PERFORMANCE AND COMPLEXITY ANALYSIS OF A REDUCED ITERATIONS LLL ALGORITHM

International Journal of Computer Networks & Communications (IJCNC) Vol.8, No.3, May 2016
DOI: 10.5121/ijcnc.2016.8309 123
PERFORMANCE AND COMPLEXITY ANALYSIS OF A
REDUCED ITERATIONS LLL ALGORITHM
Nizar OUNI1
and Ridha BOUALLEGUE2
1
National Engineering School of Tunis, SUP’COM, InnovCom laboratory, Tunisia
2
SUP’COM, InnovCom laboratory, Tunisia
ABSTRACT
Multiple-input multiple-output (MIMO) systems are playing an increasing and interesting role in the recent
wireless communication. The complexity and the performance of the systems are driving the different
studies and researches. Lattices Reduction techniques bring more resources to investigate the complexity
and performances of such systems.
In this paper, we look to modify a fixed complexity verity of the LLL algorithm to reduce the computation
operations by reducing the number of iterations without important performance degradation. Our proposal
shows that we can achieve a good performance results while avoiding extra iteration that doesn’t bring
much performance.
KEYWORDS
MIMO systems, LR-aided, Lattice, LLL, BER, Complexity.
1. INTRODUCTION
MIMO communication systems are introduced to combat fading and provide high data rate. The
MIMO system consists of transmitting multiple independent data symbols via multiple antennas.
For the reception, a MIMO decoder needs to be used to detect, estimate, and reconstruct the
received symbols. Multiple detection schemes can be used, such as the zero-forcing (ZF) or the
minimum mean square error (MMSE) criterion. Also, the maximum likelihood decoder (ML) is
considered as the optimal solution for the MIMO detection in term of Bit Error Rate (BER). But,
unfortunately the ML algorithm seems to be complex for hardware implementations. Therefore,
linear MIMO detection techniques like ZF and MMSE are better in term of complexity, but suffer
from BER performance degradation.
The lattice-reduction (LR) preprocessing technique has been proposed to be used with linear
detection in order to transform the system model into an equivalent system with better channel
matrix’s effect and so to reduce the complexity of the system. It was shown in previous studies
that LR techniques improve the BER performances significantly.
The populated LR algorithm is called Lenstra-Lenstra-Lovàsz (LLL) algorithm is the most used
one. It was called according to the name of the inventors [1]. But, the LLL algorithm brings many
challenges due to higher processing complexity and the undeterministic execution time [2].
LLL algorithm has a major limit which is the varying complexity that could be large and limits
the decoding speed of the communication system. But, it is always presenting the best
performance in term BER. The complex Lenstra-Lenstra-Lovàsz algorithm (CLLL) [3]is applying

124
the basis reduction for complex field, while the LLL is targetinga real valued matrix. The
different studies and simulation results show that CLLL requires less processing operations [4].
Effective LLL algorithm (ELLL) [5], come with a new idea that consists to change the Lovàsz
reduction condition in order to relax the related equations. Also, the FcLLL prposed by Wen [2]
reduces the number of iterations for the algorithm to fix iteration number instead of infinite
iterations. This technique improves the complexity but remains worse than LLL in term of BER
performance.
In this paper we, will focus on the FcLLL algorithm using ZF decoding technique and we propose
some modifications to the original FcLLL algorithm to keep a reduced number of loops and
targeting a good BER.
2. SYSTEM MODEL DESCRIPTION
During this paper we will consider that (. ) and (. ) denote respectively the hermission
transpose and the transpose of a matrix.
We consider the spatial multiplexing MIMO system with transmit and receive antennas
with a Rayleigh channel non variant in the time.
= . + (1)
Where = [ , , … , ] ; (s ∊ s) is the information vector with being a constellation set of
square QAM with [ ] = . and the real and imaginary parts are
{−#$% + 1, … , −1, 1, … , #$% − 1} with M) being the constellation size, . We will suppose that
the average transmit power of each antenna is normalized to one, so [ ] = . With I+ is the
m × m identity matrix.
is an × ; (N/ ≥ N1)complex channel matrix, x= [ , , … , x] is the received signal
vector, and = [ , , … , 3
]4
is the complex additive white Gaussian noise (AWGN) vector
with zero mean and covariance 5. 3
.
On the receiver side, = [ , , … , 3
]4
are the symbols at receiver’s respective antennas
which will be used to estimate transmitted e the symbols [4]. The receiver will analyze all
received information to compute the transmitted data. So, a detection, computation, equalization
and estimation of the received data will happen.
At receiver side, the linear zero forcing (ZF) detector compute the inverse of the channel matrix
to estimate the transmitted symbols which can be expressed by,
s67 = ( . )8
.9:::;:::<
=>>/?8@?A/>B? CB?DE>8 AF?/B?
. x (2)
The channel matrix is QR decomposed into two parts as = GH.

