On the Diversity-Multiplexing Tradeoff Under
Groupwise Successive Interference Cancelation
In Sook Park and Sae-Young Chung, Senior Member, IEEE
Abstract—In this paper, we study the diversity-multiplexing
tradeoff assuming groupwise successive interference cancelation (GSIC). GSIC with layered codes in input multiple-output (MIMO) can vary from the plain SIC to the maximum likelihood (ML) decoding according to the group sizes, thus is flexible in both complexity and performance. The maximum group size is a dominating factor for the computation complexity of GSIC when the processing complexity is polynomially depen-dent on the group size with a high order. In this paper, we show the tradeoff among the maximum group size, diversity gain, and multiplexing gain (GDMT) under the GSIC and under the GSIC combined with antenna selection (GSICAS).
For a given group-size bound 𝑐, we show that the number
of groups achieving the optimal DMT under the GSIC with𝛼
selected transmit antennas is equal to ⌈𝛼
𝑐⌉ and a larger group should have a higher priority in decoding order. The optimal grouping is obtained in a systematic way. Based on these results, we find a limited set of groupings that contains the grouping achieving the optimal GDMT under GSIC and under GSICAS. The optimal multiplexing gain allocation for the optimal grouping is then found systematically and efficiently within the limited set.
Index Terms—Multiple-input multiple-output (MIMO),
diver-sity, spatial multiplexing, diversity-multiplexing tradeoff (DMT), successive interference cancelation (SIC).
I. INTRODUCTION
A
multiple-input multiple-ouput (MIMO) system providestwo resources, diversity and spatial multiplexing. Diver-sity enhances the reliability of the transmitted data, whereas the spatial multiplexing increases the achievable data rate. There is a tradeoff between the diversity and multiplexing gains (DMT). The optimal DMT for the independently iden-tically distributed (i.i.d.) Rayleigh fading channel with 𝑛 transmit and 𝑚 receive antennas, when the block length 𝑙 is at least 𝑛 + 𝑚 − 1, is derived in [1] by focusing on the high SNR regime. The DMT result also holds for a wider class of channels [2]. After the result of [1], the construction of a family of codebooks achieving the optimal DMT when 𝑙 is at least 𝑛 was done for 𝑛 = 2 case in [3], [4] and for all𝑛 in [5]. The optimal DMT in [1] was derived assuming the maximum likelihood (ML) decoding. The optimal DMT’s under other decoding schemes are also studied in [6] and [7].
Paper approved by N. Al-Dhahir, the Editor for Space-Time, OFDM and Equalization of the IEEE Communications Society. Manuscript received February 28, 2010; revised November 16, 2010 and March 26, 2011.
This work was supported in part by MKE under grant no. NIPA-2011-(C1090-1111-0005).
I. S. Park was with the Dept. of Electrical Engineering, KAIST, Dae-jeon, South Korea. She is now with ETRI, DaeDae-jeon, South Korea (e-mail: insookpark@etri.re.kr; ispark@kaist.ac.kr).
S.-Y. Chung is with the Dept. of Electrical Engineering, KAIST, Daejeon, South Korea (e-mail: sychung@ee.kaist.ac.kr).
Digital Object Identifier 10.1109/TCOMM.2011.071111.100109
It was shown that the optimal DMT is also achievable by a class of lattice space time codes using a generalized minimum Euclidean distance lattice decoding [6]. The optimal DMT under a group detection is derived in [7].
A coding scheme with low decoding complexity is desired especially when𝑚 and 𝑛 are large. In general, however, there is a tradeoff between complexity and performance. Groupwise successive interference cancelation (GSIC) [8] with layered codes achieves this tradeoff depending on how the grouping is done and can be efficient in trading off performance against complexity. According to the grouping, the complexity and performance of GSIC can vary from those of the plain SIC to the ML decoding. The maximum group size can dominantly affect the complexity of GSIC if the processing complexity of each group is polynomially dependent on its group size with a high order, e.g., roughly cubic complexity in the number of antennas discussed in [9]. Thus, the optimal DMT computed over all choices of the GSIC whose maximum group size is less than or equal to a given bound can be a rough measure of the tradeoff between the performance and complexity.
In this paper, for the i.i.d. Rayleigh fading channel with𝑛 transmit and𝑚 receive antennas, we show that for a group-size bound 𝑐 the grouping that achieves the optimal GDMT under the GSIC should divide the transmit antennas into⌈𝑛
𝑐⌉
groups and moreover should give a higher priority in decoding order to the component code transmitted by a larger group. We also study GSIC combined with antenna selection (GSICAS) and show it can achieve a larger diversity gain than the plain GSIC in the low multiplexing gain regime. The optimal allocation of the multiplexing gain to the transmit antenna groups for a given grouping is derived systematically. From the characteristics of the optimal grouping in GDMT under the GSIC, we find out a very limited set of groupings which achieves the optimal GDMT.
In Section II, the system model, GSIC, and GSICAS are described. In Section III, the optimal DMT of a MIMO system under a fixed grouping is described. In Section IV we derive the optimal GDMT’s under the GSIC and then under the GSICAS. In Section V, we conclude this paper.
II. SYSTEMMODEL ANDGROUPWISESUCCESSIVE
DECODING
A. System Model
We consider a wireless communication system with 𝑛
transmit and 𝑚 receive antennas with arbitrarily distributed fading coefficients with unit variance. The fading coefficient from transmit antenna 𝑗 to receive antenna 𝑖 is denoted by
h𝑖𝑗 andH =[h𝑖𝑗]is the corresponding channel matrix.H is 0090-6778/11$25.00 c⃝ 2011 IEEE
assumed to be known to the receiver but not at the transmitter.
H is assumed to remain constant for at least a block of 𝑙
symbols.X =[x𝑖𝑗]∈ 𝒞, an 𝑛 × 𝑙 matrix, is transmitted such
thatx𝑖𝑗 is transmitted from antenna 𝑖 at time 𝑗, where 𝒞 is
the codebook ofX, which can be constructed as a Cartesian
product of many smaller codebooks for each group as will be discussed later. X is normalized such that the average
transmit power at each antenna in each symbol period is 1. LetY =[y𝑖𝑗], an𝑚×𝑙 matrix, be the received matrix where y𝑖𝑗 is the signal received from antenna𝑖 at time 𝑗. Then we
have the following relation
Y =
√ SNR
𝑛 HX + W (1)
where W is the additive noise with i.i.d. entries w𝑖𝑗 ∼
𝒞𝒩 (0, 1) and SNR is the average signal-to-noise ratio (SNR) at each receive antenna.𝒞𝒩 (0, 1) denotes the circularly sym-metric complex normal distribution with zero mean and unit variance.
B. Groupwise Successive Interference Cancelation
The GSIC with multilayered code, which is found in [8], is described in this subsection. We use the following notations. The sub matrix composed of the elements in rows𝑎 through 𝑏 and columns 𝑐 through 𝑑 of matrix A is denoted by A[𝑎 : 𝑏][𝑐 : 𝑑]. When v is a column vector, the sub-vector composed of the elements in rows𝑎 through 𝑏 of v is denoted by v[𝑎 : 𝑏].
A𝑖 represents the𝑖th column of matrix A.
The ML solution ˆX of X is the codeword in codebook 𝒞
such that ˆ X = arg min X∈𝒞 Y − √ SNR 𝑛 HX. (2)
IfX is equiprobable then ˆX is the optimal solution. But the
computation complexity of ML is too high. The complexity
can be reduced by layered coding and GSIC. Let 𝐵 denote
the number of layers or groups, which will be determined later based on how grouping is done. We let𝐼 = (𝑛1, 𝑛2, . . . , 𝑛𝐵),
which is an ordered set with0 ≤ 𝑛𝑗≤ 𝑛, ∑𝐵𝑗=1𝑛𝑗= 𝑛, and
𝑚 ≥ 𝑛1+ ⋅ ⋅ ⋅ + 𝑛𝐵−1+ 1 and implies that the number of
transmit antennas in the𝑗 th group is 𝑛𝑗.X is split according
to𝐼 into 𝐵 layers as X(𝑗) = X[𝑖𝑗−1+ 1 : 𝑖𝑗][1 : 𝑙] for 1 ≤
𝑗 ≤ 𝐵 with 𝑖0= 0 and 𝑖𝑗=∑𝑗𝑘=1𝑛𝑘. Let𝑚𝑗 ≜ 𝑚 − 𝑖𝑗−1.
We assume each X(𝑗) is encoded separately using a
DMT-optimal code, e.g., a random code as done in [1] or a lattice code as done in [6]. Therefore,X(𝑗)’s are independent from
each other by design. Without loss of generality, X(𝑗 + 1)
is decoded before X(𝑗). First, to decode X(𝐵), nullify the
signals transmitted from antennas 1 to 𝑖𝐵−1 by multiplying
(1) on the left by an(𝑚 − 𝑛 + 𝑛𝐵) × 𝑚 matrix Θ(𝐵) whose
rows are orthonormal vectors in the left null space ofH[1 :
𝑚][1 : 𝑖𝐵−1]. Then the effective channel for X(𝐵) is V(𝐵) = G(𝐵)X(𝐵) + Z(𝐵) where V(𝐵) ≜ Θ(𝐵)Y, G(𝐵) ≜ Θ(𝐵)√SNR 𝑛 H[1 : 𝑚][𝑖𝐵−1+ 1 : 𝑛], Z(𝐵) ≜ Θ(𝐵)W. Find ˆ X(𝐵) ≜ arg min X(𝐵)∥V(𝐵) − G(𝐵)X(𝐵)∥ 2
and letY(𝐵 − 1) ≜ Y −√SNR
𝑛 H[1 : 𝑚][𝑖𝐵−1+ 1 : 𝑛] ˆX(𝐵).
