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Efficient quantization and fixedpoint representation for MIMO turbodetection and turbodemapping
EURASIP Journal on Embedded Systems volume 2017, Article number: 33 (2017)
Abstract
In the domain of wireless digital communication, floatingpoint arithmetic is generally used to conduct performance evaluation studies of algorithms. This is typically limited to theoretical performance evaluation in terms of communication quality and error rates. For a practical implementation perspective, using fixedpoint arithmetic instead of floatingpoint reduces significantly implementation costs in terms of area occupation and energy consumption. However, this implies a complex conversion process, particularly if the considered algorithm includes complex arithmetic operations with high accuracy requirements and if the target system presents many configuration parameters. In this context, the purpose of the paper is to present an efficient quantization and fixedpoint representation for turbodetection and turbodemapping. The impact of floatingtofixedpoint conversion is illustrated upon the errorrate performance of the receiver for different system configurations. Only a slight degradation in the errorrate performance of the receiver is observed when implementing the detector and demapper modules which utilize the devised quantization and fixedpoint arithmetic rather than floatingpoint arithmetic.
1 Introduction
Low power consumption and reduced implementation area are vital factors to fulfill the ever increasing requirements of embedded systems. In digital communication applications, the algorithms are typically specified in floatingpoint arithmetic in order to evaluate the application performance. However, hardware architectures implementing these applications are designed using fixedpoint arithmetic to satisfy the tight constraints on implementation area and power consumption related to embedded systems. In fixedpoint representation, memory and bus widths are smaller, leading to definitively lower cost and power consumption. Moreover, floatingpoint operators are more complex, having to deal with the exponent and the mantissa, and hence, their area and latency are significantly greater than those of fixedpoint operators. In spite of that, fixedpoint arithmetic introduces an unalterable quantization error which modifies the application functionalities and degrades the desired performance. Thus, the design flow requires a floatingtofixedpoint conversion stage which optimizes the implementation cost under execution time and accuracy constraints [1]. For digital communication applications, the most commonly used criterion for evaluating the precision of fixedpoint implementation is the errorrate performance degradation. Hence, the accuracy constraint is linked to the specifications of the supported communication standards.
On the other hand, due to the rapid evolution of related standards, modern wireless digital communication systems are highly concerned about the flexibility feature. Circuits and systems adopted in this application domain must consider not only performance and implementation constraints, but also the requirement of flexibility. In this context, flexible applicationspecific hardware architectures implementing the functionalities of digital baseband components of the receiver are under design scope. Applicationspecific processors constitute a key trend in implementing definite blocks of wireless system since they provide a good solution in designing flexible architectures that can fulfill nowadays requirements in terms of low errorrate performance and high throughput and satisfy the tight constraints on implementation area and power consumption.
Recent emergent wireless communication standards, such as LTE/LTEA for mobile phones, 802.11 (WiFi) and 802.16 (WiMAX) for wireless local and wide area networks, and DVB for digital video broadcasting, support various modes and configurations related to channel coding type, modulation type and mapping style, and antenna dimension for multipleinput multipleoutput (MIMO) transmission techniques. On the other hand, iterative concept is also utilized at the receiver side to alleviate the destructive effects of the channel. Iterative processing concept (socalled turbo processing) was proposed firstly in the channel decoding [2] to achieve errorrate performance close to the theoretical limits.
The extension of turbo principle to the demapping and intersymbol interference (ISI) equalization blocks gives rise to turbodemapping [3] and turboequalization [4] concepts. These concepts are achieved when the extrinsic information at the output of the turbo decoder is fed back as a priori soft information to the input of the demapper and equalizer.
In previous work, presented in [5], flexible applicationspecific processor dedicated for turbodemapping has been proposed. The demapper implements the MaxLogMAP algorithm. It supports iterative demodulation and its flexibility is not restricted to certain types of modulation and/or mapping styles. Similarly, another flexible applicationspecific processor dedicated for minimum meansquared error (MMSE) linear equalizer has been proposed in [6]. Its flexibility is extracted from the following requirements: (1) the capability to support different MIMO schemes reaching to 4 × 4 antenna dimension, (2) the ability to maintain efficient use of hardware resources for different time diversity channel types (fast fading, quasistatic, and block fading) and (3) the possibility to execute in an iterative or noniterative modes. In fact, the techniques which were found to be effective in combating ISI are often extended to the context of MIMO detection [7, 8]. Therefore, the designed MMSE equalizer in [6] is used for iterative MIMO detection. In the remainder of this paper, iterative MIMO detection based on MMSE linear equalization is referred to as turboequalization.
