Rectangular Co-ordinate Interleaved Orthogonal
Designs
Md. Zafar Ali Khan
B.Sundar Rajan
Moon Ho Lee
Insilica Semiconductors
Frontline Grandeur, 14 Walton Road
Bangalore, India 560 001
Email: zafar@insilicasemi.com
Department of Elect. Comm. Engg.
Indian Institute of Science
Bangalore, India 560 012
Email: bsrajan@ece.iisc.ernet.in
Institute of Communication and Information
Chonbuk National University
Chonju, South Korea
Email: moonho@moak.chonbuk.ac.kr
Abstract— Space-Time block codes (STBC) from Orthogonal
Designs (OD), Quasi-Orthogonal Designs (QOD) and Co-ordinate
Interleaved Orthogonal Designs (CIOD) have been attracting
wider attention due to their amenability for fast (single-symbol
decoding for OD, CIOD and double-symbol decoding for QOD)
ML decoding, and rate-one with full-rank over quasi-static fading
channels [1]-[13]. The importance of CIOD is due to the fact
that, rate-one, full-rank, square ODs for arbitrary complex
constellations exist only for 2 transmit antennas while such a
CIOD exists for 2,3 and 4 transmit antennas with a slight restriction on the complex constellations [12], [13]. These limitations
motivate study of rectangular (non-square) designs. One way of
obtaining rectangular designs is by deleting columns from square
or non-square ODs or CIODs. In this paper, we present a new
construction of rectangular single-symbol decodable designs that
have higher maximum mutual information than those obtained
by deleting columns of CIODs and has lower peak to average
power ratio (PAPR). Simulation results are presented for three
and five transmit antennas and compared with that of OD,
QODs, CIODs to demonstrate the superiority of the proposed
rectangular designs.
GLPCOD[2], [7]
QOD [8]
QOD-RC[3]
QOD-RC[10]
CIOD[12]
Rate
3/4
1
1
1
1
Rank
4
2
4
4
4
Signal set
arbitrary
arbitrary
restricted by rotation
restricted by rotation
restricted by CPD
Decoding
SSD
DSD
DSD
DSD
SSD
TABLE II
C OMPARISON OF KNOWN DESIGNS FOR 8 T X ANTENNAS
GLPCOD [2]
QOD [8]
QOD-RC [3]
CIOD[14]
Rate
1/2
3/4
3/4
1
Rank
8
4
8
8
signal set
arbitrary
arbitrary
restricted by rotation
restricted by CPD
Decoding
SSD
DSD
DSD
DSD
both the tables form a sub-class of the class of linear STBCs
[9] any member S of which can be expressed as 1
I. I NTRODUCTION AND P RELIMINARIES
Starting from Alamouti [1], several authors have studied
Space-Time Block Codes (STBCs) obtained from Orthogonal
Designs (ODs) and their variations like quasi-orthogonal designs (QODs) and Co-ordinate interleaved orthogonal designs
(CIODs) that offer fast decoding (single-symbol decoding
(SSD) or double-symbol decoding (DSD)) over quasi-static
fading channels [1]-[13]. A scheme that trades diversity for
simpler ML decoding (double-symbol decoding) is presented
in [8] for four and eight antennas. For this scheme and the
STBCs from ODs any complex signal constellation can be
used. By sacrificing the freedom of being able to use any
complex constellation if the signal constellations are restricted
to those having certain properties then it has been shown
by several authors [3], [7], [10], [12], [14] that rate and/or
diversity can be improved from those of complex ODs for
four and eight antennas. These improvements are summarized
in Table I and Table II. In these tables QOD-RC stands for
Quasi Orthogonal Design with Rotated Constellations [3] and
CPD for Coordinate Product Distance [12]. All the designs in
0 This work was partly supported by DRDO-IISc Program on Mathematical
Engineering through a grant to B.S.Rajan.
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TABLE I
C OMPARISON OF KNOWN OD S , QOD S AND CIOD FOR 4 T X ANTENNAS
S=
K−1
A2k xkI + A2k+1 xkQ
(1)
k=0
where {Ak }2K−1
is a set of complex matrices called weight
k=0
matrices.
A (p, N, k) Generalized Linear Processing Complex Orthogonal Design ((p, N, k)-GLPCOD) is a p×N matrix Θp×N
in k complex indeterminates x1 , x2 , · · · , xk and rate R = k/p,
p ≥ N such that
• the entries of Θp×N are complex linear combinations of
0, ±xi , i = 1, · · · , k and their conjugates.
H
• Θp×N Θp×N = D, where D is a diagonal matrix whose
entries are a linear combination of |xi |2 , i = 1, · · · , k
with all strictly positive real coefficients.
When k=N =p and no entry is zero, the design is
called a Linear Processing Complex Orthogonal Design
(LPCOD). Furthermore, when the entries are only from
{±x1 , ±x2 , · · · , ±xk }, their conjugates and multiples of j
then it is called a Complex Orthogonal Design (COD).
1 Also
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referred to as a Linear Dispersion (LD) code [21].
0-7803-7974-8/03/$17.00 © 2003 IEEE
Definition 1.1 ([12]): For even integers K, N, L, a
(K, N, L)-Co-ordinate Interleaved Orthogonal Design
((K, N, L)-CIOD) of size N and rate K/L, in variables
xi , i = 0, · · · , K − 1 is a L × N matrix S(x0 , · · · , xK−1 )
(denoted simply by S), given by S =
0L/2,N/2
ΘL/2,N/2 (x̃0 , · · · , x̃K/2−1 )
0L/2,N/2
ΘL/2,N/2 (x̃K/2 , · · · , x̃K−1 )
(2)
where ΘL/2,N/2 (x0 , · · · , xK/2−1 ) is a GLPCOD of size N/2
and rate K/L, x̃i = Re{xi } + jIm{x(i+K/2)K } and where
(a)K denotes a (mod K).
Notice that in (2) all the four sub-matrices are of the same
type, i.e., all the four are L/2 × N/2 matrices. The new class
of rectangular designs defined in (11) in the beginning of
the following section are obtained by not having these submatrices to be of the same type. In the remaining part of this
section we summarize the known results on CIOD that are
used for the new class of designs. Examples of rate 1, CIODs
for N = 2, 4 are given below and
x̃0 0
,
(3)
S(x0 , x1 ) =
0 x̃1

