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Code acquisition 3.1 OPTIMUM SOLUTION In this case, the theory starts with a simple problem where, for a received signal rt= st, θ + nt, we have to estimate a generalized time invariant

Code acquisition

3.1 OPTIMUM SOLUTION

In this case, the theory starts with a simple problem where, for a received signal r(t)=

s(t, θ ) + n(t), we have to estimate a generalized time invariant vector of parameters θ (frequency, phase, delay, data, ) of a signal s(t, θ ) in the presence of Gaussian noise

n(t) The best that we can do is to find an estimate ˆθ of the parameter θ for which the aposterior probability p( ˆ θ /r) is maximum; hence the name maximum aposteriorprobability (MAP) estimate In other words, the chosen estimate based on the received

signal r is correct for the highest probability Practical implementation requires us to

locally generate a number of trial values ˜θ , to evaluate p( ˜ θ /r)for each such value and

then to choose ˜θ = ˆθ for which p( ˜θ/r) is maximum In this chapter, we focus only on code acquisition and parameter θ will include only code delay θ = {τ} and become a

scalar Analytically, this can be expressed as

Very often, in practice, evaluation of p( ˜ θ /r)in closed form is not possible By using theBayesian rule for the joint probability distribution function

p(r, ˜ θ ) = p(r)p( ˜θ/r) = p( ˜θ)p(r/ ˜θ) ( 3.2) and assuming a uniform prior distribution of θ , maximizing p( ˜ θ /r)becomes equivalent

to maximizing p(r/ ˜ θ ), a function that can be determined more easily This algorithm is

known as maximum likelihood (ML) estimation and can be defined analytically as

It is straightforward to show that in the case of Gaussian noise, the ML principle

necessi-tates the search for that value of θ that would maximize the likelihood function defined as

ISBN: 0-470-84825-1

where s(t, ˜ θ ) is the locally generated replica of the signal with a trial value ˜θ Forthe given signal power, the second term in the previous equation is a constant so thatthe maximization is equivalent to the maximization of the first term only This can beexpressed as

λ( ˜ θ )=

Instead of searching for the maximum of λ( ˜ θ )in a so-called open loop configuration, an

equivalent procedure would be to find the zero of the first derivative of λ( ˜ θ )

so-called early–late tracker

ELT ⇒ ˆθ = arg zero{E(t, ˜θ) − L(t, ˜θ)} ( 3.8)

2θ

r(t)s(t, ˜ θ − θ)dt

( 3.9)

In the case of code synchronization, θ = τ and the ML synchronizing receiver implied by

equation (3.5) should, in principle, create all possible time-offset versions of the knowncode waveform, correlate all of them with the received data and choose the ˜τ corre-

sponding to the largest correlation as its estimate, ˆτML Owing to the continuous range

of values of τ , this is not possible in practice and some type of range quantization is necessary The resulting candidate values are called cells, and the initial parameter esti-

mation problem is translated into a multiple-hypothesis problem: to locate the cell most

likely to contain the unknown offset, given this piece of data This is exactly the coarse code synchronization or code acquisition problem, the result of which is to resolve the

code phase (or the ‘epoch’) ambiguity within the size of the cell Since this remainingerror is typically larger than desired, further operations are required in order to reduce

it to acceptable levels This remaining part of the synchronization task, namely, that of

PRACTICAL SOLUTIONS 45

fine synchronization or code tracking, is performed by one of the available code-tracking

loops, which we discuss in the next chapter

Once the nature and size of these cells have been determined, the next question is how

to go about performing the search most successfully Clearly, the strategy will depend on avariety of factors such as criteria of performance, degree of complexity and computationalpower available (directly related to cost), prior available information about the location ofthe correct cell and so on A brute-force approach would try to create a bank of parallelcorrelation branches, each matched to a possible quantized value of the timing offset;

it would then process the received waveform through all of them simultaneously, pickthe largest and declare a candidate solution Unless the uncertainty region (number ofcells) is small, corresponding to either a small code period or a small initial uncertainty,

such a solution (which we may call the totally parallel solution) becomes obviously

unwieldy in complexity very quickly We note, however, that small uncertainty regionsmay be encountered in a nested design, whereby a multitude of different-period codes arecombined for precisely the purpose of aiding acquisition Furthermore, neural networkstructures are currently being explored for this purpose, where the neural network istrained for all possible such values Such a scheme would emulate the spirit (if not theexact statistical processing) of the above solutions