In the designed architectures for MaxLogMAP demapping [5] and MMSE equalization [6], fixedpoint arithmetic has been adopted. In addition, the input data, the output data, and the intermediate computational values have been quantized according to defined precisions. Due to truncation and rounding processes, quantization errors occur. These errors propagate through the computational steps of the algorithms, and they are exacerbated in iterative schemes leading to a divergence at the output. In order to maintain the numerical stability of the algorithms and to ensure that quantization errors induce only small errors in the final result, a careful numerical study has been conducted. An accurate quantization and fixedpoint representation of all parameters and computational values involved in both algorithms have been determined.
Despite quantization approach and its corresponding evaluation are considered mandatory tasks in the design flow and can take much time and effort, their presentation are rarely published in the literature. Only few number of works have illustrated the used quantization and fixedpoint representation. In [9] and [10], the implementation of a lowcomplexity turboequalizer has been presented targeting a 16b fixedpoint DSP device with two’scomplement arithmetic. The authors have focused on BPSK signal set and only presented its corresponding simulation results. In addition, the quantization of input and output values of the main modules constituting the equalizer has been given. The precise quantization of intermediate computation result values have not been shown. Moreover, the authors has focused on exploiting a given word length without illustrating the fixedpoint representation. In [11], a fixedpoint representation of MMSEbased turbo equalizer with soft cancelation (SC) has been presented. The authors have targeted specific constellation scheme (QPSK) and presented the performance comparison between a nonquantized system and quantized system for different system configuration. In [12], the previous work has been extended to 16QAM constellation scheme. With the help of extrinsic information transfer (EXIT) charts, the authors have determined the sufficient number of fractional bits.
On the other side, in [13], a quantization study of loglikelihood ratios (LLR) in bitinterleaved coded modulation (BICM) systems has been provided. The performance of LLR quantization (1, 2, and 3 b) for MIMOBICM systems has been investigated for BPSK and 16QAM constellation schemes. In [14], the quantization and fixedpoint representation of few parameters of SISO demapper algorithm have been presented without showing their effect on the demapper performance. In [15], the authors proposed an architecture that supports only 16QAM modulation scheme. The quantization of input and output has been only provided without mentioning the fixedpoint representation.
The purpose of this paper is to present the efficient data quantization and fixedpoint representation that are devised for the architectures of MMSE turboequalizer [6] and MaxLogMap turbodemapper [5]. Moreover, their influence on the receiver errorrate performance is evaluated for multitude configurations. In this regard, the contribution of this paper can be considered as an important reference. The rest of the paper is organized as follows. The system model is presented in the next section. Sections 3 and 4 describe, respectively, the adopted algorithms for turboequalization and turbodemapping, discuss the required operations to implement the algorithms, present the required fixedpoint arithmetic and data quantization, and finally show their influence on errorrate performance. At last, Section 5 concludes the paper.
2 System model
The digital wireless communication system is basically composed of three blocks: transmitter, wireless channel, and receiver. The structures of the transmitter and receiver blocks rely on the specifications of the applied wireless communication standard. In general, data processing before the transmission of source data bits into the wireless channel includes adding redundant information to the original data, and/or rearranging data stream, and/or adding diversity of data. At the receiver side, the input information is distorted by fading and other destructive channel effects. The constituent components of the receiver process the received corrupted data to retrieve the original source data by exploiting the added redundancy and/or diversity. Modern wireless communication systems adopt MIMO technology, which uses multiple antennas at both transmitter and receiver sides of the wireless system, to meet the requirement of high data rate, reliability, and bandwidth efficiency. Iterative concept is also utilized at the receiver side to alleviate the destructive effects of the channel. Passing soft information between different components in the receiver through both forward and feedback paths has shown a prominent improvement of the output over the iterations leading to errorrate performance close to theoretical limits. MIMO technology and iterative processing have been incorporated in many modern wireless communication systems. The overall system model considered in this work is presented in Fig. 1. In the following subsections, the considered models of transmitter, channel, and receiver are briefly explained.
2.1 Transmitter scheme
The transmitter chain is established by concatenating different components to provide immunity to channel effects. Initially, the source bits s, socalled systematic bits, are encoded by a turbo encoder, which concatenates in parallel two eightstate double binary circular recursive systematic convolutional (CRSC) encoders [16, 17]. The output codeword c, that is made up of the source data and parities, is then punctured to reach a desired coding rate R _{ c }. Bit interleaved coded modulation (BICM) [18, 19] is used to disperse the obtained coded binary data sequence to assure that no single coded symbol is fully destroyed while passing through a fading channel. Punctured and interleaved bit stream v is passed to the mapper. Each mbit combination is mapped to channel symbol x according to the chosen constellation (BPSK till 256QAM) formed of 2^{m} symbols. After mapping, the symbols x are transmitted using either single antenna or MIMO techniques. Signal space diversity (SSD) technique [20] can be applied against the fading events in case of singleinputsingleoutput transmission, whereas in case of MIMO transmission, spatial multiplexing (SM) is adopted to improve the transmission rate [21].
