Combining Orthogonal Space-Frequency
Block Coding and Spatial Multiplexing
in MIMO-OFDM System
Muhammad Imadur Rahman, Nicola Marchetti, Suvra Sekhar Das, Frank H.P. Fitzek, Ramjee Prasad
Center for TeleInFrastruktur (CTiF), Aalborg University, Denmark
e-mail: imr|nm|ssd|ff|prasad@kom.aau.dk; ph: +45 9635 8688
Abstract— In the present work, we have combined Orthogonal
Space-Frequency Block Coding (OSFBC) and Spatial Multiplexing (SM) in one transmission scheme for Orthogonal Frequency
Division Multiplexing (OFDM) systems. In the combined transmission scheme, both spatial diversity and multiplexing benefits
are possible to achieve. Simple Alamouti coding as the S-F coding
across spatial multiplexing branches and a simplified linear
receiver instead of a complex successive interference cancellation
receiver are used in our scheme. In the initial analysis, it is found
that SM-OSFBC-OFDM system is near to the optimum system
capacity for any 4 × 2 MIMO-OFDM system.
I. I NTRODUCTION
Multiple antennas can be used in both ends of a Multiple
Input Multiple Output (MIMO) wireless transmission system
to exploit the benefits of the spatial dimension. Two MIMO
modes can be exploited, namely Space Diversity (SD) and
Spatial Multiplexing (SM). In SD mode, Space-Time Coding
(STC) and Maximal Ratio Combining (MRC) can be used
at the transmitter side and/or receiver side respectively, to
exploit the maximum spatial diversity available in the channel.
This increases the system reliability [1]. Furthermore, SM is a
promising and powerful technique to dramatically increase the
system capacity. In rich scattering environment the independent spatial channels can be exploited to send multiple signals
at the same time and frequency, resulting in higher spectral
efficiency.
Most of the available MIMO techniques are effective in frequency flat scenarios [2]. In wideband scenarios, Orthogonal
Frequency Division Multiplexing (OFDM) can be combined
with MIMO systems, for both diversity and multiplexing
purposes. In frequency selective environments, amalgamation
of SM and OFDM techniques can be a potential source of
high spectral efficiency, thus high data rate systems can be
realized in wideband scenario. All the algorithms can be
implemented on OFDM sub-carrier level, because OFDM
converts a wideband frequency selective channel into a number
of narrowband sub-carriers. Alamouti’s remarkable orthogonal
transmission structure [3] can be applied in space-time or
space-frequency domain in OFDM systems as it is shown
in [4] and [5], to obtain higher signal quality. Similarly, SM
techniques, such as Vertical - Bell Labs LAyered Space-Time
Architecture (VBLAST) [6], can also be used in conjunction
with OFDM systems to obtain higher spectral efficiency [1].
In a cellular wireless systems, the Space-Time Block
Coded Orthogonal Frequency Division Multiplexing (STBCOFDM) [4] and Space-Frequency Block Coded Orthogonal
Frequency Division Multiplexing (SFBC-OFDM) [5] can be
used to increase the resultant Signal to Noise Ratio (SNR)
at the receiver, thus, increasing the coverage area in a cellular
system. In contrast to this, as SM-OFDM requires high receive
SNR for reliable detection, it is evident that users at farther
locations from Base Station (BS) cannot use SM techniques to
enhance the spectral efficiency. Thus, it is required to combine
both of these two techniques in one structure so that both the
diversity and multiplexing benefits can be achieved at farther
locations from transmission source.
Recently there are some approaches of incorporating the
VBLAST technique with some well known STC techniques.
One such work is described in [7], where a combination
of SD and SM for MIMO-OFDM system is proposed. We
call such systems as Joint Diversity and Multiplexing (JDM)
systems. Arguably, the performance of such a system would
be better than SD only and SM only schemes. In [7], the
SM-OFDM system uses two independent STC for two sets of
transmit antennas. Thus, an original 2 × 2 SM-OFDM system
is now extended to 4 × 2 STC aided SM-OFDM system. In
the receiver, the independent STCs are decoded first using prewhitening, followed by maximum likelihood detection. Again,
this increases the receiver complexity quite a lot, though the
system performance gets much better. In later work, Alamouti’s Space-Time Block Code (STBC) is combined with SM
for OFDM system in [8], and a linear receiver is designed for
such a combination. Following these trends, we have combined
Space-Frequency Block Code (SFBC) with SM and obtained
a linear receiver similar to [8] in this work. One advantage in
using SFBC instead of STBC is that, in SFBC, the coding
is done across the sub-carriers inside one OFDM symbol
duration, while STBC applies the coding across a number of
OFDM symbols equal to number of transmit antennas, thus,
an inherent processing delay is unavoidable in STBC.
