Embed Size (px)
Citation preview
1
Corso di Comunicazioni Mobili
Prof. Carlo Regazzoni
ASYNCHRONOUS DIRECT SEQUENCE SPREAD SPECTRUM
2
References1. R. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of Spread-Spectrum
Communications – A Tutorial”, IEEE Transactions on Communications, Vol. COM-30, No. 5, Maggio 1982, pp. 855-884.
2. K. Pahlavan, A.H. Levesque, “Wireless Information Networks”, Wiley: New York 1995.
3. A.J. Viterbi, “CDMA: Principles of Spread Spectrum Communications”: Addison Wesley: 1995.
4. J.G. Proakis, “Digital Communications”, (Terza Edizione), McGraw-Hill: 1995.
5. M.B. Pursley, “Performance Evaluation for Phase-Coded Spread-Spectrum Multiple Access Communications – Part I: System Analysis”, IEEE Trans. on Comm., Vol. 25, No. 8, pp. 795-799, Agosto 1977.
6. A. Lam, F. Olzluturk, “Performance Bounds of DS/SSMA Communications with Complex Signature Sequences”, IEEE Trans. on. Comm, vol. 40, pp. 1607-1614, Ottobre 1992.
7. D. Sarwate, M. B. Pursley, “Correlation Properties of Pseudorandom and Related Sequences”, Proceedings of the IEEE, Vol. 68, No. 5, pp. 593-619, Maggio 1980.
8. F.M. Ozluturk, S. Tantaratana, A.W. Lam: “Performance of DS/SSMA Communications with MPSK Signalling and Complex Signature Sequences”, IEEE Trans. on Comm. Vol. 43, No. 2/3/4, Febbraio 1995, pp.1127-1133.
3
IntroductionIn the previous session “TECNICHE DI TRASMISSIONE-DATI DIGITALI BASATE SUL CONCETTO DI SPREAD SPECTRUM” a Direct Sequence Spread Spectrum system with two or more users using the same band (as usual in CDMA) but different spreading codes has been partially analyzed.
The users involved in other communications are considered as interference called Crosstalk Interference whose power is related to Process Gain N. By modifying and choosing particular spreading code, their effects can be reduced.
The previous instances are main features of Code Division Multiple Access, which uses the strength of Spread Spectrum techniques to transmit, over the same band and with no temporal limitation (Asynchronous) information provided by several users.
4
Multi User DS-CDMAIn Multi-user DS-CDMA each transmitter is identified by its PN
sequence.
It is possible to detect the information transmitted through a receiver based
on a conventional matched filter. The other users, different by the
transmitting one, will be considered as Multi User Interference, MUI,
generally non Gaussian distributed.
) ( ˆ 1 t s BPSK DE-
MODULATOR
1 0 2 cos 2 t f P
PN DE-SPREADER
PN
Generator
) ( 1 t p
y(t)
The received signal after the sampling can be considered as the contribution of three components:
gnTbP
ITbP
Z 0,10,1 22
• First Term is the tx signal• η is the AWGN• I is the MUI
5
Perfomances – AWGN hp
Considering (as first case) a very simple situation where (k-1) DS-SS users are Gaussian, their power in the transmission band B is (k-1)P, where P is the transmitted power, considered equal for all users.
Its spectral density is :
The power of overall noise (MUI and AWGN) is:
With previous data it is possible to obtain the Signal to Noise Ratio at the receiver:
PKBNPKBNNTOT )1(122 00
Usually, real systems are composed by several users, so due to the central limit theorem the overall interference (MUI) can be considered as Gaussian distributed.
This hypothesis is reflected in BER computation where its Gaussian approximation is considered.
B
PKI
2
)1(
20
BPKN
E
IN
ESNR bb
out )1(000
6
Perfomances– AWGN hpBy using a BPSK modulator the transmission bandwidth is and
the BER is with Gaussian hypothesis we have:
Where is the Gaussian Error Function
cTB 2
outBPSKE SNRQP 2,
2/1
0, 2
1
4
1
NEN
KQP
bBPSKE
x
y
dyexQ 2
2
2
1ˆ)(
In a single user (k=1) and Gaussian (AWGN) scenario the DS-CDMA has the
same performance of a narrow band BPSK modulation.
7
BER Evaluation - Gaussian hpIn the last two slides a particular and usually wrong hypothesis has been considered: the MUI is modeled as white noise. In real case its spectral density is NOT flat, thus the Multi User Interference can not be considered as white noise.
To carry out a deeper analysis, the first and second order statistics of random variables (considered Gaussian) have to be computed.