x̃0
 −x̃∗1
S(x0 , · · · , x3 ) = 
 0
0
x̃1
x̃∗0
0
0
0
0
x̃2
−x̃∗3

0
0 
.
x̃3 
x̃∗2
(4)
a rate 1 STBC from CIOD for N = 3 can be obtained form
N = 4, CIOD by deleting one of the columns.
Theorem 1.1: [12] A rate 1, co-ordinate interleaved orthogonal design of size N exists if and only if N = 2, 3 or 4
Single-Symbol Decodability: Let the number of transmit
antennas be N and the number of receive antennas be M . At
each time slot t, the complex signals, sit , i = 0, 1, · · · , N − 1
are transmitted from the N antennas simultaneously. Let hij =
αij ejθij denote the path gain from the
√ transmit antenna i
to the receive antenna j, where j = −1. Assuming that
the path gains are constant over a frame length L ≥ N ,
t = 0, · · · , L − 1, the received signal vjt at the antenna j
at time t = 0, · · · , L − 1, is given by
vjt =
N
−1
hij sit + njt , j = 0, · · · , M − 1.
(5)
i=0
In matrix notation,
V = SH + N
(6)
where V ∈ CL×M (C denotes the complex field) is the
received signal matrix, the transmission matrix (also referred
as codeword matrix) S ∈ CL×N and N ∈ CL×M has entries
that are Gaussian distributed with zero mean and unit variance
and also are temporally and spatially white. In V, S and N
time runs vertically and space runs horizontally. H ∈ CN ×M
defines the channel matrix, such that the element in the ith
row and the jth column is hij . The channel matrix H and
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the transmitted codeword S are assumed to have unit variance
entries. Through out the paper, for a matrix A, AH represents
the Hermitian (conjugate transpose), AT the transpose and
|A| the determinant of A. Assuming that perfect channel state
information (CSI) is available at the receiver, the decision rule
for ML decoding is
(7)
M (S) tr (V − SH)H (V − SH) .
In general the decoding is exponential, but for STBCs from
H
ODs and CIODs (STBCs that satisfy AH
k Al +Al Ak = 0, k =
l), M (S) can be written as
2
V − (A2k xkI + A2k+1 xkQ )H +MC (8)
M (S) =
k
Mk (xk )
K−1
where
k=0 A2k xkI + A2k+1 xkQ , MC = −(K −
S =
1)tr V H V and . denotes the Frobenius norm. If xk takes
values from a signal set A, minimizing M (S) is equivalent to
min Mk (xk ), ∀k
xk ∈A
and hence single-symbol decodable. Clearly, STBCs from OD
and CIOD are single symbol decodable.
Coding Gain: If we define the coding gain as Λ =
1
minS,S det (S − S )H (S − S ) N where S, S are distinct
codewords, then for STBCs from CIODs given in (2), the
coding gain is given by two codewords that differ in a single
variable. Simple manipulations give,
ΛCIOD =
min
xk =xk ∈A
|xkI − xkI ||xkQ − xkQ |.
The metric minxk =xk ∈A |xkI − xkI ||xkQ − xkQ | is called the
co-ordinate product distance (CPD) of A [12]. The STBCs
from CIODs achieve full diversity iff the CPD of the signal
set is non-zero [12].
For constellations with CP D = 0, like QAM, we can obtain
another signal constellation with non-zero CP D by rotating
the QAM constellation. Infact for square lattice constellations,