2.2 Channel
The considered channel has a flat Rayleigh fading nature with additive white Gaussian noise (AWGN). The channel flat in frequency is a realistic model for several terrestrial mobile radio channels [22], and most works in MIMO literature assume this channel model [23, 24]. For a singleantenna transmission system, the received discrete time baseband complex signal y _{ k } can be expressed as follows [25]:
where x _{ k } is the complex signal transmitted at time k, h _{ k } is a Rayleigh distributed fading coefficient, and w _{ k } is a complex additive white Gaussian noise.
For MIMO systems with N _{ t } transmit antennas and N _{ r } receive antennas, the relation between channel, transmitted symbols and received symbols is given by the expression below:
where
where y and x represent, respectively, the received and transmitted symbol vectors, w represents the AWGN vector, H is the channel matrix whose element h _{ ij } represents the fading coefficient that characterizes the relation between the ith receive antenna and jth transmit antenna. On the other hand, the channel can be further categorized on the basis of time selectivity conditions. The time selectivity characteristic of a channel defines the variation of the channel with respect to time. It is related to the mobility of the transmitter, receiver, or the obstacles between the two depending on the nature of fading. This selectivity characterizes 3 types of channels: (1) fast fading, (2) quasistatic, and (3) block fading.
2.3 Receiver scheme
At the receiver side, the objective is to remove the channel effects to retrieve the original source data by exploiting the redundancy and diversity added to source information before transmitting data through the channel. Figure 1 shows the structure of the considered iterative receiver. It is characterized by the existence, in addition to forward paths, of feedback paths through which constituent blocks can send the information to previous blocks iteratively. On every new iteration, each block generates soft information depending on channel information and on received a priori soft information generated by other blocks in the previous iteration. The blocks constituting such receiver are referred to as softinput softoutput (SISO) processing blocks. In case of MIMO, the symbol vector y is received at the input of the MIMO equalizer, whereas in case of single antenna transmission, y symbol is received directly at the input of the demapper. For cases where SSD is used at the transmitter side, an additional latency similar to the one applied in the transmitter is required at the receiver in order to match inphase (I) and quadrature (Q) components of received symbols.
Benefiting from a priori information from the feedback path, the MMSE equalizer provides the estimated symbol vector \(\tilde {\mathbf {x}}\) and the corresponding equivalent bias vector (fading coefficient) to the demapper. The SISO demapper produces the probabilities \(\tilde {v}\) on transmit sequence in the form of log likelihood ratio (LLR), which construct after deinterleaving and depuncturing the input \(\tilde {c}\) to the decoder. The turbo decoder uses the BahlCockJelinekRaviv (BCJR) [26] decoding algorithm with MaxlogMAP approximation [27] and outputs the a posteriori information both on systematic and parity bits. This information is punctured and interleaved and then fed back to both SISO demapper and soft mapper. The latter provides a priori information to the equalizer as decoded symbol vector \(\hat {\mathbf {x}}\). This iterative process is stopped if a maximum number of iterations is reached. Then, the turbo decoder outputs the decoded bits.
3 MMSE linear equalizer
Turboequalization concept was first introduced in [4] to mitigate the detrimental effects of ISI for digital transmission protected by convolution codes. In the emerging wireless standards where MIMO techniques have been inducted, cochannel interference occurs at the receiver side. Cochannel interference is a cause of signal distortion when multiple signals are transmitted on the same frequency slots concurrently [7]. The concept of turboequalization can be used to cancel iteratively this interference caused by MIMO. One of the bestknown lowcomplexity approaches to achieve equalization in iterative MIMO systems is referred to as MMSE linear equalization (LE) [28, 29]. This approach is able to significantly lower the computational complexity compared to optimal maximumlikelihood (ML) algorithm. The use of MMSE in iterative scheme reduces the performance loss leading to errorrate results close to ML. At least, 3dB gain can be obtained for bit error rate (BER) performance, compared to a noniterative MMSE [28, 30].