125
Figure 1. MIMO system with Transmitter and Receiver antennas.
3. LATTICE REDUCTION TECHNIQUE
We can interpret the columns ℎJof the channel matrix as the basis of a lattice and assume that
the possible transmit vectors are given by ℤ+
, the m dimensional infinite integer space.
Consequently, the set of all possible undisturbed received signals is given by the lattice.
L( ) = L(ℎ , … , ℎM): = {∑ ℎJ
M
JP |ℎJ ∈ ℤ} (3)
The LR algorithm generates a lattices reduced and near-orthogonal channel matrixS = . T.
With matrix S = . T generates the same lattice as , if and only if the m × m matrix T is
unimodular [6], i.e. T contains only integer entries and VWX(T) = ±1:
L( S ) = L( ) ⇔ S = T[ VT ] ^_V]`[a (4)
Also,
S. T8
= (5)
We can find multiple bases that can be included in the space L, and the goal of the LR algorithm
is to find a set of least correlated base with the shortest basis vectors [2].Initially, an efficient (but
supposed not optimal) way to determine a reduced basis was proposed by Lenstra, Lenstra and
Lovàsz [1].Where they defined (LLL-Reduced): A basis S with QR decomposition S = Gb. Hb is
called LLL-reduced with parameter δ with (1/4 < k ≤ 1), if
mHbn,om ≤ . mHbn,nm p_a 1 ≤ < q ≤ ^ (6)
And
kmHbo8 ,o8 m ≤ mHbo,om + mHbo8 ,om p_a q = 2, … , ^ (7)
The first condition is called, size-reduced and the second one is called Lovàsz condition. The
parameter δ plays an important role to the quality of the reduced basis. We will assume δ = 3 4⁄

126
as proposed in [1]. After applying the QR decomposition of H, doing successive size-reduces if
the condition is fulfilled, the algorithm exchanges two vectors if Lovàsz condition is not fulfilled
to generateT, compute RS andQS. And so, the LLL algorithm will output QS, RS andT.
Looking to the LLL algorithm [1], one important element of its complexity is related to the fact
that the LLL algorithm is applied for the real integer vectors, it is mandatory to reformulate the
different matrices to their real-valued form, so we got:

H/?xy
= z
Real(H) −Im(H)
Im(H) real(H)
• (8)
x = z
Real(x)
Im(x)
• (9)
s = z
Real(s)
Im(s)
• and n = z
Real(n)
Im(n)
• (10)
This kind of reformulation increases the number of operations and adds more latency for the
system.
The idea behind LR-aided linear detection is to consider the equivalent system model and perform
the nonlinear quantisation on it [7]. In fact, if we combine equations (1) and (5), we can get:
= S. T8
.9;<
‚
+ (11)
With ƒ = T8
. the equivalent model and in this case S will represent a better channel quality.
And so, the detector can be represented with an equivalent model with better performance due to
the less noise enhancement increased by S. Thus, the basic idea behind approximate lattice
decoding (LD) is to use LR in conjunction with traditional low-complexity decoders. With LR,
the basis B is transformed into a new basis consisting of roughly orthogonal vectors [8].
After processing the Zero Forcing lattice reduction (ZF-LR) mechanism and by combining
equations (2) and (11), we can generate:
z…678†‡ = T8 . s…67 = S. = ƒ + S. (12)
The complex form of this algorithm was presented by Gan and Mow in [3]. But we can clearly
identify that this extension keeps the excessive number of iteration and also add more
computation latency by introducing the real and imaginary elements in the different conditions of
the algorithm. For this reasons, Vetter proposed another variety of the complex LLL, than Ling
[7] proposed a fixed complexity LLL (FCLLL). As recapitulation the modifications for the LLL
algorithm was for three points:
• Avoid the complex to real vector transformations (reduce the number of loops).
• The reduction of the number of the LLL iteration to a fixed number.
• The use of a flag to track column exchanges. When no column swap happens, the FCLLL
ends with an LLL reduced basis.
The different enhancements for the original algorithm where looking for limited iterations in term
of stopping criteria, like in [2] and [5].

