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IEICE TRANS. COMMUN., VOL.E101–B, NO.3 MARCH 2018731
PAPERLow Complexity Log-Likelihood Ratio Calculation Scheme with BitShifts and Summations
Takayoshi AOKI†a), Student Member, Keita MATSUGI†b), Nonmember, and Yukitoshi SANADA†c), Fellow
SUMMARY This paper presents an approximated log-likelihood ratiocalculation scheme with bit shifts and summations. Our previous workyielded a metric calculation scheme that replaces multiplications with bitshifts and summations in the selection of candidate signal points for jointmaximum likelihood detection (MLD). Log-likelihood ratio calculation forturbo decoding generally uses multiplications and by replacing them withbit shifts and summations it is possible to reduce the numbers of logic op-erations under specific transmission parameters. In this paper, an approx-imated log-likelihood ratio calculation scheme that substitutes bit shiftsand summations for multiplications is proposed. In the proposed scheme,additions are used only for higher-order bits. Numerical results obtainedthrough computer simulation show that this scheme can eliminate multipli-cations in turbo decoding at the cost of just 0.2 dB performance degradationat a BER of 10−4.key words: approximated log-likelihood ratio, turbo decoding
1. Introduction
Recently, mobile phones and wireless local area networks(LANs) are widely used all over the world. Wirelesscommunication systems are expected to accommodate alarger amount of data traffic. Multiple-input multiple-output(MIMO) has been proposed to realize larger capacity andbetter reliability in wireless communication systems [1]–[4].MIMO uses multiple antenna elements both at a transmitterand a receiver.
However, a limited number of receive antennas can beimplemented in a mobile terminal owing to its form fac-tor. Thus, overloaded MIMO has been investigated [5]. Inoverloaded MIMO systems there are fewer receive antennaelements than transmit antenna elements. In the receiversof overloaded MIMO systems, nonlinear detection schemessuch as maximum likelihood detection (MLD) are appliedin conjunction with error correction coding [6], [7]. The-ses schemes achieve superior performance as compared tothat of a single signal stream with higher order modulation.However, the computational complexity of MLD increasesexponentially with the number of signal streams. In order todecrease the detection complexity in overloaded MIMO sys-tems, detection schemes with lower complexity have beenrecently proposed [8]–[13]. A detection scheme with vir-
Manuscript received May 7, 2017.Manuscript revised August 22, 2017.Manuscript publicized September 19, 2017.†The authors are with the Dept. of Electronics and Electrical
Engineering, Keio University, Yokohama-shi, 223-8522 Japan.a) E-mail: [email protected]) E-mail: [email protected]) E-mail: [email protected]
DOI: 10.1587/transcom.2017EBP3180
tual channels has been proposed in [8], [9], it was combinedwith lattice reduction in [10], and a novel iterative demodu-lation scheme called a semi hard input soft output receiverwas also applied to the virtual channels in [11]. On the otherhand, [12], [13] applied slab decoding and lattice reductionto successive interference cancellation detection with pre-voting cancellation.
In [14], [15], a metric calculation scheme that replacesmultiplications with bit shifts and summations in the se-lection of candidate signal points has been proposed forjoint MLD. Furthermore, log-likelihood ratio calculation forturbo decoding generally uses multiplications and by replac-ing them with bit shifts and summations it is possible to re-duce the numbers of logic operations under specific trans-mission parameters. In this paper, an approximated log-likelihood ratio (LLR) calculation scheme is proposed. Inthe proposed scheme, multiplications are replaced by bitshifts and additions. The additions are used only for higher-order bits. The proposed scheme is possible to eliminate allthe multiplications in the turbo decoder in conjunction withsub-optional maximum a posteriori (MAP) decoding and alook-up table [16].
This paper is organized as follows. Section 2 describesthe system model. Section 3 explains the simulation results.In Sect. 4, our conclusions are presented.