3.2 PRACTICAL SOLUTIONS

In practice, most of the time total parallelism is out of the question when the number

of cells is very large (although it appears doable for smaller uncertainty regions) andsimpler solutions are necessary One of the most familiar of such approaches is thesimple technique of serial search, where the search starts from a specific cell and seriallyexamines the remaining cells in some direction and in a prespecified order until thecorrect cell is found Hence, serial search techniques do not account for any additionalinformation gathered during the past search time, which could conceivably be used toalter the direction of search toward cells that show increased posterior likelihood of beingthe correct ones A serial search starts from a cell that could be chosen totally arbitrarily(no prior information), or by some prior knowledge about a likely cell, and proceeds

in a simple and easily implementable predirected manner When the uncertainty space(collection of all possible cells) is two-dimensional (delay and frequency offset) andsearching all possible cells serially appears to be very time consuming, a speedup may

be achieved by employing a bank of filters, each matched to a possible Doppler offset.The same idea can be applied to the one-dimensional case (no frequency uncertainty),where now a bank of correlators may be employed, each starting from a different point ofthe uncertainty region This effectively amounts to dividing the search in many parallelsubsearches and therefore reducing the total search time by a proportional amount.One should be aware that although it holds true that only one cell contains the exactdelay and Doppler offsets of the incoming code, the set of desirable cells acceptable tothe receiver includes a number of cells adjacent to the exact one Indeed, the receiver willterminate acquisition and initiate tracking, the first time a cell is reached (and correctlyidentified), which is close enough to true synchronization so that the tracking loop can pull

in and perform the remaining synchronization operation successfully All these desirable

cells are collectively called hypothesis H1, and the remaining nondesirable ‘out-of-sync’

cells comprise hypothesis H0 As an example, consider the case in which the receiver

examines the code delay uncertainty in steps of half a chip time (δt = T c /2) and there

is no frequency uncertainty Then, all four cells located in the interval ( −T c , T c )around

the true delay of the incoming code are included in hypothesis H1, since some amount

of code correlation exists for each one of these cells, an amount that can initiate thecode-tracking loop

The above definition of cells and hypotheses implies that each test does not pertain

to a single value of the unknown parameter τ , but rather to a range of values It is

straightforward to show that, under mild conditions and approximations pertaining to thepseudorandom nature of the code, this reformulated hypothesis testing results in a statistic(correlation) and threshold setting that do not depend on the given (tested) value of theunknown parameter (a uniformly most powerful test) This is because the threshold value

is set by the desirable probability of false alarm per cell (see below), which is independent

decide whether the currently examined cell is a desirable (H1) one The process continuesuntil one such cell is correctly identified At that point, acquisition is terminated andtracking is initiated

3.3 CODE ACQUISITION ANALYSIS

The serial code acquisition can be represented by using the signal flow graph theory Eachcell is represented by a node of a graph and transitions between the nodes depend on theoutcome of the decision in a given cell Branches connecting the nodes characterize thesetransitions To motivate the operation in a transform domain, let us consider the simplemodel of a process represented by the graph in Figure 3.1 and evaluate the probability

p ac (t) that the process will move from a to c in exactly t seconds.

To do this, we will introduce an additional variable τ to designate the time needed for the process to move from a to b, characterized by the probability p ab (τ ) The parameter

CODE ACQUISITION ANALYSIS 47

p ac (t, τ ) represents the joint probability that the process moves from a to c in t seconds and takes τ seconds to move from a to b This probability can be represented as

