Full Diversity Unitary Precoded Integer-Forcing

1.6 MB·English

by  Amin Sakzad

1
Full Diversity Unitary Precoded Integer-Forcing
Amin Sakzad and Emanuele Viterbo†
Abstract—We consider a point-to-point flat-fading MIMO implemented at the receiver side. Other well-known linear
channel with channel state information known both at transmit- receivers [5], including zero-forcing (ZF) and minimum mean
ter and receiver. At the transmitter side, a lattice coding scheme
square estimation (MMSE) linear receivers, trade-off error
is employed at each antenna to map information symbols to
performance for reduced computational complexity. Specific
independent lattice codewords drawn from the same codebook.
Each lattice codeword is then multiplied by a unitary precoding families of full-diversity STBCs were investigated and con-
matrix P and sent through the channel. At the receiver side, structed in [6]–[11] for use with linear receivers (ZF and
5 an integer-forcing (IF) linear receiver is employed. We denote MMSE). In these schemes, full-diversity is obtained at the
1 this scheme as unitary precoded integer-forcing (UPIF). We show cost of limited rate.
0 that UPIF can achieve full-diversity under a constraint based
Precoding schemes that assume CSIR, channel state infor-
2 on the shortest vector of a lattice generated by the precoding
matrix P. This constraint and a simpler version of that provide mation at the transmitter (CSIT), and an ML decoder usually
r
p design criteria for two types of full-diversity UPIF. Type I uses make use of a singular value decomposition (SVD) to convert
A aunitaryprecoderthatadaptsateachchannelrealization.Type theMIMOchannelintoparallelsubchannels.Thenappropriate
II uses a unitary precoder, which remains fixed for all channel
precoders employed over these parallel channels to minimize

1 realizations. We then verify our results by computer simulations
the error probability of the system over different constellation
in 2×2, and 4×4 MIMO using different QAM constellations.
] We finally show that the proposed Type II UPIF outperform the sizes. Some examples are the well-known full-diversity E-
T MIMO precoding X-codes at high data rates. d precoders for 4-QAM [12], and MIMO precoding X-
min
I and Y- codes [13], which are known to provide a diversity
cs. ticeIncdoedxesT,erfumlsl-—diMveIrMsitOy.,Integer-Forcing,unitaryprecoding,lat- order of dnt,nr,ns = (cid:0)nr− n2s +1(cid:1)(cid:0)nt− n2s +1(cid:1), and the
full-diversity precoders introduced and investigated in [14].
[
Note that d shows a trade-off between full-rate and
2 I. INTRODUCTION full-diversityn.t,Innr,nosne hand, if n = n and we use a full-
t r
v Multiple-inputmultiple-output(MIMO)channelswithmul-
rate MIMO precoding X- or Y- code, then the diversity order
12 toigpyletaonatecnhnieavseaphpigeharsepdecintrathleefefiacrliyen1c9ie9s0’isnawsitrheeleksesychteacnhnneolsl-. is dnt,nr,ns = (cid:0)n2t +1(cid:1)2 < n2t (full-diversity). On the other
0 hand, if we want to ensure full-diversity for MIMO precoding
These channels are known to obtain higher capacity gains
2 X- or Y- codes, we need n = 2, which means the system
0 in comparison to single antenna point-to-point channels. For s
has very limited rate. For a detailed comparison between the
. a MIMO channel matrix H with n transmit antennas
1 nt×nr t ML-decodingcomplexityoftheabovethreeschemes,werefer
0 and nr receive antennas, a diversity gain of at most ntnr the reader to Table III of [14].
5 is achievable. If an encoder/decoder pair gives the maximum
On the other hand, in the presence of CIST and having a
1 possible diversity n n , then the entire system is said to
t r linear receiver such as MMSE linear receiver at the destina-
:
v achieve full-diversity. Let ns denote the number of symbols tion, optimal precoder designs have been extensively studied
i transmitted per channel use. If n = min{n ,n }, then the
X s r t ([15]– [18]). The design criteria for these schemes include
encoding scheme is said to have full-rate.
maximizing the rate or minimizing the error performance.
r
For MIMO channels with channel state information at the
a The MIMO linear precoding techniques such as regularized
receiver (CSIR), carefully designed space-time block-codes
ZF [19] are alternative approaches that can provide full-
(STBC) can provide full-diversity [1]. Algebraic full-diversity
diversity in conjunction with linear receivers, see [20] and
full-rate STBCs with large coding gain were later designed
references therein. The decoding complexity of such schemes
in [2], [3]. The design criteria for these codes are known
isrelativelylowincomparisontoML-decodingalgorithmsbut
as the rank and the non-vanishing determinant criteria [1],
incur some error performance degradation.
[3]. In order to take full advantage from these codes, an
In this paper, assuming CSIT and CSIR we propose a
optimal maximum likelihood (ML) detector such as a sphere
full-diversity unitary precoding scheme for n × n MIMO
decoder [4], with high computational complexity should be
(n = n = n), where the receiver is equipped with the
r t
recently proposed integer-forcing (IF) linear receiver [21].
† Amin Sakzad and Emanuele Viterbo are with Department
of Electrical and Computer Systems, Faculty of Engineering, In IF framework, the transmitter employs a lattice coding
Monash University, Clayton VIC 3800, Australia. E-mail: scheme, and sends independent lattice codewords from the
{amin.sakzad,emanuele.viterbo}@monash.edu. This work was
same codebook, across different transmit antennas (or layers).
performed at the Monash Software Defined Telecommunications (SDT)
Lab. Amin Sakzad was supported by the Australian Research Council At each receive antenna, we recover an integer linear com-
(ARC) under Discovery grants ARC DP 130100103. Emanuele Viterbo bination of the lattice codewords depending on the entries
was supported by NPRP grant NPRP5-597-2-241 from the Qatar National
of the channel matrix H. As all the lattice codewords are
ResearchFund(amemberofQatarFoundation).Asubsetofthisworkwas
presentedin[24]atITW2014,Hobart,Tasmania,Australia. drawn from the same lattice codebook, the integer linear
2
combinations are themselves guaranteed to be lattice points. our UPIF. The exact constraint is hard to verify, since it
To further estimate the original transmitted lattice codewords, is related to finding the minimum distance of a lattice
a system of linear equations with integer coefficient matrix A depending on Σ and P. Theorem 3 provides a simpler
should be solved at the receiver. In fact, the IF linear receiver constraint, based on the non-vanishing minimum product
exploits the linearity of lattices to first eliminate the noise distance of the lattice generated by P and guarantees the
and then hareness the interference between transmitted data full-diversity of UPIF.
