SISTEMI DI
RADIOCOMUNICAZIONE
ASYNCHRONOUS
DIRECT SEQUENCE
SPREAD SPECTRUM
Prof. C. Regazzoni
Sistemi di radiocomunicazione
1
References
1. R. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of Spread-Spectrum
Communications – A Tutorial”, IEEE Transactions on Communications, Vol. COM30, 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.
Sistemi di radiocomunicazione
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Introduction
In 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.
Sistemi di radiocomunicazione
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Multi User DS-CDMA
In 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.
y(t)
The received signal after the
sampling can be considered
as the contribution of three
components:
P
P
Z
Tb1,0    I 
Tb1,0  ng
2
2
BPSK DEMODULATOR
PN
DE-SPREADER
2P cos2f 0 t   1 
sˆ1 (t )
p1 (t )
PN
Generator
• First Term is the tx signal
• η is the AWGN
• I is the MUI
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Perfomances – AWGN hp
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.
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 : I 0  ( K  1) P
2
2B
The power of overall noise (MUI and AWGN) is:
NTOT  N 0 22 B  K  1P  N 0 B  ( K  1) P
With previous data it is possible to obtain the Signal to Noise Ratio at the
receiver:
E
E
SNRout 
b
N0  I0

b
N 0  ( K  1) P B
Sistemi di radiocomunicazione
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Perfomances– AWGN hp
By using a BPSK modulator the transmission bandwidth is B  2 Tc

the BER is PE ,BPSK  Q 2SNR out
PE , BPSK

and
with Gaussian hypothesis we have:
1 / 2
  K 1


1

 Q 


 4N

2 Eb N 0 


Where

Q( x) ˆ

x
1
2
e

y2
2
dy
is the Gaussian Error Function
In a single user (k=1) and Gaussian (AWGN) scenario the DS-CDMA has the
same performance of a narrow band BPSK modulation.
Sistemi di radiocomunicazione
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BER Evaluation - Gaussian hp
In 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:
var( n g )  var( )  var( I )
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:
T
   n(t ) p1 (t ) cos c t  dt
whose variance is N0T/4
0
Sistemi di radiocomunicazione
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Perfomances - Gaussian hp
I, as already explained, is the interference generated by other users.
It can be defined as out at the receiver as:
P K
I ˆ
 Z k ,1 
2 k 2
P K T

   bk (t   k ) pk (t   k ) p1 (t )dt  cos(k )
2 k =2 0

where k is the phase delay and  k is the time delay for user k
Z  0
  PrZ  0



The symbols have the same probability and Pr
b1 (1)  1
b
(
1
)


1
1



the error probability is :
1
  1 Pr Z  0
  Pr Z  0

Pe  Pr Z  0




b1 (1)  1 2 
b1 (1)  1
b1 (1)  1
2 


P   
 Pr n g 
   f n ( x)dx
where f ng (x) is the gaussian pdf of ng
2


P
T
g
2
Z
P
b1 (1)  n g
2
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Perfomances - Gaussian hp
From the previous formula the error probability becomes:
.
 P 2 


P 2



T 
T
2
2
  Q

PeG  Q( SNRout )  Q
N
T
 var ng  


0
 var( I ) 



4




where the SNR for the considered user at the receiver is:
SNR
ˆ
out

N0
 

1


  E  2  ( SNR)  1 
var(I )  var( ) 
2

1 / 2
is the multi-user interference I normalized with respect to P 2
is the spectral density of AWGN
SNR ˆ  N 0
is the signal to noise ratio in the transmitter
Sistemi di radiocomunicazione
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MUI Variance
 
2
The variance of I, var(I), or the mean square value of  ,E  , has to be
computed to obtain the final formula of Pe.. It is sufficient the mean square
value because E    0 .
E ( ) 
2
 Eb, , [bk (0)  k ,1 ( k )  bk (1) ˆ k ,1 ( k )] cos( k )2 
K
k 2

where  k ,1 ( ) ˆ  pk (t   ) p1 ( )d
o
T
and ˆ k ,1 ( ) ˆ  pk (t   ) p1 ( )d

Note: time delay and phase delay are uniformly distributed variables in [0,T)
and [0,2p) and the transmitted symbols have the same probability.
Sistemi di radiocomunicazione
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Example
Reference User
In the figures an example of
b1 (0)
asynchronous transmission
with delay
b1 (1)
is presented.
k
0
-T
t
T
Intereference User
b k ( 0)
bk (1)
-T+  k
0
Sistemi di radiocomunicazione
T+
k
t
11
MUI Variance
The previous quantities can be defined considering the a-periodic crosscorrelation between PN sequence of reference user and PN sequence of
user K.
 N l 1
0  l  N 1
  p k ( j ) p1 (j  l)
 k ,1 (l )   Njl01
  p k ( j  l ) p1 (l)
1 N  l  0
 j 0
The integrals of slide 10 can be computed as:
ˆ k ,1 ( k )   k ,1 (l k )Tc   k ,1 (l k  1)   k ,1 (l k )( k  l k Tc )
 k ,1 ( k )   k ,1 (l k  N )Tc   k ,1 (l k  N  1)   k ,1 (l k  N )( k  l k Tc )
for lk such as
l k Tc    (l k  1)Tc
Sistemi di radiocomunicazione
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MUI Variance
Using the previous values the variance of normalized MUI has been reduced
to:
E ( ) 
2
K
k 2
where

 

2
2
2
ˆ
E

(

)


(

)
E
cos
( k )
  k ,1 k
k ,1 k



1
E cos ( k ) 
2

E  k2,1 ( k )

2

ˆ k2,1 ( k )
2
2
(cos
 )d 

0

1
2
and

1T 2
   k ,1 ( )  ˆ k2,1 ( ) d
T0
This integral can be divided in a summation of all integrals in the interval
lTc , (l  1)Tc 

where 0  l  N  1 .

