Torrieri D. Principles of Spread-Spectrum Communication Systems

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Figure 6.20: Upper bound on uplink capacity per megahertz for

and

component, and to assess the power-allocation requirement of the mobile.

A base station synchronously combines and transmits the pilot and all the

signals destined for mobiles associated with it. Consequently, all the signals fade

together, and the use of orthogonal spreading sequences will prevent intracell

interference and, hence, a near-far problem on a downlink, although there will

be interference caused by asynchronously arriving multipath components. The

orthogonal sequences can be generated from the rows of a Hadamard matrix.

The orthogonality, the energy-saving sharing of the same pilot at all covered

mobiles, and the coherent demodulation of all transmitted signals are major

advantages of the downlinks. However, interference signals from other base

stations arrive at a mobile asynchronously and fade independently, thereby

significantly degrading performance.

Although there is no near-far problem on the downlinks, power control is

still desirable to enhance the received power during severe fading or when a

mobile is near a cell edge. However, this power enhancement increases inter-

cell interference. Downlink power control entails power allocation by the base

station in a manner that meets the requirements of the individual mobiles as-

sociated with it. Let denote the total power received by mobile from base

station If this mobile is associated with base station 0, then the SINR at the

mobile is

6.5.

MULTIUSER DETECTORS

349

where is the total number of base stations that produce significant power

at mobiles in a cell or sector, is the fraction of the base-station power that is

assigned to mobiles rather than to the pilot, and is the fraction of the total

power for mobiles in a cell or sector that is allocated to mobile Typically,

one might set which entails a 1-dB loss due to the pilot. Let R denote

the SINR required by network mobiles for acceptable performance. Inverting

(6-170), it is found that R is achieved by all mobiles in a cell or sector if

An outage occurs if the demands of all K mobiles in a cell or sector cannot be

met simultaneously. Thus, no outage occurs if (6-171) is satisfied and

where is the voice-activity indicator such that with probability and

with probability If the left-hand side of (6-172) is strictly less than

unity, then the transmitted power produced by base station 0 can be safely

lowered to reduce the interference in other cells or sectors. Combining (6-171)

and (6-172), a necessary condition for no outage is

The assignment of mobile to base station 0 implies the constraint that

except possibly during a soft handoff. A com-

plete performance analysis with this constraint is difficult. Simulation results

[19] indicate that the downlink capacity potentially exceeds the uplink capacity

if the orthogonal signaling is not undermined by excessive multipath.

6.5

Multiuser Detectors

The conventional single-user direct-sequence receiver of Figure 2.14 is optimal

against multiple-access interference only if the spreading sequences of all the

interfering signals and the desired signal are orthogonal. Orthogonality is possi-

ble in a synchronous communication network, but in an asynchronous network,

it is not possible to find sequences that remain orthogonal for all relative delays.

Thus, the conventional single-user receiver, which only requires knowledge of

the spreading sequence of the desired signal, is suboptimal against asynchronous

multiple-access interference. The price of the suboptimality might be minor if

the spreading sequences are carefully chosen and the noise is relatively high,

especially if an error-control code and a sector antenna or adaptive array are

used. If a potential near-far problem exists, power control may be used to limit

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its impact. However, power control is imperfect, entails a substantial overhead

cost, and is not feasible for peer-to-peer communication networks. Even if the

power control is perfect, the remaining interference causes a nonzero error floor,

which is a minimum bit error probability that exists when the thermal noise is

zero. Thus, an alternative to the conventional receiver is desirable.

A multiuser detector is a receiver that exploits the deterministic structure

of multiple-access interference or uses joint processing of a set of multiple-

access signals. An optimum multiuser detector almost completely eliminates

the multiple-access interference and, hence, the near-far problem, thereby ren-

dering power control unnecessary, but such a detector is prohibitively complex

to implement, especially when long spreading sequences are used. A more

practical multiuser detector alleviates but does not eliminate the power-control

requirements of a cellular network on its uplinks. Even if a multiuser detec-

tor rejects intracell interference from mobiles within a cell, it cannot reject

intercell interference, which arrives from mobiles associated with different base

stations than the one receiving a desired signal. Since intercell interference is

typically more than one-third of the total interference on an uplink, even ideal

multiuser detection will increase network capacity by a factor less than three.