the CP D is maximized when the angle of rotation, θ =
2
arctan(2)
√ . The
= 31.7175◦ , and is given by CP Dopt = 4d
2
5
proof is given in the appendix.
In this paper, we introduce a class of non-square designs that
are (i) single-symbol decodable, (ii) having coding gain at least
the CPD of the signal constellation used and (iii) having lower
PAPR. In particular, we present a rate 1 STBC for 3 antennas
and rate 3/4 STBCs for 5,6,7 antennas. Simulation results are
presented for 3 and 5 antennas and compared with known
STBCs to show the superiority of the new classes of codes
introduced. The rest of the material of this paper is organized
as follows: In Section II rectangular designs obtainable from
deleting columns of CIODs are studied. The conditions for
full-diversity are identified and an expression for coding
gain obtained. Another class of rectangular designs, called
asymmetric CIODs, (not obtainable from dropping columns
of CIODs) are presented and the resulting STBCs are shown
to be single-symbol decodable in Section III. The diversity
and coding gain are studied in Section IV. Simulation results
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are presented in Section V followed by concluding remarks in
Section VI.
II. D ESIGNS BY DELETING COLUMNS OF CIOD
In this section we study the rectangular designs obtainable
from CIODs of Definition 1.1 by deleting certain columns of
them. We first identify the counterpart of the CPD used for
CIOD to be the generalized CPD (GCPD) defined below for
the case of the rectangular designs.
Definition 2.1 (Generalized CPD): For any signal set A
and positive integers N1 and N2 , the generalized coordinate
product distance GCP DN1 ,N2 between any two signal points
u = uI + juQ and v = vI + jvQ , u = v, belonging to the
signal set A is defined as
GCP DN1 ,N2 (u, v) = min(a, b), where
2N1
2N2
2N2
2N1
(9)
a = |uI − vI | N1 +N2 |uQ − vQ | N1 +N2 and
b
= |uI − vI | N1 +N2 |uQ − vQ | N1 +N2
and the minimum of this value among all possible distinct
pairs of signal points in A is defined to be the GCP DN1 ,N2
of A and denoted by GCP DN1 ,N2 (A).
Remark 2.1: Observe that
1) GCP DN1 ,N2 (A) = GCP DN2 ,N1 (A)
2) when N1 = N2 then the GCP DN1 ,N2 (A) reduces to
the CPD of A.
3) The GCPD of a signal set is zero iff the CPD of the
signal set is zero.
We have,
Theorem 2.1: Let the total number of columns present after
the deletion of columns from a (K, L, N )-CIOD of Definition
1.1 be n = n1 + n2 where n1 and n2 are the number of
columns present after the deletion of columns, respectively, in
the first and second N/2 columns of the CIOD. Also, let the
design variables take values from a signal set A. The resulting
rectangular CIOD will be of full-rank iff the CPD of A is nonzero and when it is of full-rank the coding gain is equal to
GCP Dn1 ,n2 (A).
The proof is omitted due to space restrictions.
Example 2.1: consider the STBC for three transmit antennas obtained by deleting the last column of the CIOD for
N = 4 of (4), given by