127
In next section we will consider the proposal in [2] and start form the Wen’s algorithm as
described in table 1 where, Wen proposed an enhanced form of Vetter’s algorithm. The proposal
is based on an improved column traverse strategy and an enhanced termination criterion for
practical LR-aided SIC MIMO detection.
Table 1. The Fixed Complex LLL algorithm [2]
Input: H: the channel complex matrix
Output: Hb, Gb, T
1
Initialization: T = ; ; ‰ Š`[‹ = _ W (1, +
1); Œ W•; Ž••;
2 [G, H] ∶= •a( );
3 k = 3 4⁄
4 n ’ = 1
5 Œ W•n“•=1
6 ”ℎ`W ( n ’ ≤ Ž••)&& ( ]^(‰ p`[‹(2: 1: )) ≠ 0
7 Œ = Œ W•(Œ W•n“•) ;
8 ‰ p`[‹(Œ) = 0;
9 p_a Œ = 2:
10 ˜ ∶= (Hb(`, Œ) Hb(`, Œ)⁄ )
11 p ˜ ≠ 0
12 Hb(1: `, Œ) ∶= Hb(1: `, Œ) − ˜ . Hb(1:`, `)
13 T(: , Œ) ∶= T(: , Œ) − ˜ . T(: , `)
14 W V
15 W V
16 p k. Hb(Œ − 1, Œ − 1) > Hb(Œ, Œ) + Hb(Œ − 1, Œ)
17 Œ X_ Œ − 1 š_`]^ ”[› p_a Hb [ V T
18
‰_^›]X ‹ XℎW œ $[Xa :
œ = z
•ž Ÿ̅
−Ÿ •
• ”Xℎ
• =
Hb(Œ − 1, Œ − 1)
¡Hb(Œ − 1: Œ, Œ − 1)¡
Ÿ =
Hb(Œ, Œ − 1)
¡Hb(Œ − 1: Œ, Œ − 1)¡
19 Hb(Œ − 1: Œ, Œ − 1: ^) ∶= œ. Hb(Œ − 1: Œ, Œ − 1: ^)
20 Gb(: , Œ − 1: Œ) ∶= Gb(: , Œ − 1: Œ). œ4
21 ‰ p`[‹(Œ − 1: 1: Œ + 1) = 1;
22 W V
23 W V
24 n ’ = n ’ + 1
25 Œ W•n“•=Œ W•n“• +1
26 W V