Then we have Y(𝐵 − 1) =√SNR
𝑛 H[1 : 𝑚][1 : 𝑖𝐵−1]X[1 :
𝑖𝐵−1][1 : 𝑙] + W and repeat the above groupwise nulling and
canceling to have for1 ≤ 𝑗 ≤ 𝐵 − 1
V(𝑗) = G(𝑗)X(𝑗) + Z(𝑗) ˆ X(𝑗) ≜ arg min X(𝑗)∥V(𝑗) − G(𝑗)X(𝑗)∥ 2 where V(𝑗) ≜ Θ(𝑗)Y(𝑗), G(𝑗) ≜ Θ(𝑗)√SNR 𝑛 H[1 : 𝑚][𝑖𝑗−1 + 1 : 𝑖𝑗], Z(𝑗) ≜ Θ(𝑗)W, Θ(1) is the 𝑚 × 𝑚
identity matrix, and, for 𝑗 ∕= 1, Θ(𝑗) is an 𝑚𝑗 × 𝑚 matrix
whose rows are orthonormal vectors in the left null space of
H[1 : 𝑚][1 : 𝑖𝑗−1]. ⎡ ⎢ ⎣ ˆ X(1) .. . ˆ X(𝐵) ⎤ ⎥ ⎦ is a suboptimal solution of (2). Here we assume ˆX(𝑗) is found by an ML decoder. In practice,
practical algorithms such as sphere decoding [9]–[13] can be used for finding ˆX(𝑗).
The number of codewords in 𝒞 is given by 2𝑅, i.e. 𝑅
denotes the rate of 𝒞, and 𝒞 can be designed so that 𝑅
increases with SNR. Let𝒞𝑗denote the codebook forX(𝑗) and
let 𝑅𝑗 denote the rate of 𝒞𝑗, where 𝑅𝑗 increases with SNR.
Here, 𝒞 is the Cartesian product of component codebooks
𝒞𝑗, 1 ≤ 𝑗 ≤ 𝐵 and 𝑅 =∑𝑅𝑗.
C. Groupwise Successive Interference Cancelation Combined With Antenna Selection
GSIC with the grouping index set 𝐼 in subsection II-B
implies that all the 𝑛 transmit antennas are selected and grouped into 𝐵 groups and the data rate for each X(𝑗) is positive. In the low multiplexing gain regime, transmitting using only a subset of transmit antennas can result in a better diversity gain than using all the transmit antennas as will be shown in subsection IV-B. In GSICAS, a subset of𝑛 transmit antennas is selected for transmission while the other antennas are silent. The grouping index set𝐼 for GSICAS is represented by (𝑛1, 𝑛2, . . . , 𝑛𝐵) with ∑𝐵𝑗=1𝑛𝑗 = 𝛼 ≤ 𝑛 which means
that among𝑛 transmit antennas only 𝛼 antennas are selected, the 𝛼 antennas are grouped into 𝐵 groups, and the number of transmit antennas in the 𝑗 th group is 𝑛𝑗. X(𝑗) denotes
the codeword transmitted by the 𝑗 th group and the decoding process is the same as that described in subsection II-B.
III. DIVERSITY ANDMULTIPLEXINGGAINSUNDER A
FIXEDGROUPINGINDEXSET
Let 𝑃𝑒𝑟𝑟 denote the error probability of the GSIC with
a grouping index set 𝐼. Then the diversity and multiplexing gains are defined as in [14]:
Definition 1: 𝒞 is said to achieve the multiplexing gain 𝑟 and diversity gain𝑑𝐼 if
lim SNR→∞ 𝑅 log SNR ≥ 𝑟 and lim SNR→∞ log 𝑃𝑒𝑟𝑟(SNR) log SNR ≤ −𝑑𝐼.
For each𝑟, we define 𝑑∗
𝐼(𝑟) as the supremum of the diversity
gain achieved over all transmit schemes under the GSIC with𝐼 when limSNR→∞log SNR𝑅 = 𝑟. Conversely, for each diversity
gain 𝑑, 𝑟∗
𝐼(𝑑) is the supremum of the multiplexing gain
achieved over all transmit schemes attaining diversity gain𝑑 under the GSIC with𝐼. When 𝐼 = (𝑛), i.e., a set containing only𝑛, then the decoding with 𝐼 is an ML decoding, and in this case we use notations𝑑, 𝑑∗, and𝑟∗ instead of𝑑𝐼,𝑑∗
𝐼, and
𝑟∗
𝐼 respectively.
The optimal DMT 𝑑∗(𝑟) under the ML decoding is known
as follows.
Theorem 1: [1], [5] If the fading coefficients h𝑖𝑗 are
i.i.d. circularly symmetric complex normal variables and 𝑙
is assumed to be 𝑙 ≥ 𝑛, then 𝑑∗(𝑟) is the
piecewise-linear function connecting the integer points(𝑘, 𝑑∗(𝑘)), 𝑘 =
0, 1, . . . , min{𝑚, 𝑛}, where
𝑑∗(𝑘) = (𝑚 − 𝑘)(𝑛 − 𝑘). (3)
In particular the maximal diversity gain,𝑑∗
max, is𝑚𝑛 and the
maximal multiplexing gain,𝑟∗
max, ismin{𝑚, 𝑛}.
In [1] the achievability part of Theorem 1 is verified for𝑙 ≥ 𝑚 + 𝑛 − 1 and in [5] for 𝑙 ≥ 𝑛.
Actually𝑑∗(𝑟) can be represented in two alternative forms
which are easy to handle.
Proposition 1: If the channel is the i.i.d. Rayleigh fading and𝑙 ≥ 𝑛 then for 0 ≤ 𝑟 ≤ min{𝑚, 𝑛}
𝑑∗(𝑟) = (𝑚 − ⌊𝑟⌋)(𝑛 − ⌊𝑟⌋) − ¯𝑟(𝑚 + 𝑛 − 2⌊𝑟⌋ − 1) (4) and
𝑑∗(𝑟) = (𝑚 − 𝑟)(𝑛 − 𝑟) + ¯𝑟 − ¯𝑟2. (5)
where⌊𝑟⌋ is the largest integer not greater than 𝑟 and 0 ≤ ¯𝑟 = 𝑟 − ⌊𝑟⌋ < 1. For each diversity gain 𝑑, the largest achievable multiplexing gain𝑟∗(𝑑) is
𝑟∗(𝑑) = 𝑟′(𝑑) +
(
𝑚 − 𝑟′(𝑑))(𝑛 − 𝑟′(𝑑))− 𝑑
𝑚 + 𝑛 − 2𝑟′(𝑑) − 1 (6)
where 𝑟′(𝑑) is the largest integer 𝑘 such that 0 ≤ 𝑘 ≤
min{𝑚, 𝑛} and (𝑚 − 𝑘)(𝑛 − 𝑘) ≥ 𝑑.
Proof: (4) is the formula described in Theorem 1. (5) and (6) are derived directly from (4).
We use notations𝑑∗(𝑟; 𝑚, 𝑛) and 𝑟∗(𝑑; 𝑚, 𝑛) instead of 𝑑∗(𝑟)
and𝑟∗(𝑑) respectively when 𝑚 and 𝑛 are specified and H is
the i.i.d. Rayleigh fading channel.
Let 𝑟𝑗 denote the multiplexing gain achieved by 𝒞𝑗. If
limSNR→∞log SNR𝑅 = 𝑟 and limSNR→∞log SNR𝑅𝑗 = 𝑟𝑗 for
every𝑗, then we have 𝑟 =∑𝐵𝑗=1𝑟𝑗. We have the following
definitions for further development.
Definition 2: 𝑃𝑗 is defined to be the error probability of X(𝑗) on the condition that X(𝑘) are decoded correctly for
𝑗 + 1 ≤ 𝑘 ≤ 𝐵, i.e.
𝑃𝑗 = 𝑃 ( ˆX(𝑗) ∕= X(𝑗)∣ ˆX(𝑘) = X(𝑘) for 𝑗 + 1 ≤ 𝑘 ≤ 𝐵),
andX(𝑗) is said to achieve diversity gain 𝑑𝑗 when
lim
SNR→∞
log 𝑃𝑗
log SNR ≤ −𝑑𝑗.
𝑑𝑗(𝑟𝑗) denotes an achievable diversity gain of X(𝑗) when
limSNR→∞log SNR𝑅𝑗 = 𝑟𝑗, and𝑑∗𝑗(𝑟𝑗) denotes the supremum
of𝑑𝑗(𝑟𝑗). ⎛ ⎜ ⎝ X(𝑗) .. . X(𝑘) ⎞ ⎟ ⎠ is simply denoted by X(𝑗, . . . , 𝑘).
The optimal DMT under the grouping with 𝐼 is known as
follows.
Theorem 2: [7] If limSNR→∞log SNR𝑅𝑗 = 𝑟𝑗 and
limSNR→∞log SNRlog 𝑃𝑗 ≤ −𝑑𝑗 for every 1 ≤ 𝑗 ≤ 𝐵 then
we have
(i) The multiplexing gain of X(𝑗, . . . , 𝐵) is ∑𝐵𝑘=𝑗𝑟𝑘. X(𝑗, . . . , 𝐵) achieves the diversity gain of min𝑗≤𝑘≤𝐵𝑑𝑘(𝑟𝑘), i.e. lim SNR→∞ log 𝑃 ( ˆX(𝑗, . . . , 𝐵) ∕= X(𝑗, . . . , 𝐵)) log SNR ≤ −(min𝑗≤𝑘≤𝐵𝑑𝑘(𝑟𝑘)) (7)
and the supremum of its diversity gain is
min𝑗≤𝑘≤𝐵𝑑∗𝑘(𝑟𝑘) when 𝑟𝑗, . . . , 𝑟𝐵 are fixed.