2. System Description
2.1 Signal Model
The block diagram of an overloaded MIMO-OFDM systemwith joint MLD is shown in Fig. 1. Information bits are en-coded by a turbo code and coded bits are rearranged by aninterleaver. A code rate is then adjusted through puncturingin a rate matching block. After puncturing M coded bits areallocated to a 2M QAM symbol. A symbol on the lth subcar-rier, S p[l], is transmitted from the pth transmit antenna. TheOFDM signal transmitted from the pth transmit antenna isgiven by
up[n] =
N−1∑l=0
S p[l] exp(
j2πnl
N
)(1)
where n is the time index (n = 0, 1, ...,N − 1), and N isthe size of the inverse discrete Fourier transform (IDFT).A guard interval (GI) is then added by replicating the lastpart of the OFDM signal. The transmit signal from the pth
Copyright c© 2018 The Institute of Electronics, Information and Communication Engineers
732IEICE TRANS. COMMUN., VOL.E101–B, NO.3 MARCH 2018
Fig. 1 MIMO-OFDM System.
antenna is written as
vp(t) =
N−1∑n=−NGI
up[n]ptp(t − nTs) (2)
where ptp(t) is the impulse response of the transmit filter, Tsis the sampling interval of the OFDM symbol, and NGI isthe GI length. In the receiver side, the received signal at theqth receive antenna is given by
yq(t) =
NT∑p=1
yqp(t) + nq(t) (3)
where nq(t) is the noise at the qth receive antenna and yqp(t)is the received signal from the pth transmit antenna to theqth receive antenna. yqp(t) is calculated by
yqp(t) =
N−1∑n=−NGI
up[n]hqp(t − nTs) (4)
where hqp(t) is the impulse response of the channel betweenthe pth transmit and qth receive antennas. Therefore, thereceived digital signal after A/D converters are given as
yq[n] = yq(nTs). (5)
The receiver removes the GI and takes the DFT of N sam-ples. After the DFT, the signal on the lth subcarrier is ex-pressed as
Yq[l] =
N−1∑n=0
yq[n] exp(
j−2πnl
N
)
=
NT∑p=1
Hqp[l]S p[l] + Nq[l] (6)
where Hqp[l] is the frequency response between the pthtransmit antenna and the qth receive antenna and Nq[l] is thenoise through the qth receive antenna on the lth subcarrier.It can be written in a matrix form as
Y[l] = H[l]S[l] + N[l] (7)
where
Y[l] =[Y1[l] Y2[l] . . . YNR [l]
]T , (8)
H[l] =
H11[l] · · · H1NT [l]...
. . ....
HNR1[l] · · · HNRNT [l]
, (9)
S[l] =[S 1[l] S 2[l] . . . S NT [l]
]T , (10)
N[l] =[N1[l] N2[l] . . . NNR [l]
]T . (11)
2.2 Calculation of LLR for Joint MLD
An LLR is calculated for each coded bit in the systematicpart of the codeward in the joint MLD. The likelihood valuesfor the mth bit of the symbol in the pth signal are given asfollows:
D1pm =
∑S1
pm∈{S}bpm=1
exp(− 1σ2 ‖Y[l]−H[l]S1
pm‖2), (12)
D0pm =
∑S0
pm∈{S}bpm=0
exp(− 1σ2 ‖Y[l]−H[l]S0
pm‖2), (13)
where σ2 is the noise variance, S1pm or S0
pm is the vectorof candidate NT symbols in which the mth coded bit of a2M QAM symbol from the pth transmit antenna is “1” or“0”, S =
[S 1 S 2 . . . S NT
]T is the candidate symbol vector,
AOKI et al.: LOW COMPLEXITY LOG-LIKELIHOOD RATIO CALCULATION SCHEME WITH BIT SHIFTS AND SUMMATIONS733
{S}bpm=1 or {S}bpm=0 is the set of the symbol vectors in whichthe mth coded bit of the pth 2M QAM symbol, S p, is “1” or“0”, and D1
pm or D0pm is the sum of the likelihood values for
the mth coded bit of “1” or “0” in the pth symbol, respec-tively. The LLR is calculated as,
L(bpm|Y[l]) = logD1
pm
D0pm
(14)
where L(bpm|Y[l]) is the LLR of the mth bit of the symbolson the lth subcarrier transmitted from the pth antenna.