Multipath components can be accommodated as separate interference signals

or rake combining may precede the multiuser detection. Though suboptimal

compared with ideal multiuser detection, multiuser interference cancellers bear

a much more moderate implementation burden and still provide considerable

interference suppression. However, it appears that accurate power control is still

needed at least for initial synchronization and to avoid overloading the front

end of the receiver. Third-generation CDMA systems use adaptive interference

cancellation but retain a closed-loop power-control subsystem.

Optimum Detectors

Consider a DS/CDMA network with K users, each of which uses PSK to trans-

mit a block of N binary symbols. A jointly optimum detector makes collective

symbol decisions for K received signals based on the maximum a posteriori

(MAP) criterion. The individually optimum detector selects the most proba-

ble set of symbols of a single desired signal from one user based on the MAP

criterion, thereby providing the minimum symbol error probability. In nearly

all applications, jointly optimum decisions would be preferable because of their

lower complexity and because both types of decisions will agree with very high

probability unless the symbol error probability is very high. Assuming equally

likely symbols are transmitted, the jointly optimum MAP detector is the same

as the jointly optimum maximum-likelihood detector, which is henceforth re-

ferred to as the optimum detector.

For synchronous communications in the presence of white Gaussian noise,

the symbols are aligned in time, and the detection of each symbol of the desired

signal is independent of the other symbols. Thus, the optimum detector can

be determined by considering a single symbol interval Let

denote the symbol transmitted by user The customary (highly idealized)

6.5.

351

assumption is that a perfect carrier synchronization enables the receiver to

remove a common carrier frequency and phase. Thus, the composite baseband

received signal is

where is the received symbol amplitude from user is the unit-energy

spreading waveform of user and is the baseband Gaussian noise.

If it is assumed that each of the K signals has a common carrier frequency but

a distinct phase relative to the phase of the receiver-generated synchronization

signal, then each in (6-174) is replaced by where is the relative

phase of the signal from mobile

Assuming that all possible values of the symbol vector are

equally likely, the optimum detector is the maximum-likelihood detector [22],

[23], which selects the value of d that minimizes the log-likelihood function

subject to the constraint that The vector of the cross correla-

tions between and the spreading sequences is denoted by

where

Let A denote the K × K diagonal matrix with diagonal components

Let R denote the K × K correlation matrix with elements

where and because the spreading waveforms are normalized

to unit energy. Expanding (6-175), dropping an integral that is irrelevant to

the selection of d, and then substituting (6-176) and (6-177), we find that

the maximum-likelihood detector selects the value of d that maximizes the

correlation metric

subject to the constraint that or

This equation implies that the optimum detector uses a filter bank of K

parallel correlators. Correlator computes given by (6-176) and can be

implemented as the single-user detector of Figure 6.15. Equation (6-178) also

indicates that the K spreading sequences must be known so that R can be

calculated, and the K signal amplitudes must be estimated. Short spreading

sequences are necessary or R must change with each symbol. The optimum

detector is capable of making joint symbol decisions for all K signals or merely

the symbol decisions for a single signal.