(ii) X achieves the diversity gain of min1≤𝑗≤𝐵𝑑𝑗(𝑟𝑗) and
𝑑∗ 𝐼(𝑟) =∑sup 𝐵 𝑗=1𝑟𝑗=𝑟 { min 1≤𝑗≤𝐵𝑑 ∗ 𝑗(𝑟𝑗) } . (8)
Theorem 2 holds for any channel distribution but if the channel coefficients follow i.i.d. Rayleigh fading then 𝑑∗
𝐼(𝑟) has the
following form.
Corollary 1: IfH is the i.i.d. Rayleigh fading channel and
𝑙 ≥ max𝑗{𝑖𝑗− 𝑖𝑗−1} then, for each 1 ≤ 𝑗 ≤ 𝐵, for 0 ≤ 𝑟𝑗≤
𝑛𝑗
𝑑∗
𝑗(𝑟𝑗) = 𝑑∗(𝑟𝑗; 𝑚𝑗, 𝑛𝑗) = (𝑚𝑗− 𝑟𝑗)(𝑛𝑗− 𝑟𝑗) + ¯𝑟𝑗− ¯𝑟𝑗2 (9)
where ¯𝑟𝑗≜ 𝑟𝑗− ⌊𝑟𝑗⌋, and therefore
𝑑∗𝐼(𝑟) =∑𝐵max 𝑗=1𝑟𝑗=𝑟 { min 1≤𝑗≤𝐵 ( (𝑚𝑗− 𝑟𝑗)(𝑛𝑗− 𝑟𝑗) + ¯𝑟𝑗− ¯𝑟2𝑗 )} . (10) Proof: 𝑑∗ 𝑗(𝑟𝑗) = 𝑑∗(𝑟𝑗; 𝑚𝑗, 𝑛𝑗) = (𝑚𝑗− 𝑟𝑗)(𝑛𝑗− 𝑟𝑗) + ¯𝑟𝑗− ¯𝑟2𝑗 if𝑙 ≥ 𝑖𝑗− 𝑖𝑗−1, by Proposition 1. (10) is obtained by
(9) and Theorem 2 (ii).
We can obtain (10) by the the following process. For a fixed index set𝐼, 𝑑∗
𝐼(𝑟) is the piecewise-linear curve connecting the
points(𝐹𝑖, 𝐺𝑖), 0 ≤ 𝑖 ≤ 𝑀 such that 𝐹0= 0, 𝐺0= min 1≤𝑗≤𝐵𝑚𝑗𝑛𝑗 𝐹1= 𝐵 ∑ 𝑗=1 𝑠1𝑗, 𝐺1= 𝐺0 where𝑠1𝑗= 𝑟∗(𝐺1; 𝑚𝑗, 𝑛𝑗) 𝒟 = {𝐺2, 𝐺3, . . . , 𝐺𝑀} = {(𝑚𝑗− 𝑎𝑗)(𝑛𝑗− 𝑎𝑗) : 1 ≤ 𝑗 ≤ 𝐵, ⌈𝑠1𝑗⌉ ≤ 𝑎𝑗≤ 𝑛𝑗, 𝑎𝑗∈ ℤ} 𝐹𝑖= 𝐵 ∑ 𝑗=1 𝑠𝑖𝑗 with𝑠𝑖𝑗= 𝑟∗(𝐺𝑖; 𝑚𝑗, 𝑛𝑗) for 2 ≤ 𝑖 ≤ 𝑀 (11) where 𝒟 is arranged in decreasing order, i.e., 𝐺2 > ⋅ ⋅ ⋅ >
𝐺𝑀 = 0, 𝑀 = ∣𝒟∣ + 1, and 𝑠𝑖𝑗 is calculated by (6) as𝑠𝑖𝑗=
𝑘𝑖𝑗 + (𝑚𝑗𝑚−𝑘𝑗+𝑛𝑖𝑗)(𝑛𝑗−2𝑘𝑗−𝑘𝑖𝑗𝑖𝑗−1)−𝐺𝑖 with the largest integer 𝑘𝑖𝑗 such
Here,⌈𝑠1𝑗⌉ is the smallest integer greater than or equal to 𝑠1𝑗.
For a multiplexing gain𝑟, the multiplexing gain vector
r∗ (𝐼,𝑟)≜ argr 𝑑∗𝐼(𝑟) = arg r {r:∑max𝑗𝑟𝑗=𝑟}min𝑗 {(𝑚𝑗− 𝑟𝑗)(𝑛𝑗− 𝑟𝑗) + ¯𝑟𝑗− ¯𝑟 2 𝑗} (12) and𝑑∗
𝐼(𝑟) are calculated by the following simple process (P):
(1) if 𝑟 = 0 then 𝑑∗ 𝐼(𝑟) = 𝐺0, r∗(𝐼,𝑟) = (0, . . . , 0); and if 𝑟 = 𝑛 then 𝑑∗ 𝐼(𝑟) = 0, r∗(𝐼,𝑟)= (𝑛𝑗)𝐵𝑗=1 (2) if0 < 𝑟 < 𝑛, we have 𝑑∗ 𝐼(𝑟) = 𝐺𝐹𝑖− 𝐺𝑖−1 𝑖− 𝐹𝑖−1(𝑟 − 𝐹𝑖−1) + 𝐺𝑖−1 r∗ (𝐼,𝑟)= ( 𝑟∗(𝑑∗ 𝐼(𝑟); 𝑚𝑗, 𝑛𝑗)) 𝐵 𝑗=1. (13) where𝑖 ≥ 1 is the smallest integer such that 𝐹𝑖≥ 𝑟.
IV. DIVERSITY ANDMULTIPLEXINGGAINSWITH
GSICAS
As seen in the previous section, the DMT curve under a GSICAS depends on its grouping index set𝐼 = (𝑛1, . . . , 𝑛𝐵).
If the maximum group size of 𝐼, i.e., max1≤𝑗≤𝐵𝑛𝑗 is 1
then the corresponding decoder to 𝐼 is the plain
antenna-by-antenna SIC, and if the maximum group size of 𝐼 is 𝑛
then the corresponding decoder includes the ML decoding. In general, for a given bound on the maximum group size, one can try to find the best choices for selecting antennas, grouping antennas, and allocating rates across antenna groups for maximizing the diversity gain at a given multiplexing gain. Let 𝑐(𝐼) = max𝑗{𝑛𝑗}. Let ℐ(𝑐, 𝑛) ≜ {𝐼 = (𝑛1, . . . , 𝑛𝐵) :
𝑐(𝐼) ≤ 𝑐,∑𝐵𝑗=1𝑛𝑗 ≤ 𝑛} and for each 𝐼 = (𝑛1, . . . , 𝑛𝐵) let
𝐴(𝑟, 𝐼) ≜ {r = (𝑟1, . . . , 𝑟𝐵) : ∑𝑗𝑟𝑗 = 𝑟, 𝑟𝑗 ≤ 𝑛𝑗 for𝑗 =
1, . . . , 𝐵}. Then the optimal GDMT of system (1) under the GSICAS, denoted by 𝑑∗(𝑟, 𝑐), is the supremum of diversity
gain on the condition that the multiplexing gain of the transmit scheme is𝑟 and the constraint of group-sizes is less than or equal to𝑐, i.e. we have
𝑑∗(𝑟, 𝑐) = max 𝐼∈ℐ(𝑐,𝑛)𝑑 ∗ 𝐼(𝑟). (14) Here 𝑑∗ 𝐼(𝑟) = supr∈𝐴(𝑟,𝐼)𝑑∗𝐼(r) where 𝑑∗𝐼(r) ≜
min1≤𝑗≤𝐵𝑑∗𝑗(𝑟𝑗). To obtain 𝑑∗(𝑟, 𝑐), the optimal grouping
index set, denoted by 𝐼(𝑟, 𝑐), and the optimal multiplexing gain vector r∗(𝑟, 𝑐) should be found for each pair (𝑟, 𝑐).
This is a complicated problem because the cardinality of ℐ(𝑐, 𝑛) increases rapidly with 𝑛 and 𝑐, and moreover the multiplexing gain vectorr maximizing the diversity gain for
each𝐼 should be found.
ℐ(𝑐, 𝑛) is the disjoint union of 𝒥 (𝑐, 𝛼) ≜ {𝐼 = (𝑛1, . . . , 𝑛𝐵) : 𝑐(𝐼) ≤ 𝑐,∑𝐵𝑗=1𝑛𝑗= 𝛼} where 𝛼 = 1, . . . , 𝑛.
Note that𝒥 (𝑐, 𝛼) is never empty. Therefore, if we let ¯ 𝑑∗(𝑟, 𝑐; 𝛼) ≜ max 𝐼∈𝒥 (𝑐,𝛼)𝑑 ∗ 𝐼(𝑟) (15) then 𝑑∗(𝑟, 𝑐) = max 1≤𝛼≤𝑛𝑑¯ ∗(𝑟, 𝑐; 𝛼). (16)
Here, ¯𝑑∗(𝑟, 𝑐; 𝛼) is the optimal GDMT under the GSIC when
the system has𝛼 transmit and 𝑚 receive antennas.
