# Implementation of low-complexity MIMO detector and efficient soft-output demapper for MIMO-OFDM-based wireless LAN systems

- Chanho Yoon
^{1}and - Hoojin Lee
^{2}Email author

**2013**:143

https://doi.org/10.1186/1687-1499-2013-143

© Yoon and Lee; licensee Springer. 2013

**Received: **11 March 2013

**Accepted: **15 May 2013

**Published: **29 May 2013

## Abstract

In this paper, we describe a simplified soft-output demapper designed to support coded multiple-input multiple-output orthogonal frequency-division multiplexing-based system utilizing only 3-bit soft information. The IEEE 802.11n standard requires relatively high punctured convolutional code rate of *R*=5/6 for spectrally efficient high-throughput data rate settings. In order to extract soft-bit information effectively without degrading the packet error rate performance, we introduce bit-rounding and effective-bit threshold adjustment techniques to achieve such.

## Keywords

## Introduction

In many areas of modern digital communication systems, modulation and coding scheme (MCS) in conjunction with bit-interleaved coded modulation technique has been adopted. The IEEE 802.11a/g [1] and 802.11n [2] standards are such good examples. The bit-interleaved coded modulation scheme brings substantial performance enhancement not only limited to single-input single-output but also multiple-input multiple-output (MIMO) antenna system as well. In order to decode multiple layers of transmit signals propagated in the wireless fading channel, the MIMO detector takes the crucial role of extracting soft-bit signals for optimal decoding performance.

On the subject of MIMO detector, the effective performance and hardware implementation complexity are important issues. The maximum likelihood (ML) detector having log-likelihood ratio (LLR) output is known as the optimum detector. Its complexity, however, increases exponentially with the number of transmit antennas and modulation order [3]. On the other hand, applying linear MIMO detectors such as simple zero-forcing (ZF) or minimum mean squared error nulling techniques to extract soft bits require much lower implementation complexity, as a result of significant performance trade-off. This lower complexity benefit is appreciated especially when modulation order is high, such as 64-quadrature amplitude modulation (QAM). Considering linear MIMO detectors, some complimentary techniques to compensate receiver performance should be devised in the soft demapper. In addition, the size of soft-bit should be kept as small as possible for low implementation complexity. It is also expected that latency induced by heavy arithmetic operations processed in the channel decoder is proportional to resolution of soft-demapped bit resolution, and therefore, it should be minimized for implementation.

The IEEE 802.11n standard requires punctured convolutional code rate *R*=5/6 as one of mandatory high-throughput (HT) data rate settings. In order to effectively extract 3-soft-bit information without causing performance degradation induced by adopting high channel coding rate, we apply bit-rounding and effective-bit threshold adjustment techniques during the extraction of soft-bits at the demapper. The system to be investigated is based on bit-interleaved coded MIMO-orthogonal frequency-division multiplexing (OFDM) which supports IEEE 802.11n version with HT transmission mode up to MCS 15. Low-complexity ZF linear detector with extra receive antenna is assumed prior to the calculation of soft-demapped bits in this paper, as a cost-effective solution for achieving both low implementation complexity and performance [4].

We use the following notation throughout this paper. The superscripts (·)^{
T
}, (·)^{∗}, and (·)^{
H
} denote transpose, complex conjugate, and Hermitian operations, respectively. Pr(·) denotes the probability. *E*[·] stands for expectation. ℜ(*α*) denotes the real part of complex number *α*.

## Receiver system model

^{3}decimation in frequency FFT block. The output data of FFT is transferred to the MIMO detector. Finally, the output signal of MIMO detector is processed in soft demapper.

## MIMO signal detection

**r**of MIMO system with

*N*

_{ T }=2 transmit antennas and

*N*

_{ R }=3 receive antennas is given by

where **r**=[*r*_{0},*r*_{1},*r*_{2}]^{
T
} represents received symbol column vector, **H** denotes the 3×2 channel matrix which time-invariant channel per subcarrier is assumed. Element, *h*_{
i
j
}, of **H** stands for the channel gain between the *i*-th receive antenna and the *j*-th transmit antenna. **x**=[*x*_{0},*x*_{1}]^{
T
} is the transmitted symbol vector of two independent streams with total transmit power normalized to unity. Vector **n** represents additive white Gaussian noise vector generated at the receiver side with variance *σ*^{2}.