128
4.MODIFIED ALGORITHM AND STUDY OF THE EFFECT OF MAX
ITERATION ON THE FCLL ALGORITHM
In this section we will start from the FCLLL algorithm proposed by Wen [2] and we will try to do
modify it. In fact, for line 21 there is the table “ ‰ p`[‹” which is a condition for the loop as
mentioned in line 6. The summation of the elements of this “table” seems to add − 1 more
addition operations that need to be computed for each loop. So, for us, it will be better to come
back to the single element condition as mentioned in [9]. A second remark, the Lovàsz condition
such as described in line 16, is representing four complex multiplication, one addition operation
and one subtraction operation (which can be considered as addition operation). All of them are
complex and being running in loop. It will be better to use the Siegel condition which is always
fulfilling the Lovàsz condition and we can go more to show that it reduce the computing
operations [10] & [11]. The representation is below:
mHbo8 ,o8 m ≤ ¢mHbo,om (13)
Where ζ is chosen from[2, 4],
To have the Siegel condition fulfilled, we have to check if:
¤
. mHbo8 ,o8 m > mHbo,om (14)
Whatever the value of ζ, 2 or 4, we can get:
k. mHbo8 ,o8 m > mHbo,om (15)
Since δ >
¥
,
So, we can modify the algorithm in table 1 to get the below one as described in table 2.
Table 2: The proposed Modified Complex LLL algorithm
Input: H: the channel complex matrix
Outpu
t:
Hb, Gb, T
1
Initialization: T = ; = ƒW( , 2);
‰ Š`[‹ = 0; XWa$[ = 6;
2 [G, H] ∶= •a( );
3 k = 3 4⁄
4 XWa = 0
5 ”ℎ`W (‰ p`[‹ == 0) && ( XWa ≤ XWa$[ )
6 ‰ p`[‹ =1;
7 XWa = XWa + 1;
8 p_a Œ = 2:
9 ˜ ∶= a_] V(Hb(`, Œ) Hb(`, Œ)⁄ )
10 p ˜ ≥ 1
11 Hb(1: `, Œ) ∶= Hb(1: `, Œ) − ˜ . Hb(1: `, `)

129
12 T(: , Œ) ∶= T(: , Œ) − ˜ . T(: , `)
13 W V
14 W V
15 p k. Hb(Œ − 1, Œ − 1) > Hb(Œ, Œ)
16 Œ X_ Œ − 1 š_`]^ ”[› p_a Hb [ V T
17
‰_^›]X ‹ XℎW œ $[Xa :
œ = z
•ž Ÿ̅
−Ÿ •
• ”Xℎ
• =
Hb(Œ − 1, Œ − 1)
¡Hb(Œ − 1: Œ, Œ − 1)¡
Ÿ =
Hb(Œ, Œ − 1)
¡Hb(Œ − 1: Œ, Œ − 1)¡
18 Hb(Œ − 1: Œ, Œ − 1: ^) ∶= œ. Hb(Œ − 1: Œ, Œ − 1: ^)
19 Gb(: , Œ − 1: Œ) ∶= Gb(: , Œ − 1: Œ). œ4
20 ‰ p`[‹ =0;
21 W V
22 Œ ∶= Œ + 1
23 W V
24 W V
In the proposed algorithm we have modified the line 5 by avoiding “CSflag” table summation
presented in table 1 and proposed in [2]. This will help to reduce additional processing operations
which will help to “relax” the algorithm in term of complexity and decoding timing. In fact, and
as described in the previous section, the contribution of the elementof this “table” in the algorithm
doesn’t exceed the termination condition. The importance of this modification can be observed in
the next sections, especially of the gain in terms of complexity.
We can clearly observe that the max iteration number ( Ž••) is a condition to exit from the loop
and so, it can increase the number of computation operations. This means, there is an ideal max
iteration value that above it the system becomes exponential complex without large BER
enhancement. Also, the modifications in line 10 and line 15 will help to reduce the processing
operations because the algorithm will converge quicker than the initial version. In fact, the Siegel
condition helps to relax the processing operations like presented in [10].
So, it’s interesting to evaluate the effect of the Max iteration on the BER performance and also
the system complexity. For this, we tried to do the simulation of the algorithm with varying the
value of the Max iteration.
5.SIMULATION RESULTS AND EFFECT OF THE MAX ITERATION ON THE
FCLLL ALGORITHM
For our simulation, we will consider the 16QAM constellation, ZF equalization will be checked.
The MIMO model will be 4 × 4 ; means a 4 antennas at both transmitter and receiver side. We
used a frame size of 10§
. We will indicate inline any changes to the above configuration. In the
flowing figures, we tried to increase the max iteration number from 4 to 18.

