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MMSE multiuser detectors 14.1 MINIMUM MEAN-SQUARE ERROR MMSE LINEAR MULTIUSER DETECTION If the amplitude of the user’s k signal in equation 13.7 is A k, then the vector of matched filter

MMSE multiuser detectors

14.1 MINIMUM MEAN-SQUARE ERROR (MMSE)

LINEAR MULTIUSER DETECTION

If the amplitude of the user’s k signal in equation (13.7) is A k, then the vector of matched

filter outputs y in equation (13.10) can be represented as

where A is a diagonal matrix with elements A k

If the multiuser detector transfer function is denoted as M, then the minimum mean-square

error (MMSE) detector is defined as

Therefore, the MMSE linear detector replaces the transformation R−1 of the lating detector by

ISBN: 0-470-84825-1

Sync K

Sync 2

Sync 1 Matched

filter User 2

Matched filter User K

y (t )

Matched filter User 1

[R + σ 2A−2]−1,

bˆ1[i ]

bˆ2[i ]

bˆK[i ]

As an illustration for the two users case we have

MINIMUM MEAN-SQUARE ERROR (MMSE) LINEAR MULTIUSER DETECTION 493

Single-user matched filter Gaussian approximation

Single-user matched filter exact

MMSE exact & approx

filter, b – decorrelator, c – MMSE, d – minimum (upper bound), e – minimum (lower bound).

In the asynchronous case, similar to the solution in Section 13.3 of Chapter 13, the

MMSE linear detector is a K-input, K-output, linear, time-invariant filter with

trans-fer function

[RT[1]z + R[0] + σ2A−2+ R[1]z−1]−1 (14.8)

Performance results are illustrated in Figures 14.3 and 14.4 As expected, in Figure 14.3,the MMSE detector demonstrates better performance than the conventional detector deno-ted as a single-user matched filter receiver (MFR)

In Figure 14.4 bit error rate (BER) is presented versus the near–far ratio for differentdetectors One can see that MMSE shows better performance than decorrelator In thefigure signal-to-noise ratio (SNR) of the desired user is equal to 10 dB

14.2 SYSTEM MODEL IN MULTIPATH FADING

SYSTEM MODEL IN MULTIPATH FADING CHANNEL 495

is the sampled spreading sequence matrix, D = (T + Tm)/T In a single-path channel,

D = 1 due to the asynchronity of users In multipath channels, D ≥ 2 due to the path spread The code matrix is defined with several components (S (n) (0), , S (n) (D))

multi-for each symbol interval to simplify the presentation of the cross-correlation matrix

com-ponents Tm is the maximum delay spread,

Equation (14.2) now becomes

The cross-correlation matrix equation (13.70) for the spreading sequences can beformed as

MMSE DETECTOR STRUCTURES 497

and represents the correlation between users k and k, lth and lth paths, between their

nth and nth symbol intervals

14.3 MMSE DETECTOR STRUCTURES

One of the conclusions in Chapter 13 was that noise enhancement in linear Multi-user

detection (MUD) causes system performance degradation for large product KL In this

section we consider the possibility of reducing the site of the matrix to be inverted by usingmultipath combining prior to MUD The structure is called the postcombining detectorand the basic block diagram of the receiver is shown in Figure 14.5 [4]

The starting point in the derivation of the receiver structure is the cost function

E{|b − ˆb|2}

Matched filter

1, 1

Matched filter

1, L

Matched filter

K, L

Multiuser detection

Matched filter

K, 1

Multipath combining

Multipath combining

r(n)

This result is obtained by minimizing the cost function, and derivation details may be

found in any standard textbook on signal processing Here, R = STS is the signature

sequence cross-correlation matrix defined by equation (14.25) The output of the combining LMMSE receiver is

post-y[post] = (ACH

RCA+ σ2

where (SCA)Hr is the multipath [maximum ratio (MR)] combined matched filter bank

output For nonfading additive white Gaussian noise (AWGN),

L[post]= S(R + σ2(AHA)−1)−1 (14.33)

The postcombining LMMSE receiver in fading channels depends on the channel plex coefficients of all users and paths If the channel is changing rapidly, the optimalLMMSE receiver changes continuously The adaptive versions of the LMMSE receivershave increasing convergence problems as the fading rate increases The dependence on thefading channel state can be removed by applying a precombining interference suppressiontype of receiver The receiver block diagram in this case is shown in Figure 14.6 [4].The transfer function of the detector is obtained by minimizing each element of thecost function

1/Ts

Multipath combining

Multipath combining

KL × KL Multiuser detection

MMSE DETECTOR STRUCTURES 499

where

and

ˆh = LT

The solution of this minimization is [4]

0 dB

5 dB

10 dB

15 dB

and precombining LMMSE detectors in an asynchronous two-path fixed channel with different SNRs, and bit rate 16 kb s−1, Gold code of length 31, t d/T = 4.63 × 10−3 , maximum delay spread 10 chips [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis, University of Oulu, Oulu, by permission of IEEE.