x0I + jx2Q
 −x1I + jx3Q
S(x0 , · · · , x3 ) = 
0
0
x1I + jx3Q
x0I − jx2Q
0
0
Definition 3.1: For even integers K, N, L and 0 < n <
N/2, a (K, Nn , L)-Asymmetric Co-ordinate Interleaved Orthogonal Design ((K, Nn , L)-ACIOD), of rate K/L, in variables xi , i = 0, · · · , K − 1 is a L × N − n matrix
S(x0 , · · · , xK−1 ) (denoted simply by Sn ), given by Sn =
ΘL/2,N/2 (x̃0 , · · · , x̃K/2−1 )
0L/2,N/2−n
0L/2,N/2−n
ΘL/2,N/2 (x̃K/2 , · · · , x̃K−1 )
(11)
where ΘL/2,N/2 (x0 , · · · , xK/2−1 ) is a GLPCOD of size N/2
of rate K/L, x̃i = Re{xi } + jIm{x(i+K/2)K } and where
(a)K denotes a (mod K).
Notice that if we allow n = 0 the ACIOD coincides with the
CIOD given in (2). To see that any ACIOD is single-symbol
decodable we rewrite the decision metric (7) as
M (S) tr (VH V − HH SH V − VH SH + HH SH SH)
(12)
from which it is clear that since the trace of SnH Sn as well as
that of H H SnH V −V H Sn H do not contain cross-product terms
of more than one variable, Sn is single-symbol decodable. The
full-rankness is discussed along with the coding gain in the
following section. Now we present some examples of ACIOD.
Example 3.1: Let Θ be the Alamouti scheme, then for n =
1 we have


x1I + jx3Q
0
x0I + jx2Q

 −x1I + jx3Q x0I − jx2Q
0
 (13)
S=

0
x2I + jx0Q x3I + jx1Q 
0
−x3I + jx1Q x2I − jx0Q
then S is a rate 1 STBC in x0 , x1 , x2 , x3 for three transmit
antennas. Notice that


a
0
0
3
2
SH S =  0
(14)
0 ,
k=0 |xk |
0
0
b
where a = x20I +x21I +x22Q +x23Q , b = x20Q +x21Q +x22I +x23I .
Observe that there are no cross terms of the form xkI xlI , k = l
in (14) which guarantees single-symbol decodability. Also
observe that this code is quite different from that of CIODs
in that both the inphase and quadrature component of the
variables see the second transmit antenna. To calculate the
 coding gain Λ, let S, Ŝ be two codeword matrices that differ
 in only x0 . Then
.
1/3
Λ = det (S − Ŝ)H (S − Ŝ)
0
0
x2I + jx0Q
−x3I + jx1Q
(10)
=
The coding gain of this STBC is given by GCP D2,1 (A).
using |x0 − x̂0 |2 ≥ 2|x0I − x̂0I ||x0Q − x̂0Q |, we have
III. A SYMMETRIC CIOD
In this section we present a construction that gives nonsquare designs, called Asymmetric Coordinate Interleaved
Designs (ACIOD), whose coding gain is greater than the CPD
of the signal constellation used. Note that rectangular designs
are constructed in [21] for maximizing mutual information;
here we are interested in SSD STBCs only.
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[(x0I − x̂0I )2 (x0Q − x̂0Q )2 (|x0 − x̂0 |2 )]1/3
≥ 21/3 |x0I − x̂0I ||x0Q − x̂0Q | = 21/3 CP D (15)
Observe that the factor 21/3 is due to the fact that S is not
normalized.
Example 3.2: Let Θ be the rate 3/4 design COD denoted
by Θ4,4 , using (11) with n = 3 we have a rate 3/4 singlesymbol decodable STBC for 5 transmit antennas in variables
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x0 , · · · , x5 given by
Θ4,4 (x0I + jx3Q , x1I + jx4Q , x2I + jx5Q ) 04,1
S=
04,1 Θ4,4 (x3I + jx0Q , x4I + jx1Q , x5I + jx2Q )
(16)

a
0
0
S H S =  0 I3 6k=0 |xk |2 0 
0
0
b
3
3
2
2
2
2
with a =
k=0 (xkI + xk+3Q ), b =
k=0 (xkQ + xk+3I )
and I3 is the identity matrix of size 3. Observe that is also a
single-symbol decodable design as there are no cross terms in
S H S.