130
Figure 2: 4 × 4 Modified Complex LLL with 16QAM and ZF Bit error rate results. 5, 8 and 10 IterMax
results compared to ML and ordinary LLL.
Observing figure 2, the ML curve is outperforming all different curves. But we should note that
the ML scheme is extremely complex to implement. So, we are indicating it just for reference and
comparison reasons.Another quick remark is that comparing the FCLLL curves and the ordinary
LLL algorithm we can see that for IterMax ≤ 5, the LLL is better comparing FCLLL. But for
MaxIter equal to 5, the two curves are overlapping till SNR equal to 24dB and after the deviation
is minimal. Which means that in terms of performances we are still in an acceptable range and so
it will be interesting to push the analysis and also evaluate the gain in complexity and processing
operations.
In the figure 3, we increase the max iteration value from 4 to 18 to observe if any threshold value
for this parameter; that allow to reach a better result with lower iterations.
0 5 10 15 20 25 30 35
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR per receiver antenna [average in dB]
BER
ML
LLL-ZF
Modif fCLLL-ZF (5Iter)
LLL beter
than 5 Iters
8 and 10 Iter are same

131
Figure 3: BER results by varying the IterMax: curves for IterMax equal to 4,5,6,7, 8,9 and 18.
Looking to figure 3 we can observe that, staring from XWa$[ ≥ 6, the FCLLL become similar
or better than the LLL. But, starting from 8 iteration we can observe that no improvement for the
BER.
Means, the curves remain overlapping each other. This leads us to conclude that no need to
increase the XWa$[ parameter above 8 iterations. Else, the system became costly comparing to
its performance. For XWa$[ between 5 and 8, we can push the analysis. In fact, the LLL
algorithm as described in [1], will do a minimum of 2. loops; taking in consideration the fact
that the size of the channel real-valued matrix H used for the LLL algorithm is double of the
complex matrix H. Thus, for this case and with 8 IterMax we are exactly in the same condition as
the LLL algorithm. From another point of view, a XWa$[ ≤ 4 will show a BER degradation.
This is related to the fact that the algorithm will do a column swap for only half of the possible
columns of the matrix. If we consider 5, 6 and 7 as IterMax we can see that we more or less close
to the LLL algorithm, since the difference is observed only for the high SNR and the deviation is
minimal. In the case of XWa$[ = 6 the BER curve is almost overlapping the ordinary LLL
curve. From our point of view, using the XWa$[ equal to 6 seems to be the recommended value,
since it has a good complexity to performance balance.
Figures 4 and 5 show a zoom on the different curves to illustrate our analysis.
0 5 10 15 20 25 30 35
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
BER
Modif fCLLL-ZF(4Iters)
Modif LLL-ZF(18Iters)
8/9/18Iters curves
are overlapping and
7Iters one is close
to them (Red curve)

132
Figure 4: 4 × 4 Modified Complex LLL with 16QAM and ZF Bit error rate results. 5, 8 and 10 IterMax
Zoomed curves compared to ML and ordinary LLL.
Figure 5: BER results by varying the IterMax: Zoomed curves for IterMax equal to 4,5,6,7, 8,9 and 18.
We remark that starting from XWa$[ = 8 the system BER performance reach the saturation but
also the system computing operation are increasing according to the XWa$[ . Means, the
complexity continue increasing function of XWa$[ but the BER performance will saturate. Just
looking to numbers, the BER saturation is reached for 8 XWa$[ and the BER performance for 6
XWa$[ is same as the ordinary LLL. So, we got same performances as ordinary LLL with a
gain of ¼ of operations. It’s a good performance vs complexity balance to be considered…
18 20 22 24 26 28 30 32 34
10
-4
10
-3
BER
ML
LLL-ZF
LLL beter
than 5 Iters
8 and 10 Iter are same
18 20 22 24 26 28 30 32
10
-4
10
-3
BER
Modif LLL-ZF(18Iters)
8/9/18Iters curves
are overlapping and
7Iters one is close
to them (Red curve)