across different layers. The key step underpinning IF is the • Precoderdesigncriteria–TheconstraintsonPprovide
selection of A to approximate the channel matrix H. Hence, the design criteria for full-diversity-achieving precoders
a matrix B is needed such that A is full rank and BH≈A, for IF MIMO linear receivers. We define two types of
with minimum quantization error at high signal-to-noise ratio full-diversity UPIF schemes: (i) the precoder matrix P
(SNR) values. The problem of finding A based on H using depending on the channel matrix singular values (diago-
lattice reduction algorithms is addressed in [22], [23]. The nalelementsofΣ)isadaptedateachquasi-staticchannel
integer-forcing linear receiver is also shown to provide a full interval (Type I UPIF), (ii) a fixed unitary precoder P
receive diversity order n = n and a full multiplexing gain (independent of Σ) is used for all channel realizations
r
in [21], [22]. (Type II UPIF). For Type I UPIF, finding an analytical
Using perfect STBCs like the Golden code [25], [3], the optimization technique for P remains an open problem.
authorsof[26]haveshownthatunderanIFdecoder,aconstant For Type II UPIF we propose the use of full-diversity
gap to capacity is attainable. A perfect STBC code is a lin- lattice generator matrices with non-vanishing minimum
ear dispersion space-time code over a Quadrature-Amplitude product distance [32]-[37].
Modulation (QAM) constellation, which is full-rate, satisfies • Simulation results – Monte-Carlo simulations are pre-
non-vanishing determinant property, and its generator matrix sented in support the derived theoretical results. For
is a unitary matrix. In this paper, we aim at relaxing the first example, it is shown that a 2×2 full-diversity algebraic
two constraints of perfect STBCs and keeping the last one rotation matrix given in [37] in a Type II UPIF scheme
only. We analyze the diversity order of a lattice space-time with a 64-QAM constellation, can perform as close as
encoding scheme, which is precoded by a unitary matrix and 1dB away from ML decoder for the same code and 2dB
uses an IF linear receiver. First, singular value decomposition better than MIMO precoding X-codes [13].
(SVD) precoding is used to diagonalize the channel, and then
Notation. The superscripts (·)T and (·)h denote transposi-
an optimal precoder matrix is designed. Our optimization
tionandHermitiantransposition,respectively.LetG andG(cid:48) be
criterion is based on maximizing the minimum distance of
a group and its subgroup respectively, then G/G(cid:48) denotes the
the input lattice, similar to the approach of [27]. We note that
quotient group. The sets Z, C, R, and Z[i] denote the ring of
differently from [27], our precoder matrices are unitary and
rationalintegers,thefieldofcomplexnumbers,thefieldofreal
our decoder is the IF linear receiver and not the ML receiver.
numbers,andtheringofGaussianintegers,respectively,where
Related works focus on a different design criterion based on
i2 = −1. Real and imaginary parts of a complex number
maximizing the mutual information ([28], [29], [30], and [31]
z are denoted by (cid:60)(z) and (cid:61)(z), and |z| and arg(z) are
and references therein).
the modulus and the unique phase, respectively. The notation
From the receiver point of view, using IF linear receiver
(cid:107)v(cid:107) stands for the Euclidean norm of a vector v ∈ Cn. A
instead of ML decoders in slow-fading channels is advanta-
k×k matrix X = [xT,...,xT]T is formed by stacking the
geous in terms of complexity. For the IF receiver, the lattice 1 k
k−dimensional row vectors x ,...,x , and I denotes the
reduction algorithm is performed only at the beginning of 1 k k
k×k identity matrix. Finally, let v be a vector, then its j-th
each quasi-static interval for which the channel H is assumed
entry is represented by [v] .
to remain constant over multiple codeword transmissions. j
Subsequently, for each codeword transmission, only a system
of linear equations has to be solved to recover an estimate
of that codeword. On the other hand, for the ML decoding, II. SYSTEMMODEL
a sphere decoder should be used for each codeword within a
quasi-static channel interval. We first review the notion of complex lattices [38] which
We summarize the contributions of this paper as follows: are essential for the rest of the paper.
• Unitaryprecodedinteger-forcing(UPIF)–Considering Definition 1: Given a set of d-dimensional vectors
theavailabilityofCSIT,theSVDdecompositionWΣVh {(cid:96) ,...,(cid:96) } in Cd, a k-dimensional complex lattice Λ is the
1 k
for Hn×n is first performed. Using V at the transmitter set of points
and W at the receiver, the channel is reduced to a diag-
(cid:40) (cid:41)
k
onal channel matrix Σ. An additional unitary precoding (cid:88)
z (cid:96) : z ∈Z[i], (cid:96) ∈Cd .
matrix P, is then employed as VP. At the destination, m m m m
m=1
the receiver is equipped with an IF linear receiver.
• Full-diversity constraint – We find an upper bound More briefly we can write Λ = (cid:8)zL: z∈Z[i]k(cid:9), where
on the probability of error of UPIF (Theorem 1) and L = [(cid:96)T,...,(cid:96)T]T represents a basis of the lattice Λ. In
1 k
show analytically that if P satisfies a specific constraint other words, every point x ∈ Λ can be represented as a
(Theorem 2), the full diversity (n2) can be achieved by Gaussian integer linear combination of basis vectors. For any
3
point x∈Λ the Voronoi cell V(x) is We consider a quasi-static flat-fading n×n MIMO chan-
(cid:40) (cid:41) nel, where the channel state information is available at both
k
v= (cid:88) α (cid:96) : (cid:107)v−x(cid:107)≤(cid:107)v−y(cid:107), ∀y∈Λ, α ∈C . transmitter and receiver. The channel matrix is denoted by
m m m H¯ ∈Cn×n,wheretheentriesofH¯ arei.i.d.complexGaussian
m=1
random variables ∼CN(0,1). We use an n-layer lattice cod-
The Voronoi cell for the origin is denoted by V. Let L be a
ingscheme,wheretheinformationtransmittedacrossdifferent
matrix with (cid:96) as its rows, then L is called the generator
m antennas are independent. For 1 ≤ m ≤ n, the m-th layer
matrix of the lattice Λ . Throughout the paper, we only
L is equipped with a lattice encoder E : Rk → Cn which
consider full rank lattices where d = k. For example Z[i]d
maps a message ¯s ∈ Rk over the ring R into a lattice
is a lattice with the identity matrix as generator matrix. Let m
codeword x¯ ∈ Λ/Λ(cid:48) ⊂ Cn in the complex space. The
S ⊆Cn, then the linear span of S is defined as: m
matrixX¯ =[x¯T,...,x¯T]T isactuallyaspace-timecodeword,
(cid:40)(cid:88)q (cid:41) where its rows1are allnlattice codewords. This lattice space-
span(S)(cid:44) c v : q ∈N, v ∈S, c ∈C .
p p p p time codeword will then be precoded using a unitary matrix.
p=1 TheprecodingmatrixU¯ canbederivedusingthecomponents
Let B (0) = (cid:8)x∈Cd: (cid:107)x(cid:107)≤r(cid:9), be the d-dimensional ball of the channel matrix H¯ and another unitary matrix P¯, to
r
of radius r centered at 0 and dim(V) denotes the dimension be optimized later. Together with U¯, the matrix X¯ forms a
of a subspace V of Cd. For a d-dimensional lattice Λ , we space-time codeword U¯X¯ to be sent through the channel.