where
where we have used |∇x0 |2 = |x0 −x0 |2 ≥ 2|x0I −x0I ||x0Q −
x0Q |.
n
The additional factor 2 2N −n (19) is due to the additional
power transmitted on n antennas as compared to CIOD and
on normalizing the transmission matrices, vanishes.
From the equation (18) it is clear that the non-square STBCs
obtained from ACIODs will give full-diversity if none of a,
b and c is zero which is true if and only if the CPD of
the signal set is nonzero.
V. S IMULATION R ESULTS
In this section we present simulation results for 3,4,5
transmit
The total average transmit power is given by
antennas.
E{tr S H S } = L. For three transmit antennas, we compare
IV. D IVERSITY AND C ODING GAIN
where
K/2−1
a=
|x̃k |2 , b =
k=0
K
|xk |2 , c =
k=0
K−1
−3
|x̃k |2
k=K/2
x0 =x0
[|x0I − x 0I ||x0Q − x 0Q |]
n
2(N −n)
2N −n
2n
|∇x0 | 2N −n
≥ 2 2N −n min |x0I − x 0I ||x0Q − x 0Q |
=
n
x0 =x0
2 2N −n CP D
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−2
10
10
and where x̃i = Re{xi } + jIm{x(i+K/2)K } and (a)K denotes a (mod K). Observe that the total number of transmit
antennas is 2N − n.
Now consider the codeword difference matrix B(S, S ) =
S−S which is of full rank for two distinct codeword matrices
S, S , we have


0
0
aIN −n

0
0
bIn
B(S, S )H B(S, S ) = 
0
0
cIN −n
(18)
K/2−1
K
where a = k=0 |x̃k − x̃k |2 , b = k=0 |xk − xk |2 ,
K−1
2
and
k=K/2 |x̃k − x̃k | and where at least one xk differs
from xk , k = 0, · · · , K − 1. Clearly, all the three terms in the
determinant of the above matrix are minimum iff xk differs
from xk for only one k. Therefore assume, without loss of
generality, that the codeword matrices S and S are such that
they differ by only one variable, say x0 taking different values
from the signal set A.
Then, the coding gain is given by
1
Λ = min det B H (S, S )B(S, S ) 2N −n
=
R=3/4 COD with 16−QAM
R=1 QOD with 8−QAM
R=1 ACIOD with 8−QAM
R=1 CIOD with 8−QAM
−1
10
BER
In this section we show that the coding gain of the STBCs
from ACIOD is greater than the CPD (Theorem 4.1).
Theorem 4.1: The coding gain of non-square STBCs from
ACIOD with the variables taking values from a signal set, is
greater than the CPD of the signal set.
Proof: Consider S defined in (11), then