Being η and I Gaussian distributed, the pdf of ng is Gaussian with zero mean and variance given by:
because I and η are independent random variables with zero mean.
η is the output of the receiver when n(t) (the AWGN) is the input:
)var()var()var( Ing
T
c dtttptn0
1 cos)()( whose variance is N0T/4
8
Perfomances - Gaussian hpI, as already explained, is the interference generated by other users.
It can be defined as out at the receiver as:
K
2=k )cos()()()(
22ˆ
01
21, k
T
kkkk
K
kk dttptptb
PZ
PI
where k is the phase delay and is the time delay for user kk
The symbols have the same probability and
the error probability is :
TP
ng
e
dxxfP
n
bZ
bZ
bZP
g
2
111
)(2
Pr
1)1(0Pr1)1(
0Pr2
11)1(
0Pr2
1
1)1(0Pr1)1(
0Pr11 b
Zb
Z
gnbP
Z )1(2 1
where is the gaussian pdf of ng)(xfgn
9
Perfomances - Gaussian hp
)var(
4
2var
2)(0
22
ITN
TP
Qn
TP
QSNRQPg
outeG
2/1
1)(2
1
)var()var(ˆ 2
SNREIout
SNR
From the previous formula the error probability becomes:
where the SNR for the considered user at the receiver is:
.
2P
0ˆ NSNR
N0
is the multi-user interference I normalized with respect to
is the signal to noise ratio in the transmitter
is the spectral density of AWGN
10
MUI Variance The variance of I, var(I), or the mean square value of , , has to be computed to obtain the final formula of Pe.. It is sufficient the mean square value because .
2E
0E
K
kkkkkkkkb bbEE
2
21,1,,,
2 )cos()](ˆ)1()()0([)(
where
o
kk dptp )()(ˆ)( 11, and T
kk dptp
)()(ˆ)(ˆ 11,
Note: time delay and phase delay are uniformly distributed variables in [0,T)
and [0,2p) and the transmitted symbols have the same probability.
11
Example
k
-T T0t
-T+ T+0t
)1(1b
)1(kb
)0(kb
k
Reference User
Intereference User
In the figures an example of
asynchronous transmission
with delay is presented.k
)0(1b
12
MUI VarianceThe previous quantities can be defined considering the a-periodic cross-
correlation between PN sequence of reference user and PN sequence of user K.
The integrals of slide 10 can be computed as:
01 (l))(
10 l)(j)(
)( 1
01
1
01
1,
lNpljp
Nlpjp
l lN
jk
lN
jk
k
)()()1()()( 1,1,1,1, ckkkkkkckkkk TlNlNlTNl
)()()1()()(ˆ 1,1,1,1, ckkkkkkckkkk TlllTl
for lk such as ckck TlTl )1(
13
MUI Variance
)(cos)(ˆ)()( 2
2
21,
21,
2k
K
kkkkk EEE
Using the previous values the variance of normalized MUI has been reduced to:
where 2
1)(cos
2
1)(cos
2
0
22
dE k and
dT
ET
kkkkkk 0
21,
21,
21,
21, )(ˆ)(
1)(ˆ)(
This integral can be divided in a summation of all integrals in the interval
cc TllT )1(, where .10 Nl
1
0
)1(2
1,2
1,2
1,2
1, )(ˆ)(1
)(ˆ)(N
l
Tl
lTkkkkkk d
TE
c
c
14
MUI VarianceBy substituting the integral with the summation of integrals and
with the values obtained in slide 12, the variance becomes:
)(ˆ)( 21,
21, kk
K
k
N
llklklklkv babaf
NE
2
1
0,,,,3
2 )ˆ,ˆ,,(6
1
where
)1(ˆ
)(ˆ
1,,
1,,
Nlb
Nla
klk
klk
)1(ˆˆ
)(ˆˆ
1,,
1,,
lb
la
klk
klk
f x y z w x y z w xy zwv( , , , ) 2 2 2 2
15
MUI Variance - ConclusionThe last formula allow us to conclude:
• The higher the process gain N, the lower the MUI variance. This means
that by increasing the SS bandwidth the power of the Multi-User interference
will be reduced.
• A fundamental parameter is the cross-correlation function among PN
sequences. With low correlation the MUI will be reduced and the interference
can have weak effects.
In the following section these aspects will be analyzed in details
16
We assume that transmitted signal is corrupted by AWGN in the channel; received signal can be so expressed as:
where s(t) is transmitted signal and n(t) is noise with spectral density .