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CHAPTER 6. CODE-DIVISION MULTIPLE ACCESS

As an example, consider synchronous communications with K = 2 and

After the elimination of terms irrelevant to the selection, (6-178) im-

plies that the optimum detector evaluates

for the four pairs with and The pair that maximizes C provides

the decisions for and

For asynchronous communications over the AWGN channel, the derivation

of the maximum-likelihood detector is analogous but more complicated [22],

[23]. A major difference is that a desired symbol overlaps two consecutive sym-

bols from each interference signal. The optimum detector uses a filter bank of

K parallel correlators, but N symbols from each correlator must be processed

to make decisions about NK binary symbols. The vector d is NK ×

with

the first N elements representing the symbols of signal 1, the second N ele-

ments representing the symbols of signal 2, etc. The detector must estimate

the transmission delays of all K multiple-access signals, and the NK × NK

correlation matrix R has components that are partial cross correlations among

the signals. In principle, the detector must compute correlation metrics

and then select K symbol sequences, each of length

, corresponding to the

largest correlation metric. The Viterbi algorithm simplifies computations by

exploiting the fact that each received symbol overlaps at most 2(K – 1) other

symbols. Nevertheless, the computational complexity increases exponentially

with K.

In view of both the computational requirements and the parameters that

must be estimated, it is highly unlikely that the optimum multiuser detector

will have practical applications. Subsequently, alternative suboptimal multiuser

detectors are considered. All of them follow carrier removal with a filter bank

of correlators.

Decorrelating detector

The decorrelating detector may be derived by maximizing the correlation metric

of (6-178) without any constraint on d. For this purpose, the gradient of

with respect to the real-valued vector x is defined as the column

vector with components From this definition, it

follows that for column vectors x and

If A is an symmetric matrix, then expressing in component form

and using the chain rule yields

Applying (6-179) and (6-180) to the correlation metric, we find that

implies that C is maximized by where

provided that R is invertible. Since each component of the vector is a

positive multiple of the corresponding component of there is no need to

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353

Figure 6.21: Architecture of decorrelating detector and MMSE detecter. Filter

bank comprises parallel correlators.

solve for A suitable estimate of the transmitted bits is

where each component of the vector sgn(x) is the signum function of the cor-

responding component of the vector x. The signum function is defined as

and The decorrelating detector, which

implements (6-182), has the form diagrammed in Figure 6.21. For asynchro-

nous communications, each of the K correlators in the filter bank produces N

bits. The accumulator constructs the NK-dimensional vector and the linear

transformer computes The decision devices evaluate (6-182) to produce

A second derivation of the decorrelating detector assumes that the detector

has the filter bank as its first stage. If (6-174) gives the input, then the output

of this stage is

where n is the NK-dimensional noise vector. This equation indicates that the

coupling among components of d, which causes the correlation among compo-

nents of is due solely to the presence of the matrix R. The effect of this

matrix is removed by computing

If (6-182) is used to determine the NK transmitted bits, the multiple-access

interference is completely decorrelated from

A third derivation assumes the presence of the filter bank. If zero-mean,

white Gaussian noise with two-sided power spectral density enters each

correlator, then a straightforward calculation indicates that the NK × NK

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covariance matrix of is

The probability density function of given Ad is

The maximum-likelihood estimate of Ad is the estimate that maximizes (6-

186) or, equivalently, minimizes the log-likelihood function

Using (6-179) and (6-180), we again obtain the estimate given by (6-181), which

leads to given by (6-182).

Although the decorrelating detector eliminates the multiple-access interfer-

ence, it increases the noise by changing n to From (6-185), and the

symmetric character of R, it follows that the covariance matrix of the noise

vector entering the decision devices is

The variance of the noise that accompanies one of the symbols of user is

Therefore, the symbol error probability is

where is the symbol energy. The symbol error probability for single-

user detection by user in the absence of multiple access interference is

Thus, the presence of multiple-access interference requires an increase of energy

by the factor when the decorrelating detector is used if a specified

is to be maintained.

As an example, consider synchronous communications with K = 2 and

The correlation matrix and its inverse are

Equation (6-182) indicates that the symbol estimates are

and Since

6.5.

MULTIUSER DETECTORS

355

If the required energy increase or shift in each curve is less than

1.25

dB.

