Wireless Communications over MIMO Channels phần 8 ppt

248 MULTIUSER DETECTION IN CDMA SYSTEMSFigure 5.13 µ-th stage of a multistage detector for nonlinear parallel interference cancel-lation For useru, all estimates ˜a µ −1 v =u are first w

Figure 5.8 Structure of multistage detector for iterative parallel interference cancellation

soft estimates ˆa (0)

v =u = r v =u /M v,v Subtracting them fromr u leads to an improved estimate

˜a (1)

u after the first iteration The interference cancellation is simultaneously applied to allusers and repeated with updated estimates˜a (µ)

u in subsequent iterations In theµ-th iteration,

theu-th symbol becomes

iterations

The choice of the matrix M determines the kind of detector that is approximated For M= R, we approximate the decorrelator, and the coefficients M u,v = R u,v used in(5.44) equal the elements of the correlation matrix The MMSE filter is approximated

for M= R + σ2

N /σ A2 · INu Hence, the diagonal elements of M have to be replaced with

M u,u = R u,u + N0 /Es.

Convergence Behavior of Decorrelator Approximation

The convergence properties of this iterative algorithm depend on the eigenvalue distribution

of M Therefore, (5.44) is described using vector notations The matrix A= diag(diag(R))

is diagonal and contains the diagonal elements of the correlation matrix R The PIC

approx-imating the decorrelator delivers

242 MULTIUSER DETECTION IN CDMA SYSTEMS

˜a(2)

ZF2

The output after them-th iteration in (5.45) represents the m-th order Taylor series

approx-imation of R−1 (M¨uller and Verdu 2001) Rewriting it with the normalized correlationmatrix ¯R = A−1/2RA−1/2 yields

This series only converges to the true inverse of R if the magnitudes of all eigenvalues of

INu− ¯R are smaller than 1 This condition is equivalent to λmax ( ¯R) < 2 Since λmaxtendsasymptotically to(1+√β)2 (M¨uller 1998), we obtain an approximation of the maximumload below which the Jacobi algorithm will converge

Convergence Behavior of MMSE Approximation

According to the last section, we have to replace the diagonal matrix A with the matrix

To determine the convergence properties concerning the MMSE filter, (5.48) can be formed into the form of (5.46)

The first difference compared to the decorrelator is that the maximum load β depends on

the SNRσ N2/σ A2 = N0 /Es This term increases the convergence area a little bit However,for high SNRσ2

N /σ A2 becomes small and both decorrelator and MMSE filter are approached

only for low loads

This behavior is illustrated in Figure 5.9a showing the results for the first five iterationsand a load β = 0.5 Only for very low SNR (large σ2

N /σ A2) the iterative approximation

reaches the true MMSE filter For higher SNRs,β = 0.5 is beyond the convergence region

and the PIC performs even worse than the matched filter Figure 5.9b shows the results for

Eb/N0= 10 dB versus β Again, it is confirmed that convergence can be ensured only for

low load

5.2.4 Linear Successive Interference Cancellation (SIC)

The poor convergence properties of the linear PIC can be substantially improved Imaginethat the interference cancellation described in (5.44) is carried out successively for different

244 MULTIUSER DETECTION IN CDMA SYSTEMSusers starting withu = 1 and ending with u = Nu Considering the µ-th iteration for user u,

only estimates ˜a (µ −1)

v =u of the previous iterationµ− 1 are used However, updated estimates

˜a (µ)

v<u of the µ-th iteration are already available for users 1 ≤ v < u Replacing all old

estimates ˜a (µ −1)

v<u in (5.44) with their updated versions ˜a (µ)

v<u of the current iteration results

in the Gauss-Seidel algorithm

Besides improved convergence properties another advantage is the in-place implementation,

that is, updated estimates can directly overwrite old values because they are not used anylonger, thereby saving valuable memory

The analysis of the convergence behavior is not as easy as for the PIC In Golub andvan Loan (1996) it is shown that the algorithm always converges for Hermitian positive

definite matrices M Fortunately, in the context of our CDMA system M represents the correlation matrix R or R+ σ2

N /σ A2INu Hence, M can be assumed to be Hermitian and

positive definite so that the Gauss-Seidel algorithm always converges

Figure 5.10a confirms the promised convergence properties Considering a half-loadedsystem, five iterations suffice to approach the true MMSE filter At low SNRs, the perfor-mance of the MMSE filter is reached with even less iterations Figure 5.10b shows that withincreasing load more iterations are needed For loads above β = 1, the first iteration canperform even worse than the matched filter However, successive iterations substantiallyimprove the performance

Comparing the computational costs of a direct matrix inversion with the iterative imations in terms of number of multiplications, we see from (5.50) thatNu multiplicationsper iteration and user are needed For m iterations, this leads to mN2

compared to a complexity of O(N3

u) for the direct matrix inversion Hence, as long as

the number of iterations is smaller thanNu, we save computational costs.

