Int. J. Communications, Network and System Sciences, 2011, 4, 430-435
doi:10.4236/ijcns.2011.47051 Published Online July 2011 (http://www.SciRP.org/journal/ijcns)
A Design Method of Noncoherent Unitary
Space-Time Codes
Li Peng, Qiuping Peng, Lingling Yang
Department of Electronics and Information Engineering, Huazhong University of Science and Technology,
Wuhan National Laboratory for Optoelectronics, Wuhan , China
Received May 13, 2011; revised June 18, 2011; accepted June 24, 2011
Abstract
We generalized an constructing method of noncoherent unitary space time codes (N-USTC) over Rayleigh
flat fading channels. A family of N-USTCs with T symbol peroids, M transmit and N receive antennas was
constructed by the exponential mapping method based on the tangent subspace of the Grassmann manifold.
This exponential mapping method can transform the coherent space time codes (C-STC) into the N-USTC on
the Grassmann manifold. We infered an universal framework of constructing a C-STC that is designed by
using the algebraic number theory and has full rate and full diversity (FRFD) for t symbol periods and same
antennas, where M, N, T, t are general positive integer. We discussed the constraint condition that the expo-
nential mapping has only one solution, from which we presented an approach of searching the optimum ad-
justive factor αopt that can generate an optimum noncoherent codeword. For different code parameters M, N,
T, t and the optimum adjustive factor αopt, we gave the simulation results of the several N-USTCs.1
Keywords: Noncoherent Uintary Space-Time Codes (N-USTC), Coherent Space-Time Codes (C-STC),
Grassmann Manifold, Degree of Freedom, Exponenatial Map, Full Rate and Full Diversity
(FRFD)
1. Introduction
The noncoherent unitary space-time code (N-USTC) in
[1-4] provided a potential solution for the multiple an-
tennas communication in fading channel that neither
transmitter nor receiver knows the channel state imfor-
mation (CSI). This paper generalized an constructing
method for a family of the N-USTCs based on the
Grassmann manifold. The system models on noncoherent
and cohenent channel are comparatively built. Starting
from the basic theory of the Grassmann manifold [4], a
basic thought of designing the Grassmannian unitary
space-time matrix was described. That is the exponent
mapping method [5,6] from the
M
t C-STC to the
TM N-USTC for the MIMO system with
M
trans-
mit and N receive antennas, where t and T are co-
herent and noncoherent symbol periods, respectively,
and ,,,
M
NTt are general positive integer, TtM
and TMt.
In order to map the
M
t C-STC into the TM

N-USTC, firstly, one must consider how to construct the
M
t

C-STC. Many literatures [7-11] discussed multi-
farious methods of constructing the C-STCs. Enlightened
by [7-11] and other literatures (omitted in reference as
the limitation of length), we discussed a method of con-
structing the
M
t

universal C-STCs with FDFR based
on the algebraic number theory. Therefore, we created
four kinds of matrices: uncoded symbol matrix
S
, lin-
ear combinatorial matrix
L
, rotated matrix R and lin-
ear combinatorial symbol matrix Z that is =ZLSR
formed by the linear combinatorial technique of the
symbols of constellations, such as q-PSK or q-QAM, and
then we get the the coded matrix of a C-STC by trans-
forming matrix Z.
In the mapping process from the
M
t C-STC into
the TM

N-USTC, we discussed the constraint condi-
tion of the only one solution of the the exponent map,
from which we discover that the optimum codeword of
the Grassmannian N-USTC can be obtained by searching
the optimum adjustive factor opt

. Simulation tests show
that for BPSK constellation symbols, when T is un-
changed and antenna number =
M
N increases, the
1This work is supported by National Natural Science Foundation o
f
China under Grant No. 61071069.