2.3 Approximation of LLR
Sub-optional decoding by the approximation of the LLRwith the dominant terms has been presented in [16]. Theapproximated LLR is given as
LC(bpm|Y[l]) ≈ (− 1σ2 ‖Y[l]−H[l]S
1pm[l]‖2)
−(− 1σ2 ‖Y[l]−H[l]S
0pm[l]‖2)
(15)
where S1pm[l] or S
0pm[l] is one of the candidate symbol vec-
tors in {S1pm} or {S0
pm} and the one with the minimum Eu-clidean metric from Y[l] is selected as follows:
S1pm[l] = arg min
S1pm∈{S}bpm=1
NR∑q=1
{(<[E1qpm[l]])2 + (=[E1
qpm[l]])2},
(16)
S0pm[l] = arg min
S0pm∈{S}bpm=0
NR∑q=1
{(<[E0qpm[l]])2 + (=[E0
qpm[l]])2},
(17)
where<[?] is the real part of ?, =[?] is the imaginary partof ? and E1
qpm[l] or E0qpm[l] is the difference between the
coordinates of the received signal and the candidate signalpoint in which the mth coded bit of the pth signal stream is“1” or “0” and these are calculated by the followings:
E1qpm[l] = Y1
q [l] −Hqp[l]S1pm, (18)
E0qpm[l] = Y0
q [l] −Hqp[l]S0pm, (19)
where Hqp[l] =[Hq1[l] Hq2[l] . . . HqNT [l]
]T.
This approximation omits the calculation of exponen-tial terms. However, it is required to determine the dominantterms by calculating the Euclidean metric.
2.4 Summation and Subtraction Metric
The summation and subtraction metric (SSM) consists oftwo terms, ||<[E∗qpm]|+ |=[E∗qpm]|| and ||<[E∗qpm]−=[E∗qpm]||(∗ = “1” or “0”), for the selection of the candidate signalvector among {S1
pm} or {S0pm} [14], [15]. These terms are
combined with corresponding coefficients. The curves ofthe metrics are drawn in Fig. 2. The difference of the coor-dinates of the signal points is given as eiθ where θ is from
Fig. 2 Summation and Subtraction metrics.
0 to π/2. Here, the Euclidean metric is assumed to be con-stant and its value is normalized to |eiθ| = 1. Even thoughthe amplitude of the difference vector is not constant ow-ing to the noise, the error of the SSM from the Euclideanmetric is constant for the same augment of the vector if it isnormalized by the amplitude of the difference vector.
The candidate symbol vector with the minimum metricis selected as the following equations;
Sn1pm = arg min
S1pm∈{S}bpm=1
NR∑q=1
2(|<[E1qpm[l]]| + |=[E1
qpm[l]]|)
+(||<[E1qpm[l]]| − |=[E1
qpm[l]]||)},(20)
Sn0pm = arg min
S0pm∈{S}bpm=0
NR∑q=1
2(|<[E0qpm[l]]| + |=[E0
qpm[l]]|)
+(||<[E0qpm[l]]| − |=[E0
qpm[l]]||)},(21)
where Sn1pm or Sn0
pm is the candidate symbol vector from {S1pm}
or {S0pm} with the smallest metric, E1
qpm[l] and E0qpm[l] are
calculated by Eqs. (18) and (19). With the selected candi-date symbol vectors the likelihood ratio is then calculatedby Eq. (15).