A. Optimal Tradeoff of Group-size Constraint, Diversity and Multiplexing Gains Under the GSIC
The optimal GDMT under the GSIC, ¯𝑑∗(𝑟, 𝑐; 𝑛), is studied
in this subsection. For any(𝑟, 𝑐), we find a very limited subset of𝒥 (𝑐, 𝑛) which includes the optimal grouping index set to get ¯𝑑∗(𝑟, 𝑐; 𝑛) when H is the i.i.d. Rayleigh fading channel.
The optimal multiplexing gain vector to obtain𝑑∗
𝐼(𝑟) for each
pair of (𝐼, 𝑟) is obtained analytically, and thus the optimal multiplexing gain vector achieving ¯𝑑∗(𝑟, 𝑐; 𝑛) can be found.
If the channel is the i.i.d. Rayleigh fading and𝑙 ≥ 𝑛 then
¯ 𝑑∗(𝑟, 𝑐; 𝑛) = max 𝐼∈𝒥 (𝑐,𝑛)r∈𝐴(𝑟,𝐼)max 1≤𝑗≤𝐵min {(𝑚𝑗−𝑟𝑗)(𝑛𝑗−𝑟𝑗)+ ¯𝑟𝑗− ¯𝑟 2 𝑗} (17) where 𝐵, 𝑚𝑗, 𝑛𝑗, and 𝑟𝑗 depend on 𝐼. To obtain ¯𝑑∗(𝑟, 𝑐; 𝑛),
according to (17),𝑑∗
𝐼(𝑟) = maxr∈𝐴(𝑟,𝐼)
min1≤𝑗≤𝐵{(𝑚𝑗−𝑟𝑗)(𝑛𝑗−𝑟𝑗)+ ¯𝑟𝑗− ¯𝑟𝑗2} should be calculated
for all 𝐼 ∈ 𝒥 (𝑐, 𝑛) and be compared each other. For each 𝐼, 𝑑∗
𝐼(𝑟) can be obtained by the process below Corollary
1. Moreover, computing 𝑑∗
𝐼(𝑟) and r∗(𝐼,𝑟) of 𝐼’s in only a
small subset of𝒥 (𝑐, 𝑛) is sufficient to find ¯𝑑∗(𝑟, 𝑐; 𝑛) and the
corresponding multiplexing gain vectorr∗. This is proved in
the following. The optimal groupings for𝑐 = 1 and 𝑐 = 𝑛 are trivially(1, . . . , 1) and (𝑛) respectively. Therefore we assume that1 < 𝑐 < 𝑛.
First, the optimal DMT curve 𝑑∗(𝑟; 𝑚, 𝑛) satisfies the
following property.
Lemma 1: If 𝑀1≥ 𝑀2+ 𝑘 and 𝑁1≥ 𝑁2+ 𝑘 where 𝑘 is
a positive integer then for0 ≤ 𝑟 ≤ min{𝑀2, 𝑁2}
𝑑∗(𝑟 + 𝑘; 𝑀
1, 𝑁1) ≥ 𝑑∗(𝑟; 𝑀2, 𝑁2). (18)
The equality holds iff 𝑀1= 𝑀2+ 𝑘 and 𝑁1= 𝑁2+ 𝑘.
Proof: From Proposition 1 𝑑∗(𝑟 + 𝑘; 𝑀
1, 𝑁1) = (𝑀1− 𝑟 − 𝑘)(𝑁1− 𝑟 − 𝑘) + ¯𝑟 − ¯𝑟2
≥ 𝑑∗(𝑟; 𝑀 2, 𝑁2).
From Lemma 1, we have the following proposition which says that, to maximize DMT or GDMT, a component code transmitted by a group of more transmit antennas should have a higher priority in decoding order, that is, 𝑛𝑗 is
non-decreasing.
Proposition 2: For any 𝐵 ≥ 2 and 𝑛1 ≤ 𝑛2 ≤ ⋅ ⋅ ⋅ ≤
𝑛𝐵, let ℐ be the collection of the grouping index sets such
that 𝑛 transmit antennas are divided into 𝐵 groups having 𝑛1, 𝑛2, . . . , 𝑛𝐵 transmit antennas then 𝐼 = (𝑛1, 𝑛2, . . . , 𝑛𝐵)
has the maximal DMT amongℐ.
Proof: See Appendix A.
The groupings inℐ have the same group-size distribution and (𝑛1, 𝑛2, . . . , 𝑛𝐵) has the maximum GDMT among ℐ.
Proposition 3: For every 1 < 𝑐 < 𝑛, if 𝐼 = (𝑛1, 𝑛2, . . . , 𝑛𝐵) is a grouping index set such that 𝑐(𝐼) ≤ 𝑐
and 𝐵 > ⌈𝑛 𝑐
⌉
then there exists a grouping index set 𝐽 = (𝑢1, 𝑢2, . . . , 𝑢𝐾) satisfying 𝐾 =⌈𝑛 𝑐 ⌉ , 𝑐(𝐽) ≤ 𝑐, and 𝑑∗ 𝐽(𝑟) ≥ 𝑑∗𝐼(𝑟) for 0 ≤ 𝑟 ≤ 𝑛.
Proof: See Appendix B.
From Proposition 3, for a group-size constraint𝑐, the number of antenna groups of the grouping index set achieving the optimal GDMT under the GSIC is always equal to⌈𝑛𝑐⌉. Let ℱ(𝑐, 𝑛) denote the collection of the grouping index sets 𝐼 such that𝑐(𝐼) ≤ 𝑐, 𝐵 =⌈𝑛
𝑐
⌉
and𝑛𝑗 is non-decreasing, i.e.
ℱ(𝑐, 𝑛) ≜ {𝐼 = (𝑛1, . . . , 𝑛⌈𝑛 𝑐⌉) : max 1≤𝑗≤⌈𝑛 𝑐⌉ 𝑛𝑗≤ 𝑐, ⌈𝑛 𝑐⌉ ∑ 𝑗=1 𝑛𝑗= 𝑛, 𝑛1≤ ⋅ ⋅ ⋅ ≤ 𝑛⌈𝑛 𝑐⌉}.
Then from Propositions 2 and 3, for each pair of multiplexing gain𝑟 and group-size limit 𝑐, the optimal grouping index set achieving ¯𝑑∗(𝑟, 𝑐; 𝑛) is in ℱ(𝑐, 𝑛). Theorem 3: ¯ 𝑑∗(𝑟, 𝑐; 𝑛) = max 𝐼∈ℱ(𝑐,𝑛)𝑑 ∗ 𝐼(𝑟) (19)
and if 𝐼∗ is an optimal grouping index set found from (19)
then an optimal multiplexing gain vectorr∗ is r∗=(𝑟∗(𝑑∗
𝐼∗(𝑟); 𝑚𝑗, 𝑛𝑗))
𝑗 (20)
where𝑚𝑗, 𝑛𝑗 are of𝐼∗.
In (19),ℱ(𝑐, 𝑛) is very limited and has only a few members. For example, when𝑚 = 𝑛 = 10, for all 1 ≤ 𝑐 ≤ 10, ℱ(𝑐, 𝑛) are directly followed as
ℱ(10, 10) = (10), ℱ(9, 10) = {(𝑖, 10 − 𝑖) : 𝑖 = 1, 2, 3, 4, 5} ℱ(8, 10) = {(𝑖, 10 − 𝑖) : 𝑖 = 2, 3, 4, 5}, ℱ(7, 10) = {(𝑖, 10 − 𝑖) : 𝑖 = 3, 4, 5} ℱ(6, 10) = {(𝑖, 10 − 𝑖) : 𝑖 = 4, 5}, ℱ(5, 10) = {(5, 5)} ℱ(4, 10) = {(2, 4, 4), (3, 3, 4)}, ℱ(3, 10) = {(1, 3, 3, 3), (2, 2, 3, 3)} ℱ(2, 10) = {(2, 2, 2, 2, 2)}, ℱ(1, 10) = {(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)}.
To find ¯𝑑∗(𝑟, 𝑐 = 6; 10), the corresponding index set 𝐼∗, and
multiplexing gain vectorr∗, comparing DMT over only two
index sets is enough from Theorem 3, instead of comparing DMT over 412 index sets. As for ¯𝑑∗(𝑟, 𝑐 = 3; 10), instead of
194 index sets, comparing only two index sets is enough.
When 𝑐 ≥ 𝑛 2,ℱ(𝑐, 𝑛) = { (𝑖, 𝑛 − 𝑖) : 𝑛 − 𝑐 ≤ 𝑖 ≤ 𝑛 2 } and the grouping achieving the optimal GDMT is determined for each 𝑟 by a formula derived in the following. For 𝑟 = 0, if we compare only𝑑∗
𝐼(0)’s among 𝐼 having only two groups,
then according to (10)(𝑖∗, 𝑛 − 𝑖∗) where
𝑖∗≜ arg 1≤𝑖≤𝑛max{𝑑 ∗ (𝑖,𝑛−𝑖)(0)} = arg 1≤𝑖≤𝑛min{∣𝑚𝑖 − (𝑚 − 𝑖)(𝑛 − 𝑖)∣}
has the maximum value of 𝑑∗
𝐼(0). We get
𝑖∗ = ⌈2𝑚+𝑛−√4𝑚2+𝑛2
2 ⌋, which is the closest integer to 2𝑚+𝑛−√4𝑚2+𝑛2
2 . Furthermore,(𝑖∗, 𝑛 − 𝑖∗) has the following
property.