In other words, the overall probability p ac (t) is a convolution of the two intermode

transition probabilities p ab and p bc It is clear that for the graph with a large number

of nodes we will have to deal with multiple convolutions giving rise to computationalcomplexity In this case, people being involved in electrical engineering prefer to move to

a transform domain, either Laplace (s) domain for continuous variables or into z-domain for desecrate variables This leads to using z-transform for the decision process flow graph

representation and multiple convolutions will be now replaced with multiple products

making the calculus much simpler If p ij (n)is the probability for the process to move

from node i to node j in exactly n steps, then its z-transform

P i,j (z)=∞

is called the probability generating function For the analysis to follow, we will need a

few relations derived from this definition First of all, the first and the second derivative

of this function can be represented as

where T is the cell observation time that is, the time needed to create the decision variable

that will be referred to as dwell time For the variance, we start with the definition

In what follows, we will use these few relations to analyze serial search code acquisition

In order to get an initial insight into this method, we will assume that there are q cells to

be searched Parameter q may be equal to the length of the pseudonoise (PN) code to be searched or some multiple of it For example, if the update size is one-half chip, q will

be twice the code length to be searched Further assume that if a ‘hit’ (output is abovethreshold) is detected by the threshold detector, the system goes into a verification modethat may include both, an extended duration dwell time and an entry into a code loop

tracking mode In any event, we model the ‘penalty’ of obtaining a false alarm as Kτ d second and the dwell time itself as τ d second If a true hit is observed, the system hasacquired the signal, and the search is completed Assume that the false alarm probability

PFA and the probability of detection PDare given We will also assume that only one cellrepresents the synchro position Let each cell be numbered from left to right so that the

k th cell has a priori probability of having the signal present, given that it was not present

resents τ d seconds and Kτ d seconds are represented in z-transform by z K Consider node

1 The a priori probability of having the signal present is P1= 1/q, and the probability

of it not being present in the cell is 1− P1 Suppose the signal was not present Then we

advance to the next node (node 1a); since it corresponds to a probabilistic decision and not a unit time delay, no z multiplies the branch going to it At node 1a a false alarm may occur, with probability PFA= α This would require one unit of time to decide (τ d s)

and then K units of time (Kτ d s) are needed in verification mode to determine that therewas a false alarm False alarms will not occur with probability (1− α) This would take one dwell time to decide and is represented by (1 − α)z branch going to node 2.

CODE ACQUISITION ANALYSIS 49

Now consider the situation at node 1 when the signal is present If a hit occurs (that is,the signal is detected), then acquisition, as we have defined it, occurs and the process is

terminated in node F denoting ‘finish’ If there was no hit at node 1 (the integrator output

was below the threshold), which occurs with probability 1− PD, one unit of time would

be consumed for such a decision This is represented by the branch (1 − PD)zleading tonode 2 At node 2, in the upper left part of the diagram, either a false alarm occurs with

probability α and delay (K+ 1), or a false alarm does not occur with a delay of 1 unit.The remaining portion of the generating function flow graph is a repetition of the portionjust discussed with the appropriate node changes

At this stage we will assume that only Gaussian noise is present so that PFA and PD

are the same for each cell

By using standard signal flow graph reduction techniques [1], one can show that the

overall transfer function between nodes S (start) and F (finish) can be represented as

with τ d being included in the formula to translate from our unit timescale For the usual

case, when q  1, the mean acquisition time T is given by

CODE ACQUISITION IN CDMA NETWORK 51

As a partial check on the variance result, let PFA→ 0 and PD→ 1 Then we have

σ2 = (qτD)2

which is the variance of a uniformly distributed random variable, as one would expect for

the limiting case The above results provide a useful theoretical estimate of acquisition time for an idealized PN-type system In practice, two basic modifications should be

made to make the estimates reflect actual hardware or software systems First, Dopplereffects should be taken into account The result of code Doppler is to smear the relativecode phase during the acquisition dwell time, which increases or reduces the probability ofdetection depending on the code phase and the algebraic sign of the code Doppler rate TheDoppler also affects the effective code sweep rate, which in the extreme case can reduce it