L. PENG ET AL. 431
spectral efficiency increases and the performance of the
bit error rate (BER) also advances, or when =
M
N is
unchanged and T increases, so do the spectral effi-
ciency and the BER performance; for QPSK constella-
tion symbols, when =2MN and 5T, the spectral
efficiency achieves 2.4 bits/Hz/s but at the cost of sacri-
ficing the BER performance.
2. System Model and Background
Knowledge
2.1. System Model
We focus on the block fading channel model on which
the fading coefficients are assumed to be constant during
T periods of one codeword and to change independ-
ently from one codeword to the next. Under the assump-
tion of no inter-symbol interference, the noncoherent
channel model with codeword periods TM is
TNTM MNTN
TYXHW (1)
For the convenience of comparison and application
later on, we simultaneously give the coherent channel
model with codeword periods tM:
N
tNMMtNt
YHBW (2)
where Y is TN received signal matrix for nonco-
herent model or Nt matrix for coherent model,
H
is
M
N or NM fading coefficients matrix and
W is TN or Nt additive noise matrix. Elements
of
H
and W are assumed to be the independent and
identically distributed complex Gaussian random vari-
ables respectively from distribution

0,1CN and

2
0,CN

. TM
X
and
M
t
B
are noncoherent and co-
herent transmit signal matrices, respectively.
2.2. Grassmann Manifolds and Its Tangential
Space
Manifold is a topologic space which is locally homeo-
morphic to the Euclidian space. More formally, Every
point on n-dimensional manifold has a neighborhood
homeomorphic to n-dimensional Euclidian space n
R.
We consider a set of all
M
-dimension linear sub-
spaces in T-dimension complex space. This set has the
structure of manifold, called Grassmann manifold and
denoted by ,
C
TM
G, and its definition [12] is:


†
,
C
TM M
GΦΦΦ I (3)
where “†” denotes transpose for real number or conju-
gate transpose for complex number; Φ denotes the
subspace spanned by
M
column vectors in an TM

unitary matrix Φ. ,
C
TM
G can also be represented by the
quotient space of the unitary group

nU[12], i.e.

 


,
C
TM TMTMGU UU (4)
As the real dimension of the unitary group


nU is


2
dimRnn

U, one can obtain the real dimension of
,
C
TM
G:
 
2
22
,
dim 2
C
RTM TM TMMTM G
[13] according to (4). So the complex dimension of
,
C
TM
G is


,
dim C
CTM
M
TMG which means that
the N-USTCs on ,
C
TM
G have

M
TM degrees of
freedom, and the maximal symbol rate is

 
1
M
MT
[3].
Literature [5,6] introduces that the tangential space of
any a point on ,
C
TM
G forms a set of matrices as follows:
†
0
0
TM




Δ
B
QB (5)
where

M
TM
BC and the point Q can be chosen
arbitrarily, i.e., for simplified calculation, one can choose

TMM MTM M 





†
-
,BII 0as a reference subspace on
,
C
TM
G. The dimension of the tangential space defined by
(5) is also


M
TM. According to the theory of Lie-
group, i.e., the point of the tangential space on ,
C
TM
G
can be projected into the point of ,
C
TM
G by exponent
map, the point
X
of ,
C
TM
G can be denoted by the ex-
ponent form of the tangential space:
†
0
exp 0
TM TM







B
X
I
B (6)
(6) shows a complicated computing task, but it can be
simplified by the technique of the singular value decom-
pose (SVD) of matrix. B is disposed by the SVD as
follows:

†
MM
M
TMTM TM
 
ΛBUV (7)
where U and V are unitary matrices, and the form of
Λ is:
1000 0
00
000 0
M







Λ



(8)
where 1,,
M

  are the singular values of matrix B.
Putting (7) into (6), one can obtain the simplified
TM
X
:
†
†
TM
TM 