3. Complexity Reduction in LLR Calculation
3.1 Conventional LLR Calculation Schemes
In the joint MLD, the LLR of a bit in a received signal being“1” or “0” is calculated for each bit. It requires to calcu-late Euclidean distances between the received signal pointand the candidate signal points in the conventional LLRcalculation. It is necessary to calculate the Euclidean dis-tances for all the sets of candidate symbol vectors and thatrequires 2 · 2MNT NR multiplications. The number of multi-plications increases exponentially with the number of sig-nal streams. In order to reduce the calculation complexity,the candidate symbol vectors can be selected by the metriccalculation and the approximated likelihood is calculated as
734IEICE TRANS. COMMUN., VOL.E101–B, NO.3 MARCH 2018
Table 1 Example of proposed scheme.
(|∆0pm | − |∆
1pm |) = 0001110011001010
C Before Approximation After Approximation1 0001110011001010 00010000000000002 0001110011001010 00011000000000003 0001110011001010 0001110000000000
(|∆0pm | + |∆
1pm |) = 0110001011100101
(|∆0pm | + |∆
1pm |)(|∆
0pm | − |∆
1pm |)
C Before Approximation After Approximation1 0001011000111110 00001100010111002 0001011000111110 00010010100010103 0001011000111110 0001010110100010
given in Sect. 2.3. In our previous research, 2MNT NR mul-tiplications are required for the calculation of Eq. (15). Inthis paper, a new approximated LLR calculation scheme thatsubstitutes bit shifts and additions for multiplications is pro-posed.
3.2 Proposed Scheme
The proposed approximated LLR is given as follows:
LP(bpm|Y[l]) ≈ − 1σ2 |∆
1pm|
2 − (− 1σ2 |∆
0pm|
2)
= 1σ2 (|∆0
pm|2 − |∆1
pm|2)
= 1σ2 (|∆0
pm| + |∆1pm|)(|∆
0pm| − |∆
1pm|)
(22)
where ∆1pm or ∆0
pm is the SSM corresponding to Sn1pm or Sn0
pm.The value of the second parentheses, (|∆0
pm| − |∆1pm|), is fur-
ther approximated in the proposed scheme. The C bits fromthe most significant “1” bit are preserved and the rests aretreated as “0”. Table 1 shows an example of the approxi-mation process. The most significant “1” bit is the forth bitand all the bits from the (4 + C − 1) bits are set to zero. Themultiplication in Eq. (22) is then replaced by bit shifts andadditions. The value of (|∆0
pm|+ |∆1pm|) is shifted according to
the positions of the “1” bit of the approximated value. Theshifted values are summed together to obtain the approxi-mated LLR of the proposed scheme.
Generally, the value of the second parentheses, (|∆0pm|−
|∆1pm|), is smaller than that of the first parentheses, (|∆0
pm| +
|∆1pm|). Thus, this approximation should be less significant
to the LLR value.
3.3 Computational Complexity
The numbers of multiplications, summations, and bit shiftsfor each scheme are shown in Table 2. These numbers in-clude the complexity of both the candidate signal point se-lection and the likelihood value calculation. It is shown thatthe proposed scheme can reduce the numbers of multiplica-tions while it requires more numbers of summations and bitshifts as the value of C increases.
For broadband communications, high speed multipli-ers and adders are required for metric calculation. The ex-amples of a high speed adder and an array multiplier are
Table 2 Number of operations in each calculation scheme.Metrics Multipli Summations Bit
-cations Shifts
Euclidean 2 · 2MNT NR 2 · 2MNT NT NR 0Metric +4 · 2M NT NR +MNT
SSM 4 · 2M NT NR 5 · 2MNT NR 2MNT NR+2MNT NR +MNT NR
Proposed 4 · 2M NT NR 5 · 2MNT NR 2MNT NRC bits +(C − 1)MNT + 2 +CMNT
Fig. 3 16-bit Koggle-Stone adder (130 NANDs, 32 XORs, 65 NORs,255 NOTs).
