course acquisition and tracking in ds/ss systems
DESCRIPTION
Corso di Comunicazioni Mobili. Course acquisition and tracking in DS/SS systems. Prof. Carlo Regazzoni. References. 1.J.G. Proakis, “Digital Communications” (3rd Edition), McGraw-Hill: 1995. R.C. Dixon, “Spread Spectrum Systems with Commercial Applications” (3rd Edition), Wiley: 1984. - PowerPoint PPT PresentationTRANSCRIPT
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Corso di Comunicazioni Mobili
Prof. Carlo Regazzoni
Course acquisition and tracking in DS/SS
systems
2
References1. 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.
593-619, Maggio 1980.
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PN in SS Systems: Introduction
Channel Coder Modulator
Data
PN sequences Generator
Channell Decoder
De-modulator
PN sequences Generator
Data
SS Transmitter
SS Receiver
Schematic shows DS-SS system basic elements; there are two pseudo-noise 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.
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PN in SS Systems: Introduction
PN Generator
Modulo-2 Adder
Balanced Modulator
FEC Coder
Local Oscillator
Data
BPSK
Signal
bi ai
ci
To demodulate the received signal, the synchronism among the PN sequences is needed
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:
iii cba ai = 1 if bi = ci
ai = 0 if bi ≠ ci
ii cb
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PN Sequences-characteristics
1 2 3 ………………….. r – 1 r
C1 C2 C3 Cr-1 Cr
nb
Micro – switch (Ci= 1 connection)
A possible PN sequences generator is shown in figure:
bn can be written as: rnrnnn bCbCbCb .......2211
Where C1,…,Cr are micro-switch setting coefficients of generator
r is the shift register length
1Pr( 0) (1 )
2nb s 1
Pr( 1) (1 )2nb s Symbol probabilities are:
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PN Sequences - characteristics
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1ˆ
rs is the coefficient of statistical displacementWhere
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 is called also pseudo-noise length and in literature is usually indicated as N.
12 rL
The relation is true when the sequence characteristic
Polynomial is primitive and when
the initial conditions of the shift register are:
12 rNLr
r xCxCxCxP ......1)( 221
)01,......,0,0(),......,( 010 rbb
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.
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M-sequences - definition•Be M a given set of PN binary sequences;
two sequences;Mpp lk ,
•Cross-correlation peak on set M is defined as:
} ,, ,10 : )(max{)( k, llklkc ppMppNM
•Where is cross-correlation
of two sequences, and temporal shift parameter is an integer value, between 0 and N-1.
)()()()()(1
0, Njpjp xyxy
N
jhklk
Attention: is different from asynchronous transmission delay.
c
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M-sequences propertiesM-sequences, generated by a shift register of
length r, have no good correlation properties,
indeed, increasing N, cross-correlation peak
grows in a huge way.
cr N t(r)
3 7 5 5
4 15 9 9
5 31 11 9
6 63 23 17
7 127 41 17
8 255 95 33
9 511 113 33
10 1023 383 65
11 2047 287 65
12 4095 1407 129
In DS/CDMA systems, the usage of PN
sequences set characterized by a cross-
correlation value as lower as possible is
suggested, in order to decrease the MUI
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Gold sequencesBinary 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:
}2)(),(,1{)( rtrtluv ]1,0[ Nl
2/)2(21)( rrt
Where r is the shift register lenght
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Gold sequences
vTuvTuTvuvuvuvuG N 12 ,.......,,,,,),(
The Gold sequences generated by u and v (preferential couple of m-
sequences ) are composed by N + 2 sequences obtained as:
(.)kT
),( vuGG
Where
is a cyclical shift operator
is called Gold set, generated by u and v
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Gold sequences - propertiesAll 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 )()( rtGc
A Gold sequences generator is composed by two m-sequences generators with
shift register and feedback
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Preferential sequences couples
Preferential couples of m-sequences with length N = 31 (r = 5) (a) and N = 63 (r = 6)
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Preferential sequences couplesThe numbers in circles represent vectors, in octal notation, 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
),,( 1 rCC
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Kasami sequences
Optimal binary sequences generated by shift register with even length are called Kasami sequences.
Kasami sets propertiesKasami sets properties
1. Better cross-correlation properties than Gold sequences
2. More difficult to be generated
3. Extremely reduced dimensions (only available sequences)2 N
For these reasons Gold sequences are often used with shift register with even length, in order to serve several users.