L
define the m-th successive minima, for 1≤m≤d as Let H¯ = W¯ Σ¯V¯h be the singular value decomposition
(SVD) of the channel matrix where W¯ ,V¯ ∈ Cn×n are
(cid:15)m(ΛL)(cid:44)inf{r: dim(span(ΛL∩Br(0)))≥m}. (1) two unitary matrices and Σ¯ is a diagonal matrix given by
The m-th successive minima of ΛL is the infimum radius r Σ¯ =diag(σ¯1,...,σ¯n) with σ¯1 ≥···≥σ¯n all in R. A unitary
such that there are m independent vectors of Λ in B (0). precoder matrix
L r
U¯ =V¯P¯ (3)
A d-dimensional lattice Λ is called full-diversity if for
L
all distinct x,y ∈ ΛL, [x]m (cid:54)= [y]m for all 1 ≤ m ≤ d. is then employed at the transmitter where P¯ ∈Cn×n is a uni-
Theminimumproductdistanceofafull-diversitylatticeΛL is tary matrix that needs to be optimized. If X¯ =[x¯T,...,x¯T]T
1 n
denoted by dp,min(ΛL) ([35]): denotes the matrix of transmitted vectors, the received signal
(cid:89) Y¯ is given by
d (Λ )(cid:44) min |[x] |. (2) √
p,min L x∈ΛL\{0} m m Y¯ = ρ·H¯U¯X¯ +Z¯, (4)
SNR
A lattice code C ⊆ Λ is a finite set of points of Λ. One where ρ = denotes the average signal-to-noise ratio at
n
wayofconstructinglatticecodesisthroughsublattices(nested each receive antenna and the entries of Z¯ are i.i.d. distributed
lattices).AsubsetΛ(cid:48) ⊆ΛiscalledasublatticeifΛ(cid:48) isalattice as CN(0,1). Let M¯ be the generator matrix of a complex
itself. Given a sublattice Λ(cid:48), we define the quotient Λ/Λ(cid:48) as lattice ΛM¯, then by using the standard conversion from
a lattice code. The fine lattice Λ is called the coding lattice complex lattice to the equivalent real one, we have the real
and the coarse lattice Λ(cid:48) is usually referred as the shaping lattice generator matrix as:
lattice. The lattice code C = Λ/Λ(cid:48) corresponds to a finite (cid:18) (cid:60)(M¯) (cid:61)(M¯) (cid:19)
constellationoflatticepointscarvedfromthelatticeΛ,within M= −(cid:61)(M¯) (cid:60)(M¯) .
afixedVoronoicellofΛ(cid:48).Inotherwords,letV(cid:48)betheVoronoi
cell of origin for lattice Λ(cid:48), then C = Λ∩V(cid:48). Therefore, the With the above transformation, (4) can be written as
shape of such constellation is governed by V(cid:48). For example, √
Y = ρ·HUX+Z, (5)
the Voronoi region of the 4-dimensional checkerboard lattice
(cid:40) (cid:41) where all the matrices Y,H,U,X, and Z are in R2n×2n. A
4
Λ(cid:48) =D (cid:44) (x ,··· ,x )∈Z4: (cid:88) x =even , suitable block diagram is then as Fig. 1. Upon receiving Y
4 1 4 i at the destination, we multiply it by Wh to get Y(cid:48) (cid:44)WhY.
m=1
Substituting U from (3) into (5) the channel can be modeled
has exactly 24 faces, with the relevant vectors being the
as:
24 minimum Euclidian lattice points [39]. This lattice is a √
sublattice of 4-dimensional integer lattice, i.e. D ⊆ Z4 and Y(cid:48) = ρ·ΣPX+Z(cid:48), (6)
4
hence so is the scaled lattice 8D4 (cid:44) {8x: x∈D4} ⊆ Z4. where Z(cid:48) = WhZ. Note that Z(cid:48) continues to be an i.i.d.
The lattice code Λ/Λ(cid:48) = Z4/8D4 includes all 4-dimensional complex Gaussian matrix with entries ∼ CN(0,1) because
integer points inside the Voronoi region of the lattice 8D4. W is unitary.
In general, a common choice for the sublattice Λ(cid:48) of Λ is At this point, the receiver employs integer-forcing. That is
the scaled lattice gΛ for some integer g ∈ Z as in [38]. As a linear receiver architecture that creates an effective integer-
another simpler example, for Λ = Z[i]d and Λ(cid:48) = gZ[i]d, valued channel matrix. The entries of this effective channel
we have Λ/Λ(cid:48) = Z [i] for which Z is the ring of integers matrix are used to recover integer combinations of the code-
g g
modulo g. For g = 2 the cubic shaped lattice code Z [i] is words. This is in contrast to the other linear receivers and
2
simply a 4-QAM constellation. For the rest of this paper we ML-based decoders, which attempt to recover the transmitted
use Λ=Z[i]d and choose g to be a power of 2. codewordsdirectly.Afeatureofinteger-forcinglinearreceiver
4
Fig.1. RealBlockdiagramofunitaryprecodedinteger-forcing.
is in using the same lattice codes at both the transmitter ρ (instead of ρ(cid:107)a (cid:107)2) due to the fact that aTx is itself a
m m
as well as the receiver. In this framework, the sender uses codeword,whichisdecodableuptoaneffectivenoisevariance
a layered transmission scheme, and transmits independent ofρ.Specifically,forlatticecodes,itisofteneasiertothinkof
√
lattice codewords simultaneously across the layers. At the each codeword as surrounded by a ball of radius nρ. If the
receiver side, each layer is allowed to decode an integer noise falls within this ball, then decoding is successful. The
linear combination of transmitted lattice codewords. Since receivedsignal-strengthviewpointismoreappropriatefori.i.d.
any integer linear combination of lattice points is another random coding methods, where a joint typicality analysis can
lattice point, the decoded point will be another lattice point. be applied.
Assuming that the effective channel matrix is full rank, these
integercombinationscanthenbesolvedfortheoriginallattice
codewords.