UCU
XVSU (9)
where
1
cos 0
0cos
MM
M








C,

L. PENG ET AL. 432

†
1
sin 0
0
0sin
TM M
M








S.
3. Coherent Space-Time Codes
Most methods of constructing the C-STCs with FRFD
are to efficaciously combine all information symbols i
s
(1, 2,,itM ) to form the coding matrix
M
t
B
, where
all i
s
belong to one of constellations, such as q-PSK or
q-QAM, etc. If we adopt the technique of linear combi-
nation to design
M
t
B
, then the rank and determinant
properties of

M
ti
s
B is equivalent to them of


M
tiMt j
s
sBB , where


,1,ijMt and ij

. Let
r be the minimal rank available of any codeword matrix

M
ti
s
B. According to the design criterion of determi-
nate in [10], under the linear combination of all i
s
of
forming
M
t
B
, we can obtain a FRFD matrix and its rank
rM, so the maximum coding gain
11
11
rM
rM
jj
jj







can be guaranteed. Being enligh-
tened by using the algebraic number theory to construct
the C-STCs in [8,10,11], we investigate how to design
the universal coherent matrix
M
t
B
with FRFD for
,,
M
Nt being general positive integer. The applied de-
sign step is shown as follows.
a) We first create three kinds of matrices: uncoded
symbol matrix
S
, linear combinatorial matrix
L
(also
named left-multiplied matrix) and rotated matrix
R
(also named right-multiplied matrix), they have next ge-
neral forms:


11 11
22 12
2
21
221 1
12 21
1
,
1
1,
1
10 0
00
00
MtM
MtM
MM Mt
M
MM
MMM
t
ss s
ss s
ss s
jj j
jj j
 
 
 










































 



 

S
L
R
where 12 ,, ,
tM
s
ss take from the constelltions; let
t



; choosing j makes the determinate of matrix
L
be unequal to zero. Let i e

  which is an alge-
braic number [10], here 1i

 and

is a parameter
of needing the optimization design so that

is searched
in


0, π2 to maximize the coding gain.
b) For
S
left-multiplied by
L
and right-multiplied
by R, one can get the linear combinatorial symbol matrix
M
t

Z like (10) as follows:
c) In the linear combinatorial symbol matrix
M
t

Z,
circulant-right-shifting the second row one time, the third
row two times,…, the final row (i.e., the
M
th row)
1
M

times, respectively, one can get the coded matrix
like (11) as follows:

21
11 1(1)
221 1
22 2(1)
1
12 21
2
11
121 22
100
1
00
1
00
1
...
M
MMt
MM
MMt
Mt
t
MMM
MM Mt
MM
MMMM
ss s
ss s
jj j
ss s
jj j
ssss ss
 

 



 

 





 

 


 

 

 


 

 






 



ZLSR
 



 



 

1
11 21
1
1
11 21
11 11
121 22
11
11
11 21
1211 21
121 22
...
...
t
Mt Mt
M
Mt
t
Mt Mt
MM tt
MMM M
MM
Mt
tM
Mt Mt
MMMMMM
MM MM
ss
s
sjs
sjsjssjsjs
js
sjs
sj sjssj sjs
j



 


 

 


 
 
 

 





 


 








21MM
Mt
s 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(10)

L. PENG ET AL. 433

 


 


 





1
11 21
11
121 221
2
1
12 22
11)2111
12
11 11
1
11
11 21
t
Mt Mt
MM
MMM MM
Mt
t
t
Mt Mt
Mt MtMM
M
MM MM
Mt Mt Mt
tM
Mt Mt
M
ss
ssssss
s
sjs
sjs
sjsj s
js js
sjs
j

 

 



 

 

 
 
 


 


 


 









B



1
12
121
12 21
21
2
M
MM
MMM
MM
M
M
Mt
sjs
sj sjs
js
s

 






 
 
 
 
 
 
 
 
 
 
 

 

 

 
 


(11)
Several examples of the C-STCs are presented as fol-
lows. Let 2Mt, according to (10), we have:


13
22
24
12 34
1234
110
10
ss
ss j
ss ss
sjs sjs



 

  
 
 
 

 



ZLSR
If 1j, then 22 0
Z which means the linear
combinatorial form of information symbols is lost. If
1j