to zero to cause the search time to increase greatly This topic will be discussed later Thesecond refinement to the model concerns the handover process between acquisition andtracking Typically after a ‘hit’ the code-tracking loop is turned on to attempt to pull thecode into tight lock Further, often in low signal-to-noise ratio (SNR) systems in whichboth acquisition (pull-in) bandwidth and tracking bandwidth are used, multiple code loopbandwidths will be employed in order to soften the transition between acquisition andtracking modes Consequently, the probability of going from the acquisition mode tothe final code loop bandwidth in the tracking mode occurs with some probability lessthan 1 The estimation of this probability is at best a very difficult problem (although,some approximate results have been developed) At high SNRs, this probability quicklyapproaches 1, so it is not a problem At low SNRs, the above formula for acquisition

time should replace PD with PD

with PHO being the probability of handover In the S-band shuttle system, at TRW it was

found that at threshold (C/N0= 51 dB Hz) PHO varied from 0.06 to 0.5 depending upon

the code Doppler Without code Doppler PHOwas 0.25, which, if not taken into account

in the acquisition time equation, would predict the mean acquisition time to be about fourtimes too fast

3.4 CODE ACQUISITION IN CDMA NETWORK

The previous Section 3.3 is limited to the case of spread-spectrum signal in Gaussianchannel In that case, the probability of false alarm in all nonsynchro cells is the same In

a communication radio network, the interfering signal is the sum of Gaussian noise and

overall multiple access interference (MAI) In each cell, i, MAI has a different value so that P FAi = P FAj for each i = j In such a case, under the assumption of a static channel, the serial acquisition process can be modeled again by the graph from Figure 3.2 with PFA

being different for each cell We will first deal with a simpler problem in which the

proba-bility of signal detection PDdoes not depend on MAI Besides being simpler, this model isstill valid for an important class of these systems called quasi-synchronous Code Division

Multiple Access (CDMA) networks In these networks, all users are synchronized withinthe range between zero delay and the position of the first significant cross-correlationpeak Examples of such systems are described for both satellite and land mobile CDMAcommunication systems.

The average acquisition time is obtained by using the same steps as in the ous section The details are presented in Reference [2] The result, after a cumbersomemanipulation of very long equations can be expressed as

By inspection, we can see from equation (3.29) that the minimum average acquisition time

is obtained for large values of parameter α Besides PFA, this parameter also depends

on the position of the cells with high P FAi within the code delay uncertainty region The

set of P FAi , representing the probability distribution function of PFA, will be called MAI

pattern or MAI profile From equation (3.31), one can see that for a large α ,the products

iP FAi should be large This means larger P FAi for larger i That means that hopefully, synchronization will be acquired before we get to the region with high PFAor in the case

of multiple sweep of the uncertainty region, we will have smaller numbers of sweeps ofthe region

In an asynchronous network, MAI takes on different values in all cells including the

synchro cell so that, in general, PD is different In such a case, the average acquisitiontime becomes [2]

CODE ACQUISITION IN CDMA NETWORK 53

Table 3.1 Mean acquisition time for different distributions of PFA and PD. Reproduced from Katz, M and Glisic, S (2000) Modelling of code acquisition process in CDMA

networks-asynchronous networks IEEE J Select Areas Commun., 18(1), 73 – 86, by permission of

It is interesting to compare the expression for mean acquisition time with previous results

Table 3.1 summarizes the results obtained for Case#1, constant PFA and PD, Case#2, valued PFA and a constant PD in quasi-synchronous networks and Case#3, q-valued PFAand q-valued PDin asynchronous networks

q-The form of the three expressions provides an easy insight into the major differences inaverage acquisition times for the three cases In the expression for case#2, when compared

with case#1, PFA should be replaced by PFA and PDin the numerator should be modified

by a factor α given by equation (3.30) The first factor takes into account the average

PFA and the second modification takes into account the position of the initial search cell

with respect to the distribution of P FAi In the expression for case#3, when compared with

case#2, PD should be replaced by ˜PDin addition to a new term  that should be added

to the numerator This term can be expressed as  = 2k(PFA− PRP˜D) .

A first observation is that a sufficient condition for  to be zero is that PFA or PD orboth of them have a constant distribution, that is, at least one of the following conditions is

met: P FAi = PFA, i = 1, 2, , q or P Di = PD, i = 1, 2, , q The proof for it is forward from the definitions of PFA, PR, ˜ PDand  Since PFA≤ PRand ˜PD≤ 1, the sign

straight-for  cannot be determined without knowing the particular distributions of PFA and PD.From the definition of ˜PD, one can see that ˜PD→ PDas long as P Di ≈ 1, i = 1, 2, , q However, it is enough that at least one PDis small to cause a considerable reduction of thefinal value of ˜PD The variation of ˜PDalso depends on the number of cells q.