The goal of integer-forcing linear receiver is to project ΣP A. Decoding Complexity Advantage of UPIF
(by left multiplying it with a receiver filtering matrix B) onto
a non-singular integer matrix A. In order to uniquely recover We highlight the decoding complexity advantage of UPIF
theinformationsymbols,thematrixAmustbeinvertibleover in comparison to ML decoder techniques such as sphere
the ring R. Thus, we have decoder[4]inslow-fadingMIMOchannels.Inthesechannels,
√ the phases and the amplitudes of the channel coefficients can
Y(cid:48)(cid:48) =BY(cid:48) = ρ·BΣPX+BZ(cid:48). (7)
be assumed approximately constant over a long period. One
For the IF receiver [21] formulation, the model is given by such period is called a quasi-static channel interval or simply
√ √ an interval. Let the channel coefficients remain unchanged for
Y(cid:48)(cid:48) = ρ·AX+ ρ·(BΣP−A)X +BZ(cid:48) a time period t, which is much larger than the codeword
(cid:124) (cid:123)(cid:122) (cid:125)
quantization noise term length 2n, so that many codewords can be transmitted in
√ this interval. For example, let m = t ∈ Z be the number
(cid:44) ρ·AX+E , (8) 2n
of feasible codeword transmissions within one such quasi-
√
where ρ · AX is the desired signal component, and the static channel interval. In IF, the matrices A and B are
effective noise is computed once, based on H and estimated ρ in the first
√ codeword transmission. In the subsequent m − 1 codeword
ρ·(BΣP−A)X+BZ(cid:48). (9)
transmissions within the same interval, only a system of
We further denote the effective noise term along the m-th linear equations, with integer coefficients A, will be solved
layerbye andbystackingtheserowvectors,wegetE.The to recover the original information symbols at the receiver.
m
average energy of the effective noise along the m-th row of Referring to Table II of [22], the overall complexity of using
Y(cid:48)(cid:48) is defined as IF in an interval is dominated by the complexity of finding
the matrix A as the m systems of 2n linear equations with
G(am,bm)(cid:44)ρ(cid:107)bmΣP−am(cid:107)2+(cid:107)bm(cid:107)2, (10) 2n unknown variables can be solved using Gauss-Jordan
elimination process with computational complexity of order
where a and b denote the m-th row of A and B,
m m m×(2n)3. This is in contrast to ML decoding algorithms in
respectively. Note that in order to increase the m-th layer
slow-fading MIMO channels. In these scenarios, an algorithm
effective signal-to-noise ratio [21]
such as sphere decoder should be employed for each and
ρ
SNR = (11) all m codeword transmissions inside one quasi-static channel
eff G(a ,b )
m m interval. Therefore, the overall computational complexity for
the term G(a ,b ) has to be minimized, for each m, by an interval is m times the complexity of the sphere-decoder
m m
appropriately selecting the matrices A and B. We refer to the algorithm [4]. This makes IF linear receiver computationally
above signal model as unitary precoded integer-forcing. Note more efficient than ML decoding algorithms in slow-fading
that the signal strength in the numerator of (11) is simply channels.
5
III. DIVERSITYANALYSIS where em is the m-th row of E defined in (8).
The following Theorem provides an upper bound on
Wefirstrecallthedefinitionofdiversityandaweakversion P (cid:0)m,ΣP,Z2n(cid:1).
of Woodbury identity [40], which are used in the rest of this e
Section. Theorem 1 (Upper Bound on Probability of Error): For
all 1 ≤ m ≤ 2n, the term P (cid:0)m,ΣP,Z2n(cid:1) as in (20), is
Definition 2: In an 2n×2n MIMO system and at a high e
upper bounded as
signal-to-noise ration ρ, if the average probability P , that
e
a transmitted symbol vector is wrongly decoded, is approx- P (cid:0)m,ΣP,Z2n(cid:1)≤exp(cid:16)−c(cid:15)2 (Λ )(cid:17), (21)
imated by (c.ρ)−δ, then δ is called the diversity order (or e 2n−m+1 L−p1
diversity gain) and c is called the coding gain. For a MIMO where c is a constant independent of ρ and (cid:15)2 (Λ )
system with precoding, if δ = (2n)2, then, we say that the is the (2n−m+1)-th successive minima of2thne−mla+tt1iceLw−pi1th
precoder achieves full-diversity order. generator matrix L−1 defined in (19).
p
Lemma 1 (Woodbury Identity): Let M and M be two
1 2 Proof: The proof is given in Appendix A. Note that in
d×d invertible matrices, then the following identity holds:
the proof we assume the use of dithering which makes x
m
I −M (I +M M )−1M =(I +M M )−1. a random variable distributed uniformly within the shaping
d 1 d 2 1 2 d 1 2
region.
We now proceed to analyze the diversity order of UPIF. The We are interested in P (cid:0)2n,ΣP,Z2n(cid:1) which is an upper
eqfufaencttiizvaetinoonisneoiisne(t9er)mis. NnootteGthaautssmiaantridxis(cid:0)trIibut+edρd·uΣehtΣo(cid:1)t−h1e bound for Pe(cid:0)m,ΣP,Ze2n(cid:1), for all 1 ≤ m ≤ 2n. We define
2n the error probability for unitary precoded integer forcing over
is a positive definite matrix as it is a diagonal matrix with
Z2n as
positive entries, hence it admits a Cholesky decomposition P (cid:0)ΣP,Z2n(cid:1)(cid:44)P (cid:0)2n,ΣP,Z2n(cid:1). (22)
e e
(cid:0)I +ρ·ΣhΣ(cid:1)−1 =LLh. (12)
2n Based on (21), we have
The optimum value of b that minimizes (10) given a (cid:16) (cid:17)
m m P (ΣP,Z)≤exp −c(cid:15)2(Λ ) . (23)
is [21] e 1 L−p1
(cid:16) (cid:17)−1 Let the average probability
b =ρ·a ΣPh I +ρ·ΣP(ΣP)h . (13)
m m 2n
P =E (P (ΣP,Z)),
e H e
By defining
S(cid:44)I +ρ·ΣP(ΣP)h, where the expectation is taken over all channel matrices H.
2n
Theorem 2: Let the precoding matrix P be such that
we re-write (13) as
[Pv] (cid:54)=0,wherev∈Z2nisthevectorsatisfying(cid:15)2(Λ )=
bm =ρ·amΣPhS−1. (14) (cid:107)L−p11v(cid:107)2, then the unitary precoded integer-forcing1exLpl−pa1ined
through (5) to (10) achieves full-diversity (2n)2.