3.1 Algorithmic overview
The inputs to the MMSE equalizer are the received symbol vector y of size N _{ r }, channel matrix H of size N _{ r }×N _{ t }, and the variance of the AWGN vector \(\sigma ^{2}_{w}\). Using this information, the equalizer generates the estimated symbol vector \(\tilde {\mathbf {x}}\). The equalizer considers that a symbol of the vector x is distorted by the N _{ t }−1 other symbols of the vector and by the noise channel and it tries to combat both. Equation (2) can be written in the following form:
where j∈{0,N _{ t }−1}, h _{ i }, h _{ j } are the ith and jth columns of H matrix and w is the AWGN noise vector of size N _{ r }. One of the lowcomplexity techniques to achieve the equalization function is the use of filterbased symbol equalization [29]. An estimation of the symbol x _{ j } can be carried out through a linear filter which minimizes the mean square error (MSE) between the transmitted symbol x _{ j } and the output of the equalizer \(\tilde {x}_{j}\). Using the Wiener filter \(\mathbf {a}^{H}_{j} = \lambda _{j}.\mathbf {P}^{H}_{j}\), the estimation of x is given by [28]:
where j∈{0,N _{ t }−1}, \(\hat {x}_{j}\) is the jth element of vector \(\hat {\mathbf {x}}\), h _{ j } is the jth column of H matrix, and (.)^{H} is the Hermitian operator. P _{ j } and λ _{ j } are defined as follows:
where
and \(\sigma ^{2}_{x}\), \(\sigma ^{2}_{\hat {x}}\), and \(\sigma ^{2}_{w}\) are variances of transmitted symbols, decoded symbols, and noise, respectively. I is the identity matrix of size N _{ r }×N _{ r }.
where
Equation (4) can be written as:
where
During the first iteration of turboequalization process, no a priori information is presented (\(\hat {\mathbf {x}}\) is a null vector and \(\sigma ^{2}_{\hat {x}}=0\)) and the symbols are equiprobable. The estimated symbol becomes as following:
where
and
From the second iteration, the a priori information, which is provided by channel decoder about transmitted symbols, improves gradually over the iterations and approaches to asymptotic performance. Asymptotic performance is achieved when the a priori data is perfect, i.e., becomes equal to the transmitted data (\(\hat {x}_{j}=x_{j}\)).
3.2 MMSE algorithm towards implementation
The abovelisted expressions exhibit three main computations steps:

1.
Detection vector computation referred by P in (5)

2.
Equalization coefficients computation referred by β, λ, and g in (8), (7), and (10)

3.
Estimated symbols computation referred by \(\tilde {\mathbf {x}}\) (9)
A closer look at the expressions required in MMSE algorithm ((4) to (10)) reveals the serial nature of the implied elementary computations. Firstly, one need to compute serially the detection vector (P) and the equalization coefficients (β, λ, and g) due to their related dependency, and then symbols are estimated using these coefficients. Furthermore, the expressions performed to fulfill the equalization tasks of computing the detection vector and coefficients and estimating symbols have similar arithmetic operations. But since the computed coefficients are involved in symbol estimation process, the two tasks are executed at different times.
Furthermore, at each iteration, new value of decoded symbols variance \(\sigma ^{2}_{\hat {x}}\) (6) is delivered to the equalizer imposing the recomputation of β, λ, g, and P for all channel selectivity types. In fact, these values also depend on the channel matrix H (6), which entries change according to the time selectivity of the channel. Hence, the time diversity of the channel decides how frequent the computations of detection vector and equalization coefficients are required. These computations are recomputed repeatedly for each received vector in case of fast fading channel, once for a set of received vectors for which channel matrix is considered as constant in case of block fading channels and once for all received vectors of the frame in case of quasistatic channel. Thus, the channel type (fast fading, quasistatic, or block fading) specifies the computation overhead per iteration. To ensure efficiency and flexibility related to time selectivity of the channel, hardware operators are shared among all required computations in order to take into account the required treatment of data flow for each channel type.
Another flexibility requirement is related to antenna dimension. To cope with diverse configurations which are imposed by the emerging communication standards, different MIMO schemes are supported. In order to maintain efficiency and to meet the requested flexibility requirement, the hardware implementation considers the lowest complex configuration (2×2) and applies a hardware resource sharing technique to support the other highorder configurations. To manage variable size complex matrix operations that are involved in the MMSE equalization algorithm, complex matrix operations are decomposed into basic real arithmetic operations. The required operations to perform coefficient computations and symbol estimation can be categorized into complex number operations and complex matrix operations.
Complex number addition, subtraction, and negation are performed using real operators as shown in Fig. 2. Complex number multiplication is reformulated to reduce the number of required multiplication operations. Figure 3 shows the real operators used in complex number multiplication operation.
Complex matrix operations involved in MMSE such as matrix addition, subtraction, conjugation, and multiplication are broken down into basic complex number operations. The Hermitian of a complex matrix can be viewed as matrix conjugation followed by a transposition (swapping columns for rows in the matrix). As an example of complex matrix operations decomposition, Fig. 4 shows the required operators to perform 2×2 complex matrix multiplication operation. In the figure, each complex multiplier and each complex adder integrates the real operators presented in Figs. 3 and 2 a respectively.