**W**is defined by

**H**

^{ H }

**H**can be expressed as channel norm and correlation term,

The correlation term, interpreted as interference among transmit signals over the air interface, can be expressed as $C={h}_{00}^{\ast}{h}_{01}+{h}_{10}^{\ast}{h}_{11}+{h}_{20}^{\ast}{h}_{21}$.

**H**

^{ H }

**H**)

^{−1}should be normalized by determinant $\Delta =det\left({\mathbf{H}}^{H}\mathbf{H}\right)={N}_{0}^{2}{N}_{1}^{2}-{\left|C\right|}^{2}$. However, due to fixed-point implementation complexity and numerical instability issues, division operation by the determinant should be avoided if possible. In fact, it is unnecessary to normalize the estimated transmit symbols during the matrix inversion step, but normalization can be deferred to the soft demapper. Then, the scaled ZF filter coefficient matrix

**W**can be rewritten as

**W**is multiplied by received signal vector

**r**to estimate the transmitted signal vector $\widehat{\mathbf{x}}$. The estimated symbol vector $\widehat{\mathbf{x}}$ is a scaled version of the transmitted signal, and it is scaled exactly by the determinant Δ. The resulting estimated transmit vector $\widehat{\mathbf{x}}$ can be written as

As observed, note that noise vector **n** is boosted by **W**^{
H
}, and covariance matrix of noise vector is **R**=*σ*^{2}(**H**^{
H
}**H**)^{−1}. In other words, the noise is generally correlated [6] after ZF equalization.

**w**and Δ to following equations:

where vector **W** is simply channel impulse response per subcarrier per receive antennas and Δ is the channel norm. The transmitted symbol estimation method in this case is simply maximal ratio combining.

Knowing the theoretically mapped point of estimated QAM signals, we consider the scaled effect of Δ at the soft demapper during the extraction of soft-bits. This will be described in the next section.

## Soft-output demapper

At the demapper, the extraction algorithm of soft-output bits in [7] is complex so that sub-optimal solutions [8] need to be adopted. As extraction of soft-bit information is concerned, we first review conventional soft-demapping techniques briefly and then focus on simplified sub-optimal approach for generating multi-level modulation cases.

### Log-likelihood ratio-based bit metric

*k*-th subcarrier symbol at the

*j*-th bit can be defined as

*i*indicates spatial stream.

*S*

^{0}represents the subset of $\frac{1}{2}{M}^{{N}_{T}}$ vectors

**s**for which the

*j*-th bit of the corresponding symbol is equal to bit 0. The above metric can be simplified as max-sum approximation which eliminates calculation of logarithm [9].

As the above bit metric is the procedure for exact soft ML MIMO detection, total computation of ${M}^{{N}_{T}}\times {N}_{R}$ Euclidean distances ar required.

### ZF equalizer output-based bit metric

where **x**^{
i
} is the ZF equalized output in Eq. (5). Here, *S*^{0} indicates subset of $\frac{1}{2}M$ vectors **s** for which the *j*-th bit of the corresponding symbol is equal to bit 0. Above equation dramatically reduces computations required to estimate the minimum Euclidean distances which is a trade-off between complexity and performance. Note that at high signal-to-noise ratio (SNR) region, Eq. (10) is a piecewise linear function of real or imaginary part of ZF equalized output $\widehat{\mathbf{x}}$, as suggested in [8]. Furthermore, much simpler bit metric algorithm is suggested by [9] where complexity of the demapper is maintained at almost the same level for all the multi-level modulation modes.

### Noise power weighting-based bit metric

**n**is affected by ZF filter matrix

**W**. In Eq. (5), the variance

*σ*

^{2}of nulling vector and noise vector product can be expressed as

Thus, by multiplying the inverse of norm of the ZF filter coefficient $1/{\left|\left|{\mathbf{W}}_{i}^{H}\right|\right|}^{2}$ to the estimated symbol $\widehat{{\mathbf{x}}_{i}}$, the exact variance of colored noise affected to soft-bit information is reflected on the channel decoder, delivering enhanced channel decoding performance.