Optimal receiver is, for definition, receiver which select bit sequence:
Which is the most probable, given received signal r(t) observed during a temporal period 0 t NT+2T, i.e.:
Optimal Receiver: Asynchronous Transmission
)t(n)t(s)t(r
0N21
KkNnnbk 1,1),(ˆ
TNTttrnbPnb k
nbk
k 2,0),()(maxarg)(ˆ
)(
17
Optimal Receiver: Asynchronous Transmission
Two consecutive symbols from each user interfere with desired signal.
Receiver knows energies of signals and their transmission delays.
Optimal receiver evaluates the following likelihood function:
kE k
K
k
K
l
N
i
N
j
TNT
llkklklk
K
k
N
i
TNT
kkkk
TNT
TNT K
k
N
ikkkk
dtjTtciTtcjbib
dtiTtctribdttr
dtiTtcibtr
1 1 1 1
2
0
1 1
2
0
2
0
2
2
0
2
1 1
)()()()(EE
)()()(E2)(
)()(E)()(
b
Where b represents the data sequences received from K users
18
Optimal Receiver: Asynchronous Transmission
First integral:
doesn’t depend on K, so can be ignored in maximization while the second integral:
k
k
Ti
iTkkk dtiTtctr(i)r
1
)()( Ni1
represents correlator o matched filter outputs for K-th user in each signal interval.
k
k
iTTNT
iTlklk
TNT
llkk
dtjTiTtctc
dtjTtciTtc
2
2
0
)()(
)()(
Third integral can be easily decomposed in terms regarding cross-correlation:
TNT
dttr2
0
2 )(
19
Optimal Receiver: Asynchronous Transmission
Indeed can be written:
)()( lkklkl for k l
)(lk for k > l
can be expressed as a correlation measure (one for each K identifier sequences) which involves the outputs:
of K correlators or matched filters.
NiKk(i)rk 1,1,
)(b
20
Optimal Receiver: Asynchronous Transmission
By using vectorial notation can be shown that NK outputs of correlators
or matched filters can be expressed in form:
where
(i)rk
nbRr N
tttt N )( )2( )1( rrrr t(i)r(i)(i) rri K )( 21r
tttt N )( )2( )1( bbbb
t)i(b)i(b)i(b)i( KK2211 E E Eb
tttt )N()2()1( nnnn t(i)n(i)(i) nn)i( K21 n
21
Optimal Receiver: Asynchronous Transmission
)()(
)()()(
)()()(
)()(
ta
ta
ta
ta
ta
ta
ta
ta
ta
ta
N
010000
101000.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.0.....0101
0..........010
RR
RRR
RRR
RR
R
)m(aR is a KxK matrix which elements are:
dtmTtctcmR llkkkl )()()(
22
Optimal Receiver: Asynchronous Transmission
Gaussian noise vector n(i) is zero mean and its autocorrelation matrix is:
Vector r constitutes a set of statistics which are sufficient for estimation of
transmitted bits .
The maximum likelihood detector has to calculate 2NK correlation measures to
select the K sequences of length N which correspond to the best correlation
measures.
The computational load of this approach is too high for real time usage
)jk(N2
1)j()k(E a0
t Rnn
)i(bk
23
Optimal Receiver: Alternative Approach
Considering maximization of (b) like a problem of forward dynamic programming can be possible by using Viterbi algorithm after matched filters bench.
Viterbi algorithmViterbi algorithm
Each transmitted symbol is overlapped with no more than 2(K-1) symbols
When the algorithm uses a finite decision delay (a sufficient number of states), the performances degradation becomes negligible
b1(i)
b2(i-1) b2(i)
bK(i-1) bK(i)
24
Optimal Receiver: Alternative Approach
The previous consideration points out that there is not a singular method to
decompose .
Some versions of Viterbi algorithm for multi-user detection, proposed in the
state of the art, are characterized by 2K states and computational complexity
O(4K/K) which is still very high.
This kind of approach is so used for a very little number of users (K<10 ).
When number of users is very high, sub-optimal receivers are considered
)(b
25
Sub-optimal Receivers: Conventional Receiver
The conventional receiver for single user is a demodulator which:
1. Correlates received signal with user’s sequence.
2. Connect matched filter output to a detector which implements a
decision rule.
Conventional receiver for single user suppose that the overall noise (channel
noise and interference) is white Gaussian
26
Sub-optimal Receivers: Conventional Receiver
The conventional receiver is more vulnerable to MUI because is impossible to
design orthogonal sequences, for each couple of users, for any time offset.