For matrix inversion, the analytical method is used for 2×2 matrix inversion which is given by:
In case of 4×4 matrix, the matrix is first divided into four submatrices and then inverted in a blockwise manner by using the following formula:
where
and A, B, C, D, W, X, Y , and Z are 2×2 matrices. In case of 3×3 matrix, the matrix can be extended firstly to a 4×4 matrix and then inverted by applying the same formula derived above for 4×4 matrix inversion. The extending is done by copying all three rows of 3×3 matrix into first three rows of 4×4 matrix and then putting zeros in all elements of fourth row and fourth column except in their intersection where one should be placed. The final result lies in the first three elements of first three rows and first three columns.
Based on the positivedefinite property of the matrix resulting from the multiplication of the MIMO channel matrix H by its Hermitian H ^{H}, β _{ j } values and the matrices determinants (Δ _{ E }), (Δ _{ A }), and (\(\phantom {\dot {i}\!}\Delta _{\mathbf {D  CA^{1}B}}\)) are proved to be positive real numbers. Hence, there is no need to implement the computationally demanding complex number inversion operations required in the computations of determinants and λ _{ j } values (7). Real inversion operations can be applied instead.
Inversion process is preferably replaced by lookup table (LUT). LUT is appraised as an efficient implementation of inversion process by using memory instead of large numbers of logical elements. Both resource utilization and propagation delay are reduced at the cost of accuracy. The utilized LUT should contain all possible inverse values. The value x intended to be inverted is used directly as the LUT index (address) to retrieve the inverse value \(\frac {1}{x}\).
3.3 Quantization and fixedpoint arithmetic
The aim of quantization and fixedpoint arithmetic is to minimize the implementation cost. However, a minimum computational accuracy must be guaranteed to maintain the application performance. A careful numerical study has been conducted to determine the accurate quantization and fixedpoint representation of all parameters and computational values involved in the algorithm. The implementation cost is minimized as long as the equalizer performance is fulfilled.
A fixedpoint data is made up of integer part and fractional part. The number of bits required for integer part is defined from the dynamic range of the data in order to avoid the occurrence of an overflow [31]. The reusability and sharing of resources implies that the allocated registers and operators deal with multiple computational values which have different dynamic ranges and variable precisions. So, the bitwidth for all components have to be fixed and their precisions vary according to the data requirements by choosing different bits for integer and fractional parts. Long numerical simulations have been conducted for different configurations to find the required data width and accurate precisions for fixedpoint representation of all parameters involved in MMSE LE algorithm. Utilizing 16b two’s complement representation with different bits for integer and fractional part in different computation steps shows low performance degradation. Using fixedpoint representation demands establishing a virtual decimal point placed in between two bit locations for a given length of data. Figure 5 and Table 1 illustrate the devised quantization and fixedpoint representation of different parameters for MMSE algorithm and matrix inversion.
For different modulation types, the quantization values are shown in Table 1 in signed two’s complement representation using the notation Q[I].[F] where [I] and [F] designate the number of bits for integer part and fractional part, respectively. In all algorithmic steps, fixedpoint arithmetic is used. First of all, input data that is presented in less than 16b is extended to 16b by adding zeros in the added lower bits. Secondly, for all addition/subtraction operations, the operands (addends/subtrahend and minuend) should have the same precision. In case of overflow/underflow, the total/difference is directly set to the most positive/most negative value. Finally, after each multiplication, the doubleprecision product is converted to 16b by eliminating the m least significant bits (LSB) and 16−m most significant bits (MSB) where m=F _{ a }+F _{ b }−F _{ c } and F _{ a }, F _{ b }, and F _{ c } represent the number of fractional part of the multiplicand, multiplier, and product, respectively. Figure 6 illustrates an example for the adopted technique to quantize the product value. The multiplicand and multiplies are represented in Q[10].[6] and Q[5].[11], respectively, whereas, the product is represented in Q[6].[10]. In fact, the multiplication of these two 16b values results in 32b product value that can be represented in Q[15].[17]. From the expression above, we have m=7, and thus to accommodate the product in the target quantization representation (Q[6].[10]), the 7 LSB are truncated as well as the 9 MSB.
An overflow/underflow is detected if the multiplicand and multiplier have same/opposite signs, and the product is greater/smaller than the most positive/most negative value. In such case, the product is fixed to the most positive/most negative value. Last of all, the inversion operation of real numbers is achieved by the assist of a single \(\frac {1}{x}\)LUT instead of undergoing expensive computations. The LUT contains 16b positive values. At each index, the stored value represents the quantized inverse of the index value.