, the linear combinatorial operation is retained.
For

π4
2ei
  , we can get the 22 C-STC matrix
which is same as those in [5,6]


12 34
22
34 12
s
sss
ss ss

 


 






B
Again let 3M

, 4t

, we have
2
14710
22
342 5 8 112
22
36912 3
10 00
1000
100 0
100 0
ssss
j jssss
jjssss
 
 


























ZLSR
when
π
6
4ei
  ,
2π
3
ei j, 34 Z is full rank. Thus we can get the 34

C-STC matrix like (12) as follows:






 
 
222232
1234567891011 12
322 2222222
341011121 234567 89
222 3222222
789101112123 456
ssssssssss ss
s
jsj ss jsj ssjsj ssjsj s
s jsjssjsjssjs jss jsjs

 


 
 
 
 
  
 
 
 
 
B (12)
Similarly, we can get 23
B
, 24
B
, 25
B
and 33

B
which and all above will be applied to simulation testing
later on.
4. Noncoherent Space-Time Codes
Literatures [5,6] introduce the design criterion of the
Grassmann N-USTC. Let i
 and
j
 denote the sub-
spaces spanned by the column vectors of i
X
and
j
X
,
respectively. Let

12
,,,
M

  denote the principal
angles between i
 and
j
, then the chordal product
distance between two points i
X
and
j
X
on ,
C
TM
G is:
2
,
1
sin
M
ij m
m



 (13)
The design criterion of the Grassmann N-USTC C is
to make the minimal chordal product distance achieve
the maximum, i.e. , max min
ij
ij
XXC C
. It is known from the
expression (6) that the product ,0i Θ of the chordal dis-
tances between the subspace i  and the reference sub-
space 0

is equal to the product of all singular values
in matrix i
B
whose
M
column vectors span the sub-
space i

. The design criterion of a C-STC is to maxi-
mize its coding gain, which is equal to maximizing the
minimum product of singular values of codeword matrix.
Therefore we can use the matrix
M
t
B
of (11) to design
the matrix B in (6).
The exponential map from
M
t
B
to TM
X
must be the
monotone and reversible, which requires that the expo-
nential map of (6) is the reversible map, i.e., (9) exists
the reversible matrix. So cos m

and sin m

in (9)
should be the monotone function, then the constraint

L. PENG ET AL. 434
condition of m

is:

max π/2,0,1, ,1
mMt
mBmM
  (14)
where

mMt
B  is the mth singular value of any
codeword
M
t
B which is equal to the mth principal
angle between any  and 0

. Therefore, a conceiv-
able skill is that taking a scale

, called the adjustive
factor which multiplies the codeword matrix
M
t
B, can
guarantee the map to be monotone and reversible. Thus
(6) can be rewritten as follows:

†
0
exp 0
TM TM




 





B
X
I
B (15)
Obviously,

only affects the singular value of the
matrix
M
t
B. Let

max
0,  denote a range of

values,
the method of optimum searching

is described as
follows.
Let , ,C
ij TM
XXG be two distinct N-USTC code-
words, the SVD of the matrix †
j
i
X
X is ††
ji
Σ
X
XUV,
where Σ is a diagonal matrix formed by the singular
values 1,,
M

 . The
M
principal angles between the
subspaces i  and
j
 spanned respectively by i
X
and
j
X
are 1 cos
mm



, 0,1,,1mM, substi-
tuting 1 cos
mm



 for the expression (13), we can
get:

2
,
1
1
M
ij m
m


 
 (16)
Under the condition of making ,ij  maximize, by
searching

in

max
0,  , we can get the optimum
opt

.
Now we design TM
X
by mapping exponentially
M
t
B
to ,
C
TM
G. The design step is:
(a) For
M
transmit antennas and t coherent peri-
ods, according to (10) and (11), design the C-STC matrix
M
t
B
from the information symbol 12 ,, ,
tM
s
ss.
(b) Substitute
M
t
B
for B in (15), search the opti-
mum adjustive factor