Proposition 4: We assume that𝑛 > 1. For all 𝑛 > 𝑖 > 𝑖∗,
𝑑∗ (𝑖,𝑛−𝑖)(𝑟) ≤ 𝑑∗(𝑖∗,𝑛−𝑖∗)(𝑟) for 0 ≤ 𝑟 ≤ 𝑛. (21) If𝑖∗≤ 𝑖1< 𝑖2< 𝑛 then 𝑑∗ (𝑖1,𝑛−𝑖1)(𝑟) ≥ 𝑑 ∗ (𝑖2,𝑛−𝑖2)(𝑟) for 0 ≤ 𝑟 ≤ 𝑛. (22) Proof: See Appendix C.
Notice that𝑖∗≤ ⌈𝑛
2⌋, and if we think about the optimal DMT
curve of (1) under GSIC’s with {(𝑖, 𝑛 − 𝑖) : 1 ≤ 𝑖 < 𝑛} only it is sufficient to consider the index sets (𝑖, 𝑛 − 𝑖) with 𝑖 ≤ 𝑖∗ from Propositions 2 and 4. Let 𝐼2
𝑟 be the index set
grouping the transmit antennas into two groups that achieves max1≤𝑖≤𝑛𝑑∗(𝑖,𝑛−𝑖)(𝑟). Then 𝐼𝑟2 varies according to 𝑟, and 𝐼𝑟2
and𝑑∗ 𝐼2
𝑟(𝑟) are derived in the following proposition.
Proposition 5: Let𝑎(𝑖) be defined for 0 ≤ 𝑖 ≤ 𝑖∗ as
𝑎(𝑖) ≜ ⎧ ⎨ ⎩ 0 if𝑖 = 𝑖∗ 𝑠1(𝑖) + 𝑠2(𝑖) if1 ≤ 𝑖 ≤ 𝑖∗− 1 𝑛 if𝑖 = 0
where𝑠1(𝑖) and 𝑠2(𝑖) are the multiplexing gains such that
𝑑∗(𝑠
1(𝑖); 𝑚, 𝑖+1) = 𝑚𝑖, 𝑑∗(𝑠2(𝑖); 𝑚−𝑖−1, 𝑛−𝑖−1) = 𝑚𝑖.
Then we have 𝐼2
𝑟 = (𝑖, 𝑛 − 𝑖) for 𝑎(𝑖) ≤ 𝑟 ≤ 𝑎(𝑖 − 1). (23)
Proof: See Appendix D. Thus, we have𝑑∗ 𝐼2 𝑟(𝑟) = 𝑑 ∗ (𝑖,𝑛−𝑖)(𝑟) for 𝑎(𝑖) ≤ 𝑟 ≤ 𝑎(𝑖 − 1), and𝑑∗ 𝐼2 𝑟(𝑟) ≥ 𝑑 ∗
𝐼(𝑟) for 0 ≤ 𝑟 ≤ 𝑛 for any 𝐼 ∈ {(𝑛1, 𝑛−𝑛1) :
1 ≤ 𝑛1 ≤ 𝑛}. In Proposition 5, 𝑎(𝑖)’s are easily calculated
for given 𝑚 and 𝑛 and hence 𝑑∗
𝐼2
𝑟(𝑟) is easily derived. In
particular, when 𝑚 = 𝑛, 𝑎(𝑖)’s are given as 𝑎(𝑖) = 𝑚 − 𝑖 − 𝑖 𝑚 + 𝑖 − ⌈√ 𝑚𝑖⌉+ ⌈√ 𝑚𝑖⌉2− 𝑚𝑖 2⌈√𝑚𝑖⌉− 1 for1 ≤ 𝑖 ≤ 𝑖∗− 1. Now, from Propositions 4 and 5, we have
the following result. Corollary 2: If 𝑛 2 ≤ 𝑐 ≤ 𝑛 − 𝑖∗ then 𝐼∗= (𝑛 − 𝑐, 𝑐) for all 0 ≤ 𝑟 ≤ 𝑛; and if𝑛 − 𝑖∗< 𝑐 < 𝑛 then 𝐼∗= { (𝑗, 𝑛 − 𝑗) if 𝑖∗≥ 𝑗 > (𝑛 − 𝑐) and 𝑎(𝑗) ≤ 𝑟 ≤ 𝑎(𝑗 − 1) (𝑛 − 𝑐, 𝑐) if 𝑟 > 𝑎(𝑛 − 𝑐). ¯
𝑑∗(𝑟, 𝑐; 𝑛) and r∗ are calculated as𝑑∗
𝐼∗(𝑟) and r∗(𝐼∗,𝑟)
respec-tively by process (P).
Therefore, when𝑚 = 𝑛 = 10, for all 0 ≤ 𝑟 ≤ 10 𝐼∗= (4, 6)
for𝑐 = 6 and 𝐼∗ = (5, 5) for 𝑐 = 5. Then 𝑑∗
𝐼∗(𝑟) and r∗(𝐼∗,𝑟)
are computed by process (P). If 𝑐 = 7 then 𝐼∗= { (4, 6) if 0 ≤ 𝑟 ≤ 188 143 (3, 7) if 188 143 < 𝑟 ≤ 10 and𝑑∗
𝒢𝛼≜ {{ (𝑛1, . . . , 𝑛⌈𝛼 𝑐⌉) ∈ ℱ(𝑐, 𝛼) : 𝑛⌈𝛼𝑐⌉= 𝑐, 𝑚𝑛1> (𝑚 − (⌈ 𝛼 𝑐⌉ − 1)𝑐)𝑐 } if 𝑐 < 𝛼 ≤⌊𝑛 𝑐 ⌋ 𝑐 ℱ(𝑐, 𝛼) if 𝛼 = 𝑐 or⌊𝑛 𝑐 ⌋ 𝑐 < 𝛼 ≤ 𝑛 (24)
B. Optimal Tradeoff for Group-size Constraint, Diversity and Multiplexing Gains Under the GSIC Combined With Antenna Selection
GSIC combined with antenna selection has more flexibility and is better in GDMT than GSIC, which is presented below and descriptively in examples. The optimal GDMT under the GSICAS,𝑑∗(𝑟, 𝑐), is obtained systematically by applying
The-orem 3. It is assumed that𝑛 ≥ 𝑐, since if 𝑛 < 𝑐 then obviously (𝑛) is the optimal grouping and 𝑑∗(𝑟, 𝑐) = 𝑑∗(𝑟; 𝑚, 𝑛). We
define𝒢𝛼for𝑐 ≤ 𝛼 ≤ 𝑛 as in (24). We also define 𝒢(𝑐, 𝑛) as
𝒢(𝑐, 𝑛) ≜ ∪
𝑐≤𝛼≤𝑛
𝒢𝛼. (25)
The following shows the main result in this section, which says that we only need to consider𝐼(𝑟, 𝑐) in 𝒢(𝑐, 𝑛) for each (𝑟, 𝑐) and that the optimal multiplexing gain vector is simply given by that achieved at𝑑∗
𝐼(𝑟,𝑐)(𝑟).
Theorem 4: The optimal GDMT under the GSICAS is given by
𝑑∗(𝑟, 𝑐) = max 𝐼∈𝒢(𝑐,𝑛)𝑑
∗
𝐼(𝑟) (26)
and the optimal multiplexing gain vectorr∗(𝑟, 𝑐) is given by r∗(𝑟, 𝑐) =(𝑟∗(𝑑∗
𝐼(𝑟,𝑐)(𝑟); 𝑚𝑗, 𝑛𝑗))
𝑗 (27)
where𝑚𝑗 and𝑛𝑗 are of𝐼(𝑟, 𝑐).
Proof: From Theorem 3, 𝑑∗(𝑟, 𝑐) = max 1≤𝛼≤𝑛𝐼∈ℱ(𝑐,𝛼)max 𝑑 ∗ 𝐼(𝑟). (28) If 𝛼 ≤ 𝑐 then ℱ(𝑐, 𝛼) = {(𝛼)} and 𝑑∗ (𝛼)(𝑟) = 𝑑∗(𝑟; 𝑚, 𝛼) ≤ 𝑑∗ (𝑐)(𝑟). If 𝑐 < 𝛼 ≤ ⌊𝑛𝑐⌋𝑐 then for any (𝑛1, . . . , 𝑛⌈𝛼 𝑐⌉) ∈ ℱ(𝑐, 𝛼) such that 𝑛⌈𝛼𝑐⌉ < 𝑐
we can see that 𝑑∗
(𝑛1,...,𝑛⌈ 𝛼𝑐 ⌉−1,𝑐)(𝑟) ≥ 𝑑(𝑛1,...,𝑛⌈ 𝛼𝑐 ⌉)(𝑟) and (𝑛1, . . . , 𝑛⌈𝛼 𝑐⌉−1, 𝑐) ∈ ℱ(𝑐, ∑⌈𝛼 𝑐⌉−1 𝑗=1 𝑛𝑗 + 𝑐). If (𝑛1, . . . , 𝑛⌈𝛼 𝑐⌉) ∈ ℱ(𝑐, 𝛼), 𝑚𝑛1 ≤ (𝑚 − (⌈𝛼𝑐⌉ − 1)𝑐)𝑐, and 𝑐 < 𝛼 ≤ ⌊𝑛 𝑐 ⌋ 𝑐 then 𝐼 = (𝑐, . . . , 𝑐) *+ , ⌈𝛼 𝑐⌉ 𝑐’s ) has a better DMT than (𝑛1, . . . , 𝑛⌈𝛼 𝑐⌉) i.e. 𝑑 ∗ 𝐼(𝑟) ≥ 𝑑∗(𝑛1,...,𝑛⌈ 𝛼𝑐 ⌉)(𝑟);
because for any multiplexing gain vector (𝑟1, . . . , 𝑟⌈𝛼 𝑐⌉) of
grouping (𝑛1, . . . , 𝑛⌈𝛼
𝑐⌉) the multiplexing gain allocation
(𝑟2, . . . , 𝑟⌈𝛼
𝑐⌉, 𝑟1) for 𝐼 has larger diversity gain. In detail,
𝑚 + 𝑛1 > 𝑚 − (⌈𝛼𝑐⌉ − 1)𝑐 + 𝑐 and we have from (5)
𝑑∗(𝑟
1; 𝑚, 𝑛1) ≤ 𝑑∗(𝑟1; 𝑚 − (⌈𝛼𝑐⌉ − 1)𝑐, 𝑐) for any 0 ≤ 𝑟1 ≤
𝑛1. For𝑘 = 1, . . . , ⌈𝛼𝑐⌉ − 1 we have∑𝑘𝑗=1𝑛𝑗 ≥ (𝑘 − 1)𝑐 and
thus𝑑∗(𝑟𝑘+1; 𝑚−∑𝑘
𝑗=1𝑛𝑗, 𝑛𝑘+1) ≤ 𝑑∗(𝑟𝑘+1; 𝑚−(𝑘−1)𝑐, 𝑐)
for any0 ≤ 𝑟𝑘+1≤ 𝑛𝑘+1.