presented in Figs. 3 and 4 [17], [18]. It is assumed that the8x8 multiplier consists of seven 5-bit full adders and four4x4 multipliers, the 4x4 multiplier consists of three 5-bitfull adders and four 2x2 multipliers, and a Koggle-Stoneadder (KSA) is used for all those full adders. The struc-ture of a bit shifter with 4 input bits, {S 0, S 1, S 2, S 3}, isalso shown in Fig. 5. With the use of these circuits, thenumber of logic operations required for each scheme is pre-sented in Table 3. As for the candidate signal point se-lection, the number of the multiplications for the proposedscheme is equivalent to that of the SSM. However, the dom-inant part of the multiplications given with a coefficient of(4 · 2MNT NR) is for the calculations of {Hqp[l]S p[l]} whereS p[l] is the candidate symbol of the pth antenna on the lthsubcarrier. These calculations can be carried our prior to themetric calculations. The summations given with a coeffi-
AOKI et al.: LOW COMPLEXITY LOG-LIKELIHOOD RATIO CALCULATION SCHEME WITH BIT SHIFTS AND SUMMATIONS735
Fig. 4 16x16 array multiplier (3476 NANDs, 1320 XORs, 1942 NORs,8290 NOTs).
Fig. 5 Bit shifter (180 NANDs, 90 NORs, 360 NOTs).
cient of (5 ·2MNT NR + MNT NR) are for the calculations of theSSM metrics. Regarding the LLR calculations, 2MNT NRmultiplications and MNT NR summations in the conventionalSSM scheme are replaced with (C−1)MNT summations andCMNT bit shifts in the proposed scheme. This implies about(3476 ·2+130)NR/(130 · (C−1)+180 ·C) times reduction ofNAND logics (more reduction factors for the other logics).
The numbers of logic operations required for differentschemes with the specific system parameters as an exam-ple are presented in Table 4. The number of coded bits foreach symbol is M = 2, the number of transmit antennas isNT = 4, and the number of receive antennas is NR = 2.From the table, it is clear that the proposed scheme reducesthe numbers of logic operations by from 0.85 to 0.89 fordifferent values of C. From Table 3, the proposed schemecan effectively reduce the numbers of logic operations whenMNT ≤ 8.
4. Numerical Results
4.1 Simulation Conditions
The simulation conditions are shown in Table 5. A turbo
Table 3 Number of logic operations.Metrics Logic Number of logic operations
Euclidean NAND: 3476(2 · 2MNT NR + 4 · 2M NT NR)Metric +130(2 · 2MNT NT NR + MNT )
XOR: 1320(2 · 2MNT NR + 4 · 2M NT NR)+32(2 · 2MNT NT NR + MNT )
NOR: 1942(2 · 2MNT NR + 4 · 2M NT NR)+65(2 · 2MNT NT NR + MNT )
NOT: 8290((2 · 2MNT NR + 4 · 2M NT NR)+255(2 · 2MNT NT NR + M)
SSM NAND: 3476(4 · 2M NT NR + 2MNT NR)+130(5 · 2MNT NR + MNT NR)+180 · 2MNT NR
XOR: 1320(4 · 2M NT NR + 2MNT NR)+32(5 · 2MNT NR + MNT NR)
NOR: 1942(4 · 2M NT NR + 2MNT NR)+65(5 · 2MNT NR + MNT NR)+90 · 2MNT NR
NOT: 8290(4 · 2M NT NR + 2MNT NR)+255(5 · 2MNT NR + MNT NR)+360 · 2MNT NR
Proposed NAND: 3476(4 · 2M NT NR)C bits +130(5 · 2MNT NR + (C − 1)MNT + 2)
+180(2MNT NR + CMNT )XOR: 1320(4 · 2M NT NR)
+32(5 · 2MNT NR + (C − 1)MNT + 2)+180(2MNT NR + CMNT )
NOR: 1942(4 · 2M NT NR)+65(5 · 2MNT NR + (C − 1)MNT + 2)+90(2MNT NR + CMNT )
NOT: 8290(4 · 2M NT NR)+255(5 · 2MNT NR + (C − 1)MNT + 2)+360(2MNT NR + CMNT )
Table 4 Example of number of logic operations (M = 2,NT = 4,NR =
2).Metrics Logic Number of