Our work aims to achieve contemporarily the multiplexing
gain (via two SM branches) and the diversity gain (via SFBC
codes), keeping the complexity low (through the receiver
linearity). A possible scenario where such an hybrid scheme
would be useful could be the intermediate region of the cell, in
fact while close to the BS the SM mode is more advantageous
m11
m1
IFFT
SM branch will be
mp,1
m(1)
=
p
(2)
mp,2
mp
=
CP
SFBC
IFFT
CP
2
m
m1
CP
rem
FFT
CP
rem
FFT
z1
Linear
RX
SM
m
1
2
IFFT
CP
IFFT
CP
z
z2
SFBC
m2
m22
Fig. 1.
Scheme
Simplified System Model for SM-OSFBC-OFDM Transmission
and close to the cell edge the SD mode is more suitable, it
can be seen that the proposed scheme will give benefits in
between.
The rest of this paper is organized as follows. The SMSFBC-OFDM system model is presented in Section II. Capacity analysis, simulations and discussions are provided in
Section III. The conclusion is presented in Section IV.
II. SM-OSFBC-OFDM T RANSMISSION S CHEME
In this section, we will explain the transmission structure of the JDM scheme based on combining SM and Orthogonal Space-Frequency Block Code (OSFBC). Following
this, we propose a linear two-stage receiver, which is an
extension of Least-Square (LS) receiver in [8], where the
linear reception technique is used for Spatially-Multiplexed
Orthogonal Space-Time Block Coded Orthogonal Frequency
Division Multiplexing (SM-OSTBC-OFDM) system based on
Zero Forcing (ZF) criterion. In this part, we investigate the
two-stage linear receiver with both ZF and Minimum Mean
Square Error (MMSE) criterion.
A. Joint Diversity and Multiplexing based Transmitter
We denote the number of SM branches at the transmitter
side and number of receive antennas as P and Q respectively.
We have N number of sub-carriers in the system.
Figure 1 explains the basic transmitter architecture. At
first source bits are Forward Error Correction (FEC) coded
and bit interleaved. The interleaved bit stream is baseband
modulated using an appropriate constellation diagram, such
as Binary Phase Shift Keying (BPSK), Quadrature Amplitude
Modulation (QAM) etc. We denote this baseband modulated
symbols as mk . The sequences of mk is demultiplexed into
m1 , . . . , mP vectors. mp is transmitted via pth spatial channel.
For every pth SM branch, we implement a block coding
across the sub-carriers, thus SFBC is included in the system.
For pth SM branch, we have ∆p number of antennas where
SFBC can be implemented. When ∆p = ∆, ∀p, then we have
∆ ∗ P number of transmit antennas at the transmission side.
When ∆ = 2, we can use well-known Alamouti coding [3]
across the sub-carriers.
(δ)
For pth SM branch, mp is coded into two vectors, mp ; δ =
1, 2. Thus, the output of the SFBC encoder block of the pth
−m∗p,2
m∗p,1
Following this, we define
mp,1 mp,3
mp,o =
mp,2 mp,4
mp,e =
...
...
mp,N −1
mp,N
−m∗p,N
mp,N −1
mp,N
m∗p,N −1
. . . mp,N −3
. . . mp,N −2
(1)
(2)
(3)
(4)
(1)
Using these equations, we can write that mp,o = mp,o ,
(1)
(2)
(2)
mp,e = −m∗p,e , mp,o = mp,e , mp,e = m∗p,o .
After SM and SFBC operations, IFFT modulation is performed and Cyclic Prefix (CP) is added before transmission via
respective transmit antenna. Transmitted time domain samples,
(δ)
(δ)
(δ)
(δ)
xp , can be related to mp as, xp = FH {mp }.