ℐ(𝑐, 𝑛) is a very large set and there are many groupings to compare their DMT’s to obtain (14), but𝒢(𝑐, 𝑛) is very limited and the process to obtain the optimal GDMT becomes much simpler. Since each 𝑑∗
𝐼(𝑟) is obtained analytically by (11)
and comparison among𝑑∗
𝐼(𝑟)’s can be done by using process
(P), 𝑑∗(𝑟, 𝑐) can be obtained analytically. From Theorem 4,
0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6
Spatial multiplexing gain r=R/log SNR
Diversity gain d \bar{d}^*(r,1;2) d*(r,1) \bar{d}^*(r,2;3) d*(r,2) (1, 2) (1/2, 1)
Fig. 1. 𝑑∗(𝑟, 1) and ¯𝑑∗(𝑟, 1; 2) for MIMO Rayleigh fading channel with
𝑚 = 𝑛 = 2. 𝑑∗(𝑟, 2) and ¯𝑑∗(𝑟, 2; 3) for the MIMO Rayleigh fading channel with𝑚 = 𝑛 = 3. 𝑑∗(𝑟, 1) is the line connecting points (0, 1), (1
2, 1), (2, 0)
and the curve of ¯𝑑∗(𝑟, 1; 2) is the line connecting points (0, 2), (1
2, 1), (2, 0);
𝑑∗(𝑟, 2) is the line with (0, 3), (1/3, 3), (1 + 2/3, 1), (3, 0) and ¯𝑑∗(𝑟, 2; 3) is the line with points(0, 6), (1, 2), (1 + 2/3, 1), (3, 0).
for each (𝑟, 𝑐) the data rate for the 𝑗 th antenna group of 𝐼(𝑟, 𝑐) can be determined as 𝑟∗(𝑑∗
𝐼(𝑟,𝑐)(𝑟); 𝑚𝑗, 𝑛𝑗)log SNR
to achieve𝑑∗(𝑟, 𝑐).
For some constant𝑟0 depending on𝑚, 𝑛, and 𝑐, 𝑑∗(𝑟, 𝑐) >
¯
𝑑∗(𝑟, 𝑐; 𝑛) for 𝑟 < 𝑟
0 and 𝑑∗(𝑟, 𝑐) = ¯𝑑∗(𝑟, 𝑐; 𝑛) for 𝑟 ≥ 𝑟0.
From the following examples, we can see quantitatively the advantage of the GSICAS over the GSIC.
Example 1: When 𝑚 = 𝑛 = 2 and 𝑐 = 1, then ℱ(1, 2) = {(1, 1)}, 𝒢(1, 2) = {(1), (1, 1)}, and 𝑑∗(𝑟, 1) = { 𝑑∗ (1)(𝑟) for0 ≤ 𝑟 < 12 𝑑∗ (1,1)(𝑟) for 12 ≤ 𝑟 ≤ 2 (29) 𝑑∗ (1)(1/2) = 𝑑∗(1,1)(1/2) = 1. We have ¯𝑑∗(𝑟, 1; 2) = 𝑑∗ (1,1)(𝑟) for 0 ≤ 𝑟 ≤ 2. 𝑑∗(𝑟, 1) > ¯ 𝑑∗(𝑟, 1; 2) for 0 ≤ 𝑟 < 1 2 and𝑑∗(𝑟, 1) = ¯𝑑∗(𝑟, 1; 2) for 12 ≤ 𝑟 ≤ 2.
For the case of𝑚 = 𝑛 = 3 and 𝑐 = 2, ℱ(2, 3) = {(1, 2)}, 𝒢(2, 3) = {(2), (1, 2)}, and 𝑑∗(𝑟, 2) = { 𝑑∗ (2)(𝑟) for0 ≤ 𝑟 < 1 𝑑∗ (1,2)(𝑟) for1 ≤ 𝑟 ≤ 3 (30) 𝑑∗ (2)(1) = 𝑑∗(1,2)(1) = 2.
Here, 𝑑∗(𝑟, 2) > ¯𝑑∗(𝑟, 2; 3) for 0 ≤ 𝑟 < 1 and 𝑑∗(𝑟, 2) =
¯
𝑑∗(𝑟, 2; 3) for 1 ≤ 𝑟 ≤ 3.
In Fig. 1,𝑑∗(𝑟, 𝑐) and ¯𝑑∗(𝑟, 𝑐; 𝑛) for the cases of (𝑚 = 𝑛 =
2, 𝑐 = 1) and (𝑚 = 𝑛 = 3, 𝑐 = 2) are compared visually. For larger𝑚, 𝑛, and 𝑐, we consider the following cases.
Example 2: For𝑚 = 𝑛 = 8 and 𝑐 = 5, we have ℱ(5, 8) = {(3, 5)}, 𝒢(5, 8) = {(5), (1, 5), (2, 5), (3, 5)},
0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 35 40
Spatial multiplexing gain r=R/log SNR
Diversity gain d
\bar{d}^*(r,5;8)
d*(r,5)
(2+11/16, 12.5)
Fig. 2. 𝑑∗(𝑟, 5) and ¯𝑑∗(𝑟, 5; 8) for the MIMO Rayleigh fading channel with𝑚 = 𝑛 = 8. The curve of 𝑑∗(𝑟, 5) is the line linking the following points (0, 24), (1
9, 24), (1.8, 16), (2 + 27, 14), (3.625, 9), (4.6, 6), (5 +
1/3, 4), (8, 0), and the curve of ¯𝑑∗(𝑟, 5; 8) is the line with points (0, 40), (1, 28), (2, 18), (2 + 11 16, 12.5), (3.625, 9), (4.6, 6), (5 + 1/3, 4), (8, 0). and 𝑑∗(𝑟, 5) = { 𝑑∗ (5)(𝑟) for0 ≤ 𝑟 < 2 +1116 𝑑∗ (3,5)(𝑟) for2 +1116 ≤ 𝑟 ≤ 8 (31) with𝑑∗ (5)(2 +1116) = 𝑑∗(3,5)(2 + 1116) = 12.5. Here 𝑑∗(𝑟, 5) is larger than ¯𝑑∗(𝑟, 5; 8) = 𝑑∗ (3,5)(𝑟) for 0 ≤ 𝑟 < 2 + 1116 and is
much larger for near𝑟 = 0.
Consider another case of 𝑚 = 𝑛 = 10 and 𝑐 = 4,
then ℱ(4, 10) = {(2, 4, 4), (3, 3, 4)} and 𝒢(4, 10) = {(4), (3, 4), (4, 4), (1, 4, 4), (2, 3, 4), (2, 4, 4), (3, 3, 4)}. We can easily see that ¯𝑑∗(𝑟, 4; 10) = 𝑑∗
(2,4,4)(𝑟) for all
0 ≤ 𝑟 ≤ 10. 𝑑∗(𝑟, 4), on the other hand, is
𝑑∗(𝑟, 4) = ⎧ ⎨ ⎩ 𝑑∗ (4)(𝑟) for0 ≤ 𝑟 < 1 +113 𝑑∗ (4,4)(𝑟) for 1 + 113 ≤ 𝑟 < 5 −2025197 𝑑∗ (2,4,4)(𝑟) for 5 − 2025197 ≤ 𝑟 ≤ 10 (32) with 𝑑∗ (4) ( 1 + 3 11 ) = 𝑑∗ (4,4) ( 1 + 3 11 ) = 24 and 𝑑∗ (4,4) ( 5 − 197 2025 ) = 𝑑∗ (2,4,4) ( 5 − 197 2025 ) = 7 + 43 45.