Let
L (cid:44)PhL, (15) Proof: The proof is given in Appendix B.
p
Based on the proof of the above theorem, to achieve full-
then the average energy of the effective noise term along the
diversity, all we need is to select a unitary precoder matrix
m-th layer is given by
P such that if v ∈ Z2n be the vector, which arises to the
G(am,bm) = ρ(cid:107)bmΣP−am(cid:107)2+(cid:107)bm(cid:107)2 minimumdistanceofthelatticeΛL−p1,thenitsatisfies[Pv]1 (cid:54)=
= ρ·a (I −(ΣP)hS−1ΣP)ah (16) 0. However, enforcing this condition seems to be unpractical
m 2n m (cid:16) (cid:17)
= ρ·a (cid:16)I +ρ·(ΣP)hΣP(cid:17)−1ah(17) because it involves computation of dmin ΛL−p1 which is an
m 2n m NP-hard problem. For that reason, we choose to work with a
= ρ·amPh(cid:0)I2n+ρ·ΣhΣ(cid:1)−1Pahm stronger condition that is dp,min(ΛP) (cid:54)= 0. It is clear that if
d (Λ ) (cid:54)= 0 then [Pv] (cid:54)= 0 for every vector 0 (cid:54)= v ∈
= ρ·a PhLLhPah (18) p,min P 1
m m Z2n.
= ρ·a L Lhah, (19)
m p p m Theorem 3: Let the precoding matrix P be such that
where (16) holds because of (14) and equation (9) in [22], d (Λ ) (cid:54)= 0, then the achievable diversity of the unitary
p,min P
(17) is true because of the Woodbury identity in Lemma 1 for precoded integer-forcing explained through (5) to (10) is
√ √ (2n)2.
d=2n, M = ρ·(ΣP)h, M = ρ·ΣP,
1 2
According to the above theorem, if lattice codes along with
and (18) and (19) are obtained based on (12) and (15), a non-zero minimum product distance precoder are used at
respectively. We denote the probability of error for decoding the transmitter of a MIMO channel and integer-forcing is
the m-th layer in the integer lattice Z2n by Pe(cid:0)m,ΣP,Z2n(cid:1). employed at the receiver, full (transmit and receive) diversity
It follows that canbeachieved.Wenextprovidetwodifferentdesigncriteria
√
(cid:18) (cid:19)
P (cid:0)m,ΣP,Z2n(cid:1)=Pr |e |≥ ρ , (20) for obtaining unitary precoders suitable for integer-forcing
e m 2 linear receivers.
6
IV. OPTIMALDESIGNOFFULL-DIVERSITYUNITARY
1
PRECODERS
Based on the above two theorems, we next design classes 0.9
of full-diversity unitary precoders for integer-forcing linear
receivers. We first propose a method of designing optimal 0.8
precoders based on Theorem 2 and then we provide our
0.7
optimal precoders based on Theorem 3.
θn()
ta0.6
A. Design of Type I UPIF
0.5
Based on (23) and Theorem 2, the optimal Type I UPIF is
as follows:
0.4
P =arg max min (cid:107)L−1Pv(cid:107)2, (24)
1,opt
P∈O2nv∈Z2n\{0} 0 0.2 0.4 0.6 0.8 1
[Pv]1(cid:54)=0 tan(η)
where O is the orthogonal group of all 2n×2n orthogonal
2n
matrices with matrix multiplication operation. In other words, Fig. 2. The variation of tanθ based on the variation of tanη in a 2×2
weshoulddesignaprecodermatrixPsuchthattheminimum complexMIMOChannelusingrealTypeIUPIF.
distance of the lattice Λ is maximized, while [Pv] (cid:54)= 0
L−p1 1
for v∈Z2n and (cid:15)2(Λ )=(cid:107)L−1v(cid:107)2.
1 L−p1 p
Remark 1: If we relax the constraint on P ∈ O to be
2n
(cid:0) (cid:1)
Tr PPT ≤ ρ , where Tr(.) denotes the trace of a matrix,
0 distance of the corresponding lattice Λ is considered as a
aonpdtimreimzaotivoenthperocbolnemstr,iawnthi[cPhv]i1s (cid:54)=the0,mwaeingestuabjmecotreofgesntuerdayl 2×2 real approximately optimal percoLd−pe1r matrix P(R) to be
1,opt
in [42]. The optimal solution for this problem is derived to be employed for both real and complex 2×2 UPIF. Using (18),
the hexagonal lattice A [43] for 2×2 real MIMO channels the matrix L−1 is given by
2
andSchlaflilatticeD [43]for2×2complexMIMOchannels. (cid:18) (cid:112) (cid:19) (cid:18) (cid:19)
4 1+ρσ2 0 cosη 0
However,thegeneratormatrixoftheselatticesarenotunitary. L−1 = 1 (cid:112) =ξ ,
0 1+ρσ2 0 sinη
In this paper, we focus on solving the optimization problem 2
(cid:114)
in (24), which is slightly different from that of [42]. (cid:16)(cid:112) (cid:17)2 (cid:16)(cid:112) (cid:17)2
where ξ = 1+ρσ2 + 1+ρσ2 =
Theproblemstatedin(24)seemstobeahardcombinatorial 1 2
(cid:112)
problem, which we could not solve analytically. However, a 2+ρ(σ2+σ2) and
1 1
systematic approach of finding P is to perform an exhaus-
1,opt (cid:32)(cid:112) (cid:33)
tivesearchinO2n.AconvenientparametrizationofO2n using η =tan−1 1+ρσ22 .
(2n)2 parameters is presented in [44]. Such a parametriza- (cid:112)1+ρσ2
1
tion uses the fact that an orthogonal transformation P can
It follows that L−1 is equal to
be composed from elementary orthogonal transformations in p
two-dimensional sub-spaces. The induced matrices using the (cid:18) (cid:19)(cid:18) (cid:19)
cosη 0 cosθ sinθ
algorithm given in [44] will be uniformly distributed with L−1P(θ)=ξ 0 sinη −sinθ cosθ .
respect to Haar measure.
In Fig. 2, we plot the variation of tanθ based on tanη for
Forthe2×2MIMOsystem,sinceweimposedtheorthogo-
105 samples of 2×2 complex MIMO systems employing real
nalityconstraintonP,weparameterizeitusingasingleangle
2×2 precoders P(θ). In particular,
θ as follows:
(cid:18) cosθ sinθ (cid:19) (cid:112)1+ρσ2 1
P(θ)= , (25) tanη = 2 = ,
−sinθ cosθ (cid:112)1+ρσ12 βL−p1
It can be easily seen by the symmetry of integers that it where β is the condition number of L−1. For large values
suffices to consider θ ∈[0,π/4] for the maximization in (24). L−p1 p
Hence, we numerically search for of ρ, tanη ≈ σσ21 = β1H, where βH is the condition number of
the channel H. The generator matrix of the lattice Λ can
P(R) =arg max min (cid:107)L−1P(θ)v(cid:107)2, (26) be considered now as the equivalent channel√matrix. ILt−pca1n be
1,opt
P(θ)∈O2[P(θ)v]1(cid:54)=0 seenthatiftheconditionnumberislessthan 3,thenthebest
where O is the group of all 2×2 orthogonal matrices with precoder obtains by putting θ = π/4. In Fig. 3, we plot the
2
matrix multiplication operation. This exhaustive search can variation of the coding gain of the lattice ΛL−p1 versus tanη.
be done by discretizing θ ∈ [0, π/4] using fine steps of The coding gain formula is:
0.001radiansandconsequentlyfindingtheminimumdistance (cid:15)2(Λ )
orefalΛpLr−pe1coudsininggmGaatruixss, wrehdiucchtiroensualtlsgoinritthhme h[i4g5h]e.stTmhein2im×um2 γ(ΛL−p1)= det1(cid:0)LL−−p11(cid:1)22n. (27)
p
7
B. Design of Type II UPIF
1.16
In previous Subsection, we introduced P as a suitable
1,opt
1.14
precoder matrix for UPIF. Equations (25) and (26) show that
1.12 Type I UPIF precoder matrices P and P(R) are adapted
1,opt 1,opt
based on L−1 and consequently H and ρ. In this Subsection,
1.1
however we introduce Type II UPIF, where the optimal pre-
21.08 coder matrix P is fixed and it does not change by varying
Λ−L1p the channel mat2r,oipxt H.