Logic Operations
Euclidean NAND: 4537872Metric XOR: 1651968
NOR: 2503944NOT: 11322360
Ratio toEuclidean Metric
SSM NAND: 983200 0.22XOR: 293632 0.18NOR: 524240 0.21NOT: 2167600 0.19
Ratio to SSMProposed NAND: 871588 0.89C = 1 bits XOR: 250944 0.85
NOR: 461906 0.88NOT: 1849630 0.88
Ratio to SSMProposed NAND: 874068 0.89C = 2 bits XOR: 251200 0.86
NOR: 463146 0.88NOT: 1853750 0.88
Ratio to SSMProposed NAND: 876548 0.89C = 3 bits XOR: 251456 0.86
NOR: 464386 0.89NOT: 1857870 0.88
code with 8 state memory is applied as the error-correctioncode [19]. The base rate of the code is 1/3 and the code rateis selected from 1/3, 1/2, 5/6 through circular buffer ratematching. The interleaver size is fixed to 4800. The numberof the transmit antennas is selected from 2 to 4 while thenumber of the receive antennas is fixed to 2. Each signal
736IEICE TRANS. COMMUN., VOL.E101–B, NO.3 MARCH 2018
Table 5 Simulation conditions.Error Correction Coding Turbo CodeCode Rate 1/3, 1/2, 5/6Interleave Size 4800Modulation Scheme QPSK
/OFDMNumber of Signal Streams 2∼4Number of 2∼4Transmit AntennasNumber of 2Receive AntennasChannel Bandwidth 2.5 MHzSubcarrier Spacing 15 kHzNumber of Subcarriers 256Number of Data Subcarriers 151Sampling Frequency 3.84 MHzGuard Interval 5.21 µs (1st symbol)
4.69 µs (2nd-7th symbols)A/D Converter 12 bitsOutput of DFT 16 bitsDecoding Algorithm Log-MAP, Max-Log-MAPNumber of 8Decoding IterationsChannel Model 6 taps GSM-TU
18 taps Rayleigh FadingChannel Estimation IdealNumber of Bits 9.6 × 106
transmitted for each plot 9.6 × 107 in Figs. 6 and 7
stream is modulated with QPSK and multiplexed throughOFDM. The coded bits are mapped to QPSK symbols byGray mapping. The channel bandwidth is 2.5 MHz and sub-carrier spacing is 15 kHz. The number of subcarriers is 256while the number of data subcarriers is 151. The samplingfrequency is then 3.84 MHz. The guard interval of the 1stsymbol is 5.21 µs, and the guard intervals of the 2nd-7thsymbols are 4.69 µs. The resolution of the A/D convertersis set to 12 bits and the resolution of the DFT outputs isset to 16 bits. The approximated LLR calculation is car-ried out with the resolution of 16 bits. The average outputof the A/D converters is adjusted to the fourth bit from themost significant bit. Six taps GSM-TU and 18 taps Rayleighfading are assumed as channel models [20]. The 18 tapsRayleigh fading channel assumes a uniform delay profilewith a maximum path delay of 4.42 µs. Channel estimationin the receiver is assumed to be ideal. The decoding algo-rithm is Log-MAP unless it is specified as MAX-Log-MAP.The number of decoding iterations is eight. For each plot,9.6 × 106 bits are transmitted (9.6 × 106 bits in Figs. 6 and7). For comparison, the performance curves of three con-ventional schemes are included in the following sections.“Joint MLD” is the scheme described in Sect. 2.2. “Eu-clidean Metric” represents the scheme that selects candidatesymbol vectors according to the Euclidean metrics and cal-culates the approximated LLRs by Eq. (15). “SSM” repre-sents the scheme that selects candidate symbol vectors fromthe summation and subtraction metric and the LLR is calcu-lated by Eq. (15). The number of bits to be presented in theproposed scheme is set to C = 1, 2, or 3 bits.