Besides parallel and SIC strategies, there exist further iterative approaches like theconjugate gradient method and a general polynomial series expansion of the inverse (M ¨uller1998) These approaches are not pursued here

All linear techniques described so far do not reach the SUB, that is, interference remains

in the system after filtering The information theoretic analysis in Section 4.3 showed thatthe optimum detector performs much better than linear techniques Therefore, we have tolook for nonlinear approaches that come closer to the optimum solution These techniquesexploit the finite signal alphabet to improve the MUD

A major drawback of the previously introduced linear detectors is not exploiting the discretenature of the transmit signals This shortcoming can be easily overcome by introducingnonlinear devices into the multistage structure to exploit the discrete alphabets This meansthat the signals ˜a (µ)

v =u in (5.44) or (5.50) are passed through a suited nonlinear device before

they are used for interference cancellation For simplicity, we restrict the analysis to anormalized BPSK, that is, we transmit x = ±1 An extension to quaternary phase shiftkeying (QPSK) that treats real and imaginary parts separately is straightforward whileschemes with more levels need more sophisticated methods

becomes worse Therefore, more sophisticated functions taking into account the reliability

of the signals should be preferred A selection analysed by K¨uhn et al (2002) is depicted

in Figure 5.11

To keep the influence of wrong decisions as small as possible, it is advantageous not

to decide on unreliable small samples but to keep them small Obviously, interference isgenerally not perfectly cancelled by these approaches, but the error made by wrong decision

is remarkably reduced The simplest form that follows this strategy is the clipper or limiter

It has a linear shape for|y| ≤ 1 and outputs ±1 for larger inputs |y| > 1

246 MULTIUSER DETECTION IN CDMA SYSTEMS

α

Figure 5.11 Examples for nonlinear devices

than 1 For small values, the reliability is low and the interference can only be partlyreduced In case of a wrong sign, the degradation is not as large as for the hard decision

A smooth version of the clipper is obtained with the tanh-function avoiding sharp edges

We know from Section 3.4 on page 110 that the expectation of a bit is obtained from itslog-likelihood ratio L by tanh(L/2) However, the LLR can be determined only if the

signal to interference plus noise ratio (SINR) is perfectly known This represents a bigdifficulty because we do not know the exact interference level in each iteration Therefore,

we introduce a parameterα according to

that depends on the SNR as well as the effective interference and has to be optimized withrespect to a minimum error rate Figure 5.12 compares the tanh-function for different α

with the hard decision and the clipper For smallα, the tanh is very smooth and its output

is pretended to be unreliable even for large inputs On the contrary,α= 1 comes close tothe clipper in the nearly linear area around the origin and large α > 1 approach the hard

−4 −3 −2 −1 0 1 2 3 4

−1

−0.5

00.5

Figure 5.12 Comparison of tanh for differentα with hard decision and clipper

values and allows interference reduction only above a thresholdα

(5.55)

Obviously, it reduces to a simple clipper forα= 0

Finally, we will look at coded CDMA systems If the computational costs do notrepresent a restriction, the channel decoder can be used as a nonlinear device (Hagenauer1996a) Since it exploits the redundancy of the code, it can increase the reliability of theestimates remarkably Again, we have to distinguish between hard-output and soft-outputdecoding For convolutional codes presented in Chapter 3, hard-output decoding can beperformed by the Viterbi algorithm while soft-output decoding can be carried out by theBCJR or Max-Log-MAP algorithms

5.3.2 Uncoded Nonlinear Interference Cancellation

Uncoded Parallel Interference Cancellation

First, we have to optimize the parameterα for the nonlinear functions NL 1, NL 2, and

tanh We start our analysis with the PIC whose structure for the linear case in Figure 5.8has to be extended Figure 5.13 shows theµ-th stage of the resulting multistage receiver.