11.06
γ By further expanding (cid:15)2(Λ ) in (23), we get:
1.04 1 L−p1
(cid:15)2(Λ ) = (cid:107)L−1v(cid:107)2
1.02 1 L−p1 p
= vhL−hL−1v=vh(cid:0)L Lh(cid:1)−1v
1 p p p p
0.98 = vhPh(cid:16)(cid:0)I2n+ρ·ΣhΣ(cid:1)−1(cid:17)−1Pv (28)
0 0.2 0.4 0.6 0.8 1
tan(η) = vhPh(cid:0)I +ρ·ΣhΣ(cid:1)Pv, (29)
2n
Fcoigm.p3l.exTMheIMvaOriaCtihoannnoeflγu(sΛinLg−pr1e)albTaysepdeoInUtPheIFv.ariationoftanηina2×2 wdehfienreite(2m8)atfroixll,owwsefcraonmp(e1r9fo).rmSinaceCIh2onle+skρy·ΣdehcΣomispaospitoisointivtoe
obtain I +ρ·ΣhΣ=DDh. With this, (29) can be written
2n
as (cid:107)DPv(cid:107)2. Hence, we get
(cid:15)2(Λ ) = (cid:107)DPv(cid:107)2
8000 1 L−p1
2n
(cid:88)
7000 = [D]2 [Pv]2
m,m m
m=1
6000 (cid:32) (cid:33) 1
2n 2n 2n
(cid:89) (cid:89)
5000 ≥ 2n [D]2m,m [Pv]2m (30)
m=1 m=1
4000 (cid:32) (cid:33) 1
2n 2n 2n
= 2n (cid:89) (cid:0)1+ρσ2 (H)(cid:1) (cid:89) [Pv]2
3000 m m
m=1 m=1
2000 2n
≥ 2n (cid:89) (cid:0)ρσ2 (H)(cid:1)21n (cid:0)d2 (Λ )(cid:1)21n (31)
m p,min P
1000
m=1
1 1
00.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 = 2nρ(det(H))n dpn,min(ΛP), (32)
where(30)followsfromCauchy–Schwarzinequality,and(31)
is true because 1+ρσ2 (H)>ρσ2 (H), for all 1≤m≤2n
Fusigin.g4.realTThyepehiIstUogPrIaFm. of γ(ΛL−p1) in a 2×2 complex MIMO Channel and d2p,min(ΛP)≤(cid:81)2mmn=1[Pu]2m fmor all u∈Z2n.
Based on (32) and since almost surely det(H) (cid:54)= 0, the
optimalTypeIIUPIFcanbeobtainedbysolvingthefollowing
optimization problem:
1
tThheeicnotedgienrglgaattiincemZea2snuwreisththγe(cid:0)inZc2rne(cid:1)as=ei1n.dBeansseitdyoonfΛFiLg−p.13o,vweer P2,opt =argPm∈Oax2ndpnmin(ΛP), (33)
Thesolutionforthemaximizationin(33)isprovidedin[32]–
can safely say that the equivalent precoded channel matrix
L−1 is at least perform as good as an integer 2×2 channel [36] using algebraic number theoretic lattices. A list of full-
p
diversityalgebraicrotationsisavailablein[37].Allthelattices
matrix. In other words, the precoder matrix will change the
provided there satisfy non-vanishing minimum product dis-
bad channels to equivalent channels as good as an integer
tance criterion too. Hence, thanks to [37], the optimal Type II
one. Fig. 4 shows the histogram of γ(Λ ) for the same
L−p1 UPIF along with their corresponding d are available. We
105 channel samples. There is a probability mass at around p,min
now proceed to verify this result by conducting simulations.
1, which states that unitary matrices has cleared all the bad
channels with γ(Λ ) < 1 to convert them to an equivalent
H
channel matrix L−1 with γ(Λ ) ≥ 1. Note also that the V. SIMULATIONRESULTS
p L−p1
maximum attainable coding gain (Hermite’s constant [43]) The UPIF scheme and MIMO precoding X-codes and Y-
for 2-dimensional lattices is √2 . As it is clear from Figs. 3 codes[13]sharesimilar properties,whichmakethemsuitable
3
and 4, this quantity has been achieved by few lattices in our for comparison: (i) both schemes use SVD decomposition
simulations too. technique to transform the channel matrix into a diagonal
8
100 100
Type I UPIF Type II UPIF
X−Precoders−ML X−Codes−ML
Type I UPML Type II UPML
10−1
10−1
Rate Rate10−2
Codeword Error 10−2 64−QAM Codeword Error 10−3 4−QAM 16−QAM 64−QAM
4−QAM 16−QAM
10−3
10−4
10−45 10 15 20 25 30 35 40 10−55 10 15 20 25 30 35 40 45
Es/N0=ρ(dB) Es/N0=ρ(dB)
Fig.5. TypeIUPIFincomparisonwith,X-Precodersdecodedwithsphere Fig.6. TypeIIUPIFincomparisonwithX-CodesandTypeIIUPMLina
decodingalgorithm,andTypeIIUPMLina2×2complexMIMOChannel. 2×2complexMIMOChannel.
one, (ii) the precoder matrices in both systems must be
X-precoders cannot be captured. This will be more visible in
unitary/orthogonal matrices, (iii) both the detectors at the
the next subsection, where Type II UPIF outperform X-codes
receiverside,i.e.latticereductionbasedIFlinearreceiver[22]
in MIMO channels with larger number of antennas employed
andacombinationoftwo2-dimensionalMLdecoders,provide
at larger constellation sizes.
full receive diversity in 2×2 MIMO.