B. Two-Stage Linear Receiver
In [9], a two stage interference cancellation receiver scheme
for STBC is presented. This receiver treats one of the branches
as the interfering source for the other one. This receiver is
used to derive a linear reception technique for SM-OSTBCOFDM system in [8]. In this work, we adopt a similar
receiver structure for our Spatially-Multiplexed Orthogonal
Space-Frequency Block Coded Orthogonal Frequency Division Multiplexing (SM-OSFBC-OFDM) system.
We consider P = 2, ∆ = 2 and Q = 2. We assume
perfect time and frequency synchronization is achieved in
the system. Thus, we can represent the system in frequency
domain notations. We can write the equivalent system model
as the following:
zk = Hk mk + nk
(5)
where k ∈ [1, . . . , N2 ], Hk is defined as
 (1)
(2)
(1)
(2)
h11,o h11,o h12,o h12,o
(2)
(1)
(2)
 h(1)
 21,o h21,o h22,o h22,o
Hk =  (2)∗
(1)∗
(2)∗
 h11,e −h11,e h12,e −h(1)∗
12,e
(2)∗
(1)∗
(2)∗
(1)∗
h21,e −h21,e h22,e −h22,e





(6)
k
∗
∗ T
and zk
=
[z1,o z1,e z2,o
z2,e
]k , mk
=
T
∗
∗ T
[m1,o m1,e m2,o m2,e ]k , nk = [n1,o n2,o n1,e n2,e ]k .
We denote coherence bandwidth and sub-carrier spacing
as Bc and ∆f respectively. We define severely frequencyselective scenario when coherence bandwidth is smaller than
a pair of sub-carrier bandwidth, i.e. ∆f < Bc < 2∆f . In this
case, we use a tool called ’Companion Matrix’ explained in
Appendix I.
We can represent (5) as
z = Hi | Hj m + n
(7)
with
"
Hi =
(1)
h1,o
(2)∗
h1,e
(2)
h1,o
(1)∗
−h1,e
#
"
& Hj =
(1)
h2,o
(2)∗
h2,e
(2)
h2,o
(1)∗
−h2,e
#
ei
We denote the companion matrices of Hi and Hj as H
e j respectively. We define a new matrix H
e = [H
ei H
e j ]T
and H
with
"
"
#
#
(1)H
(2)T
(1)H
(2)T
h1,e
h1,o
h2,e
h2,o
e
e
Hi =
& Hj =
(2)H
(1)T
(2)H
(1)T
h1,e
−h1,o
h2,e
−h2,o
Now, at the beginning of the receiver, we can filter the
received signal z like following:
"
#
e
H
i
e =
e
Hi | Hj m + Hn
z0 = Hz
(8)
ej
H
Now, (8) can be written as
0
α1 I G12
0
e
z =
m + Hn
(9)
0
G21 α2 I
0
0
(1)H (1)
(2)T (2)∗
where α1
=
h1,e h1,o + h1,o h1,e , α2
=
(1)H (1)
(2)T (2)∗
h2,e h2,o + h2,o h2,e
and G12 , −G21 , shown in
Eq. (10) form an orthogonal pair as defined in Appendix I.
Now we define an LS receiver W as
0
1
α2 I
−G12
(11)
W= 0
0
γ
−G21
α1 I
TABLE I
OFDM S IMULATION PARAMETERS
Parameters
System bandwidth, B
Carrier frequency, fc
User mobility,v
OFDM sub-carriers, N
Subcarrier spacing, ∆f = B/N
CP length, NCP
Total samples in OFDM Symbol
with CP, Ns = N + NCP
Symbol duration, Ts = Tu + TCP
OFDM symbols/frame, Nf
Frame duration, Tf = Nf Ts
Data Symbol mapping
Channel coding scheme
Indoor
Indoor
20MHz
5.4 GHz
3 kmph
200 kmph
64
256
312.5 kHz
78.13 kHz
16
100
80
356
4.0 µs
17.8 µs
16
64.0 µs
284.8 µs
QPSK
1
-rate convolutional coding
2
B. Theoretical Capacity Analysis
The theoretical outage capacity of SFBC-OFDM, STBCOFDM, SM-OFDM and SM-SFBC-OFDM systems are evaluated in this section via a semi-analytical Monte-Carlo simulation approach. This is done primarily for indoor environment.