Fig. 6 BER vs. Eb/N0 (two signal streams, code rate of 1/3, six tapsGSM-TU).
Fig. 7 BER vs. Eb/N0 (two signal streams, code rate of 1/2, six tapsGSM-TU).
4.2 BER Performance on Fading Channels
The bit error rate (BER) versus Eb/N0 with two signalstreams for code rates of 1/3, 1/2, or 5/6 on the six tapsGSM-TU channel is shown in Figs. 6-8, respectively. Theperformance with the SSMs deteriorates by 0.1 dB in Fig. 6,by 0.05 dB in Fig. 7, and by 0.025 dB in Fig. 8 at a BERof 10−4 as compared to those without LLR approximation.This is because the wrong selection of a candidate symbolvector occurs when the approximation by the SSM is used.The BER performance of the proposed scheme (C = 3 bits)further deteriorates by 0.1 dB in Fig. 6, by 0.025 dB in Fig. 7,and by 0.05 dB in Fig. 8 at a BER of 10−4 as compared tothose with the SSM. The performance degradation occursowing to the further approximation of the LLR in the pro-posed scheme.
AOKI et al.: LOW COMPLEXITY LOG-LIKELIHOOD RATIO CALCULATION SCHEME WITH BIT SHIFTS AND SUMMATIONS737
Fig. 8 BER vs. Eb/N0 (two signal streams, code rate of 5/6, six tapsGSM-TU).
Fig. 9 BER vs. Eb/N0 (two signal streams, code rate of 1/3, 18 tapsRayleigh fading).
The BER versus Eb/N0 with two signal streams for acode rate of 1/3 on the 18 taps Rayleigh fading channel isshown in Fig. 9. The performance with the SSM deterioratesby 0.1 dB in Fig. 9 as compared to that without LLR ap-proximation. The BER performance of the proposed scheme(C = 3 bits) further deteriorates by 0.05 dB in Fig. 9 as com-pared to that with the SSM. The reason is the same as thecase of the six taps GSM-TU channel.
The BER versus Eb/N0 with two signal streams for acode rate of 1/3 on the six taps GSM-TU channel is shownFig. 10. As the decoding algorithm, MAX-Log-MAP is ap-plied. The performance curves with Log-MAP given inFig. 6 are also presented by dashed lines as references. Theperformance joint MLD deteriorates by 0.2 dB in Fig. 10as compared to that with the Log-MAP algorithm given inFig. 6.
Fig. 10 BER vs. Eb/N0 (two signal streams, code rate of 1/3, six tapsGSM-TU, MAX-Log-MAP algorithm).
Fig. 11 BER vs. Eb/N0 (four signal streams, code rate of 1/3, six tapsGSM-TU).
4.3 BER Performance of Overloaded MIMO
The BER versus Eb/N0 with four signal streams for coderates of 1/3 and 5/6 on the six taps GSM-TU channel isshown in Figs. 11 and 12, respectively. The performancewith the SSMs deteriorates by 0.2 dB in Fig. 11 at a BERof 10−4 as compared to those of Euclidean Metric. This isbecause the wrong selection of a candidate symbol vectoroccurs when the approximation by the SSM is used. TheBER performance of the proposed scheme (C = 3 bits) fur-ther deteriorates by 0.05 dB in Fig. 11 at a BER of 10−4 ascompared to those with the SSM. The performance degrada-tion occurs owing to the further approximation of the LLRin the proposed scheme. When the number of signal streamsis four and the code rate is 5/6, the BER performance gaps
738IEICE TRANS. COMMUN., VOL.E101–B, NO.3 MARCH 2018
Fig. 12 BER vs. Eb/N0 (four signal streams, code rate of 5/6, six tapsGSM-TU).