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Acquisition - introductionIn 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:
ci
T
ii
T
iii Tbdttgcbdttgcbvcc
12)(1212)(12120
22
0
22 1,0ib i-th bit transmitted value
1,0ic i-th chip of PN sequence value
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PN AcquisitionDS/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
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PN Acquisition
header header
t
t
0
0
T
T
Received signal (Hypothesis: the received signal is cyclical)
PN sequence replica generated by receiver (4 chip delayed)
data
N=7
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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 , and properly quantized.cs Tt
Performances evaluation criteria for PN acquisition Performances evaluation criteria for PN acquisition methodsmethods
•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.
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Matched filter correlator
T1 T2 T3 T4 T5 T6 T7
C7 C6 C5 C4 C3 C2 C1
Y(nTc)
r(nTc)
•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•Result is compared with a suitable threshold, empirically chosen on channel noise
•Correlator operates convolution between received signal and PN sequence replicaFigure 1 acquisition diagram:
•matched filter to a 7 chip-PN sequence g(t)
•Single chips binary values [c1,c2,…,c7]
THcnTrnTTgnTrnTYN
n
nNc
N
n
ccc
1
0
1
0
)(
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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:
cc TllTl )1( : Coarse acquisitionCoarse acquisition: Time resolution = chip time Tc •Over sampling received signal by a factor and increasing delay line taps of the same factor, it’s possible to obtain a coarse acquisition with Tc/ 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)
sec/27.1sec/10*127.10/127 77 GopsopssopsKMF
•If N = 127 and Rc = 10Mc/s
.sec/.sec/ opsNRTNopsK ccMF •Having chip rate Rc and spreading sequence length N, number of elementary operations (sums and multiplications) required is:
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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 TH.
THdttgtryIT
ll
0
)(
c
ll
lT 1
11..,2,1
N
T
Tl
c
00
TTI
Figure 2: Sliding correlator
•Sliding correlator operates a “step-by-step” correlation between received signal and generated PN sequence.
yl
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Serial search (sliding correlator)•When received signal delay is equal to sequence replica pre-set delay (under Tc secondssliding 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
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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
cll T 1
chip rate = Rc
process gain = N
number of elementary operations required:
MFcbSC KopsRopsNRTsNopsK sec/sec/./ 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.
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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.
•
•1• •2• •3• •…………………..• •r •–• 1 • •r•
•C•1• •C•2• •C•3• •C•r•-•1• •C•r•
•
•nb•
•Micro •–• switch• •(C•i•= 1 connection) •
•Close loop switch•And open input•Switch when•Register is full•start •generation •
•PN received signal •(delayed) •
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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.
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Rapid sequential search (RASE)
•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
•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.
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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:
duu
IePTH
u
202
2det
2
2
1
4ˆ 02 ITN
22
12
ˆ
cT
PTI
cT
0where Is deviation of coarse-estimated acquisition delay value from effective value
)(0 xI First type Bessel modified function, zero order (defined in numerical form)
TH: detection threshold, P: transmitted signal power
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Correct and false synchronization probability
•False synchronization probability is given by:
22THfals eP
4ˆ 02 ITN
•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
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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.
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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/, 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)
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Delay-Locked-Loop
DLL Tracking Scheme
d()
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Delay-Locked-Loop algorithm•Correlation between received signal and two versions of generated replica: one delayed of Tc/ seconds and one anticipated of 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.
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Delay-Locked-Loop algorithm
22)(cc
RRPd
elsewhere 0
se 1
se 1
2
2
2
dcdc
d
ddcc
d
dT
T
-TT
Rc
Composite correlations
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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.
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Tau-Dither-Loop algorithm•TDL algorithm starts shifting forward and backward of dTc/ 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:
)()()()(1 tntctstw d )()()()(2 tntctstw d
•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:
TDTcDither signals
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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:
)()()()()()( 2211 tntwtqtwtqtw
•Square-law envelope detector output is alternatively equal to:
)()()( 21 tntPRtu dc
)()()( 22 tntPRtu dc
•It’s possible to write u(t) as:
)()()()()()()()()()()( 22
212211 tntPRtqtPRtqtntutqtutqtu dd cc
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Tau-Dither-Loop algorithm•This expression is similar to composite correlation in DLL systems; if q(t) as carrier signal for u(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:
)()()()()(
)()()()()()()()()()()()(2
22
1
22112211
tntRtPqtRtPq
tntutqtutqtntutqtutqtqtz
dd cc
)()(
2)(
2
)())(())(()(
22
22
21
tnRP
RP
tnRtqERtqEPd
dcdc
cc
•VCC control signal is similar to the ones obtained in DLL scheme, but has half power; this the reason of performances decrement.