To demonstrate analytically the advantage of the decorrelating detector,

consider synchronous communications and a receiver with a filter bank of K

conventional detectors. Each conventional detector is a single-user matched

filter. If perfect carrier synchronization removes a common phase shift of all

the signals and produces the baseband received signal of (6-174), then (6-176)

implies, that the output of detector is

The set of K symbols is estimated by

By symmetry, we can assume that in the evaluation of the symbol error

probability. Let denote the (K – 1)-dimensional

vector of all the

Conditioning on and calculating that we find that the

conditional symbol error probability for user is

where

If all symbol sets are equally likely, then the symbol error probability for user

where is the choice of the vector which can take values.

For K= 2 with and (6-195) to (6-197)

yield the symbol error probability for user 1:

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The symbol error probability for user 2 is given by (6-198) with the roles of

and interchanged. The second term in (6-198) is usually negligible compared

with the first one if Thus, if

then a comparison of (6-198) with (6-192) indicates that decorrelating detector

usually outperforms the conventional detector. However, if is sufficiently

small, then the conventional detector gives a lower than the decorrelating

detector.

In a more realistic model of the decorrelating and conventional detectors,

the received signal in Figure 6.21 is passband. Correlator uses a synchronized

carrier to remove carrier at the common frequency Since each carrier has

a distinct phase

the elements of the correlation matrix are

if For synchronous communications with K = 2, (6-192) and (6-198)

with then represent the conditional symbol error probability

given the value of Averaging over is necessary to obtain

and

Compared with the optimum detector, the decorrelating detector offers

greatly reduced, but still formidable, computational requirements. There is no

need to estimate the signal amplitudes, but the transmission delays of asynchro-

nous signals must still be estimated. The inversion of the correlation matrix

R in real time is not possible for asynchronous signals with practical values

of NK. Suboptimal partitioning and short spreading sequences are generally

necessary and degrade the theoretical performance given by (6-189).

Minimum-Mean-Square-Error Detector

The minimum-mean-square-error (MMSE) detector is the receiver that results

from a linear transformation of by the K × K matrix L such that the metric

is minimized. Let denote the solution of the equation

Let denote the trace of the matrix B. Since

for a vector

6.5.

MULTIUSER DETECTORS

357

Substitution of this equation into (6-201) and the application of (6-202) yields

which proves that minimizes M. If the data symbols are independent and

equally likely to be +1 or –1, then where I is the identity matrix.

Using this result, (6-183), (6-185), E[n] =

and the independence of d and n,

we obtain

Substitution of these equations into (6-202) yields

provided that the inverses exist. Since A and, hence, are diagonal matrices

with positive diagonal components if all

signals are active, (6-206) may be

simplified to the linear transformation matrix

without any change in the MMSE estimate of the transmitted symbols:

The MMSE detector has the structure of Figure 6.21.

The MMSE and decorrelating detectors have almost the same computational

requirements, and they both have equalizer counterparts, but they differ in

several ways. The MMSE detector is near-far resistant, but does not obliterate

the multiple-access interference and, hence, does not completely eliminate the

near-far problem. However, it does not accentuate the noise to the degree

that the decorrelating detector does. As and the MMSE

estimate approaches the decorrelating detector estimate. As increases, the

MMSE estimate approaches that of the conventional detector given by (6-194),

and the symbol error probability generally tends to be lower than that provided

by the decorrelating detector. A disadvantage of the MMSE detector is that

the signal amplitudes must be estimated so that A in (6-207) can be computed.

For either the MMSE or decorrelating linear detectors to be practical, it

is necessary for the spreading sequences to be short. Short sequences ensure

that the correlation matrix R is approximately constant for significant time

durations if the communication channel and the amplitudes of the interference

signals are slowly varying. The price of short sequences is a security loss and the

occasional but sometimes persistent performance loss due to a particular set of

relative signal delays. Even with short spreading sequences, adaptive versions

of the MMSE detector are much more practical than the nonadaptive versions

of either linear detector.