Corso di Comunicazioni Mobili
Course acquisition and
tracking in DS/SS systems
Prof. Carlo Regazzoni
1
References
1. J.G. Proakis, “Digital Communications” (3rd Edition), McGraw-
Hill:
1995.
2. R.C. Dixon, “Spread Spectrum Systems with Commercial Applications”
(3rd Edition), Wiley: 1984.
3. A.J. Viterbi, “CDMA: Principles of Spread Spectrum Communications”:
Addison Wesley: 1995.
4. D. Sarwate, M. B. Pursley, “Correlation Properties of Pseudorandom and
Related Sequences”, Proceedings of the IEEE, Vol. 68, No. 5, pp. 593619, Maggio 1980.
2
PN in SS Systems: Introduction
Data
Channel Coder
Modulator
PN sequences
Generator
SS Transmitter
Channell
Decoder
Demodulator
Data
PN sequences
Generator
SS Receiver
Schematic shows DS-SS system basic elements; there are two pseudonoise sequences generators .
In the modulator a PN sequence is multiplied by transmitted signal; in
demodulator received signal is newly multiplied by a PN sequence replica.
3
PN in SS Systems: Introduction
bi
ai
PN
Generator
Modulo-2
Adder
Balanced
Modulator
BPSK
Signal
ci
Data
FEC Coder
Local
Oscillator
To demodulate the received signal, the synchronism among the PN
sequences is needed
bi  ci
A practical method for multiplying transmitted signal and PN sequence is a
“module-2” adder.
If bi represents i-th bit of the PN sequence and ci is the corresponding bit from
codec, the “module-2” sum is:
ai  bi  ci
ai = 1 if bi = ci
ai = 0 if bi ≠ ci
4
PN Sequences-characteristics
A possible PN sequences generator is shown in figure:
1
C1
2
C2
3
…………………..
C3
r–1
r
Cr-1
bn 
Cr
Micro – switch
(Ci= 1 connection)
bn can be written as:
bn  C1bn 1  C 2 bn  2  ....... C r bn  r
Where C1,…,Cr are micro-switch setting coefficients of generator
r is the shift register length
Symbol probabilities are:
Pr(bn  0) 
1
(1  s)
2
Pr(bn  1) 
1
(1  s)
2
5
PN Sequences - characteristics
Where
s ˆ
1
2
r 1
is the coefficient of statistical displacement
Increasing r, s becomes negligible, and the sequence becomes more and
more random.
PN sequences, obtained by the previous generator, are periodical with period:
L  2 r  1 L is called also pseudo-noise length and in literature is
usually indicated as N.
L  N  2r  1
is true when the sequence characteristic
P( x)  1  C1 x  C2 x 2  ...... Cr x r
Polynomial
is primitive and when
the initial conditions of the shift register are: (b10 ,......,br 0 )  (0,0,......,
01)
The relation
In this case PN sequence is called maximum length sequence (m-sequence)
NB: r is the length of the shift register and NOT of the sequence.
6
M-sequences - definition
•Be M a given set of PN binary sequences;
pk , pl  M
two sequences;
•Cross-correlation peak on set M is defined as:
 c ( M )  max{  k ,l ( ) : 0    N  1, pk , pl  M , pk  pl }
•Where
N 1
 k ,l ( )   pk ( j ) ph ( j   )   xy ( )   xy (  N )
is cross-correlation
j 0
of two sequences, and temporal shift parameter
between 0 and N-1.

is an integer value,
Attention:  is different from c asynchronous transmission delay.
7
M-sequences properties
M-sequences, generated by a shift register of
r
N
c
t(r)
length r, have no good correlation properties,
3
7
5
5
indeed, increasing N, cross-correlation peak
4
15
9
9
5
31
11
9
6
63
23
17
7
127
41
17
8
255
95
33
9
511
113
33
correlation value as lower as possible is
10
1023
383
65
suggested, in order to decrease the MUI
11
2047
287
65
12
4095
1407
129
grows in a huge way.
In DS/CDMA systems, the usage of PN
sequences set characterized by a cross-
8
Gold sequences
 Binary sequences
 Best ratio between number of available sequences and optimal values of
cross-correlation
 Based on shift register generation scheme like m-sequences
 Generated by a preferential couple of m-sequences u and v with the
following properties:
uv (l ) {1,t (r ),t (r )  2}
l [0, N  1]
t (r )  1  2( r 2) / 2
Where r is the shift register lenght
9
Gold sequences
The Gold sequences generated by u and v
(preferential couple of m-
sequences ) are composed by N + 2 sequences obtained as:


G(u, v)  u, v, u  v, u  Tv, u  T 2v,.......,u  T N 1v
Where
T k (.)
is a cyclical shift operator
G  G(u, v) is called Gold set, generated by u and v
10
Gold sequences - properties
All the Gold sequences in set G have the same correlation properties of u and v
This means that the Cross-correlation peak of Gold set is the same of u and v
couple  c (G)  t (r )
A Gold sequences generator is composed by two m-sequences generators with
shift register and feedback
11
Preferential sequences couples
Preferential couples of m-sequences with length N = 31 (r = 5) (a) and
N = 63 (r = 6)
12
Preferential sequences couples
The numbers in circles represent vectors, in octal notation, (C1 ,, Cr ) which
elements are binary coefficients for micro switches setting of a shift
register m-sequences generator.
Each circle represents a well defined sequence.
The couples of circles linked by the segment represent m-sequences
preferential couples which can be used to generate Gold sets
It can be proved that the correlation properties are optimal for binary
sequences whose generators have odd length
13
Kasami sequences
Optimal binary sequences generated by shift register with even length are
called Kasami sequences.
Kasami sets properties
1. Better cross-correlation properties than Gold sequences
2. More difficult to be generated
3. Extremely reduced dimensions (only  N  2
available sequences)
For these reasons Gold sequences are often used with shift register with even
length, in order to serve several users.
14
Acquisition - introduction
In a DS/SS system, PN-code replica generated by receiver must be
synchronized with received signal within a little fraction of chip time Tc=1/W,
where W is transmitted signal band (for simplicity, we suppose to use a
BPSK modulation).
De-spreading operation is possible only if two signals are strictly
synchronized. Synchronization phase in DS/SS systems is called PN
acquisition.
When synchronization is reached, matched filter output, sampled every t =
Tc, is:
Tc
vi 
Tc
 2b 12c 1 g (t )dt  2b 12c 1  g (t )dt  2b 1T
2
i
i
2
2
i
0
2
i
i
c
0
bi  0,1 i-th bit transmitted value
ci  0,1 i-th chip of PN sequence value
15
PN Acquisition
DS/SS systems use, generally, very stable hi-frequency digital clocks to
reduce transmitter-receiver asynchronism caused by hardware problems.
In case of synchronous transmission, initial temporal displacements
between transmitted PN signal and receiver replica are due to non perfect
knowledge about tx and rx distance.
A synchronization header is generally used, at the beginning of data block,
containing one or more “1” symbols, spreaded by PN sequence.
So the receiver can have a non modulated and delayed PN sequence to
compare with its replica
16
PN Acquisition
header
0
header
T
data
t
Received signal
(Hypothesis:
the
received signal is
cyclical)
N=7
0
t
T
PN sequence replica
generated by receiver
(4 chip delayed)
17
Fundamental acquisition methods
•Matched filter correlators
•Serial search: sliding correlators
•Sequential search
•Parallel acquisition
PN acquisition is a totally digital operation, indeed is done on PN signal
received from the channel, BPSK demodulated with a coherent system,
sampled with t s  Tc, and properly quantized.
Performances evaluation criteria for PN acquisition
methods
•Acquisition mean time
•Computational complexity
•Correct synchronization probability
•False synchronization probability
False synchronization probability is the real error probability of this kind of
algorithms because a false acquisition produces a wrong demodulation
process.
18
Matched filter correlator
•Theoretically optimal (without multiple access interference)
•Theoretical acquisition time: only one bit time T
•Constituted by a tapped delay line
•Every Tc seconds, delay line content is scalar multiplied by PN sequence
mirrored replica
r(nTc)
T1
T2
T3
T4
T5
T6
T7
•Result is compared with a
suitable threshold, empirically
chosen on channel noise
•Correlator
operates
C7
C6
C5
C4
C3
C2
C1
convolution between received
signal and PN sequence

replica
Figure 1 acquisition diagram:
•matched filter to a 7 chip-PN
Y(nTc)
sequence g(t)
N 1
N 1

•Single chips binary values
Y (nTc ) 
r nTc g T  nTc   r nTc cN n TH

n 0
n 0
[c1,c2,…,c7]
19


Matched filter correlator
•If transmitted signal delay is an integer multiple l of chip time Tc maximum
value of convolution product, equal (without noise) to bit energy, is correctly
detected in lTc seconds
•In case of generic delay is still detected in lTc seconds:
l : lTc    (l  1)Tc
Coarse acquisition: Time resolution = chip time Tc
•Over sampling received signal by a factor b and increasing delay line taps of
the same factor, it’s possible to obtain a coarse acquisition with Tc/b seconds
resolution
•Delay line needs a precise shift every 50-100 nsec., so it’s difficult to realize
with low cost IC components.
•matched filter correlator is computationally heavy, even in software
development on modern DSP architecture (clock: 720 MHz/1 GHz)
•Having chip rate Rc and
spreading sequence length K
MF  Nops / Tc sec .  NRc ops / sec .
N, number of elementary
•If N = 127 and Rc = 10Mc/s
operations
(sums
and
7
7
K

127
ops
/
10
s
.