For comparison we show the performance of UPIF precod-
ing when the IF receiver is replaced by an ML decoder. We B. Type II UPIF versus X-Codes
will denote this with the acronym UPML. In the following
We have conducted simulations for 2×2 and 4×4 MIMO
subsections, we simulate and compare our proposed Type I/II
UPIF channel over 4/16/64-QAM constellations. For 2×2
UPIF decoded using lattice reduction based IF receiver, with
MIMO channels, the matrices A and B for IF linear receiver
Type I/II UPML, and MIMO X-Precoders/X-Codes.
werefoundusingAlgorithm2presentedin[22]whileforlarger
MIMOs we have employed Algorithm1. We have chosen our
A. Type I UPIF versus X-Precoders TypeIIUPIFsbasedonthedesigncriteriongivenin(33).The
Weshowsimulationsfora2×2MIMOUPIFchannelover 2×2 and 4×4 full-diversity algebraic rotation [37] matrices
4/16/64-QAM constellations. The matrices A and B for IF which has been used for simulations have the following
1
linearreceiverwerefoundusingAlgorithm2presentedin[22]. minimum product distances (dn ) 0.668740, 0.438993, and
p,min
We have chosen our Type I UPIF by running an exhaustive 0.289520. For reference purposes, we also presented MIMO
search based on the design criterion given in (26). For ref- X-codes [13] decoded under sphere decoding algorithm [4].
erence purposes, we also presented MIMO X-precoder [13] For 4-QAM constellation the optimal angle for a X-Code is
scheme decoded under sphere decoding algorithm [4]. The 26.6degreeswhilefor16-QAMitisequalto15degrees[13].
step size for our brute force search for both the cases (Type We also found the optimal angle for 64-QAM to be equal
I UPIF and X-precoders) is 0.001 radians. Fig. 5 shows the to 8 degrees. In Figs. 6 and 7, we plot the codeword error
codeword error rate [22] curves versus signal-to-noise ratio performanceofTypeIIUPIFversusMIMOX-codesandType
ρ of the different schemes. Since the performance of UPIF II unitary precoders decoded using a maximum likelihood
decreases by growing ρ parallel to ML decoded curve, full- decoder (Type II UPML) such as sphere decoding algorithm
diversity of UPIF is guaranteed. In particular, at codeword versus signal-to-noise ratio E /N = ρ. As it is clear from
s 0
error rate 10−3 over 4/16/64-QAM constellations, the gaps Figs. 6 and 7, the Type II UPIF outperform MIMO X-codes
of 2.2dB, 2.15dB, and 2.1dB between Type I UPIF and X- forlargerconstellationsizesinhigherMIMOdimensions.The
precoder are observed. This shows a decrease in coding loss main reason is that the Type II UPIF precoder matrices are
by increasing the constellation size. The 2×2 X-precoders designedtoachievefull-diversityandthediversitygainshows
outperform the other schemes because of two main reasons: itself when the number of antennas increases. In particular,
(i) these precoders are designed for the particular lattice the diversity order [13] of the optimal MIMO X-codes (in
constellation (for example, 4-QAM, 16-QAM, or 64-QAM) terms of diversity not the error performance) is known to be
while Type I UPIF are designed for the (infinite) lattice Z2n: d = (cid:0)n − ns +1(cid:1)(cid:0)n − ns +1(cid:1), where n and n
nt,nr,ns r 2 t 2 r t
(ii) since we are only able to simulate Type I UPIF over a denote the number of receive and transmit antennas, respec-
2×2 MIMO channel, the diversity gain of Type I UPIF over tively, and n denotes the number of symbols. In our setting,
s
9
100 100
Type I UPIF
X−Precoderes−ML
Type I UPML
10−1 10−1 Type II UPIF
X−Codes−ML
Type II UPML
e e
at at
or R10−2 or R10−2
Err Err
d d
Codewor10−3 4−QAM 64−QAM Codewor10−3
10−4 10−4
Type II UPIF
X−Codes−ML
Type II UPML
10−5 10−5
0 5 10 15 20 25 30 35 40 25 30 35 40
Es/N0=ρ(dB) Es/N0=ρ(dB)
Fig.7. TypeIIUPIFincomparisonwithX-CodesandTypeIIUPMLina Fig.8. TypeIversusTypeIIUPIFandUPMLschemesina2×2complex
4×4complexMIMOChannel. MIMOChannel.
have access to limited feedback over a delay-free link from
for 2×2 and 4×4 MIMO X-codes, we have d =4 and
2,2,2
the IF receiver [46], [47]. In this case, designing a suitable
d = 9 in comparison with full-diversity orders 4 and 16
4,4,4
codebook of unitary precoding matrices attaining higher rates
for Type II UPIF, respectively.
andobtaininghighercodinggainsinerrorperformanceseems
to be a promising research topic.
C. Type I UPIF versus Type II UPIF
In this subsection, we compare Type I and Type II UPIF APPENDIXA
together. Fig. 8 shows the codeword error performance of PROOFOFTHEOREM1
TypeIandTypeIIUPIFincomparisontotheircorresponding For the proof of this we follow the lines of the proof of
UPML. It is obvious that the Type II UPIF and UPML Theorem 7 in [41]. Since the minimum Euclidean distance
√
outperform Type I UPIF and UPML precoders by no more of Z is unity, an error is declared if e ≥ ρ. Therefore,
than 0.5dB and 1dB, respectively. P (cid:0)m,ΣP,Z2n(cid:1) equals to m 2
e
The sub-optimality of unitary precoder matrices obtained (cid:18) √ (cid:19) (cid:18) √ (cid:19)
ρ ρ
usingquantizedexhaustivesearchalgorithmisthemainreason =Pr |e |≥ =2Pr e ≥ (34)
m m
2 2
for having Type II UPIF performing better than Type I UPIF.
E(exp(te ))
Furthermore,wenotethatthedesigncriteriagivenin(26)and ≤2min (cid:16)√ m(cid:17) (35)
(33) are different and we cannot say which one can perform t>0 exp ρt
2
betterthantheotherbyjustlookingattheseequations.Hence, E(exp(cid:0)t√ρ·(cid:104)b ΣP−a ,x (cid:105)+t·(cid:104)b ,z(cid:48) (cid:105)(cid:1))
considering the simplicity of Type II UPIF in comparison to =2min m (cid:16)√m (cid:17)m m m ,(36)
Type I and their better error performance, we suggest using t>0 exp ρt
2
of Type II UPIF for practical purposes.
where (34) follows from the symmetry of effective noise
around zero, (35) follows from Chernoff’s bound. Since x
VI. SUMMARYANDDIRECTIONSFORFUTUREWORK m
and z(cid:48) are independent, the equation (36) can be written as
m
Aunitaryprecodingscheme(UPIF)hasbeenintroducedfor √
E(exp(t ρ·(cid:104)b ΣP−a ,x (cid:105)))E(exp(t·(cid:104)b ,z(cid:48) (cid:105)))
transmissionoveraflat-fadingMIMOchannelwithbothCSIT min m m (cid:16)√m (cid:17) m m .