Prior to the interference cancellation, the interference reduced signals ˜r (µ −1)

v of the previousiteration are scaled with coefficientsM v,v−1 The application of the nonlinear function nowyields estimates for all signals

˜a (µ −1) = QM−1· ˜r (µ −1)

248 MULTIUSER DETECTION IN CDMA SYSTEMS

Figure 5.13 µ-th stage of a multistage detector for nonlinear parallel interference

cancel-lation

For useru, all estimates ˜a (µ −1)

v =u are first weighted with the correlation coefficientsM u,v =u,then summed up and finally subtracted from the matched filter outputr u

Figure 5.14 shows the performance of the nonlinearities NL 1 and NL 2 versus thedesign parameterα Looking at NL 2, we observe that αNL 2

opt = 0 is always the best choiceregardless of the number of iterations Hence, NL 2 reduces to a simple clipper Withregard to NL 1, the optimumα depends on the iteration In the first stage, the minimum

BER is also delivered by a clipper obtained with αNL 1

opt = 1 For the fifth stage, 0.3 ≤

αNL 1

opt ≤ 0.4 is the best choice Moreover, the comparison of NL 2 with NL 1 shows that

NL 1 is at least as good as NL 2 and generally outperforms NL 2 (Zha and Blostein2003)

The same analysis has been performed for the tanh-function From Figure 5.15 werecognize that 1≤ α ≤ 2 is an appropriate choice for a large variety of loads With growing

β, the optimum α becomes smaller and approaches 1 for β = 1.25 This indicates that the

SINR is small for largeβ However, the differences are rather small in this interval Only

very low values ofα result in a severe degradation because no interference is cancelled

for α = 0 leading to the matched filter performance If α is chosen too large, the tanh

function saturates for most inputs and the error rate performance equals that of a harddecision

Figure 5.16a now compares all proposed nonlinearities for a fully loaded OFDM-CDMAsystem withβ = 1 and five iterations The tanh-function with optimized α shows the best

performance among all schemes NL 1 and clipper come closest to the tanh The harddecision already loses 2 dB compared to the tanh Although the nonlinearities consider thefinite nature of the signal alphabet and all nonlinearities clearly outperform the matched

Figure 5.14 PIC optimization for NL 1 and NL 2 in an uncoded OFDM-CDMA systemwith a 4-path Rayleigh fading channel andEb/N0= 8 dB (solid lines: NL 1, dashed lines:

Figure 5.15 PIC optimization for tanh in an uncoded OFDM-CDMA system with a 4-pathRayleigh fading channel andEb/N0= 8 dB

filter, we observe an error floor that the SUB cannot be reached Figure 5.16b illustratesthis loss versusβ At a load of β= 1, the error rate is increased by one decade compared tothe single-user case; forβ = 1.5, only the tanh can achieve a slight improvement compared

to a simple matched filter Therefore, we can conclude that nonlinear devices taking intoaccount the finite nature of the signal alphabet improve the convergence behavior of PIC.The SUB is approximately reached up to loads ofβ = 0.5 For higher loads, performance

degrades dramatically until no benefit to the matched filter can be observed

250 MULTIUSER DETECTION IN CDMA SYSTEMS

Figure 5.16 PIC performance comparison of different nonlinearities with optimizedα in

an uncoded OFDM-CDMA system with 4-path Rayleigh fading channel

Uncoded Successive Interference Cancellation

From linear interference cancellation techniques, we already know that SIC according tothe Gauss-Seidel algorithm converges much better than the PIC Consequently, we nowanalyze on the nonlinear SIC Figure 5.17 illustrates the influence of the parameterα for

NL 1 and NL 2 on the SIC performance As already observed for PIC, αNL 2

opt = 0 is thebest choice regardless of the load and the considered iteration, and reduces nonlinearity

Figure 5.17 SIC optimization for NL 1 and NL 2 in an uncoded OFDM-CDMA systemwith a 4-path Rayleigh fading channel andEb/N0= 8 dB (solid lines: NL 1, dashed lines:

NL 2)

Figure 5.18 SIC optimization for tanh in an uncoded OFDM-CDMA system with a 4-pathRayleigh fading channel andEb/N0= 8 dB

NL 2 to a simple clipper However, the influence of α on the error rate performance is

much larger than for PIC For αNL 2→ 1 which leads to a large interval of magnitudeswhere no interference is cancelled, the error rate tends to 0.5 for all iterations while theloss was quite moderate for the PIC

