Transcript Carrier Tracking Error Effects in RAKE Reception of

Multipath Resolution Effects in
Wideband CDMA Transmission
Rodger E. Ziemer
Electrical and Computer Engineering Dept.
University of Colorado at Colorado Springs
Colorado Springs, CO 80933
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 Bathtub Doppler power spectrum no longer valid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
2
resolution negated by phase/timing errors?
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.)
3
Well Known Diversity Result
Proakis; Diversity reception in context of RAKE
(L = no. fingers; k = Ave. SNR in kth finger; rr
= 0 for FSK and -1 for BPSK):
1 L
P2 =   k
2 k =1

 k (1  r r )   2 L  1 L
1
1 


L
2   k (1  r r )  
 k =1 2 k (1  r r )

k
where  k = 
i =1, i  k 1   k
L
Flat Rayleigh channel; says to resolve multipath
as as much possible (BEP versus L
monotonically decreases for any Eb/N0)
4
The Two Issues of This Talk
First Issue: W-CDMA for finer
resolution of multipath with diversity
combining by RAKE
Second Issue: Wideband achieved by
multicarrier spread spectrum
5
RAKE Receiver Structure
6
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:


p( y) =  k1 exp   y /  k  K  I 0 2 Ky /  k , y  0

where  k is the SNR for the kth bundle (e.g., RAKE finger)
Model for tracking loop phase (e.g., RAKE finger):
p(k ) =
exp  k ,loop cos k 
2 I 0 ( k ,loop )
, k   ,  k ,loop
Rb
=k
BL
7
Decision Statistic: RAKE Receiver
Adapting from Proakis:
F
I
U ReG
2 E   cos   cos  N J
H
K
Given a and  , U is a Gaussian RV (drop Re).
L
1
2
b
k
k 1
k
k
L
k
k
k
k
k 1
1
Its moment generating function is
e
(s) E exp(
sU1 ) Ek ' s,k ' s exp 
smU1 0.5s 
2
Average of exp( ) sum becomes product of
averages
8
2
U1
j
Ricean Envelope; Tikonov
Phase
Again,L from Proakis: L
mU1 2 Eb  k cos k 2 Eb k cos k and 
2
k 1
k 1
L
2
U1
2 Eb N0 k cos2 k
k 1
Laplace transform of the detection statistic pdf is
L
( s) =  E k ,k   k ( s ) , where
k =1

 k ( s) = exp 2 Eb k cos k s  Eb N 0 k cos 2 k  s 2

The k’s are assumed Ricean distributed; make
integrand of average look like Ricean pdf with
additional factors outside integral.
9
Laplace Transform of Detection
Statistic
Average over k:
exp 
KB
B
k / 1
k
k ( s| k ) 
1 B
a
d
f,
i
B Eb 2 cos k s 
N 0 cos 2 k s 2
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
10
Gauss-Chebyshev Quadrature to Get
BEP from MGF of Decision Statistic
G-C formula
from Biglieri, et al.
/ 2
b
b
g
g

where  tan a
2k 
1f/ a
2f and E  0 as  
1
P(  0)  Re  c jck k Im  c jck E
k 1
k

c affects the number of nodes necessary to
achieve a desired accuracy


A recommendation in Biglieri, et al is the value
minimizing D(c)
Or else 1/2 the smallest real part of the poles of
D(s)
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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
Eb Rb
Eb Rb Ppdp  k 
SNR  k  =
Ppdp  k  =
2
 int2
N 0 BL   int
N 0 BL
1
N 0 BL
where BL is the tracking device bandwidth, Ppdp  k 
2
is the power delay profile for the fading, and  int
is
in variance of the internal noise
Typically, by minimizing phase jitter due to
2

external and internal noise, int / N0 BL  1
12
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
13
Pb versus Eb/N0; various orders of diversity, L;
Ricean fading, K = 6 dB; σint2/N0BL = 1; Rb/BL = 15
dB; expon. PDP
14
Pb vs. L; Ricean fading, K = -6, 0, 6 dB, Eb/N0 = 7
dB; σint2/N0BL = 1; Rb/BL = 15 dB; expon. PDP;
opt. L values: 37, 34, and 26
15
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 probability
Finer 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-SS
Have L channels (carriers) to be combined at
receiver. For simplification assume


Equal gain combining
DPSK modulation
Follow same procedure as before:




Pb, Ray
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
=
 L  1!2L
1  1  r   c 


1




c
L
n
 L  n  1! 1  1  r   c  ,  r = corr. due to Doppler


n
!


2 1  c
n =0
  = per channel SNR

  c
L 1


18
Results for fdTb =
-5
10
(r = 1)
0
10
-1
10
-2
P
b
10
DPSK in AWGN
-3
10
-4
10
K
K
K
K
K
-5
10
-6
10
0
=
=
=
=
=
-10 dB
0 dB
5 dB
10 dB
20 dB
5
10
15
Eb/N0, dB
20
25
30
19
Moderate Doppler Spread;
Nearly Rayleigh
0
10
fdTb = 0.02
K = -20 dB
-2
10
-4
b
10
P
DPSK in
AWGN
-6
10
L=
L=
L=
L=
L=
L=
L=
L=
-8
10
-10
10
0
1
2
3
4
5
6
7
8
10
20
30
Eb/N0, dB
40
50
60
20
Higher Doppler Spread; Ricean;
Uniform power across carriers
0
10
fdTb = 0.04
K = 10 dB
-10
10
DPSK in AWGN
-20
P
b
10
-30
10
-40
10
L=
L=
L=
L=
-50
10
0
1
2
4
8
10
20
30
Eb/N0, dB
40
50
60
21
BEP versus L; K = 10 dB, and fdTb
= 0.04 for uniform power profile
0
10
-2
P
b
10
-4
10
-6
10
Lmin = 8; P b,min = 1.6514e-008
-8
10
2
4
6
8
10
L
12
14
16
18
20
22
Summary
Have an optimum number of paths
Nonoptimum, equal gain combining
used to simplify analysis
DPSK modulation exhibits error floor
due to Doppler spread
23
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
1999
T. 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, 1997
R. E. Ziemer and T. B. Welch, “Equal-Gain Combining of Multichannel DPSK in DopplerSpread Ricean Fading,” IEEE Veh. Tech. Transactions, Vol. 49, pp. 1846-1859, Sept. 2000
S. Kondo and L. G. Milstein, “Performance of Multicarrier DS CDMA Systems,” IEEE Trans. on
Commun., Vol. 44, pp. 238-246, Feb. 1996
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