First, the indoor channel is simulated using the exponential
model mentioned above. Then the instantaneous channel capacity is obtained using the simulated CTF based on the
0
0
0
where γ = α1 α2 −[G12 (1, 1)G12 (2, 2) − G12 (1, 2)G12 (2, 1)]. following equation:
Thus, the estimated symbol vector can be written as
N −1
h
i
ρ
1 X
e
b = Wz0 = m + WHn
m
(12)
log2 det IQ + Hk H∗k
(14)
C=
N
P
k=0
In relation to severely frequency-selective scenario, we define moderately frequency-selective scenario when Bc > 2∆f , where ρ is the transmit SNR and Hk is the equivalent effective
th
and in that case we can easily say that neighboring sub-carriers CTF of k sub-carrier. Equivalent CTF means the CTF at
the particular sub-carrier at the receiver, as shown in (5).
have identical channel frequency response.
The MMSE receiver can be implemented in the same simple The above instantaneous capacity is derived for each channel
way. Defining the new constants then we can rewrite (12) as realization and then the Cumulative Distribution Function
(CDF) of the instantaneous channel capacity is plotted in
0
1
β2 I
−G12
b = 0
(13) Figure 2 for outdoor scenario. For a large number of random
m
0
δ
−G21
β1 I
channels, the outage and mean capacity can be determined
where β1 = α1 + σn , β2 = α2 + σn , with σn noise from these figures. In our case, we have simulated 5,000
0
0
0
variance on one receive antenna, and δ = β1 β2 − random channels and obtained the CDFs.
We have compared the system capacity of diversity only
[G12 (1, 1)G12 (2, 2) − G12 (1, 2)G12 (2, 1)].
schemes, multiplexing only schemes and hybrid diversityIII. A NALYSIS , S IMULATIONS AND D ISCUSSIONS
multiplexing schemes. For diversity only schemes, 2×1 SFBC
A. System Parameters
and STBC are presented. For multiplexing schemes, 2 × 2 and
We have used two simulation scenarios as explained in 4 × 2 multiplexing schemes are used. Obviously our scheme
Figure I. For all our analysis and simulations, we have becomes 4 × 2 hybrid scheme.
confined ourselves to the case of dual transmit and receive
We define ’10% outage capacity’ as the system capacity
antenna MIMO system with 2 antennas per spatial multiplex- in bits/second/Hz (bps/Hz) above which the system capacity
ing branches (i.e. Q = 2, P = 2 and ∆ = 2). We assume remains at least 90% of the connection time. According to
that perfect time and frequency synchronization is established. Figure 2, diversity only schemes (i.e. STBC and SFBC) have
We also assume that perfect channel estimation values for similar outage capacity characteristics, approximately at 1.6
each sub-carrier for both the spatial channels are available bps/Hz. In contrast to this, 2 × 2 spatial multiplexing only
at the receiver. We use exponential channel model to generate scheme has 10% outage capacity of 4.2 bps/Hz, compared to
corresponding Channel Impulse Response (CIR) and Channel our 4 × 2 hybrid schemes at 6.2 bps/Hz. The maximum outage
Transfer Function (CTF) of the channel. In our exponential capacity of any 4 × 2 ’open loop’ MIMO schemes can be 6.9
model, power delay profile of the channel is exponentially bps/Hz. The last outage capacity value is an upper bound for
distributed with decay between the first and last impulse as any 4 × 2 ’open loop’ MIMO scheme. This is achievable with
best available source, channel and S-F coding. In our case,
-40dB.
"
G12 =
(1)H
(1)
(2)T
(2)∗
h1,e h2,o + h1,o h2,e
(2)H (1)
(1)T (2)∗
h1,e h2,o − h1,o h2,e
(1)H
(2)
(2)T
(1)∗
h1,e h2,o − h1,o h2,e
(2)H (2)
(1)T (1)∗
h1,e h2,o + h1,o h2,e
"
#
; G21 =
k
(1)
(2)T
(1)H
(2)
(2)T
(1)∗
h2,e h1,o − h2,o h1,e
(2)H (2)
(1)T (1)∗
h2,e h1,o + h2,o h1,e
(10)
0
10
ZF−BLAST
MMSE−BLAST
ML
SM−OSFBC, ZF−Lin
SM−OSFBC, MMSE−Lin
−1
FEP
10
−2
10
2
CDF of the correponding capacity of at 10 dB SNR
4
6
8
1
10
12
SNR, dB
14
16
18
20
Fig. 4. FER performance of diversity only and hybrid schemes in outdoor
scenario
0.9
0.8
4x2 SM−SFBC−OFDM
2x1 SFBC−OFDM
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0.9065 0.907 0.9075 0.908 0.9085 0.909
b/s/Hz
0
1.8208
0.7
0.6
0.5
Probability that quantity < abscissa
Probability that quantity < abscissa
(2)∗
Modulation:QPSK, Coded FER, Outdoor
even though we have simple convolutional code as the FEC
code and simple Alamouti scheme as the S-F code, it can
be seen that the capacity performance is very close to the
optimum boundary.