127
*
10
ops/ sec  1.27Gops/ sec
MF
multiplications) required is:
20
Serial search (sliding correlator)
•It’s the most used algorithm for real applications; theoretical efficiency is not
high but the reduced computational complexity allows an easy implementation
•PN received signal is periodically multiplied by PN sequence replica, cyclically
shifted by a delay time which is incremented every time of a pre-setted value.
Product is than integrated for TI time and result is compared with a threshold
yl
TH.
Figure 2: Sliding correlator
TI

yl  r (t ) g t   l dt TH

0

 l   l 1 
lTc
b
0  0
 T

l  1,2..,
 1  bN  1
 Tc b 
TI  T
•Sliding correlator operates a “step-by-step” correlation between received
signal and generated PN sequence.
21
Serial search (sliding correlator)
•When received signal delay is equal to sequence replica pre-set delay (under
Tc/b seconds sliding correlator output correlation yl should overcome
synchronization threshold.
•Integration time TI can be properly chosen to reduce false and missed
synchronization probability.
Figure 3:
Analytical correlation (a)
“step-by-step” correlation (b)
calculated by sliding correlator
22
Serial search (sliding correlator)
•Serial search is a parametrical method (TI, l).
•Parameter values influence algorithm performances (false and missed
synchronization probability, computational complexity)
•If:
TI = T
 l   l 1  Tc
chip rate = Rc
process gain = N
number of elementary operations required:
KSC  Nops/ Ts.  NRbops/ sec  Rcops/ sec  KMF
Rb = 1/T (bit-rate)
•Every search step requires at least one bit-header to correlate shifted
sequence and received signal. So at least NT seconds are necessary to find
the right shift time to impose to PN sequence replica.
23
Rapid sequential search (RASE)
•PN signals initial delay acquisition method, studied by Ward (1965) as
alternative to matched filter correlator (computationally too much complex) and
sliding correlator (too much slow)
•Thought for spreading sequences generated by single shift register devices
(m-sequences). It’s applicability to double register devices (Gold, Kasami) is
not simple.
•
•PN received signal
•(delayed) •
•1•
•2•
•3•
•…………………..•
•r•–• 1 •
•r•
•
 b•n
•
•C•1•
•Close loop switch
•And open input
•Switch when
•Register is full
•start •generation •
•C•2•
•C•3•
•C•r••-1•
•C•r•
•Micro •–• switch•
•(C•i•= 1 connection) •
24
Rapid sequential search (RASE)
•We suppose to receive a delayed binary sequence without any additional
noise. Let sequence chip flow into the shift register until it’s full: so we’ll have
generator initialization. PN sequence generation starts from the following chip
after latest chip loaded into the shift register.
•RASE effective scheme
starts from received and
coherent-demodulated
signal, sampled with a
sample period equal to
chip time, quantized on
two levels (+1 and -1)
by an hard limiter
•Into the shift register a PN sequence initial phase is loaded, but it’s not
effective, but it’s a noisy estimation.
25
Rapid sequential search (RASE)
•When register is full, generator loop is closed and the obtained PN sequence
is correlated with received PN signal. If correlation overcomes a pre-set
threshold, we assume that synchronism is acquired, else another initial phase
is loaded.
•RASE search scheme is quicker than serial search: time required for
acquisition is typically reduced by a factor equal to 20.
•RASE computational complexity is practically the same than serial search.
•RASE algorithm performances are heavily conditioned by initial phase
estimation effectiveness, done demodulating and quantizing signal on two
levels, before every de-spreading operation.
•RASE scheme is vulnerable to jamming noise and multiple access
interference. The scheme is so useful in single user ambient an on AWGN
channel
26
Correct and false synchronization probability
•In case of serial search is possible to obtain analytical expressions for
correct synchronization probability (Pdet) and false synchronization probability
(Pfals).
•Correct synchronization probability is given by:

Pdet 
 2
1
2
e
u    2 2
TH
where
0  
I 0 ( x)
Tc
b
 u 
I 0  2 du
  
 2 ˆ
N 0TI
4
PTI2   
 ˆ
1  
2  Tc 
2
Is deviation of coarse-estimated acquisition delay
value from effective value
First type Bessel modified function, zero order
(defined in numerical form)
TH: detection threshold, P: transmitted signal power
27
Correct and false synchronization probability
•False synchronization probability is given by:
Pfals  e
TH 2 2
N 0TI
 ˆ
4
2
•It’s possible to obtain suitable performances, from correct and false
synchronization probabilities point of view, imposing the right values of
integration time TI and detection threshold TH
•In a signal recognition system, false alarm probability and correct detection
probability are related one another and they are in a trade-off condition:
imposing a too low false alarm probability could lead to an unsatisfying correct
detection probability
28
Parallel acquisition
•Former schemes suffer of some problems, which condition their applicability,
like computational complexity (matched filter), very long acquisition time (serial
search), restricted performances (RASE).
•To obviate some of these problems, parallel acquisition strategy is used. It
consists in correlating received PN sequence in parallel with a certain number
of generated PN sequence cyclically delayed replica.
•Correlators outputs are combined together by algorithms finalized to
minimize false alarm probability.
•Computational complexity is very high, but robustness and acquisition
velocity, even in real conditions, make these algorithms very interesting for
future applications.
29
Tracking: introduction
•While acquisition is called coarse acquisition too, PN sequence tracking is
called also fine acquisition.
•When coarse acquisition ends, acquiring received signal initial delay with a
time resolution equal to Tc/b, tracking phase begins, to keep synchronism
between received PN signal and receiver-generated PN sequence. Tracking
phase tracks also carrier phase, in case of coherent demodulation.
•Tracking algorithm must compensate temporization jitter of generated PN
sequence compared to ideal synchronism. Jitter values are fractions of chip
time.
•In DS/SS systems two basic schemes are generally used:
•Delay-Locked-Loop (DLL)
•Tau-Dither-Loop (TDL)
30
Delay-Locked-Loop
d()
DLL Tracking Scheme
31
Delay-Locked-Loop algorithm
•Correlation between received signal and two versions of generated replica:
one delayed of d Tc/2 b seconds and one anticipated of d seconds.
•These two correlations are band-pass filtered, processed by a square-law
envelope detection block and subtracted one another.
•Subtraction result is filtered by a loop-filter (generally low-pass): it generates a
pilot-signal d() for VCC. This signal is called composite correlation. VCC
generates clock signal for receiver PN sequence generator.
•When synchronism is not correct, correlators outputs are not equal and VCC is
piloted by a non-zero signal which shifts forward or backward generated PN
sequence.
•Trade-off point is obtained when correlators outputs are equally spaced
compared to peak value, so sequence generator is exactly synchronized.
32
Delay-Locked-Loop algorithm
Composite correlations


d ( )  P Rc2   d   Rc2   d 
     2
d
 se -Tc   d     d
1 
T

c


2
R c    d        2
d
 se   d    Tc   d
1 
T
c


0 elsewhere
33
Tau-Dither-Loop
•TDL system is simpler to implement than DLL because it has only one
correlation branch. From tracking precision point of view, performances are a
little lower than DLL system.
34
Tau-Dither-Loop algorithm
•TDL algorithm starts shifting forward and backward of d Tc/2 b seconds PN
sequence replica.
•Received signal s(t) is multiplied by two versions with different phase of PN
sequence replica; only one branch is used and the two signals are so
summed.
•Result signal w(t) is, alternatively, equal to:
w1 (t )  s(t )c(t     d )  n(t )
w2 (t )  s(t )c(t    d )  n(t )
•Two dither signals q1(t) and q2(t)
are used; they are obtained by
square wave generator q(t) in the
way shown in figure:
Dither signals
TDTc
35
Tau-Dither-Loop algorithm
•Dither signals are used to displace phase to PN sequence alternatively with
rate fD = 1/TD where TD >> Tc. Using these two signals is possible to write:
w(t )  q1 (t )w1 (t )  q2 (t )w2 (t )  n(t )
•Square-law envelope detector output is alternatively equal to:
u1 (t )  PRc2 (t  d )  n(t )
u2 (t )  PRc2 (t  d )  n(t )
•It’s possible to write u(t) as:
u(t )  q1 (t )u1 (t )  q2 (t )u2 (t )  n(t )  q1 (t )PRc2 (t  d )  q2 (t )PRc2 (t  d )  n(t )
36
Tau-Dither-Loop algorithm
•This expression is similar to composite correlation in DLL systems; if q(t) as
carrier signal for u(t):
z (t )  q(t )q1 (t )u1 (t )  q2 (t )u2 (t )  n(t )  q1 (t )u1 (t )  q2 (t )u2 (t )  n(t ) 
 Pq1 (t ) Rc2 (t   d )  Pq2 (t ) Rc2 (t   d )  n(t )
•Under multiplicative factors given by q1(t) and q2(t) signals, composite
correlation is obtained, like in DLL systems. If loop filter has narrow band
compared to fD, q1(t) and q2(t) are mediate and following control signal is
obtained:


d ( )  P E (q1 (t ))Rc2   d   E (q2 (t ))Rc2   d   n(t ) 

P 2
P
Rc (   d )  Rc2 (   d )  n(t )
2
2
•VCC control signal is similar to the ones obtained in DLL scheme, but has half
power; this the reason of performances decrement.
37