RAKE receiver and the precombining LMMSE (LMMSE-RAKE) receiver with a different

spreading factor (G) in a two-path Rayleigh fading channel with maximum delay spreads of 2µs

for G= 4, and 7 µs for other spreading factors The average signal-to-noise ratio is 20 dB, the data modulation is BPSK, the number of users is 2, the other user has 20-dB higher power Data rates vary from 128 kb s−1 to 2.048 Mbit s−1; no channel coding is assumed [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis,

University of Oulu, Oulu, by permission of IEEE.

The illustration of LMMSE-RAKE receiver performance in near–far environment isshown in Figure 14.8 [5] Considerable improvement compared to conventional RAKE

SPATIAL PROCESSING 501

Multipath combining

Multiuser detection

(a)

Spatial combining

1/Ts

rI(n)

Multipath combining

Multiuser detection

MFK,L

Multipath combining

Multipath combining

Multiuser detection

Multipath combining

Spatial combining

Spatial combining

Postcombining interference suppression receivers with spatial signal processing (c) SMT receiver (d) MST receiver Precombining interference suppression receivers with spatial signal processing.

Multiuser detection

1/ Ts

r1( n )

Spatial combining

1/ Ts

rI( n )

Multipath combining

Multipath combining

r1( n )

Multipath combining 1/ Ts

rI(n)

Multiuser detection

Multiuser detection

(d)

Spatial combining

Spatial combining

k,l is the complex attenuation factor of the kth user’s lth path,

τ k,l,i is the propagation delay for the ith sensor, ε i is the position vector of the ith sensor with respect to some arbitrarily chosen reference point, λ is the wavelength of the carrier, e(φ k,l ) is a unit vector pointing to direction φ k,l (direction-of-arrival) and .,  indicates

the inner product

SINGLE-USER LMMSE RECEIVERS FOR FREQUENCY-SELECTIVE FADING CHANNELS 503

Assuming that the number of propagation paths is the same for all users, the channelimpulse response can be written as

where C is the channel matrix defined in equation (14.19).◦ is the Schur product defined

as Z= X◦Y ∈ C x ×y, that is, all components of the matrix X∈ C x ×y are multiplied

ele-mentwise by the matrix Y∈ C x ×y and

i = diag( ˜φ i )⊗ INb with ˜φ i = diag(φ1, , φ K ),

φ k = [φ k,1, , φ k,L]Tis the matrix of the direction vectors

14.5 SINGLE-USER LMMSE RECEIVERS

FOR FREQUENCY-SELECTIVE FADING

CHANNELS

14.5.1 Adaptive precombining LMMSE receivers

In this case, Mean-Square Error (MSE) criterion E{|h − ˆh|2} requires that the

refer-ence signal h = CAb is available in adaptive implementations For adaptive single-user

receivers, the optimization criterion is presented for each path separately, that is,

J k,l = E{|(h) k,l − (ˆh) k,l|2} (14.45)

The receiver block diagram is given in Figure 14.10, [9–17]

*

*

Channel estimator

Adaptive FIR wkl(n)

LMS

Channel estimator

Adaptive FIR wkl(n)

(14.47)

The filter coefficients w are derived using the MSE criterion (E[ |e (n)

k,l|2]) This leads to

the optimal filter coefficients w[MSE]k,l = R−1R

rd where Rrd is the cross-correlation

SINGLE-USER LMMSE RECEIVERS FOR FREQUENCY-SELECTIVE FADING CHANNELS 505

vector between the input vector r and the desired response d k,l and R r is the inputsignal cross-correlation matrix Adaptive filtering can be implemented by using a number

of algorithms

The steepest descent algorithm

In this case we have

From this equation and assuming that M > 1, the least mean square (LMS) algorithm for

updating the filter coefficients results in

To combiner

bˆkPilot

is the fixed spreading sequence of the kth user with the delay τ k,l In this case everybranch from Figure 14.10 can be represented as shown in Figure 14.11

In this case equation (14.53) gives

k,l e ∗(n) k,l r(n)

µ (n) k,l = µ/(r H(n)r(n) ); 0 < µ < 1 (14.54)

e (n) k,l = d (n) k,l − y (n) k,l

The reference signal is

exam-SINGLE-USER LMMSE RECEIVERS FOR FREQUENCY-SELECTIVE FADING CHANNELS 507

RAKE and the adaptive LMMSE-RAKE in a two-path fading channel for the vehicle speeds

40 km h−1with different numbers of users [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis, University of Oulu, Oulu, by permission

of IEEE.

of length 11 symbols was used in a conventional channel estimator Perfect channel mation and ideal truncated precombining LMMSE receivers were used in the analysis

esti-to obtain the lower bound for error probability The receiver-processing window is three

symbols (M= 3) unless otherwise stated The adaptive algorithm used in the simulationswas normalized LMS with