Results for the normalized average acquisition time (Tacq/T i )are presented in Figure 3.3

Tacq1 is obtained by using the exact results (Case #3 in Table 3.1), Tacq2 is the

approx-imation where the standard expression for T is used (Case #1 in Table 3.1) with

in CDMA networks-asynchronous networks IEEE J Select Areas Commun., 18(1), 73 – 86, by

WIRELESS NETWORKS WITH MULTIPATH

AND TRANSMITTER DIVERSITY

The serial acquisition process of a RAKE receiver consists of two main steps The firststep, called initial acquisition, is defined as the process required to acquire the first path,corresponding to any of the available signal paths The subsequent process required toacquire the remaining paths is referred to as postinitial acquisition

The code delay uncertainty region will be divided into a number of cells in such away that the delay between two adjacent cells is equal to a chip interval The channel

multipath profile will be characterized by a vector D (delays) as

MODELING OF THE SERIAL CODE ACQUISITION PROCESS 55

Figure 3.4 Overall decision process flow graph.

where d l is the probability of having a multipath signal component l chip intervals after

the first signal component (front end of the signal) has been received

In order to simplify the notation, we will assume that there are v− 1 nonsynchro

cells so that all together, with S potential synchro cells (multipath spread), the total number of cells is v + S − 1 The overall decision process flow diagram is shown in Figure 3.4, where nonsynchro cells are represented by v− 1 nodes with corresponding

transfer functions H 0i (z), i = 1, 2, , v − 1 Owing to MAI, H 0i (z)is different for eachcell of code delay uncertainty region

If MAI is approximated as Gaussian noise, then H 0i (z) = H0(z) The vth cell represents

S substates, which are potential synchro states, and its overall transfer function is H1(z)

Figure 3.5 depicts the decision process flow graphs for the synchro cell v, including the

first and last nonsynchro cells

The theory for this case is available in Reference [3] and here we discuss some practical

results First of all, let us assume that the number of cells is much larger than the multipath

spread, that is, v  S In this case, the average acquisition time can be approximated by

and (1 − P d ) represents the probability of missing one of the L available signal paths.

Here we have assumed that the initial acquisition time is much longer than the postinitial

acquisition time If L0 fingers are available, then each finger can search only v/L0 cells,

so reducing further this acquisition time by a factor 1/L

Figure 3.5 Decision process flow graph for the synchro cell (vth cell) and nonsynchro cells (e.g first and (v – 1)th cells shown) [3] Reproduced from Glisic, S and Katz, M (2001) Modeling of code acquisition process for RAKE receiver in wideband CDMA wireless networks with multipath

and transmitter diversity IEEE J Select Areas Commun., 19(1), 21 – 32, by permission of IEEE.

For macro diversity, the model is still valid with v= 1 so that all cells are included

within S cells of the model from Figure 3.5 Within these S cells, there will be in general

LM synchro cells, where M is the number of transmitters If we assume L= 1 (no

multipath), for L0 available RAKE fingers, the initial search will start by partitioning the

uncertainty region into L0 segments When one finger is synchronized, the uncertaintyregion will be partitioned again into equal segments among the remaining fingers Underthese conditions the average acquisition time will be approximated by

In practice, in all existing standards on CDMA a special synchronization channel (SCH)

is used for code acquisition In wideband cdma2000, wideband IS-665 and IS-95, a pilotchannel is used for these purposes This is an unmodulated signal spread by relativelyshort code, which is transmitted continuously This model is applicable directly to thesystems mentioned above For European Telecommunications Standards Institute (ETSI)Universal Mobile Telecommunication System (UMTS), a discontinuous transmission in

TWO-DIMENSIONAL CODE ACQUISITION IN SPATIALLY AND TEMPORARILY WHITE NOISE 57

wideband CDMA wireless networks with multipath and transmitter diversity IEEE J Select.