1
E  k2,1 ( k )  ˆ k2,1 ( k ) 
T
N 1(l 1)Tc
   k2,1 ( )  ˆ k2,1 ( )d
l 0 lTc
Sistemi di radiocomunicazione
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MUI Variance
By substituting the integral with the summation of integrals and  k2,1 ( )  ˆ k2,1 ( )
with the values obtained in slide 12, the variance becomes:
 
E
2
where

1 K N 1
ˆ )
ˆ

f
(
a
,
b
,
a
,
b


v
k
,
l
k
,
l
k
,
l
k ,l
3
6 N k 2 l 0

a k ,l ˆ  k ,1 (l  N )
aˆ k ,l ˆ  k ,1 (l )
bk ,l ˆ  k ,1 (l  N  1)
bˆk ,l ˆ  k ,1 (l  1)
f v ( x, y, z, w)  x 2  y 2  z 2  w2  xy  zw
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MUI Variance - Conclusion
The 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
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Optimal Receiver: Asynchronous
Transmission
We assume that transmitted signal is corrupted by AWGN in the channel;
received signal can be so expressed as:
r( t )  s( t )  n( t )
where s(t) is transmitted signal and n(t) is noise with spectral density1 N0
2
Optimal receiver is, for definition, receiver which select bit sequence:
.
bˆ (n),1  n  N ,1  k  K 
k
Which is the most probable, given received signal r(t) observed during a
temporal period 0  t  NT+2T, i.e.:
bˆ (n) arg max P b (n) r(t),t  0, NT  2T 
k
k
bk ( n )
Sistemi di radiocomunicazione
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Optimal Receiver: Asynchronous
Transmission
Two consecutive symbols from each user interfere with desired signal.
Receiver knows energies of signals
 
Eand
k  their transmission delays
k
Optimal
receiver evaluates the following likelihood function:
(b)  
NT  2T
0

K
K
2
K
N


r (t )   E k  bk (i)ck (t  iT   k ) dt 
k 1
i 1


NT  2T
0
K
N
NT  2T
i 1
0
r (t )dt  2 E k  bk (i) 
2
k 1
N
N
  E k E l  bk (i)bl ( j ) 
k 1 l 1
.
i 1 j 1
NT  2T
0
r (t )ck (t  iT   k )dt 
ck (t  iT   k )cl (t  jT   l )dt
Where b represents the data sequences received from K users
Sistemi di radiocomunicazione
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Optimal Receiver: Asynchronous
Transmission

First integral:
NT  2T
0
r 2 (t )dt
doesn’t depend on K, so can be ignored in maximization while the second
integral:
i 1T  k

rk (i)  r (t )ck (t  iT   k )dt
1i  N
iT  k
represents correlator o matched filter outputs for K-th user in each signal
interval.
Third integral can be easily decomposed in terms regarding cross-correlation:
NT  2T
 c (t  iT   )c (t  jT   )dt 
 c (t )c (t  iT  jT     )dt

k
k
l
l
0
NT  2T  iT  k
k
 iT  k
l
k
Sistemi di radiocomunicazione
l
18
Optimal Receiver: Asynchronous
Transmission
Indeed can be written:
kl (  )  kl (  k   l )
for k  l
lk (  )
for k > l
 (bcan
) be expressed as a correlation measure (one for each K identifier
sequences) which involves the outputs:
 rk(i),1  k  K ,1  i  N 
of K correlators or matched filters.
Sistemi di radiocomunicazione
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Optimal Receiver: Asynchronous
Transmission
By using vectorial notation can be shown that NK outputs rkof
(i) correlators
matched filters can be expressed in form:
where


r  rt (1) rt (2)
rt ( N) t

b  bt (1) bt (2)
b( i )   E1b1( i ) E 2 b2 ( i )

n  n (1)n ( 2 )
t
t
t

n (N)
t
or
r  RN b  n
r(i)  r1(i) r2(i)
rK (i) t

bt ( N) t
E K bK ( i ) 
t
n( i )  n1(i) n2(i)
Sistemi di radiocomunicazione
nK (i)t
20
Optimal Receiver: Asynchronous
Transmission
R ta( 0 ) R ta( 1 )
0
.....
.....
0 
 t

t
t
R
(
1
)
R
(
0
)
R
(
1
)
0
.....
0
a
a
 a

.
.
.
.
. 
 .
RN   .
.
.
.
.
. 