The MSE criterion now gives

An implementation example can be seen in Reference [21] The stochastic approximation

of the gradient of equation (14.60) for the MOE criterion gives

If we want to keep the useful signal autocorrelation unchanged, equation (14.61) should

be constrained to satisfy sTk,lx(n) k,l = 0 The orthogonality condition is maintained at eachstep of the algorithm by projecting the gradient onto the linear subspace orthogonal to

sTk,l In practice, this is accomplished by subtracting an estimate of the desired signalcomponent from the received signal vector An implementation can be seen in Reference[22] So we have

is a block diagonal matrix of sampled spreading sequence vectors Effectively M separate

filters are adapted

SINGLE-USER LMMSE RECEIVERS FOR FREQUENCY-SELECTIVE FADING CHANNELS 509

In practice, the energy of multipath components (E[ |c k,l|2]) is not known and must

be estimated

Constant modulus algorithm

In this case the optimization criterion is E[( |y k,l|2− ω)2] where ω is the so-called constant modulus (CM), set according to the received signal power, that is, ω = E[|c k,l|2] or

ω (n) = |c (n)

k,l|2 By using the CM algorithm, it is possible to avoid the use of the datadecisions in the reference signal in the adaptive LMMSE-RAKE receiver by taking the

absolute value of the estimated channel coefficients ( |ˆc (n)

k,l |) in adapting the receiver In

the precombining LMMSE receiver framework, the cost function for the BPSK datamodulation is

Constrained LMMSE-RAKE, Griffiths’ algorithm and constant modulus algorithm

The adaptive LMMSE-RAKE, the Griffiths’ algorithm (GRA) and the constant modulus

algorithm contain no constraints By applying the orthogonality constraint sTk,lx(n) k,l = 0 to

each of these algorithms, an additional term sTk,lx(n) k,l s k,l is subtracted from the new x(n k,l +1)

update at every iteration The constrained LMMSE-RAKE receiver becomes [23, 24]

x(n k,l +1)= x(n)

k,l + 2µ (n)

k,l ( ˆc (n) k,l ˆb (n)

k − y (n) k,l ) r(n)− sT

k,lx(n) k,lsk,l (14.70)

The GRA and the constant modulus algorithm can also be defined in a similar way

14.5.3 Blind least squares receivers

All blind adaptive algorithms described in the previous section are based on the gradient

of the cost function In practical adaptive algorithms, the gradient is estimated, that is, theexpectation in the optimization criterion is not taken but is replaced in most cases by somestochastic approximation In fact, the stochastic approximation used in LMS algorithms

is accurate only for small step-sizes µ This results in rather slow convergence, which

may be intolerable in practical applications

Another drawback with the blind adaptive receivers presented above is the delay mation Those receiver structures as such support only conventional delay estimation based

esti-on matched filtering (MF) The MF-based delay estimatiesti-on is sufficient for the downlinkreceivers in systems with an unmodulated pilot channel since the zero-mean multiple-access interference (MAI) can be averaged out if the rate of fading is low enough IfCode Division Multiple Access (CDMA) systems do not have the pilot channel, it would

be beneficial to use some near–far resistant delay estimators

14.5.4 Least square (LS) receiver

One possible solution to both the convergence and the synchronization problems is based

on blind linear least square (LS) receivers Cost function in this case is

sample-SINGLE-USER LMMSE RECEIVERS FOR FREQUENCY-SELECTIVE FADING CHANNELS 511

14.5.5 Method based on the matrix inversion lemma

The general relation

(A + BCD)−1= A−1− A−1B(DA−1B + C−1)−1DA−1 (14.76)becomes

In time-variant channels, the old values of the inverses must be weighted by the so-called

forgetting factor (0 < γ < 1), which results in

It is sufficient to initialize the algorithm as ˆR−1(0)r = I.

For illustration purposes, a number of numerical examples are shown in Figures 14.13

to 4.20 [5] and in Table 14.1 System parameters are shown in the figures

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

blind adaptive receivers in a two-path fading channel with vehicle speeds of 40 km h−1, the

number of active users K = 10, SNR = 20 dB, µ = 10−1 [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis, University of Oulu,

Oulu , by permission of IEEE.

A B C 0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

B – Griffiths’ algorithm

C – blind adaptive MOE

D – adaptive LMMSE-RAKE

blind adaptive receivers in a two-path fading channel with vehicle speeds of 40 km h−1, the

number of active users K = 10, SNR = 20 dB, µ = 100−1 [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis, University of Oulu,

Oulu, by permission of IEEE.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

blind adaptive receivers in a two-path fading channel with vehicle speeds of 40 km h−1, the

number of active users K = 20, SNR = 20 dB, µ = 10−1 [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis, University of Oulu,

Oulu, by permission of IEEE.

SINGLE-USER LMMSE RECEIVERS FOR FREQUENCY-SELECTIVE FADING CHANNELS 513

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

blind adaptive receivers in a two-path fading channel with vehicle speeds of 40 km h−1, the

number of active users K = 20, SNR = 20 dB, µ = 100−1 [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis, University of Oulu,

Oulu, by permission of IEEE.

receiver spans of one (M = 1) and three symbol intervals (M = 3) in a two-path fading channel

at an SNR of 20 dB [5] Reproduced from Latva-aho, M (1998) Advanced Receivers for Wideband CDMA Systems Ph.D Thesis, University of Oulu, Oulu, by permission of IEEE.