With regard to NL 1, α has nearly no influence at the first iteration In subsequent

stages, for example, the fifth iteration, the influence increases with growing loadβ and the

lowest error rate is obtained forαNL 1

opt = 0.4 Again, NL 1 with optimum α shows a better

performance than NL 2

Figure 5.18 depicts the optimization for the tanh-function Astonishingly, the results inthe first iteration differ from those of the PIC The lowest error probability is obtained for

αtanh

opt = 2 regardless of the load β Also, in the subsequent stages this choice of α represents

a very good solution and it coincides with the results of the PIC

As can be seen from Figure 5.19, NL 1 outperforms all other schemes and representsthe best nonlinearity under consideration Forβ = 1, the SUB is reached within a gap of0.5 dB for all SNRs The clipper (NL 2 withα= 0) and the tanh come closest to NL 1while hard decisions lose remarkably From Figure 5.19b we see that, compared to theSUB, NL 1 and the tanh are able to keep the loss quite low up to a load ofβ = 1.5 Even

for this high load, the gain over the matched filter is significant Hence, we can concludethat the considered nonlinearities with optimum design parameters improve the performancefor both PIC and SIC However, SIC still shows a better convergence behavior and comesclose to the SUB even for high loads

Performance of Nonlinear SIC for QPSK Modulation

If we change from BPSK to QPSK, the effective interference is doubled (cf Chapter 4).Only slight changes are necessary to adapt the presented algorithms to QPSK All nonlin-earities have to be applied separately to the real and imaginary parts of the signals Because

252 MULTIUSER DETECTION IN CDMA SYSTEMS

Figure 5.19 Performance comparison of different nonlinearities with optimized α for

SIC in an uncoded OFDM-CDMA system with 4-path Rayleigh fading channel (fiveiterations)

of the doubled interference, the results we obtain forβ = 0.75 and QPSK are nearly the

same as forβ = 1.5 and BPSK.

Figure 5.20 analyzes the influence of the parameter α of the nonlinearities for an

OFDM-CDMA system with β= 1 and a 4-path Rayleigh fading channel For mediumSNRs like Eb/N0= 8 dB, the results coincide with those already obtained for BPSK.Nearly no influence can be observed in the first stage For further iterations and larger

Figure 5.21 SIC performance for nonlinearities with optimizedα in an uncoded

OFDM-CDMA system with QPSK and 4-path Rayleigh fading channel (β = 1)

SNR, for example, 20 dB, NL 1 requires a largerα to perform optimally Because of the

high interference, the estimates are less reliable and the step toward±1 occurs at higheramplitudes With reference to the tanh,αtanh

opt = 2 still represents a very good choice.Figure 5.21 shows the BER performance for 5 and 10 iterations and optimized α for

NL 1 and tanh While the hard decision does not gain from additional iterations, the linear function NL 1, the tanh-function, and the clipper enhance the error rate remarkably.The tanh with optimized α is still the best choice However, for this high load there

non-remains a large gap to the SUB (bold line) that roughly amounts to 4 dB at an error rate

of 2· 10−3.

5.3.3 Nonlinear Coded Interference Cancellation

Resuming the way from linear multistage receivers to nonlinear interference cancellationschemes, it is straightforward to incorporate the channel decoder into the iterative structuresfor coded CDMA systems Again, we restrict to BPSK and QPSK schemes for notationalsimplicity The structure of the transmitter is already known from Figure 5.1 The cor-responding receiver for PIC is depicted in Figure 5.22 After the matched filter bank, theobtained signalsr u, 1≤ u ≤ Nu, are de-interleaved and FEC decoded The decoders deliver

either soft-outputsL( ˆb u ) of the code bits like log-likelihood ratios or hard estimates ˆb u outputs can be generated by the BCJR or the Max-Log-MAP decoder while hard-outputsare obtained by the Viterbi algorithm

Soft-Next, the outputs are interleaved and processed by a nonlinear function This is necessaryfor soft-outputs because log-likelihoods are generally not limited in magnitude while thetrue code bits are either+1 or −1 From Chapter 3, we know that the expectation of a bitcan be calculated with its log-likelihood ratio by tanh(L/2) (see (3.36) on page 110) This

is exactly the reason for using the tanh Finally, the interfering signals are weighted with