*i*refers to

*i*-th row vector here. Then, the simplified noise power weight soft-bit LLR metric related to real part in case of 64-QAM modulation is defined as

Since ${\widehat{x}}_{i}$ and ||**W**_{
i
}||^{2} are both scaled, soft-bit weighting of noise enhancement should be $\Delta /{\left|\left|{\mathbf{W}}_{i}\right|\right|}^{2}$ instead of 1/||**W**_{
i
}||^{2}, and their scaling effect is effectively gone. In summary, only one division operation is necessary to normalize the estimated transmit signal ${\widehat{x}}_{i}$. In addition, we found that the ratio of $\Delta /{\left|\left|{\mathbf{W}}_{i}\right|\right|}^{2}$ is typically in the range of 0 to 8. Therefore, only 3 bits are enough to express the colored noise weighting as well as normalization factor.

## Hardware architecture

where “ampi_trk" is amplitude tracking result of the *i*-th spatial stream. This tracking coefficient reflects variation of average magnitude of pilot tones allocated to every data field OFDM symbols. Although time selectivity of wide-band nomadic systems such as wireless local area network (WLAN) is considered negligible, average channel power may change for long packet format in case of long packet aggregation feature is enabled.

After the multiplication of incoming ZF equalized signal by amplitude tracking result , 6 least significant bits (LSBs) corresponding to floating point are cut off, making scaled signal as 20 bits. During that time, higher order modulation threshold values corresponding to decision boundary of 16-QAM and 64-QAM signals are calculated to let soft demapper extract sub-optimal LLR values. Note that these decision boundaries are scaled by Δ.

**List of variables used in the analysis**

Δ | Channel determinant | 11 bits |

ampx_trk | Scale for amplitude tracking | 7 bits |

ratefield_rx | Modulation order indicator | 1 bit |

${\widehat{x}}_{i}$ | Estimated transmit symbol of | 18 bits |

|| | Norm of pseudo inverse matrix of | 17 bits |

## Three-bit extraction/quantization

Based on colored noise power weight bit metric discussed in the previous section, the soft-bit calculation procedure (Eq. 13) is a piecewise linear function [9] which is simple to implement. In contrast, the 3-bit quantization process, however, is a non-linear function. During the extraction, the magnitude of soft-bit values are analyzed for a given specific range.

At this point, 3-bit LLR is examined whether its absolute value is greater than or equal to 3. If it holds true, the final 3-bit output is set to either 3 or -3, limiting 3-bit signed number to seven levels instead of eight (i.e., from -4 to 3). If not, bit-rounding scheme, equivalent to adding 0.5 bit, is applied to mitigate problems related to negative-value biased signal due to fixed-point quantization effect on 2’s compliment conversion. We later find that this biased signal affects the soft-input Viterbi decoding performance significantly, especially for high code rate *R*=5/6. Note that the above procedure can be applied similarly to finding the 3-bit value of $\Delta /{\left|\left|{\mathbf{W}}_{i}\right|\right|}^{2}$. In this case, *b*_{0} of Figure 3 can be replaced with $|\Delta -{\left|\left|{\mathbf{W}}_{i}\right|\right|}^{2}|,\Delta \ge {\left|\left|{\mathbf{W}}_{i}\right|\right|}^{2}$ or $|\Delta -{\left|\left|{\mathbf{W}}_{i}\right|\right|}^{2}|/4,\Delta <{\left|\left|{\mathbf{W}}_{i}\right|\right|}^{2}$ since both Δ and ||**W**_{
i
}||^{2} are positive values.

## Simulation results

**H**in 3×2 matrix form in the MIMO detector. Simulation parameters are given in Table 2, and packet size is fixed to 1,000 bytes, as suggested in [10].

**Simulation parameters**

Carrier frequency | 5 GHz |

System bandwidth | 20 MHz |

Channel model | Exp. 50-ns rms delay spread |

Modulation & coding set | MCS12 to MCS15 |

Packet length | 1,000 bytes |

Freq. / sampl. offset | 40/40 ppm |

## Conclusions

We have proposed a linear MIMO detector-based soft-demapping metric as well as its hardware architecture that is simple to implement. With a combining technique of limiting 3-bit effective quantization to seven-level bit-rounding and effective-bit threshold adjustment, a considerable gain can be realized in coded MIMO-OFDM-based system in high data rate transmission modes, especially for high code rates. The proposed soft-bit demapper has been tested/verified with Xilinx Virtex II XC2V8000 FPGAs operating at 80 MHz.

## Declarations

### Acknowledgments

This research was funded by the MSIP (Ministry of Science, ICT & Future Planning), Korea in the ICT R&D Program 2013. The work of Hoojin Lee was financially supported by Hansung University. The content of this paper was partially presented in [11].

## Authors’ Affiliations

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## Copyright

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