In terms of mean capacity, we see that the schemes obtain
1.9374, 1.7482, 8.1031, 7.5242 and 5.5859 bps/Hz respectively. These values are obtained by finding the mean value
of simulation data that are used in Figure 2. Thus the hybrid
scheme gains 1.9383 bps/Hz of mean capacity compared to
2 × 2 spatial multiplexing only schemes. This is achieved
by introducing 2 more antennas and by incorporating SFBC
across each spatial multiplexing branch.
(1)H
h2,e h1,o + h2,o h1,e
(2)H (1)
(1)T (2)∗
h2,e h1,o − h2,o h1,e
0.4
0.3
2x1 Alamouti−SFBC
2x1 Alamouti−STBC
4x2 SM−OFDM
4x2 SM−SFBC−OFDM
2x2 SM−OFDM
0.2
0.1
0
0
2
4
6
8
Capacity in bps/Hz
10
12
14
Fig. 2. CDF of theoretical capacity of corresponding MIMO-OFDM systems
Modulation:QPSK, Coded FER, Indoor
0
10
ZF−BLAST
MMSE−BLAST
ML
OSFBC
SM−OSFBC, ZF−Lin
SM−OSFBC, MMSE−Lin
1.821
1.8212
b/s/Hz
1.8214
1.8216
Fig. 5. CDF of spectral efficiency of corresponding MIMO-OFDM systems
C. FER Analysis
−1
FEP
10
−2
10
0
2
4
6
8
10
SNR, dB
12
14
16
18
20
Fig. 3. FER performance of diversity only and hybrid schemes in indoor
scenario
Figures 3 and 4 show the Frame Error Rate (FER) performance of the JDM schemes in indoor and outdoor scenario
respectively. In indoor scenario, a frame consists of N ∗ Lf ∗
M ∗ P ∗ Rc = 64 ∗ 16 ∗ 2 ∗ 2 ∗ 0.5 = 2048 source bits, while in
outdoor scenario, it is 256∗16∗2∗2∗0.5 = 8192 source bits in
one frame. All the schemes use Quadrature Phase Shift Keying
(QPSK) as the baseband modulation scheme. As a reference,
performance of optimum ML receiver for 2 × 2 SM scheme is
plotted along with the other schemes.
For various transmit antenna configurations, the total transmit power was kept constant, thus, the SNR at the x-axis
reflects total SNR of the systems. We note that 2 × 2 Spa-
#
k
tial Multiplexed Orthogonal Frequency Division Multiplexing
(SM-OFDM) performs worse in terms of FER compared to
2 × 1 SFBC-OFDM system. In SM-OFDM system, we get
a higher rate, but we lose in diversity. Considering this, we
can see that 4 × 2 SM-OSFBC-OFDM performs better than
SFBC-OFDM system in terms of FER. In this case, not only
the diversity gain is achieved, but also spatial multiplexing is
realized. This clearly shows the benefits obtained by adding
spatial dimensions at the transmitter and using SFBC in the
SM branches. For instance, SM-OSFBC MMSE-Lin achieves a
gain of 2dB, compared to MMSE-BLAST at an FER of 10−3
in indoor scenario as seen in Figure 3.
Similar trend is also noted in outdoor scenario. Including
more antennas for transmitter SFBC offers immense benefit.
But, of course, it is clear that outdoor channel is more
frequency selective, thus all the systems require more SNR
compared to indoor scenario for any FER reference point.
D. Spectral Efficiency Analysis
It is expected that the achievable spectral efficiency of the
system appears to be as close as possible to the upper bound.