0
0
aIN −n

0
0
bIn
(17)
SH S = 
0
0 cIN −n )
(19)
−4
10
5
10
15
ρ
20
25
Fig. 1. The BER performance of STBCs from OD, QODs and the design
of this paper at 3 bits/sec/Hz in quasi-static Rayleigh fading channel.
the STBC obtained from ACIOD with rate 3/4 Complex
Orthogonal design (GLPCOD) and rate 1 QOD obtained by
deleting one column of the 4 antenna code given in [11] at a
rate of 3 bits/sec/Hz in Fig. 1. The 8-QAM were appropriately
rotated for both ACIOD and QOD to achieve full diversity.
Observe that the ACIOD performs 2 dB better than OD and 0.1
dB better than QOD at BER=10−4 . However ACIOD allows
single-symbol decoding and hence lower receiver complexity
while QOD has double-symbol decoding (similar observations
hold for N = 4, 5 and hence, for clarity, the curve for QOD
[3], [10] N = 4 has been omitted in Fig. 2).
Fig. 2 gives the comparison of rate 1 CIOD for four transmit
antennas and rate 3/4 ACIOD for five transmit antennas, with
known STBCs at a rate of 2 bits/sec/Hz. The CIOD and
ACIOD uses appropriate QPSK constellations, while rate 3/4
GLPCOD for four transmit antennas uses 6-PSK for a rate
of 1.94 bits/sec/Hz. The rate 1/2 GLPCOD for five transmit
antennas uses the 16-QAM. Also compared is the rate 1
STBC for four transmit antennas obtained by constellation
rotation (STBC-CR) which maximizes coding gain [19] and
hence is better than DAST [18]. Observe that while STBCCR has higher coding gain for four transmit antennas it has
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MMI of CIOD is given by
R=1/2 COD
R=3/4 COD
R=1 CIOD
R=1 STBC−CR
R=1/2 COD
R=3/4 CIOD
R=3/4 QOD
R=3/4 ICIOD
−2
P , Probability of bit error
b
10
−3
10
CD (N, M, ρ)
=
=
1
{C1,O + C2,O }
2
1
{CO (N1 , M, ρ) + CO (N2 , M, ρ)}
2
where N1 + N2 = N, N2 < N1 . Observe the scaling of ρ
in the above equation
due to the trace constraint on transmit
power, i.e. tr S H S = L. The above result follows from the
fact that the CIOD is block diagonal with each block being a
GLPCOD.
Proceeding, similarly we have the MMI of STBCs from
ACIOD as
−4
10
CA (N, M, ρ) = CO (N1 , M, ρ)
(22)
−5
10
6
10
8
12
14
Eb/N0 (dB)
16
18
20
22
Fig. 2. The BER performance of the CIOD scheme for 4 transmit and 1
receive antenna compared with STBC-CR, rate 1/2 GLPCOD and rate 3/4
GLPCOD and the BER performance of rate 3/4 ACIOD for 5 transmit and
1 receive antennas compared with rate 3/4 QOD and rate 1/2 GLPCOD at a
throughout of 2 bits/sec/Hz in Rayleigh fading.
where CA is the MMI of STBCs from ACIOD for N transmit
and M receive antennas at a SNR of ρ and N = 2N1 − n.
Comparing with CD in (22) it is easily seen that these codes
have higher MMI as compared to the corresponding STBCs
obtained by deleting columns of CIOD. For example for N =
3, N1 = 2, N2 = 1 and hence CD < CA . The increase in
MMI for M > 1 is obvious.
higher multiplicity and hence performs 1 dB inferior to CIOD.
Comparison of rate 3/4 CIOD and the STBC from ACIOD for
N = 5 shows that ACIOD performs 3 dB better than CIOD.
TABLE III
C OMPARISON OF KNOWN RECTANGULAR GLPCOD S
Tx. Antennas
VI. D ISCUSSION
In this paper we have presented a new construction of nonsquare single-symbol decodable STBCs that have better coding
gain and lower PAPR as compared to the non-square STBCs
obtained from CIODs [12] by deleting columns as it has
lesser number of zeros. The coding gain (CPD) for rotated
lattice constellations is also maximized. Table III gives the
comparison of rates for ODs and ACIODs.
Another important property of the STBCs from ACIOD is
that they have higher Maximum Mutual Information (MMI)
as compared to the corresponding CIODs.
Towards this end observe that the STBCs from ACIOD
consists of two ODs of size N that are separated in time.
The MMI in bits per channel use of GLPCOD for N transmit
and M receive antennas at a SNR of ρ can be written as [16]
CO (N, M, ρ) =
K
ρ
log2 1 + H2
L
N
(20)
HH
observe that H is a N × M matrix. Since H2 = H