Figs. 2 and 3 show these two examples graphically. As shown in the above examples, the difference between 𝑑∗(𝑟, 𝑐) and ¯𝑑∗(𝑟, 𝑐; 𝑛) becomes larger for small 𝑟, particularly
near𝑟 = 0, for larger 𝑚, 𝑛, and 𝑐. V. CONCLUSION
The DMT under the GSICAS is studied with the constraint on the maximum group size of the transmit antenna groups. When the channel of a MIMO system is the i.i.d. Rayleigh fad-ing, we showed analytic ways of obtaining the optimal GDMT under the GSIC and under the GSICAS. For a given group-size bound𝑐, the grouping achieving the optimal GDMT under the GSIC is shown to split the𝑛 transmit antennas into ⌈𝑛
𝑐⌉
groups and to give a higher priority in decoding order to the component code transmitted by a larger group. From this, for a given group-size bound, a very limited collection of
0 2 4 6 8 10 0 5 10 15 20 25 30 35 40
Spatial multiplexing gain r=R/log SNR
Diversity gain d
\bar{d}^*(r,4;10)
d*(r,4)
(5−197/2025, 7+43/45)
Fig. 3. 𝑑∗(𝑟, 4) and ¯𝑑∗(𝑟, 4; 10) for the MIMO Rayleigh fading channel with𝑚 = 𝑛 = 10. The curve of 𝑑∗(𝑟, 4) is the line with points (0, 16), (1+
91
99, 16), (3+2377, 12), (4+37, 9), (6+1145, 5), (6+3445, 4), (8+3145, 1), (10, 0),
and the curve of ¯𝑑∗(𝑟, 4; 10) is the line with points (0, 40), (1, 27), (1 +
3
11, 24), (2 + 89, 16), (3 + 19, 15), (4 + 89, 8), (5 −2025197, 7 + 4345), (6 + 11
45, 5), (6 +3445, 4), (8 +3145, 1), (10, 0).
grouping index sets is shown to achieve the optimal GDMT under the GSICAS. For a fixed grouping index set the optimal multiplexing gain allocation to the transmit antenna groups in diversity is obtained analytically, and thus the corresponding data rate allocation for the grouping is derived. Therefore, the optimal GDMT under the GSICAS, the corresponding grouping, and rate allocation are obtained systematically.
APPENDIXA PROOF OFPROPOSITION2
First, for the case of𝐵 = 2 with ℐ = {(𝑛1, 𝑛2), (𝑛2, 𝑛1)}
we show that 𝑑∗
(𝑛1,𝑛2)(𝑟) ≥ 𝑑∗(𝑛2,𝑛1)(𝑟) for 0 ≤ 𝑟 ≤ 𝑛 . (33) From (10), we have𝑑∗
(𝑛2,𝑛1)(0) = min{𝑚𝑛2, (𝑚 − 𝑛2)𝑛1} = (𝑚−𝑛2)𝑛1. Let𝑠1and𝑠2be the multiplexing gains such that
𝑑∗ (𝑛1,𝑛2)(𝑠1+ 𝑠2) = 𝑑 ∗(𝑠 1; 𝑚, 𝑛1) = 𝑑∗(𝑠 2; 𝑚 − 𝑛1, 𝑛2) = (𝑚 − 𝑛2)𝑛1.
Then𝑠2= 𝑛2− 𝑛1and𝑠1 is some positive real number since
𝑚𝑛1> (𝑚 − 𝑛2)𝑛1. For0 ≤ 𝑟 ≤ 𝑛2− 𝑛1+ 𝑠1 we have
𝑑∗
(𝑛1,𝑛2)(𝑟) ≥ 𝑑∗(𝑛2,𝑛1)(0) ≥ 𝑑∗(𝑛2,𝑛1)(𝑟). (34) For𝑛2− 𝑛1+ 𝑠1< 𝑟 ≤ 𝑛, 𝑟 is split into 𝑡1 and𝑡2 such that
𝑟 = 𝑡1+ 𝑡2 and
𝑑∗
(𝑛1,𝑛2)(𝑟) = 𝑑
∗(𝑡
1; 𝑚, 𝑛1) = 𝑑∗(𝑡2; 𝑚 − 𝑛1, 𝑛2),
where𝑡1> 𝑠1 and𝑡2> 𝑛2− 𝑛1. By Lemma 1
𝑑∗(𝑡 2; 𝑚 − 𝑛1, 𝑛2) = 𝑑∗(𝑡2− (𝑛2− 𝑛1); 𝑚 − 𝑛2, 𝑛1) and 𝑑∗(𝑛 2− 𝑛1+ 𝑡1; 𝑚, 𝑛2) = 𝑑∗(𝑡1; 𝑚 − (𝑛2− 𝑛1), 𝑛1) < 𝑑∗(𝑡 1; 𝑚, 𝑛1).
Therefore𝑑∗
(𝑛2,𝑛1)(𝑟) = 𝑑∗(˜𝑡2; 𝑚 − 𝑛2, 𝑛1) where ˜𝑡2 is some positive number such that ˜𝑡2> 𝑡2− (𝑛2− 𝑛1) and
𝑑∗ (𝑛2,𝑛1)(𝑟) < 𝑑∗(𝑡2− (𝑛2− 𝑛1); 𝑚 − 𝑛2, 𝑛1) = 𝑑∗(𝑛1,𝑛2)(𝑟). (35) By (34) and (35) we conclude (33). Then, for 𝐵 > 2, 𝑑∗ (𝑛1,𝑛2,...,𝑛𝐵)(𝑟) = max𝐼∈ℐ𝑑∗𝐼(𝑟) is
proved by applying (33) repeatedly. APPENDIXB
PROOF OFPROPOSITION3
We let
𝑏(𝑞) ≜ 𝐵 −⌈𝑛𝑐⌉+ 𝑞 for 1 ≤ 𝑞 ≤⌈𝑛𝑐⌉ then 𝑏(1) ≥ 2 and 𝑏(⌈𝑛
𝑐⌉) = 𝐵. There exist nonnegative
integers𝑡𝑞 for1 ≤ 𝑞 ≤ ⌈𝑛𝑐⌉ such that
𝑛𝑏(𝑞)+ 𝑡𝑞≤ 𝑐 and ⌈𝑛 𝑐⌉ ∑ 𝑞=1 (𝑛𝑏(𝑞)+ 𝑡𝑞) = 𝑛.
Let𝑢𝑞 ≜ 𝑛𝑏(𝑞)+𝑡𝑞for1 ≤ 𝑞 ≤ ⌈𝑛𝑐⌉ and 𝐽 ≜ (𝑢1, . . . , 𝑢⌈𝑛 𝑐⌉).
Then
𝑑∗
𝐽(𝑟) ≥ 𝑑∗𝐼(𝑟) for 0 ≤ 𝑟 ≤ 𝑛, (36)
which is proved in the following.
Let 𝑚1 ≜ 𝑚 and 𝑚𝑖 ≜ 𝑚 −( ∑𝑖−1𝑗=1𝑛𝑗 ) for2 ≤ 𝑖 ≤ 𝐵, then 𝑑∗ 𝐼(𝑟) =∑𝐵max 𝑗=1𝑟𝑗=𝑟 min 1≤𝑗≤𝐵𝑑 ∗(𝑟 𝑗; 𝑚𝑗, 𝑛𝑗). Let 𝑣1 ≜ 𝑚 and 𝑣𝑞 ≜ 𝑚 −( ∑𝑞−1𝑗=1𝑢𝑗 ) for2 ≤ 𝑞 ≤ ⌈𝑛 𝑐 ⌉ , then 𝑑∗ 𝐽(𝑟) =∑⌈ 𝑛max 𝑐 ⌉ 𝑞=1𝑟𝑞=𝑟 min 1≤𝑞≤⌈𝑛 𝑐⌉ 𝑑∗(𝑟 𝑞; 𝑣𝑞, 𝑢𝑞). Since𝑚𝑏(𝑞) = 𝑣𝑞−∑⌈ 𝑛 𝑐⌉ 𝑝=𝑞𝑡𝑝,𝑣𝑞 ≥ 𝑚𝑏(𝑞)+𝑡𝑞for1 ≤ 𝑞 ≤⌈𝑛𝑐⌉.
For any given 0 ≤ 𝑟 ≤ 𝑛, let r = (𝑟1, . . . , 𝑟𝐵) be a
multiplexing gain vector such that ∑𝐵𝑗=1𝑟𝑗 = 𝑟 and each
𝑟𝑗 is allocated to the group of 𝑛𝑗 transmit antennas then
∑𝑏(1)−1 𝑗=1 𝑟𝑗 ≤∑⌈ 𝑛 𝑐⌉ 𝑞=1𝑡𝑞, since∑𝑏(1)−1𝑗=1 𝑛𝑗 =∑⌈ 𝑛 𝑐⌉ 𝑞=1𝑡𝑞.
There-fore, there exist𝑠1, . . . , 𝑠⌈𝑛
𝑐⌉such that ∑𝑏(1)−1 𝑗=1 𝑟𝑗 =∑⌈ 𝑛 𝑐⌉ 𝑞=1𝑠𝑞
with𝑠𝑞 ≤ 𝑡𝑞 for𝑞 = 1, . . . , ⌈𝑛𝑐⌉, and we have
min 1≤𝑗≤𝐵𝑑 ∗(𝑟 𝑗; 𝑚𝑗, 𝑛𝑗) ≤ min 1≤𝑞≤⌈𝑛 𝑐⌉ 𝑑∗(𝑟 𝑏(𝑞); 𝑚𝑏(𝑞), 𝑛𝑏(𝑞)) ≤ min 1≤𝑞≤⌈𝑛 𝑐⌉ 𝑑∗(𝑟 𝑏(𝑞)+ 𝑡𝑞; 𝑣𝑞, 𝑢𝑞) ≤ min 1≤𝑞≤⌈𝑛 𝑐⌉ 𝑑∗(𝑟 𝑏(𝑞)+ 𝑠𝑞; 𝑣𝑞, 𝑢𝑞)
where the second inequality follows from Lemma 1 and ∑⌈𝑛
𝑐⌉
𝑞=1𝑟𝑏(𝑞)+ 𝑠𝑞 = 𝑟. This concludes (36).