The solution can be the use of sequences with good correlation properties to
contain MUI (Gold, Kasami).
The situation is critical when other users transmit signals with more power than
considered signal (near-far problem).
Practical solutions require a power control method by using a separate channel
monitored by all users.
The solution can be multi-user detectors
27
Sub-optimal Receiver: De-correlating Detector
The correlator output is:
Likelihood function is:
Where
nbRr N
)()()( 1 bRrRbRrb NNKN
)()(
)()()(
)()()(
)()(
ta
ta
ta
ta
ta
ta
ta
ta
ta
ta
N
010000
101000.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.0.....0101
0..........010
RR
RRR
RRR
RR
R
28
Sub-optimal Receiver: De-correlating Detector
But (see slide 27)nbRr N
nRbb 10 N
So is an unbiased estimation of b.
The interference is so eliminated.
0b
It can be proved that the vector b which maximize maximum likelihood function is:
This ML estimation of b is obtained transforming matched filters bench outputs.
rRb 10 N
29
Sub-optimal Receiver: De-correlating Detector
The solution is obtained by searching linear transformation:
Where matrix A is computed to minimize the mean square error (MSE)
Arb 0
)()(E
)()(E)(J
t
0t0
ArbArb
bbbbb
It can be proved that the optimal value A to minimize J(b) in asynchronous case is:
rIRb 10N
0 )N2
1(
10N
0 )N2
1( IRA
30
Sub-optimal Receiver: Minimum Mean Square Error
DetectorThe output of detector is:
When is low compared to other diagonal elements in , minimum
MSE solution approximate ML solution of de-correlating receiver.
When noise level is high with respect to signal level in diagonal elements in
matrix approximate identical matrix (under a scale factor ).
So when SNR is low, detector substantially ignore MUI because channel noise
is dominant.
Minimum MSE detector provides a biased estimation of b, then there is a
residual MUI.
)sgn(ˆ 0bb
0N21
NR
0ANR
0N21
31
Sub-optimal Receiver: Minimum Mean Square Error
DetectorTo obtain b a linear system is to be computed:
An efficient solving method is the square factorization(*) of matrix:
With this method 3NK2 multiplications are required to detect NK bits.
Computational load is 3K multiplications per bit and it is independent from block length N and increase linearly with K.
* Proakis, appendix D
rbIR )N2
1( 0N
IR 02
1NN
32
Conclusions: BER EvaluationFor an asynchronous DS/CDMA system, BER expression can be written (partially reported in slide 14)[5] as:
N
Kbabaf
NE
K
k
N
llklklklkv 3
1)ˆ,ˆ,,(
6
1ˆ
2
1
0,,,,3
2
It leads to: 2/1
2
2
1
3
12/11)(
2
1
)var()var(ˆ
SNRN
KSNRE
IoutSNR
•If stochastic PN sequences are considered: N
KE
3
12
This formulation is wrong for “few users”
2/1
2
1
3
1
SNRN
KQPE
whereas can be used for large number
of users. It is useful for a simple evaluation of DS/CDMA system performances
33
Conclusions: BER EvaluationFrom PE expression can be derived an evaluation of CDMA system capacity, in terms of number simultaneous users served with a certain Quality of Service (QoS)
For high values of x:x
xxQ
2
)2exp()(
2
Considering admissible PE 10-3 (sufficient for vocal applications)
31011.3 Q 1
2
1
11.3
13
02
NENK
b
Considering the right side of equation as upper bound:
1
2
1
11.3
13
02
NENK
b
34
Conclusions: BER Evaluation
For high values of signal-to-noise ratio an approximation is possible:
3
NK
A simple guidance, about a DS/CDMA system, to estimate system capacity is
that more than N/3 asynchronous users can’t be served, where N is the
process gain, with a probability error lower than 10-3.
35
Example: numerical results
K = 2
K = 4
K = 6
K = 2
K = 4
K = 6
)( 0N )( 0N
K = 2
K = 4
K = 6
7N 31N
127N
•BER Gaussian evaluation for DS/CDMA systems
•BPSK modulation
•Gold sequences
•K = number of users
36
Comments
BER Gaussian evaluation is only an approximation of real BER.
For SNR < 10 dB, Gaussian noise is predominant and BER is barely influenced
by new users.
For very high SNR MUI is predominant and the higher the number of users, the
lower are performances, if process gain is low.
Increasing SNR over a certain threshold, BER saturates: this is the bottle-neck
given by MUI presence.
To increase performances, a higher process gain is needed; this fact involves
an expansion of transmission band, at the equal bit-rate.