3.4 Performance evaluation
One of the most critical parts of the quantization process is the evaluation of the degradation of the application performance. A software model of the MMSE equalizer has been developed to examine the impact of the devised quantization and the adopted fixedpoint arithmetic on the errorrate performance. The model with quantization and fixedpoint specifications is simulated for different system configurations, and the corresponding errorrate performance is measured. Figure 7 presents the obtained frame errorrate (FER) performance for 4×4 MIMO SM with QPSK, 16QAM, and 64QAM. In addition, the obtained FER results are compared to corresponding FER of a reference floatingpoint model. The analysis of the results has shown a performance loss below 0.2 dB for 64QAM and below 0.1 dB for 16QAM and QPSK at FER = 10^{−3}. Note that the FER values are recorded for 100 erroneous frames for each \(\frac {E_{b}}{N_{0}}\) value.
4 MaxLogMAP demapper
Iterative demapping was proposed firstly in [3] based on bit interleaved coded modulation (BICM) with additional soft feedback from the SISO convolutional decoder to the constellation demapper. For a system with convolutional code, BICM and 8PSK modulation, 1 and 1.5dB gains for BER performance were reported for Rayleigh flat fading channels and channels with AWGN, respectively. In [32], the impact of different mapping styles on the performance of BICM with iterative demapping for Rayleigh fading channels have been investigated. Iterative demapping has provided significant coding gains for several mapping schemes of QAM constellations. In [33], only a small gain of 0.1 dB was observed when the convolutional code was replaced by a turbo code. This result makes iterative demapping with turbolike coding solutions unsatisfactory even though the added complexity is relatively small. On the other hand, SSD technique was introduced in [20] to improve the performance gains. An improvement exceeding 0.8dB gain is observed at BER lower than 10^{−7} at the price of a relatively small added complexity without sacrificing the iterative process convergence. In [34], the use of iterative demapping shows performance improvement of 1.2 dB at BER of 10^{−6} for QAM BICM scheme with LDPC channel decoder over flat fading Rayleigh channel with 15% of erasures. The symbolbysymbol maximum a posteriori (MAP) algorithm is the optimal algorithm for obtaining the outputs of the demapper. The MAP algorithm is likely to be considered of high complexity for hardware implementation in a real system basically because of the numerical representation of probabilities, nonlinear functions, and mixed multiplications and additions of these values [27]. To avoid the number of complicated operations, certain simplifications are applied. Implementing the MAP algorithm in its logarithmic domain instead of probabilistic form reduces the computational complexity. Operating in logarithmic domain eliminates exponential operations and transforms multiplication/division operations into addition/subtraction operations. MaxLogMAP demapping algorithm is a suboptimal direct transformation of the MAP algorithm into logarithmic domain; hence, values and operations are easier to handle.
4.1 Algorithmic overview
Depending on the transmitter configuration and propagation conditions, the input from the wireless channel can be either directly delivered to the demapper or passed through a channel equalizer as shown in Fig. 1. To reduce the computational complexity, the demapper works in logarithmic domain and produces probabilities \(\tilde {v}\) on received sequence in the form LLRs, where v represents the binary mapping of the transmitted sequence. The demapper computes the LLRs using the following expression [35]:
where m is the number of bits per symbol, i=0,1,…,m−1, \(L\left (\tilde {v}_{t}^{i}\right)\) is the LLR of ith bit of transmitted symbol at time t, \(\mathcal {X}_{0}^{i}\) and \(\mathcal {X}_{1}^{i}\) are the symbol sets of constellation for which symbols have their ith bit equals b∈{0,1}, ρ _{ t } is the channel fading coefficient and σ ^{2} is the AWGN variance, and \(P\left (\hat {v}_{t}^{l}\right)\) is the probability of lth bit of symbol x computed through a priori information. To reduce the complexity, maxlog approximation [27] is applied by using the following formulas:
The expression in (17) becomes:
where
and
where v ^{l} is the lth bit of each received modulated symbol.
In the case of noniterative demodulation, no a priori information is provided to the demapper. The expression of LLRs in (21) becomes:
Moreover, for Gray mapped constellations, I and Q components are independent from each other; hence, the Euclidean distance is calculated in one dimension. In case where m is even, further simplification can be applied. The expression in (24) can be transformed in this case into the following expressions [36]:
and
where
and \(\mathcal {X}(I)_{b}^{i}\) and \(\mathcal {X}(Q)_{b}^{j}\) are the constellation point sets on Iaxis and Qaxis with ith and jth bits of symbol x that have a value equals to b. Applying this simplification, \(2^{\frac {m}{2}}\) onedimensional Euclidean distances are computed instead of 2^{m} twodimensional Euclidean distances for each LLR.