Areas Commun., 19(1), 21 – 32, by permission of IEEE.

the synchro channel (both primary and secondary) is used and signal detection, based

on code-matched filters, is expected to be used For these applications, the models will

be discussed later in this chapter Figure 3.6 presents the relative error in percentages

defined as ε(%) = (Tacq− Tacq)/Tacq× 100, where Tacq is the exact result [3] and Tacq

the approximation equation (3.36)

3.6 TWO-DIMENSIONAL CODE ACQUISITION IN

SPATIALLY AND TEMPORARILY WHITE NOISE

Code acquisition discussed so far dealt with the search through a discrete number ofpossible relative delay positions between codes, with each position being referred to as a

delay cell In this section, the problem is further extended to consider also the direction (or

angle) of arrival of the received signal A single-antenna receiver can resolve the signal

in the delay domain while, with an antenna array, separation in the angular domain is also

possible An angular cell can be regarded as a 360◦/m angle covered by a directional beam

of antenna array Assuming that the uncertainty region has q delay cells and m angular

Cell 1 Cell 2

q

−1q

q delay cells

m angular cells TS: Transmitting station

RS: Receiving station

Gi : Array gain

Angular cells Delay cells

Figure 3.7 Principle of two-dimensional code acquisition.

cells, the total number of cells to be searched is Q = qm This spatial and temporal

partitioning of the uncertainty region is illustrated in Figure 3.7

We assume that a priori probability of the synchro cell is uniformly distributed in the Q

cell arrangement defining the uncertainty region There is a direct correspondence betweenthe spatial distribution of interference and the interference observed in the angular cells.For instance, a uniform spatial distribution of interferers will be mapped into angular cellswith equal amounts of associated interfering power

The angular division can be carried out by well-established and relatively simple

beam-forming techniques Given an antenna array with m elements, an analog beamformer (e.g Butler matrix) can be used to generate a set of m spatially orthogonal beams in fixed

angular directions A similar result can be achieved by a digital beamformer with a set

of appropriate complex weighting vectors corresponding to preferred steering directions

Note that to achieve m nonoverlapping beams covering the entire spatial uncertainty

region being served, a corresponding number of antenna elements is required In order

to simplify the problem formulation, it will be assumed that within each angular cell the

array gain G i corresponds to the maximum array gain, G i = m, i = 1, 2, , m.

As an initial step, it is also assumed that the angular spread of the signal ing on the antenna array is smaller than the beamwidth generated by the array It canthen be assumed that the impact of the received signal is seen only from one angularcell, occupying only one delay cell (e.g single-path channel) The extension dealing withmultipath channels is available in Reference [4] The discussion is limited to the situa-tion in which interference is both temporarily and spatially white The interference power,

imping-uniformly distributed within the angular uncertainty region, is denoted by σ I2 The

interfer-ence power in the ith sector is σ I i2, i = 1, 2, , m, where σ2

TWO-DIMENSIONAL CODE ACQUISITION IN SPATIALLY AND TEMPORARILY WHITE NOISE 59

where the indexes 1D and 2D correspond to one-dimensional and two-dimensional search

domains, A is the signal amplitude received by one antenna element and S is the

cor-responding signal power For simplicity, from now on we consider SIR as SNR, andinterference as noise

3.6.1 Performance in a single-path channel

The available Q cells are serially searched in the angular and delay domains A single synchro cell is associated with the single-path channel signal (L= 1) Basically, thecells could be searched by following either a fix angle/sweep delay (FASD) or a fixdelay/sweep angle (FDSA) procedure In the former approach, the search is carried out

by serially searching (sweeping) through the q delay cells of a given angular cell This

(time-domain) procedure is repeated on each consecutive angular cell until the synchro cell

is detected In the latter case, a given delay cell is searched first through the m angular cells

and the process is similarly repeated in the consecutive delay cells Figure 3.8 illustratesthe principles of FASD and FDSA search strategies

dc1 dc2 dc3

dc

q

ac m

dc

q

ac m

FDSA (Fix delay/Sweep angle)

.

Figure 3.8 Principle of FASD and FDSA search strategies.