254 MULTIUSER DETECTION IN CDMA SYSTEMS

tanh

tanhPIC/SIC stage

FECdec

FECdec

Figure 5.22 Single stage of a nonlinear PIC receiver in coded CDMA systems

the correlation coefficientsM u,v, summed up, and subtracted from the matched filter output

r u The obtained estimate represents the input of the next stage

For the following simulation results, an OFDM-CDMA system with a half-rate lutional code of constraint length Lc= 7 is considered As in a mobile radio channel, a4-path Rayleigh fading channel with uniform power delay profile is employed Moreover,BPSK and QPSK are alternatively chosen and on an average each user has the same SNR.Figure 5.23a shows the performance of the coded PIC After four PIC iterations, theSUB is obtained even for β = 1.5, which is equivalent to a spectral efficiency of η =

convo-Rc· β = 0.75 We see that decoding helps improve the convergence of iterative interference

cancellation schemes The reliability of the estimated interference is enhanced, leading to

a better cancellation step At low SNR, a gap to the SUB occurs that grows for increasingload Forβ = 2, the PIC scheme does not converge anymore

Figure 5.23b compares hard- and soft-decision outputs at the decoder The upper boldsolid line denotes the matched filter performance and the lower bold solid curve representsthe SUB Naturally, the performances for hard- and soft-outputs after the first decoding(single-user matched filter (SUMF), upper solid line) are the same For subsequent iterations,the soft-output always outperforms the hard-decision output However, the differences arerather small and amount at the most to 0.5 dB For this example, both hard- and soft-outputdecoding reach the SUB at error rates below 10−4 Nevertheless, for extremely high loads,convergence may be maintained with soft-output decoding while hard-decision decodingwill fail

Next, parallel and SIC are compared in Figure 5.24a The upper bold solid line sents the matched filter performance without interference cancellation, and the lower boldsolid curve represents the SUB After three PIC iterations, both SIC and PIC reach the SUB.However, SIC converges faster and in this example needs one iteration less than the PICscheme Hence, the benefits of SIC are preserved when coding is applied In Figure 5.24b,the load is increased toβ = 2, that is, the spectral efficiency η = 1 bit/s/Hz of such a sys-

repre-tem is twice as high as for half-rate coded TDMA or FDMA sysrepre-tems (K¨uhn 2001a,c) We

of PIC and SIC b) comparing PIC and SIC for various loads

see that the SIC scheme performs as much for β = 2 as the PIC approach performs for

β = 1.5 For a doubly loaded system, the PIC scheme does not converge anymore.

Sorted Nonlinear Successive Interference Cancellation

There exists a major difference between parallel and SIC Owing to the problem of errorpropagation, the order of detection is crucial for SIC This dependency is illustrated in

256 MULTIUSER DETECTION IN CDMA SYSTEMS

to a faster convergence Especially, at low SNR the gap to the SUB can be decreased.Therefore, sorting is always applied for SIC in subsequent parts

Figure 5.26 now compares the performance of SIC for BPSK and QPSK modulations

We know from Section 4.1 that the use of QPSK in the uplink doubles the effective ference The spectral efficiency is also doubled because we transmit twice as many bits

Figure 5.26 SIC performance for OFDM-CDMA system with a half-rate convolutionalcode (L = 7) and 4-path Rayleigh fading channel (bold line: single-user bound)

per symbol as for BPSK,ηQPSK= 2ηBPSKholds After four PIC iterations, we observe thatthe SUB has been reached for QPSK andβ = 1 This is equivalent to β = 2 for BPSK.

No convergence is obtained forβ = 2 and QPSK because the initial SINR is too low forachieving reliable estimates from the decoders

Influence of Convolutional Codes

Finally, the influence of different convolutional codes is analyzed Figure 5.27 comparestwo half-rate convolutional codes: the already usedLc= 7 code and a weaker Lc= 3 code.Looking at Figure 5.27a, we see that three or four iterations suffice to reach the SUB forBPSK With regard to QPSK, even 10 iterations cannot close the gap of approximately

4 dB On the contrary, convergence starts earlier with the weak Lc= 3 code as shown

in Figure 5.27b Although the Lc= 3 code has a worse SUB, it performs better than thestrong convolutional code and reaches its SUB even forβ = 2 and QPSK Although thedifference between the two SUBs amounts to 2 dB in favor of the stronger convolutionalcode, theLc= 3 code now gains 2 dB compared to the Lc= 7 code

The explanation for this behavior can be found by observing the SUB curves for bothcodes in Figure 5.27 We see that the Lc= 3 code has a slightly better performance atlow SNR For larger SNR, the curves intersect and the Lc= 7 code becomes superior.However, the first interference cancellation stage suffers from noise as well as from severeinterference that was not yet cancelled For increasing loads, the SINR at the decoderinputs becomes smaller and smaller until it reaches the intersection of both curves Forhigher loads, the weak code now performs better In our example, parameters were cho-sen such that the strong code cannot achieve convergence while the Lc= 3 code stillreaches its SUB Therefore, we can conclude that strong error control codes are not