In case of passing the received symbols through SISO equalizer (Fig. 1), symbol y in expressions (21), (24), (25), and (26) is replaced with \(\tilde {x}\) (4). Also, the fading factor ρ and variance σ ^{2} in the upmentioned expressions are replaced with g (10) and \(g(1g)\sigma ^{2}_{x}\), respectively [29].
4.2 MaxLogMap demapping algorithm towards implementation
The simplified expression in (21) exhibits four main computation steps:

1.
Euclidean distance computation to find D

2.
A priori LLRs summation to calculate A p _{ i }

3.
Minimum operations referred by the min functions.

4.
Subtraction operation of minimum values to determine \(L\left (\tilde {v}_{t}^{i}\right)\) values
To determine output LLRs related to each received symbol y, computations of Euclidean distances and a priori LLR summation are repeated consecutively for all symbols of target constellation. Performing concurrently, these computations enhances the demapper execution performance. In a constellation with m bits per modulated symbol, 2^{m} Euclidean distances, and 2^{m} a priori LLR summation operations are needed. Their corresponding 2^{m} resultant differences are fed to m minimum finder operations to determine the minimum values relative to each bit. m subtractors are needed to determine the final LLR values. Thus, the complexity of the demapper implementation varies significantly with respect to m.
Recent emergent standards specify different mapping types starting from BPSK till 256QAM as shown in Table 2. To meet with different standards specifications, the demapper supports the implementation of all required computations for variable modulation orders where m can range from 1 to 8. In general, the allocated hardware resources are not shared among different computational tasks (computing of Euclidean distance, a priori LLR summation, finding the minimum values, and subtraction operations to determine the final LLR values) to achieve the best execution performance. Furthermore, for operations depending on constellation size, sufficient resources are instantiated to suit the highestorder target constellation (256QAM). A simple way to cope with modulation order variety is to store the constellation information (x ^{I}, x ^{Q}, and the binary mapping μ) in LUT, which contents are rewritten when system configuration changes. The size of the LUT, socalled Constellation LUT, varies according to the modulation order and mapping style. In fact, the depth of Constellation LUT equals the number of constellation points involved in determining the LLRs associated to one input symbol, whereas the width is constant and it is determined by the total number of bits representing the constellation information. Figure 8 shows the structure and organization of Constellation LUT for 16QAM modulation scheme. The LUT in Fig. 8 b contains the needed information of 16QAM constellation presented in Fig. 8 a when Gray mapping simplifications of expressions (25) and (26) are applied. The LUT in Fig. 8 c represents the constellation information required when using the general MaxLogMAP demapping algorithm expressed in (21).
Furthermore, while exploring expressions (21), (25), and (26), one can notice that they share common arithmetic operations in computation of onedimensional or twodimensional Euclidean distances. In fact, computing of one twodimensional distance is equivalent to compute two separate onedimensional distances. Hence, same hardware resources can be used for different mapping styles. Figure 9 shows the operators used in Euclidean distance computation while targeting the highest parallelism. A separate operator is allocated for each required operation and the inversion operation is achieved using \(\frac {1}{2x}\)LUT instead of undergoing expensivencomputations.
Moreover, supporting iterative demapping requires the implementation of operators that perform a priori LLR summation. To accommodate all target constellations, hardware implementation is set to meet with the requirements of highestorder target constellation (256QAM). Figure 10 a shows the operators used in the summation of a priori LLRs in case of 256QAM modulation scheme, whereas Fig. 10 b shows the eight subtraction operators used to realize D _{ i } values expressed in the following equation:
where D _{ i } represents the subtraction of summation of a priori LLRs corresponding to bit v ^{i} from the computed Euclidean distance. In case of lowerorder modulation schemes, the usage rate of hardware resources involved in a priori LLR summation decreases. For example, in case of 16QAM modulation scheme only half of the hardware resources related to this operation will be activated.
Similarly to a priori LLR summation, the requirements of 256QAM modulation scheme is adopted to implement hardware resources capable to perform minimum operations referred by the min functions and the subtraction operation of minimum values to determine \(L\left (\tilde {v}_{t}^{i}\right)\) values expressed in (21). Sharing resources for different minimum operations leads to decreased throughput especially for highorder modulation schemes. Figure 11 presents the Minimum Finder operational unit for one LLR corresponding to bit v ^{i}. Updating the minimum value depends on the value of v ^{i} and the sign S of the resultant value of subtracting available minimum value from new D _{ i }. The sign S represents the most significant bit (MSB) value of the difference.