133
6. COMPLEXITY ANALYSIS
In this section we will discuss the complexity aspect of our proposal and show the profits and
benefits of our proposal.
First we will give some details about the operation done by the algorithm. In [12] it was presented
that a real matrix multiplication of A ( × $) and B (ª × $) leads to matrix C ( × $) and
the overall operations are (ª − 1)$ addition operation and ª$ multiplication operations. It
is also known that a complex addition is equivalent to two real addition operations. In fact, for the
complex case we will add the real and imaginary parts separately. For the complex multiplication
it is different and the operation can be written as below:
([ + «) ∗ (š + V) = ([š − «V) + («š + [V)
= ([š − «V) + q-([ + «)(š + V) − [š − «V® (16)
The first options can be done in four multiplications and two additions (assuming that a
subtraction can done via an addition operation). The second option can be done in three
multiplications and five additions. But the first option is almost used. So, we will consider it.
Also, in [13] it was shown that the different arithmetical operation requires different FLOPS. In
the table below we present the number of FLOPS needs for each operation (for real values) [13].
Table 3: FLOPS vs operations
Operation Add Mult Sqrt Div
Nombre of FLOPS 1 1 8 8
• The size reduction require ( Ž•• − 2) × {1 × (¯°) + 2 × ($]`X + ±VV)}
• The lovàsz condition require {4 × ($]`X) + 2 × (±VV)}
• Colum swap require × {3 × (±VV)}
• The Givens rotation matrix computation require {2 × (¯°) + 2 × ($]`X) + 1 × (±VV) + 1 ×
( •]aX)}
• The rotation operation for R (matrix multiplication) require 2 × {2 × ($]`X) + 1 × (±VV)} ×
• The rotation operation for Q (matrix multiplication) require 2 × {2 × ($]`X) + 1 × (±VV)} ×
2
• The CSflag condition sum require (±VV)
Also, the complex division and square root operations consists of many real operations.
• A square root of complex value require {1 × (¯°) + 3 × ($]`X) + 2 × (±VV) + 3 ×
( •aX)} of real values.
• A complex value division require {1 × (¯°) + 8 × ($]`X) + 4 × (±VV)}
All these operation will be running in loop for Ž•• iterations for MIMO 8 × 8 scheme.

134
Table 4: Complexity gain
FLOPS Performance Comments
LLL algorithm
Ž•• = ∞
12200 <
Wen’s algorithm
[5] Ž•• = 18
< 9300 Outperform LLL
(11dB at 108³
)
Our algorithm
Ž•• = 18
< 9100 Gain 2dB at 108³
vs LLL
The most important point is that we
reach same performance with ~31%
of FLOPS gainOur algorithm
Ž•• = 8
< 6200 Gain 2dB at 108³
vs LLL
Our algorithm
Ž•• = 6
< 5800 Loose 2dB at 108³
vs LLL
Gain ~36% of FLOPS and the
performance degradation is minimal
(2dB)
The table above shows that with our proposal we can reach approximately the same performances
as the LLL algorithm with reducing 36% of FLOPS. This is important in term of decoding delay.
In fact, we can avoid some decoding delay and achieve the same performance with limited
iteration number ( Ž•• = 8).
7. CONCLUSIONS
In this paper we proposed some modifications to the FcLLL algorithm proposed by Wen [2].
Simulation results show that for 4 × 4 MIMO system, there is min and max values for the
XWa$[ (5 to 8) where the BER performances seems to be good (more or less near to the
original LLL results) and also the system complexity remains reasonable. Outside these limits the
complexity vs performance balance become undesirable. And the extra iterations don't enhance
the performance. Thus, to implement this algorithm we recommend an ideal value of XWa$[ =
6 which allows having a BER quite same as the original LLL and limits the iterations loop. In
fact, with this recommended value we can gain ~36% of operations and the BER degradation will
be ~2dB at 108³
. The challenge of our proposal was to not bring many changes to the original
algorithm, but to identify the possible points that we can enhance in order to relax the processing
operations and complexity while keeping good performance results (nearest to the original
algorithm). Such study and the presented results aim to help the industry using a low complexity,
low cost and high performance solution based on the LLL decoding technique.
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[4] C. P. Schnorr and M. Euchner, (1994) “Lattice Basis Reduction: Improved Practical Alorithms and
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135
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