and CSIR, and an IF linear receiver. The diversity gains of t>0 1exp ρt
2 2
the proposed approach have been analyzed both theoretically (37)
and numerically. Two different design criteria for two types We continue to use moment generating functions of uniform
of unitary precoders, Type I UPIF and Type II UPIF, were and Gaussian distributions to further upper bound the two
given. For both cases, an SVD is performed first to transform exponential terms in (37). Since the components of z(cid:48) are
m
the channel to a diagonal one. Type I UPIF includes precoder distributed based on a Gaussian distribution with zero mean
matricesadaptingbasedonthechannelmatrixsingularvalues, and unit variance, we have
while Type II UPIF consists of precoding matrices, which (cid:18)t2(cid:107)b (cid:107)2(cid:19)
remain fixed by changing the channel matrix. E(exp(t·(cid:104)b ,z(cid:48) (cid:105)))≤exp m , (38)
m m 2
Designingfull-diversityunitaryprecodersormoregenerally
space-time block codes for IF receivers without CSIT would for the second term in (37). We now investigate its
√
be of interest. Another direction is to let the transmitter first term. Let q (cid:44) t ρ · (b ΣP − a ). The
m m m
10
entries of x ∈ Z2n are uniformly distributed over where c = (2n)3+(3n)2. Using inequality of (46) in (44),
the hypercumbe (cid:2)−t√ρ(cid:107)q (cid:107),t√ρ(cid:107)q (cid:107)(cid:3)2n, therefore the prob0ability of error for decoding the m-th layer is upper
E(cid:0)exp(t√ρ·(cid:104)q ,x (cid:105))(cid:1) equamls to m bounded as
m m
= Eexpt√ρ·(cid:88)2n[qm]j[xm]j Pe(cid:0)m,ΣP,Z2n(cid:1)≤exp(cid:16)−c(cid:15)22n−m+1(ΛL−p1)(cid:17), (47)
where c= 1 . This completes the proof.
j=1 4c0
(cid:89)2n √
= E(exp(t ρ·[q ] [x ] )) (39)
m j m j
j=1 APPENDIXB
(cid:89)2n sinh(cid:0)t√ρ|[qm]j[xm]j|(cid:1) PROOFOFTHEOREM2
≤ √ (40)
t ρ|[q ] [x ] |
m j m j
j=1
Toward the proof of this theorem, we make use of the
≤ (cid:89)2n exp(cid:18)t2ρ|[qm]j|2(cid:19) (41) following two lemmas. The first lemma gives the expression
6 forthefirst-orderexpansionofthemarginalprobabilitydensity
j=1
(cid:18)t2ρ(cid:107)q (cid:107)2(cid:19) function (PDF) of eigenvalues of H.
≤ exp m , (42) Lemma 2: LettheentriesofthematrixHbei.i.d.complex
2
Gaussian random variables with zero mean and unit variance.
where (39) follows from the independence of components of
The first-order expansion of the marginal PDF of the k-th
x , (40) follows from the moment generating function of a
m largest eigenvalue λ of the complex central Wishart matrix
k
uniform distribution, and (41) follows from the inequality HHh is given by
sinh(y) (cid:18)y2(cid:19) (cid:16) (cid:17)
≤exp , F (λ )=q λdk +o λdk ,
y 6 λk k k k k
for every y ∈ R. Combining (42) and (38) in (37), we get as λk →0, with dk =(2n−k+1)2−1 and qk being positive
P (m,ΣP,Z) less than or equal to constants.
e √
(cid:16) (cid:17) (cid:16) (cid:17) Lemma 3: Forascalarchannelmodeledbyy = SNRβx+
exp t2ρ(cid:107)bmΣ2P−am(cid:107)2 exp t2(cid:107)b2m(cid:107)2 n , where n ∼ N(0,1), and E[|x|2] = 1 and α = β2
≤ 2mt>i0n exp(cid:16)√ρt(cid:17) is a nonnegative random variable whose probability density
2 function (PDF) F (α) is such that as
(cid:16) (cid:17) α
= 2minexp t2G(cid:16)(√a2m,b(cid:17)m) (43) Fα(α)=qαt+o(αt), (48)
t>0 exp ρt the average word error probability (WEP) P , which is given
2 e
(cid:18) −ρ (cid:19) by
= 2exp , (44)
4G(a ,b ) (cid:90) ∞
m m P =E(P (α))= exp(−kαSNR)F (α)dα,
√ e e α
where (44) obtained by optimizing t, that is t= ρ . 0
2G(am,bm) is such that as SNR→∞
Now since G(am,bm) needs to be minimized, am and bm P = 2√tqΓ(t+ 23)(kSNR)−(t+1)+o(cid:16)SNR−(t+1)(cid:17).
e
should be chosen appropriately as in [21] and [22], thus we π(t+1)
get
Now we proceed to give the proof of Theorem 2. Let v be
G(a ,b )
mρ m =(cid:15)2m(ΛLhp), the Gaussian vector for which
where (cid:15)2m(ΛLhp) denotes the m-th successive minimum of the (cid:15)21(ΛL−p1) = (cid:107)L−p1v(cid:107)2
lattice = vhL−hL−1v=vh(cid:0)L Lh(cid:1)−1v
Λ =(cid:8)LhPd | ∀d∈Z2n(cid:9). p p p p
Lhp = vhPh(cid:16)(cid:0)I +ρ·ΣhΣ(cid:1)−1(cid:17)−1Pv (49)
HereL−1 isageneratoroftheduallatticeΛ∗ .Thuswehave 2n
the relaption (see Lemma 4 in [21]) Lhp = vhPh(cid:0)I2n+ρ·ΣhΣ(cid:1)Pv
(cid:15)2 (Λ )≤ (2n)3+(3n)2 = (2n)3+(3n)2 , (45) (cid:44) (cid:107)DPv(cid:107)2 ≥[D]21,1[Pv]21 (50)
m Lhp (cid:15)22n−m+1(Λ∗Lhp) (cid:15)22n−m+1(ΛL−p1) =≥ (cid:112)(1ρ+λρ(σH12()H[P))v[]P2v]21 ≥ρσ12(H)[Pv]21((5521))
where(cid:15)2 (Λ∗ )isthe(2n−m+1)-thsuccessiveminima 1 1
2n−m+1 Lh
of the lattice Λ∗ . Tpherefore, we have where (49) follows from (19), (50) follows from Cholesky
Lh
p decomposition of I + ρ · ΣhΣ and the fact that D is a
2n
ρ ≥ (cid:15)22n−m+1(ΛL−p1), (46) diagonal matrix, (51) is true because 1+ρσ12(H)>ρσ12(H),
G(am,bm) c0 and (52) follows from the fact that σ12(H)=λ1(H).