APPENDIXC PROOF OFPROPOSITION4
For each𝑖 such that 𝑛 ≥ 𝑖 > 𝑖∗we have𝑚𝑖 > (𝑚−𝑖)(𝑛−𝑖)
and𝑑∗
(𝑖,𝑛−𝑖)(0) = (𝑚−𝑖)(𝑛−𝑖). Both 𝑚𝑖∗and(𝑚−𝑖∗)(𝑛−𝑖∗)
are greater than or equal to(𝑚 − 𝑖)(𝑛 − 𝑖) and 𝑑∗
(𝑖∗,𝑛−𝑖∗)(𝑟) ≥ 𝑑∗(𝑖,𝑛−𝑖)(𝑟) for 0 ≤ 𝑟 ≤ 𝑖 − 𝑖∗+ 𝑠1
where 𝑠1 is the multiplexing gain such that 𝑑(𝑠1; 𝑚, 𝑖∗) =
(𝑚 − 𝑖)(𝑛 − 𝑖), and 𝑑∗ (𝑖∗,𝑛−𝑖∗)(𝑖 − 𝑖∗+ 𝑠1) = 𝑑(𝑠1; 𝑚, 𝑖∗) = 𝑑(𝑖−𝑖∗; 𝑚−𝑖∗, 𝑛−𝑖∗). For 𝑖−𝑖∗+𝑠 1< 𝑟 ≤ 𝑛, 𝑑∗(𝑖∗,𝑛−𝑖∗)(𝑟) = 𝑑∗(˜𝑠 1; 𝑚, 𝑖∗) = 𝑑∗(𝑖−𝑖∗+˜𝑠2; 𝑚−𝑖∗, 𝑛−𝑖∗) for some ˜𝑠1≥ 𝑠1 and˜𝑠2> 0. Here 𝑟 = ˜𝑠1+ 𝑖 − 𝑖∗+ ˜𝑠2.𝑑∗(˜𝑠2; 𝑚 − 𝑖, 𝑛 − 𝑖) = 𝑑∗(𝑖 − 𝑖∗+ ˜𝑠2; 𝑚 − 𝑖∗, 𝑛 − 𝑖∗) and 𝑑∗(𝑖 − 𝑖∗+ ˜𝑠1; 𝑚, 𝑖) = 𝑑∗(˜𝑠1; 𝑚 − 𝑖 + 𝑖∗, 𝑖∗) < 𝑑∗(˜𝑠1; 𝑚, 𝑖∗) by Lemma 1. Hence 𝑑∗ (𝑖,𝑛−𝑖)(𝑟) < 𝑑∗(˜𝑠2; 𝑚 − 𝑖, 𝑛 − 𝑖) = 𝑑∗(𝑖∗,𝑛−𝑖∗)(𝑟). Therefore we have (21). Next, if 𝑛 > 𝑖2 > 𝑖1 > 𝑖∗ then 𝑚𝑖1 > (𝑚 − 𝑖1)(𝑛 − 𝑖1), 𝑚𝑖2> (𝑚−𝑖2)(𝑛−𝑖2), and 𝑑∗(𝑖1,𝑛−𝑖1)(𝑟) > 𝑑 ∗ (𝑖2,𝑛−𝑖2)(𝑟) for 0 ≤ 𝑟 ≤ 𝑖2 − 𝑖1+ 𝑡1 where 𝑡1 = 𝑟∗((𝑚 − 𝑖2)(𝑛 − 𝑖2); 𝑚, 𝑖1) > 0. For 𝑖2 − 𝑖1 + 𝑡1 < 𝑟 ≤ 𝑛, by Lemma 1 𝑑∗ (𝑖1,𝑛−𝑖1)(𝑟) ≥ 𝑑∗(𝑖2,𝑛−𝑖2)(𝑟). Therefore 𝑑∗(𝑖1,𝑛−𝑖1)(𝑟) ≥ 𝑑∗
(𝑖2,𝑛−𝑖2)(𝑟) for 0 ≤ 𝑟 ≤ 𝑛. Combining this with (21), (22) holds for𝑛 > 𝑖2> 𝑖1≥ 𝑖∗.
APPENDIXD
PROOF OFPROPOSITION5
For any 𝑖 such that 2 ≤ 𝑖 ≤ 𝑖∗, we have for all1 ≤ 𝑗 < 𝑖
𝑑∗ (𝑖,𝑛−𝑖)(𝑟) ≥ 𝑑∗(𝑗,𝑛−𝑗)(𝑟) for 0 ≤ 𝑟 ≤ 𝑎(𝑖 − 1), (37) since 𝑑∗ (𝑗,𝑛−𝑗)(𝑟) ≤ 𝑑∗(𝑗,𝑛−𝑗)(0) = 𝑚𝑗 ≤ 𝑚(𝑖 − 1) = 𝑑∗ (𝑖,𝑛−𝑖) ( 𝑎(𝑖 − 1))≤ 𝑑∗ (𝑖,𝑛−𝑖)(𝑟)
for0 ≤ 𝑟 ≤ 𝑎(𝑖 − 1). In addition, for 2 ≤ 𝑖 ≤ 𝑖∗,
𝑑∗
(𝑖,𝑛−𝑖)(𝑟) ≤ 𝑑∗(𝑖−1,𝑛−𝑖+1)(𝑟) for 𝑎(𝑖 − 1) ≤ 𝑟 ≤ 𝑛 (38)
which is proved as follows. By Lemma 1,𝑑∗(1+𝑠
2(𝑖−1); 𝑚−
𝑖+1, 𝑛−𝑖+1) = 𝑚(𝑖−1) but 𝑑∗(1; 𝑚, 𝑖) < 𝑚(𝑖−1), and thus
1 + 𝑠2(𝑖 − 1) > 𝑎(𝑖 − 1) and 𝑑∗(𝑖,𝑛−𝑖)(𝑟) ≤ 𝑑∗(𝑖−1,𝑛−𝑖+1)(𝑟) =
𝑚(𝑖 − 1) for 𝑎(𝑖 − 1) ≤ 𝑟 ≤ 1 + 𝑠2(𝑖 − 1). For any 𝑟 such
that1 + 𝑠2(𝑖 − 1) < 𝑟 ≤ 𝑛, 𝑑∗(𝑖−1,𝑛−𝑖+1)(𝑟) = 𝑑∗(1 + 𝑡2; 𝑚 − 𝑖 + 1, 𝑛 − 𝑖 + 1) for some 𝑡2> 𝑠2(𝑖 − 1). 𝑑∗(1 + 𝑡2; 𝑚 − 𝑖 + 1, 𝑛 − 𝑖 + 1) = 𝑑∗(𝑡 2; 𝑚 − 𝑖, 𝑛 − 𝑖) but 𝑑∗(𝑟 − 𝑡2; 𝑚, 𝑖) < 𝑑∗(𝑟 − 1 − 𝑡 2; 𝑚, 𝑖 − 1). Hence 𝑑∗(𝑖,𝑛−𝑖)(𝑟) ≤ 𝑑∗(𝑖−1,𝑛−𝑖+1)(𝑟) for1 + 𝑠2(𝑖 − 1) < 𝑟 ≤ 𝑛.
From (37), (38), and Proposition 4, for each 1 ≤ 𝑖 ≤ 𝑖∗,
we have for𝑎(𝑖) ≤ 𝑟 ≤ 𝑎(𝑖 − 1) 𝑑∗
(𝑖,𝑛−𝑖)(𝑟) ≥ 𝑑∗(𝑗,𝑛−𝑗)(𝑟) for all 1 ≤ 𝑗 ≤ 𝑛. (39)
ACKNOWLEDGMENT
The authors would like to thank anonymous reviewers for their helpful comments.
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In Sook Park received the B.S. degree from Kyung
Hee University, Seoul, Korea, in 1997, and the M.S. and Ph.D. degrees from the Korea Advanced Institute of Science and Technology (KAIST), Dae-jeon, Korea, in 2000 and 2005, respectively, all in mathematics.
From January 2005 to December 2006, she was a Postdoctoral researcher at the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. From January 2007 to December 2008, she was a Postdoctoral Researcher in the Department of Electrical Engineering and Computer Science at KAIST. From March 2009 to February 2010, she was a Postdoctoral Researcher in the Department of Electrical Engineering at KAIST. Since July 2010, she has been with ETRI, where she now works as a Senior Member of Research Staff. Her research interests lie in interference alignment, cooperative communication in wireless mesh networks, detection for multiple-input multiple-output systems, and applied analysis.
Sae-Young Chung (S’89-M’00-SM’07) received
the B.S. and M.S. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1990 and 1992, respectively. He received the Ph.D. degree from the Department of Electrical Engineer-ing and Computer Science, Massachusetts Institute of Technology, Cambridge, MA in 2000.
From September 2000 to December 2004, he was with Airvana, Inc. Since January 2005, he has been with KAIST, Daejeon, Korea, where he is now a KAIST Chair Associate Professor in the Department of Electrical Engineering. He will serve as a TPC co-chair for the 2014 IEEE International Symposium on Information Theory (ISIT). He also served as a TPC co-chair of WiOpt 2009. His research interests include network information theory, coding theory, and their applications to wireless communications.
Dr. Chung is an Editor for the IEEE TRANSACTIONS ONCOMMUNICA