In addition, the figure shows the subtractor operator required to perform the subtraction operation of minimum pairs corresponding to symbol sets \(\mathcal {X}_{0}^{i}\) and \(\mathcal {X}_{1}^{i}\).
4.3 Quantization and fixedpoint arithmetic
As for the equalizer module, all computational values are quantized according to defined precisions. Detailed analysis and long numerical simulations have been conducted for different configurations to find the required data width and accurate precisions for fixedpoint representation of all parameters involved in MaxLogMAP demapping algorithm. As discussed in previous subsection, the demapper implementation does not adopt sharing of hardware resources among different computational operation types. Hardware components are considered to deal with the same type of data. Hence, quantized computational parameters may have different data widths. Accordingly, bitwidths of each computational parameter is carefully selected to ensure least performance degradation. A tradeoff between performance and implementation costs has been conducted.
On the other hand, fixedpoint representation is used by placing virtual decimal point in between two bit locations to separate the number of bits representing integer and fractional parts. The square operation in calculating the Euclidean distance (22) implies the definite positivity of resultant parameters. This criterion is exploited to classify computational parameters into signed and unsigned numbers. Two’s complement representation is used to represent signed values, whereas unsigned numbers are represented in binary representation which is considered simpler and does not impose extra bits. Figure 12 represents the devised quantization of different parameters used in all computational operations of the MaxLogMAP demapping algorithm. Table 3 shows the values of parameter quantization in fixedpoint representation. The notation Q[I].[F] is used where [I] and [F] designate the number of bits for integer and fractional parts, respectively. The prefixes “US” and “S” indicate whether the parameter is considered unsigned binary number or signed binary number represented in two’s complement representation.
Furthermore, the operands of addition and subtraction operations are prior adjusted to the same fixedpoint representation. Sign extending or zero padding (adding zeros to lower or upper bits) techniques are applied based on the quantization characteristics of parameters prior and post the adjustment. Before performing addition or subtraction operations, the operands are 1b signextended to avoid underflow or overflow occurrence. The inversion operation of variance σ ^{2} is achieved using a LUT instead of undergoing expensive computations. The LUT contains 8b positive values which are required to represent the inverse values \(\left (\frac {1}{2\sigma ^{2}}\right)\). At each index, the stored value represents the quantized \(\frac {1}{2x}\) value of the index value x.
4.4 Performance evaluation
In order to evaluate the efficiency of the quantization parameters, the application performance is verified. Also, the computation accuracy due to adopted fixedpoint arithmetic is evaluated. To measure the impact of quantization errors on the demapper performance, a methodology based on bittrue simulation of the fixedpoint application has been utilized. For various system configurations, a software model implementing the devised quantization and fixedpoint specifications is used to simulate the demapper functionality. Accordingly, the corresponding frame errorrate (FER) performances of the receiver are recorded. Figure 13 presents the obtained FER curves compared to the reference floatingpoint curves. The analysis of the results has shown a performance loss below 0.05 dB for QPSK and 16QAM at FER=10^{−2}. Note that the FER values are recorded for 100 erroneous frames for each \(\frac {E_{b}}{N_{0}}\) value. The obtained results shows a slight degradation in errorrate performance of the receiver. Thus, the effect of the quantization errors on the generated output LLR values is insignificant.
5 Conclusions
Fixedpoint arithmetic and data quantization affect the performance of algorithmic implementation. In this paper, related issues to the fixedpoint arithmetic of MMSE MIMO linear turboequalization and MaxLogMap demapping are discussed for all algorithmic parameters and steps. An efficient quantization and fixedpoint representation have been presented. Their impact is illustrated upon the FER performance for different system configurations. Only a slight degradation in the FER performance of the receiver is observed when implementing the equalizer and demapper modules which utilize the devised quantization and fixedpoint arithmetic rather than floatingpoint arithmetic.
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MR investigated the algorithms and conducted the numerical study to determine the quantization and fixedpoint representation of all parameters and computational values involved in the algorithms, evaluated the impact of devised quantization and fixedpoint arithmetic on the errorrate performance, and wrote the paper. AB scientifically supervised the work, contributed in determining the quantization and fixedpoint representation, and participated in writing the paper. MJ, YM, and YA academically supervised the work for IMT Atlantique and Lebanese University. All authors read and approved the final manuscript.
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Rizk, M., Baghdadi, A., Jézéquel, M. et al. Efficient quantization and fixedpoint representation for MIMO turbodetection and turbodemapping. J Embedded Systems 2017, 33 (2017). https://doi.org/10.1186/s136390170081y
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DOI: https://doi.org/10.1186/s136390170081y