MC-SS)Kondo & Milstein (1996) showed that for equal bandwidths, W-CDMA and MC-SS give same diversity gain under ideal conditions (maximal ratio combining, etc.)

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Well Known Diversity Result

Proakis; Diversity reception in context of RAKE (L = no. fingers; = Ave. SNR in kth finger; rfor FSK and -1 for BPSK):

Flat Rayleigh channel; says to resolve multipath as as much possible (BEP versus L monotonically decreases for any Eb/N0)

k

21 1

1,

2 1(1 )1 11

2 2 (1 ) 2 (1 )

where 1

LLk r

kk kk r k r

Lk

ki i k k

LP

L

5

The Two Issues of This Talk

First Issue: W-CDMA for finer resolution of multipath with diversity combining by RAKESecond Issue: Wideband achieved by multicarrier spread spectrum

6

RAKE Receiver Structure

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Model for Fine Resolution

Resolution increases (chip duration decreases): Multipath reflections are from smaller patches or

include smaller “bundles” of rays

A model for envelope of multipath components:

Model for tracking loop phase (e.g., RAKE finger):

10( ) exp / 2 / , 0

where is the SNR for the th bundle (e.g., RAKE finger)

k k k

k

p y y K I Ky y

k

,loop

,loop0 ,loop

exp cos( ) , ,

2 ( )k k b

k k k kk L

Rp

I B

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Decision Statistic: RAKE Receiver

Adapting from Proakis:

Given and U1 is a Gaussian RV (drop Re). Its moment generating function is

Average of exp( ) sum becomes product of averages

U E Nb k k k k kk

L

k

L

1

2

11

2 FHG

IKJ

Re cos cos

( ) exp( ) exp .' , 's E sU E sm sk k Us s U 1

2 2

1 10 5 e j

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Ricean Envelope; Tikonov Phase

Again, from Proakis:

Laplace transform of the detection statistic pdf is

The k’s are assumed Ricean distributed; make integrand of average look like Ricean pdf with additional factors outside integral.

m E E E NU b k kk

L

b k kk

L

U b k kk

L

1 12 2 2

2

1 1

20

2

1

cos cos cos and

,

1

2 20

( ) ( ) , where

( ) exp 2 cos cos

k k

L

kk

k b k k b k k

s E s

s E s E N s

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Laplace Transform of Detection StatisticAverage over k:

Can’t get a closed form for the average over k with respect to a Tikonov phase pdf: For given s carry out the average numerically; do

product Use numerical technique of Biglieri, et al., Elec.

Letters, Feb. 1, 1996, pp. 191-192, to get probability of error

k kk k

b k k

sKB B

B

B E s N s

( | )exp /

,

cos cos

1

1

2 02 2

a f

d i

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Gauss-Chebyshev Quadrature to Get BEP from MGF of Decision Statistic

G-C formula from Biglieri, et al.

c affects the number of nodes necessary to achieve a desired accuracy A recommendation in Biglieri, et al is the

value minimizing (c) Or else 1/2 the smallest real part of the

poles of (s)

P c jc c jc E

k E

kk

k k

k

( ) Re Im

tan /

/

0

1

2 1 2 0

1

2

b g b ga fafwhere and as

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More Practical Case: Internal Noise in Phase Tracking Device

Generalize to the signal-to-noise ratio, SNR(k), in the kth finger of the RAKE receiver being

Typically, by minimizing phase jitter due to external and internal noise,

pdppdp 22

int0 int 0

0

pdp

2int

SNR1

where is the tracking device bandwidth,

is the power delay profile for the fading, and is

in variance of the internal noise

b b b b

L L

L

L

P kE R E Rk P k

N B N BN B

B P k

2int 0/ 1LN B

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Pb versus Eb/N0; Ricean fading with K = 0 dB; loop SNR 20 dB above Eb/N0 = 0 dB; L = no. of

RAKE fingers; constant PDP

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Pb versus Eb/N0; various orders of diversity, L; Ricean fading, K = 6 dB; σint

2/N0BL = 1; Rb/BL = 15 dB; expon. PDP

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Pb vs. L; Ricean fading, K = -6, 0, 6 dB, Eb/N0 = 7 dB; σint

2/N0BL = 1; Rb/BL = 15 dB; expon. PDP; opt. L values: 37, 34, and 26

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Pb versus L; Ricean fading, K = 6 dB; Eb/N0 = 5, 7, & 9

dB; σint2/N0BL = 1; Rb/BL = 15 dB; exp. PDP; Opt. L

values: 18, 26, & 41 for Eb/N0 = 5, 7, & 9 dB, respectively

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Summary – RAKE Phase Tracking

An optimum number of paths exists, giving a minimum bit error probabilityFiner multipath resolution, through wider spread bandwidth, buys improved performance The majority of this improvement is

obtain for a few RAKE fingers combined (say five or so)

It is less dramatic as the number of fingers goes beyond 10 or 15.

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Next: MC-SSHave L channels (carriers) to be combined at receiver. For simplification assume Equal gain combining DPSK modulation

Follow same procedure as before: Obtain MGF of single carrier MGF of sum is product of separate MGF’s Use G-C integration to obtain bit error probability Can obtain closed form result for Rayleigh fading

1

, Ray

0

corr. due to Doppler1 1 1 ! 1 11,

1 !2 !1 per channel SNR 2 1

nLL

c cb L

nc cc

L nP

L n

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Results for fdTb = 10-5 ( 1)

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

0, dB

Pb DPSK in AWGN

K = -10 dBK = 0 dB K = 5 dB K = 10 dB K = 20 dB

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Moderate Doppler Spread; Nearly Rayleigh

0 10 20 30 40 50 6010

-10

10-8

10-6

10-4

10-2

100

Eb/N

0, dB

Pb

L = 1L = 2L = 3L = 4L = 5L = 6L = 7L = 8

fdT

b = 0.02

K = -20 dB

DPSK inAWGN

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Higher Doppler Spread; Ricean; Uniform power across carriers

0 10 20 30 40 50 6010

-50

10-40

10-30

10-20

10-10

100

Eb/N

0, dB

Pb

DPSK in AWGN

L = 1L = 2L = 4L = 8

fdT

b = 0.04

K = 10 dB

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BEP versus L; K = 10 dB, and fdTb = 0.04 for uniform power

profile

2 4 6 8 10 12 14 16 18 2010

-8

10-6

10-4

10-2

100

L

Pb

Lmin

= 8; Pb,min

= 1.6514e-008

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Summary

Have an optimum number of pathsNonoptimum, equal gain combining used to simplify analysisDPSK modulation exhibits error floor due to Doppler spread

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References

R. E. Ziemer, B. R. Vojcic, L. B. Milstein, and J. G. Proaki s, “Effects of Carrier Tracking in RAKE Reception of Wide-Band DSSS in Ricean Fading,” vol. 47, no. 6, pp. 681-686, June 1 1999T. B. Welch, Analysis of Reduced Complesity Direct-Sequence Code-Division Multiple-Access Systems in Doubly Spread Channels, Ph. D. Dissertation, University of Colorado at Colorado Springs, 1997R. E. Ziemer and T. B. Welch, “Equal-Gain Combining of Multichannel DPSK in Doppler-Spread Ricean Fading,” IEEE Veh. Tech. Transactions, Vol. 49, pp. 1846-1859, Sept. 2000S. Kondo and L. G. Milstein, “Performance of Multicarrier DS CDMA Systems,” IEEE Trans. on Commun., Vol. 44, pp. 238-246, Feb. 1996