.
.
.
.
. 
 .
 0
0
0
R ta( 1 ) R ta( 0 ) R ta( 1 ) 


t
t
0
0
0
R a( 1 ) R a( 0 )
 0
Ra ( m )
is a KxK matrix which elements are:

Rkl (m) 
 ck (t   k )cl (t  mT   l )dt
Sistemi di radiocomunicazione
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Optimal Receiver: Asynchronous
Transmission
Gaussian noise vector n(i) is zero mean and its autocorrelation matrix is:


1
E n( k ) nt ( j )  N0 Ra ( k  j )
2
Vector r constitutes a set of statistics which are sufficient for estimation of
transmitted bits bk ( i ) .
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
Sistemi di radiocomunicazione
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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 algorithm
Each transmitted symbol is overlapped with no more than 2(K-1) symbols
b1(i)
b2(i-1)
b2(i)
bK(i-1)
bK(i)
When the algorithm uses a finite decision delay (a sufficient number of states),
the performances degradation becomes negligible
Sistemi di radiocomunicazione
23
Optimal Receiver: Alternative Approach
The previous consideration points out that there is not a singular method to
decompose (b
.)
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
Sistemi di radiocomunicazione
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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
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25
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
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26
Sub-optimal Receiver:
De-correlating Detector
The correlator output is:
r  RN b  n
Likelihood function is:
(b)  (r  R N b)R NK1 (r  R N b)
Where
R ta( 0 ) R ta( 1 )
0
.....
.....
0 
 t

t
t
R
(
1
)
R
(
0
)
R
(
1
)
0
.....
0
a
a
a


.
.
.
.
. 
 .
RN   .
.
.
.
.
. 


.
.
.
.
. 
 .
 0
0
0
R ta( 1 ) R ta( 0 ) R ta( 1 ) 


0
0
0
R ta( 1 ) R ta( 0 )
 0
Sistemi di radiocomunicazione
27
Sub-optimal Receiver:
De-correlating Detector
It can be proved that the vector b which maximize maximum likelihood function
is:
b 0  R N1r
This ML estimation of b is obtained transforming matched filters bench outputs.
But
(see slide
27)
rR
Nb  n

b 0  b  R N1n
b 0an unbiased estimation of b.
So is
The interference is so eliminated.
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28
Sub-optimal Receiver:
De-correlating Detector
The solution is obtained by searching linear transformation:
b0  Ar
Where matrix A is computed to minimize the mean square error (MSE)


 E ( b  Ar )t ( b  Ar )
J ( b )  E ( b  b0 )t ( b  b0 ) 
It can be proved that the optimal value A to minimize J(b) in asynchronous case
is:

1
A0  ( RN  N0 I )1
2
1
b0  ( RN  N0 I )1 r
2
Sistemi di radiocomunicazione
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Sub-optimal Receiver: Minimum Mean
Square Error Detector
The output of detector is:
bˆ  sgn( b0 )
When 1 N0 is low compared to other diagonal elements in RN,
2
minimum
MSE solution approximate ML solution of de-correlating receiver.
When noise level is high with respect to signal level in diagonal elements in RN
matrix A0 approximate identical matrix (under a scale factor 1 2 N
).0
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.
Sistemi di radiocomunicazione
30
Sub-optimal Receiver: Minimum Mean
Square Error Detector
To obtain b a linear system is to be computed:
1
( RN  N0 I )b  r
2
An efficient solving method is the square factorization(*) of matrix:
RN 
1
N0I
2
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
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Conclusions: BER Evaluation
For an asynchronous DS/CDMA system, BER expression can be written
(partially reported in slide 14)[5] as:


1 K N 1
ˆ )  K 1
ˆ
E   ˆ
f
(
a
,
b
,
a
,
b
 v k ,l k ,l k ,l k ,l 3N
6 N 3 k  2 l 0
2
It leads to:
SNR
ˆ
out

1


  E  2  ( SNR)  1 
var( I )  var( ) 
2

 
•If stochastic PN sequences are considered:
1/ 2
 
E 2 
1 
 K 1



2SNR 
 3N
1 / 2
K 1
3N
1/ 2


K

1
1


This formulation is wrong for “few users”
PE  Q 

 
3
N
2
SNR
 

whereas can be used for large number
of users. It is useful for a simple evaluation of DS/CDMA system performances
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Conclusions: BER Evaluation
From 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:
exp(  x 2 2)
Q( x ) 
2 x
Considering admissible PE  10-3 (sufficient for vocal applications)
Q3.11  10
3
Considering the right side of
equation as upper bound:
 1

1
  1
K  3N 

2
 3.11 2 Eb N 0 
 1

1

  1
K  3N 

2
 3.11 2 Eb N 0 
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Conclusions: BER Evaluation
For high values of signal-to-noise ratio an approximation is possible:
N
K
3
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.
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Example: numerical results
N  31
N 7
K=6
K=6
K=4
K=4
K=2
K=2
( N 0 )
( N 0 )
N  127
K=6
K=2
K=4
•BER Gaussian evaluation
DS/CDMA systems
•BPSK modulation
•Gold sequences
•K = number of users
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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.
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