In our case, the upper bound is shown in Figure 2 as 4 × 2
SM-OFDM system capacity. We have simulated the spectral
efficiency in the following way. For every frame realizations,
we simulate the channel CTF, and we run the simulations for
1000 times with different AWGN contents. Then we find out
the FER that can be used according to following equation to
find our instantaneous spectral efficiency,
Nb (1 − F ER)/Tf
Br
=
(15)
B
B
where Es is the spectral efficiency, Br is the data rate, Nb
is the number of source bits. For ’Outdoor’ parameters, we
have obtained the spectral efficiency curves for SFBC-OFDM
and SM-SFBC-OFDM as it can be seen in Figure 5. The
10% outage capacity is seen to be 0.9 bps/Hz and 1.8 bps/Hz
respectively. It has to be noted that the theoretical capacity (as
shown in Figure 2) is the upper bound achievable if one uses
the best channel coding, the best space-frequency coding and
optimum receiver in terms of error rate performances.
Here we show (Figure 5) that the spectral efficiency achievable with SM-SFBC is nearly one-quarter of the theoretical
capacity optimum (1.8212 bps/Hz against 8.1031 bps/Hz),
therefore by using better channel coding (e.g. Turbo-codes
or LDPC) or a space-frequency coding optimized for the
combination with SM (i.e. Alamouti is optimized for each SM
barnch separately, what we need is a spatial code optimum for
both the SM branches together), it should be possible to further
increase the spectral efficiency.
Es =
IV. C ONCLUSION
A combination of OSFBC and SM in one transmission
scheme for OFDM systems has been presented, such that
both spatial diversity and multiplexing benefits are possible
to achieve. It is found that SM-OSFBC-OFDM system is near
to the optimum system capacity for any 4 × 2 MIMO-OFDM
system. Our scheme is compared via simulations with OSFBC
and VBLAST based SM techniques.
It can be interesting to study this hybrid MIMO schemes
for multi-user scenario. Future works that will extend the
present single-user link-level analysis to a multi-user system,
will provide more insights about the achievable advantages
of the proposed hybrid scheme when multiuser diversity is
present in the system.
R EFERENCES
[1] A.J. Paulraj, R. Nabar & D. Gore, Introduction to Space-Time Wireless
Communications, 1st ed. Cambridge University Press, September 2003.
[2] M. I. Rahman et al., “Multi-antenna Techniques in Multi-user OFDM Systems,” Aalborg University, Denmark, JADE project Deliverable, D3.2[1],
September 2004.
[3] S. M. Alamouti, “A Simple Transmit Diversity Technique for Wireless
Communications,” IEEE JSAC, vol. 16, no. 8, October 1998.
[4] K.F. Lee, & D.B. Williams, “A Space-time Coded Transmitter Diversity
Technique for Frequency Selective Fading Channels,” in IEEE Sensor
Array and Multichannel Signal Processing Workshop, Cambridge, USA,
March 2000, pp. 149–152.
[5] ——, “A Space-Frequency Transmitter Diversity Technique for OFDM
Systems,” in IEEE GLOBECOM, vol. 3, November-December 2000, pp.
1473–1477.
[6] P.W. Wolniansky et al., “V-BLAST: An Architecture for Realizing Very
High Data Rates Over the Rich-Scattering Wireless Channel,” in Proc.
IEEE-URSI International Symposium on Signals, Systems and Electronics, Pisa, Italy, May 1998.
[7] Y. Li et al., “MIMO-OFDM for Wireless Communications: Signal Detection with Enhanced Channel Estimation,” IEEE Trans. Comm., vol. 50,
no. 9, September 2002.
[8] X. Zhuang et al., “Transmit Diversity and Spatial Multiplexing in FourTransmit-Antenna OFDM,” in Proc. of ICC, vol. 4, May 2003, pp. 2316
– 2320.
[9] A. Stamoulis, Z. Liu & G.B. Giannakis, “Space-Time Block-Coded
OFDMA With Linear Precoding for Multirate Services,” IEEE Trans.
on Signal Processing, vol. 50, no. 1, pp. 19–129, January 2002.
A PPENDIX I
C OMPANION M ATRIX
Let us define a matrix H as
h11 h12
H=
h21 h22
e as
We can define another pair matrix H
−hT22 hT12
e
H=
hT21 −hT11
(16)
(17)
e is an orthogonal pair. hij , ∀i, j,
The matrix pair H and H
e are matrices
is a column vector of size m × 1, thus H and H
of sizes 2m × 2 and 2 × 2m respectively.