Linear detectors

number of antennas. To solve the problem the SD algorithm has been proposed that reduces significantly the search space of possible solutions while still providing the ML solution. For a few good examples refer to [61], [62] and [63]. The SD algorithm gives an efficient solution to the Integer Least Squares (ILS) problem and it turns out that there is an analogy between the search for the optimal solution and the bounded tree search methods. In Sec. 4.6 a detailed overview is given on the SD algorithm and it is discussed how the solution of the ILS problem can be transformed to a tree-search problem.

4. The complexity of SD algorithm can be further reduced by introducing some ap-proximations such as (i) early termination of the tree-search, (ii) introducing con-straints on the maximum number of nodes that the SD algorithm is allowed to visit or (iii) on the run-time of the detector. Each of these strategies introduces some ap-proximations which prevents these detectors from achieving the ML performance.

In case of the non-ML detectors a trade-off has to be made between the computa-tional complexity and the quality of detection. Usually, when approximations are introduced some preprocessing methods are applied such as: matrix regularization, lattice reduction, ordering, improved decompositions. These methods increase the overall computational complexity, however, the benefit is significant and near-ML BER is achieved. Papers [20], [57], [56], [58], [59] and [60] focus on finding a near-ML solution with a significant decrease in computational complexity. In Sec. 4.7 the most effective non-ML tree-search based methods are discussed.

4.3 Linear detectors

4.3.1 Zero-forcing detection

The ZF detector aims to find the least-squares solution, i.e., it searches for the un-constrained vector˜s∈Cⁿthat minimizes the squared Euclidean distance to the received symbol vector ˜yaccording to

˜s_ZF = arg min

˜s∈Cⁿ

k˜y−H˜˜sk² (4.1)

The problem looks very similar to the ML solution, however in this case the search space is not restricted to a finite alphabet. Consequently, in order to arrive to the final solution a clipping of the unconstrained solutionˆs=Q{˜s_ZF}to a valid symbol vector is required.

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4.3. LINEAR DETECTORS

The optimization problem is solved by setting the derivative of the squared Euclidean distance to zero, as follows

∂ the channel matrix based on the SVD the expected value of post-detection noise power can be evaluated as

Eq. 4.4 shows that linear detection suffers from a significant noise enhancement when the condition number of the channel matrix is large, because this means that the minimum singular value is small. As a result, the post-detection noise power E^hkˆ˜v_ZFk²₂ⁱ, i.e., the BER, is mainly determined by the minimum singular value of the channel matrix

E^hkˆ˜v_ZFk²₂ⁱ=

As a conclusion the advantage of the ZF detection is that it completely eliminates the

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4.3. LINEAR DETECTORS

interference with low complexity computations, however, the BER degradation caused by the noise enhancement of the inverse channel matrix multiplication is the main drawback of this approach.

4.3.2 Minimum mean square error detection

It was shown in Sec. 4.3.1 that ZF detection completely eliminates interference, how-ever, the noise power is enhanced significantly. The aim of MMSE detection is to maxi-mize post-detection signal-to-interference plus noise (SINR) ratio, namely, MMSE tries to find a weight matrix W˜_MMSE that minimizes the following criteria

W˜_MMSE= arg min

W∈˜ R^M×N

E^hkW˜ ^H˜y−˜s_tk²ⁱ. (4.6)

The solution is found by setting the partial derivative to zero as follows

∂

∂W˜ E^hkW˜ ^H˜y−˜s_tk²ⁱ= 0

∂

∂W˜ E^hW˜ ^H˜y˜y^HW˜ −˜s_t˜y^HW˜ −W˜^H˜y˜s^H_t +˜s_t˜s^H_t ⁱ= 0

∂

∂W˜ (W˜ ^HR_yyW˜ −R_syW˜ −W˜^HR_ys+R_ss) = 0 W˜ ^HR_yy−R_sy = 0

W˜ ^H=R_syR_yy⁻¹

(4.7)

whereR_ysis the cross-covariance matrix of the received symbol vector˜yand the symbol vector sent ˜s_t, and R_yy is the covariance matrix of the received symbol vectors. As a result, the MMSE weight matrix is equal to

W˜^H_{M M SE} =R_syR⁻¹_yy. (4.8)

In order to determine the covariance matrices several assumptions are made that are common in the most of the communication systems:

• successive noise samples are uncorrelated and AWGN channel is assumed, thus, the noise covariance is R_nn =σ²I_m;

• the symbols of the transmitted symbol vectors_tare statistically independent, thus, the symbol vectors covariance is R_ss=σ_s²I_n;

• the noise samples are independent of the symbols in the symbol vector sent, thus, the covariance of the symbols and noise is R_sn=0.

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4.3. LINEAR DETECTORS

With the above assumption covariance matricesR_sy and R_yy are determined as follows:

R_yy=E^hy˜˜y^Hi

=E^h(H˜˜s_t+n)(˜ H˜˜s_t+n)˜ ^Hi

=E^hH˜˜s_t˜s^H_t H˜^H+H˜˜s_tn˜^H+˜n˜s^H_tH˜^H+n˜˜n^Hi

=σ_s²H ˜˜H^H+σ²I_m

(4.9)

R_sy =E^h˜s_t˜y^Hi

=E^h˜s_t(H˜˜s_t+˜n)^Hi

=E^h˜s_t˜s^H_tH˜^H+˜s_tn˜^Hi

=σ²_sH˜^H

(4.10)

Based on the above results the MMSE weight matrix is defined as follows:

W˜_{M M SE}^H =σ²_sH˜^H(σ_s²H ˜˜H^H+σ²I_m)⁻¹ =H˜^H(H ˜˜H^H+σ²

σ_s²I_m)⁻¹ (4.11) Complexity reduction is achieved with the following equivalent transformation

W˜^H_{M M SE} = (H˜^HH˜ +σ²

σ_s²I_n)⁻¹H˜^H. (4.12)

˜s_{M M SE} =W˜_{M M SE}^H ·y˜

=ˆ˜s_t+ (H˜^HH˜ + σ²

σ_s²I_n)⁻¹H˜^Hv˜

=ˆ˜s_t+ˆ˜v_{M M SE}

(4.13)

In Eq. 4.13 weight matrix W˜ _{M M SE} is applied for detection. Unlike ZF detection it is shown that inˆ˜s_tthe interference is not perfectly suppressed, however, the post-detection noise power can be much lower in certain situations. In order to compare the average post-detection noise power of MMSE detectionE^hkˆ˜v_{M M SE}kⁱwith the ZF post-detection noise power, unit signal power is assumed σ_s² = 1, thus E^hkˆ˜v_{M M SE}kⁱ is evaluated as follows:

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4.3. LINEAR DETECTORS

Figure 4.2: Bit error rate performance comparison of linear detectors for 4×4 MIMO systems with 16 and 64-QAM symbol constellations.

E^hkvˆ˜_{M M SE}k²₂ⁱ=E^hk(H˜^HH˜ +σ²I_n)⁻¹H˜^H˜vk²ⁱ

The noise enhancement caused by the minimum singular value in case of MMSE detection is given as follows

E^hkˆ˜v_{M M SE}k²₂ⁱ= Comparing the noise enhancement of the MMSE and ZF detection based on Eq. 4.15 and Eq. 4.5 when σ²_min σ² it is visible that MMSE detection is less critical. In case when σ² σ_min² the performance of the two detectors becomes the same.

DOI:10.15774/PPKE.ITK.2015.010

In document Design and Implementation of High-Performance Computing Algorithms for Wireless MIMO Communications (Pldal 45-50)