where H is the N M ×1 vector formed by stacking the columns
of H, we have
CO (N, M, ρ)
=
K
C(M N, 1, M ρ).
L
(21)
For STBCs obtained from CIODs by deleting columns, recollect that it consists of two GLPCODs, Θ1 , Θ2 of rate K/L.
Let C1,O , C2,O be the MMI of Θ1 , Θ2 respectively. Then the
GLOBECOM 2003
3
5
6
7
9
10
11
12
Orthogonal design compared
Compared design
Delay
Rate
[2], [6]
4
3/4
[4]
11
7/11
[4]
30
3/5
[2], [6], [9]
8
1/2
[6]
16
1/2
[2]
32
1/2
[6]
32
1/2
[2]
64
1/2
[6]
64
1/2
[2]
128
1/2
[6]
128
1/2
[2]
256
1/2
AND
ACIOD S
ACIOD
Delay
Rate
4
1
8
3/4
8
3/4
8
3/4
22
7/11
22
7/11
60
3/5
60
3/5
A PPENDIX
Theorem 1.1: Consider a lattice constellation A, with signal
points from the square lattice (2k − 1 − Q)d + j(2l − 1 − Q)d
where k, l ∈ Z and d is chosen so that the average energy of
the constellation is 1, rotated by an angle θ so as to maximize
=
CP D. The CP D of A is maximized at θ = arctan(2)
2
31.7175◦ and is given by
4d2
CP Dopt = √ .
(23)
5
Proof: The proof is in three steps. First we derive
the optimum value of θ for 4-QAM, denoted as θopt ( the
corresponding CP D is denoted as CP Dopt ). Second, we
show that at θopt , CP Dopt is in-fact the CP D for all other
lattice constellations. Finally, we show that for any other value
of θ ∈ [0, π/2], CP D < CP Dopt completing the proof.
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Step 1: Any point P(x, y) ∈ 2 rotated by an angle θ ∈
[0, 90◦ ] can be written as
xR
cos(θ)
sin(θ)
x
=
.
(24)
yR
− sin(θ) cos(θ)
y
R
Let P1 (x1 , y1 ), P2 (x2 , y2 ) be two distinct points in A such
that x = x1 − x2 , y = y1 − y2 . Observe that x, y =
0, ±2d, · · · . We may write x = ±2md, y = ±2nd, m, n ∈
Z but both x, y cannot be zero simultaneously, as P1 , P2
are distinct points in A. Since, rotation is a linear operation,
xr
x
=R
,
(25)
yr
y
where xr = x1 R − x2 R , yR = y1 R − y2 R . The
CP D(P1 , P2 ) is given by
CP D(P1 , P2 )
=
=
|xr ||yr |
2
2
xy cos(2θ) + (x) − (y) sin(2θ) .
2
For 4-QAM, possible values of CP D(P1 , P2 ) are
CP D1 = 2d2 | sin(2θ)|, CP D2 = 4d2 | cos(2θ)|.
(26)
As sine is an increasing function and cosine a decreasing
function of θ in the first quadrant, equating CP D1 , CP D2
gives the optimal angle of rotation, θopt . Let CP D(θ) be the
CP D at angle θ and CP Dopt = maxθ CP D(θ). It follows
that θopt = arctan(±2)
= 31.7175◦ , 58.285◦ and CP Dopt =
2
2
2
2d sin(2θopt ) = 4d cos(2θopt ).
Step
2: Substituting the optimal values of
sin(2θopt ), cos(2θopt ) in (26) we have
4d2 CP D(P1 , P2 ) = √ ±nm + n2 − m2 5
(27)
where n, m ∈ Z and both n, m are not simultaneously zero
and Z is the set of integers. It suffice to show that
| ± nm + n2 − m2 | ≥ 1∀n, m
provided both n, m are not simultaneously zero. The quadratic
equations in n, | ± nm + n2 − m2 | has roots
√
m
n = {±1 ± 5}.
2
Since n, m ∈ Z, | ± nm√+ n2 − m2 | ∈ Z and is equal to zero
2
2
only if n = 0, m
2 {±1± 5}. Necessarily, |±nm+n −m | ≥
1 for n, m ∈ Z and both n, m are not simultaneously zero.
Therefore at θopt the CP D(θopt ) = CP Dopt .
Step 3: Next, observe that for any value of θ other than
θopt either CP D1 or CP D2 is less than CP Dopt . It follows
that CP D(θ) ≤ CP Dopt with equality iff θ = θopt .
Observe that Theorem 1.1 has application in all schemes where
the performance depends on the CP D such as the schemes
in [22], etc. and the references therein. Also note that the
essence of Theorem 1.1 was presented in [23], however our
proof is simpler. Finally, a note of caution in comparing the
coding gains of CIOD and other STBCs. The average transmit
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power constraint for the different STBCs should be satisfied
for fair comparison. For example, the coding gain of Alamouti
scheme is 4d2 /2 for lattice constellations and for the CIOD
for N = 2 is 4d2 √25 , implying a coding gain of 0.4 dB for
Alamouti code.
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