Fig. 13 Required Eb/N0 at BER=10−4 vs. code rate (three signalstreams, six taps GSM-TU).
between the conventional SSM scheme and the proposedschemes become wider as Eb/N0 increases. This is owingto the degradation of the decoding capability according tothe increase of the code rate.
The required Eb/N0 at BER=10−4 versus the code ratewith three signal streams on the six taps GSM-TU chan-nel is shown in Fig. 13. It is shown that the amount of thedegradation on the required Eb/N0 increases as the code rategrows in Fig. 13. Different from the other schemes, the re-quired Eb/N0 values of the proposed scheme is larger thanthat without approximation. The reason is that the accu-racy of the proposed LLR does not improve even though thevalue of Eb/N0 increases. Since the decoding performanceof the turbo code is degraded as the code rate increases, the
Fig. 14 Required Eb/N0 at BER=10−4 vs. number of signal streams(code rate of 1/3, six taps GSM-TU).
difference of the required Eb/N0 enlarges.The required Eb/N0 at a BER of 10−4 versus the num-
ber of signal streams with a code rate of 1/3 on the six tapsGSM-TU channel is shown in Fig. 14. It is shown that theamounts of the performance degradation in the proposedscheme enlarge as the number of signal streams increases.The reason is that the error of the approximated LLR effectsmore on the decoding performance as the number of candi-date signal points increases.
5. Conclusions
In this paper, an approximated log-likelihood ratio (LLR)calculation scheme is proposed. In the proposed scheme,multiplications are replaced by bit shifts and additions.The numerical results obtained through computer simulationhave shown that the BER of the proposed scheme (C = 3bits) for two signal streams deteriorates only by about 0.1dB as compared to that with the SSM and by 0.2 dB ascompared to that without approximation. For the overloadedcase, as the code rate increases to 5/6, the BER performancegaps between the conventional SSM scheme and the pro-posed schemes become wider as Eb/N0 increases. The pro-posed scheme approaches the performance without approx-imation as the number of signal streams is decreased and/orthe code rate lowered. It has also been shown that the pro-posed scheme can effectively reduce the numbers of logicoperations when MNT ≤ 8.
Acknowledgments
This work is supported in part by a Grant-in-Aid for Scien-tific Research (C) under Grant No.16K06366 from the Min-
AOKI et al.: LOW COMPLEXITY LOG-LIKELIHOOD RATIO CALCULATION SCHEME WITH BIT SHIFTS AND SUMMATIONS739
istry of Education, Culture, Sport, Science, and Technologyin Japan.
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Takayoshi Aoki was born in Fukuoka, Japanin 1992. He received his B.E. degree in elec-tronics engineering from Keio University, Japanin 2015. Since April 2015, he has been a gradu-ate student in School of Integrated Design Engi-neering, Graduate School of Science and Tech-nology, Keio University. His research interestsare mainly on MIMO-OFDM system.
Keita Matsugi was born in Ehime, Japan in1994. He has been B.E. student in School of In-tegrated Design Engineering, Graduate Schoolof Science and Technology, Keio University.His research interests are mainly on MIMO-OFDM system.
Yukitoshi Sanada was born in Tokyo in1969. He received his B.E. degree in elec-trical engineering from Keio University, Yoko-hama, Japan, his M.A.Sc. degree in electri-cal engineering from the University of Victoria,B.C., Canada, and his Ph.D. degree in electricalengineering from Keio University, Yokohama,Japan, in 1992, 1995, and 1997, respectively. In1997 he joined the Faculty of Engineering, To-kyo Institute of Technology as a Research As-sociate. In 2000, he joined Advanced Telecom-
munication Laboratory, Sony Computer Science Laboratories, Inc, as anassociate researcher. In 2001, he joined the Faculty of Science and En-gineering, Keio University, where he is now a professor. He received theYoung Engineer Award from IEICE Japan in 1997. His current research in-terests are in software defined radio, cognitive radio, and OFDM systems.