QPSKQPSK

Figure 5.27 Sorted SIC performance for half-rate coded OFDM-CDMA system withβ = 2,

a 4-path Rayleigh fading channel and different convolutional codes (bold line: single-userbound)

258 MULTIUSER DETECTION IN CDMA SYSTEMSalways the best choice and that the coding scheme has to be carefully adapted to thedetector.4

5.4.1 BLAST-like Detection

As we saw in the previous results, the first detection stage suffers severely from tiuser interference Hence, its error rate will dominate the performance of subsequentdetection steps because of error propagation The overall performance and the conver-gence speed can be improved by a linear suppression of the interference prior to the firstdetection stage The Bell Labs Layered Space-Time (BLAST) detection of the Bell Labs(Foschini 1996; Foschini and Gans 1998; Golden et al 1998; Wolniansky et al 1998)pursues this approach for multiple antenna systems It can be directly applied to CDMAsystems since both systems have similar structures and the same mathematical description

Next, the symbol ˆa1 = Q(˜a1 ) can be decided with improved reliability because less

interfer-ence disturbs this decision Instead of performing a hard decision, other nonlinear functions

as analyzed in Section 5.3 can be used After detecting ˆa1, its influence onto the remaining

signals can be removed by subtracting its contribution s1ˆa1 from the received vector y

(interference cancellation)

where the vector s1 represents the first column of the system matrix S (see Figure 4.7 or

(4.68)) The residual signal ˜y2 is then processed by a second filter w2 It is obtained by removing s1 from S and calculating the ZF or MMSE filter W2 for the reduced system

matrix ˜S2 =s2 · · · sNu



The first column of W2 denotes the filter w2 that is used forsuppressing the interference of the second user by˜a2= wH

2 · ˜y2 This procedure is repeated

until all users have been detected

To determine the linear filters in the different detection steps, the system matricesdescribing the reduced systems have to be inverted This causes high implementation costs.However, a much more convenient way exists that avoids multiple matrix inversions Thisapproach leads to identical results and is presented in the next section

5.4.2 QL Decomposition for Zero-Forcing Solution

In this subsection, an alternative implementation of the BLAST detector is introduced Itsaves computational complexity compared to the original detector introduced in the last

4 It has to be mentioned that this conclusion holds for uniform power distribution among the users If different power levels occur (near-far effects), strong codes can have a better performance than weak codes (Caire et al 2004).

section As for the linear multiuser detectors, we can distinguish the ZF and the MMSEsolution We start with the derivation of the QL decomposition for the linear ZF solution.5

Going back to the model y = Sa + n, the system matrix S can be decomposed into

anNs× Nu matrix Q with orthogonal columns qu of unit lengths and anNu× Nu lower

triangular matrix L (Golub and van Loan 1996)

with ˜n still representing white Gaussian noise because Q is unitary To clarify the effect

of multiplying with QH, we consider the matched filter outputs r = SHy = Ra + SHn.

Performing a Cholesky decomposition (Golub and van Loan 1996) of R = LHL with the

lower triangular matrix L, results in

r = LH

La + SH

n ⇒ ˜r = La + L−HSH

The comparison of (5.61) with (5.62) illustrates that the multiplication of y with QH can

be split into two steps First, a matched filter is applied, providing the colored noise vector

SHn with the covariance matrix = σ2

NR= σ2

NLHL Therefore, the second step represents

a multiplication with −1/2= L−H which can be interpreted as whitening.

Because of the triangular structure of L in (5.61), the received vector ˜y has been partly

freed of interference, for example, ˜y1 depends only on a1 disturbed by the noise term

˜n1 Hence, it can be directly estimated by appropriate scaling and the application of a

nonlinearityQ(·)

ˆa1 = Q/L−11,1 · ˜y10. (5.63)The obtained estimate can be inserted in the second row to subtract interference from ˜y2

and so on We obtain theu-th estimate by

ˆa u = Q

!1

The linear filtering with Q has to cope partly with the same problems as the decorrelator.

For the first user, all Nu− 1 interfering signals have to be linearly suppressed Since all

5 Throughout the subsequent derivation, the QL decomposition will be used Equivalently, the QR

decomposi-tion of S can often be found in publicadecomposi-tions.