, and construct the exponential
mapping matrix

†
0
0
M
t
Mt







B
EB.
(c) According to (7) and the above
E
matrix, apply-
ing the SVD to

opt Mt
 B, we get the noncoherent
codeword TM
X
like (9), and TMt. □
5. Examples and Numerical Simulation
Results
According to the above presented method, this section
gives several examples of the N-USTCs whose numeri-
cal simulation curves are shown as Figure 1. Let

DMTM denote the degree of freedom. Suppose
modulation is q-PSK with symbol number p. So the
spectral efficiency of the N-USTC is


2
log
TM pDT

 bits/Hz/s.
Example 1: Compare two curves of solid line with
black dot and dash line with circle in Figure 1. For sys-
tem 2MN

 and QPSK modulation with 4p

, let
2t

, we construct the C-STC 22
B
,where

π4
2ei
  , 1j

. When 3t, we get 23

B
,
where

π4
3ei
  , 1j

. As TMt, having
4T

for 2t

and 5T

for 3t, corresponding
to 4D

and 6D

, we compute 42 2

 bits/Hz/s
and 52 2.4



bits/Hz/s, respectively. We map the
C-STC 22

B
into the N-USTC 42
X
on 4,2
C
G and
23

B
into 52

X
on 5,2
C
G which correspond to the op-
timum adjustive factor 42
,0.29 opt Q

 and 52
,0.25 opt Q

,
respectively. At 5 10

bit error rate (BER), 52

X
out-
perform 42

X
about 3 dB. Obviously, under all para-
meter being same except T increasing, the N-USTC
BER performance and the spectral efficiency are im-
proved.
Example 2: Compare four curves of solid line with
black square, dash line with white square, solid line with
black diamond and dash line with white diamond in Fig-
ure 1. For system 2MN

 and BPSK modulation
with 2p

, let 2, 3, 4, 5t

, get 4,5, 6, 7T and
4, 6,8,10D

, so 421



, 52 1.2

, 62 1.33



and
72 1.43



bits/Hz/s, respectively. We map 22

B
,
23

B
, 24

B
and 26

B
into 42
X
, 52
X
, 62

X
and
72

X
whose factors are 42
,0.41
opt B

, 52
,0.36
opt B

,
62
,0.32 opt B

 and 72
,0.29 opt B

. At 5
10  BER, the
performance of the N-USTC improve about 0.5 - 1.0 dB
and the spectral efficiency increases along with T in-
creasing.
Example 3: Compare two curves with solid line with
black triangle and dash line with white triangle in Figure
1. For system 3MN

 and BPSK modulation with
2p

, let 3,4t

, get 6, 7T and 9,1 2D

, so
63 1.5



and 73 1.71



bits/Hz/s, respectively. We
map 33

B
and 34

B
into 63
X
and 73
X
whose fac-
Figure 1. Performance comparison of several N-USTCs.

L. PENG ET AL. 435
tors are 63
,0.24 opt B

 and 73
,0.22 opt B

. At 5
10

BER,
the performance of the N-USTC improve about 0.8 dB
from 6T to 7T, and the spectral efficiency also
increases 0.2 bits/Hz/s.
6. Conclusions
A specific step that maps the coherent space-time matrix
into the noncoherent space-time matrix by means of the
exponent form of the tangential space of Grassmann
manifold was summed up for designing the N-USTCs.
Especially, our work makes the structural parameters
,,,
M
NTt with regard to both the N-USTC based on the
Grassmann manifold and the C-STC based on the alge-
braic number theory be able to be designed more flexibly.
We also discovered that in the discussed family of
Grassmannian N-USTC, the optimum codeword can be
obtained by searching the optimum adjustive factor opt

.
It is noticed that the design of the parameter j in left-
multiplied matrix
L
is open problem, we will track this
problem in the future.
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