Simplified Decoding for Some Non–Coherent
Codes over the Grassmannian
∗ Ecole
Antonio M. Cipriano∗ , Inès Kammoun† and Jean–Claude Belfiore∗
Nationale Supérieure des Télécommunications, Dept. ComElec, Paris, France, {cipriano,belfiore}@enst.fr
† ISECS, Route Menzel Chaker km 0.5, B.P. 868, 3018 Sfax, Tunisia, [email protected]
Abstract— We consider multiple–antenna non–coherent communications over Rayleigh flat fading channels and single–
antenna non–coherent systems with unknown channel phase.
For both systems, we propose some sub–optimal simplified
decodings for the class of unitary space–time codes obtained via
the exponential map [1]. To this aim, we use the geometrical
interpretation of these codes as sets of points over the Grassmann
manifold GT,M . We compare these codes with other propositions
in the literature, for which a simplified decoder exists.
I. I NTRODUCTION
The design of space–time codes for multiple–antenna channels with no channel state information at the receiver (CSIR)
has received less attention than the case with full CSIR, despite
the high information rates achievable by these systems [2],
[3]. In fact, even if the parameters which control some useful
performance indicators (e.g. the pairwise error probability) are
well known [4], [5], the lack of appropriate mathematical tools
to manipulate them constitutes one of the major difficulties
code designers have to cope with. An approach to solve this
problem is the so–called training–based strategy [6], [7]. Part
of the frame period is dedicated to send a pilot signal so that
a rough channel estimation is possible. Then, coding methods
developed for the coherent space–time codes can be applied.
Another approach, developed in this article, focuses on unitary
codebooks, since they are optimal from both a rate and an error
probability perspective at high SNR [3], [5]. In this case, a
parameterization of the set of unitary matrices must be chosen
[1], [8]. The parameterization allows us to control to some
extent code parameters, but the map is highly non–linear, so
that finding an effective decoding strategy is often difficult.
In this work we discuss some simplified decodings for
unitary codes obtained via the exponential parameterization
of the Grassmannian GT,M , the manifold of M –dimensional
subspaces of CT with T ≥ 2M , as in [1]. Based on a
geometric interpretation of the coding procedure, we implement a threshold detector and some more complex decoding
strategies based on Schnorr–Euchner and Pohst algorithms.
We compare them with the Generalized Likelihood Ratio Test
(GLRT) performances obtained by exhaustive search. Finally,
in the case of one and two transmit and receiver antennas, we
compare the performance of our codes with other propositions
for which a simplified decoding exists [6]–[9].
This work was supported by the Italian Ministry of University and Research
(MIUR) within the framework of the PRIMO project FIRB RBNE018RFY.
II. S YSTEM M ODEL
Consider a block Rayleigh flat fading channel, with M
antennas at the transmitter, N antennas at the receiver. The
transmission is made by blocks of T symbol periods. Inside
the block, the fading is assumed constant. The received signal,
named also received super–symbol is
YT ×N = XT ×M HM ×N + M/(ρT )WT ×N ,
(1)
where the subscripts indicate matrix dimensions. Channel
entries and noise entries are i.i.d. circularly complex Gaussian
variables with unit variance, ρ is the average SNR per receive
antenna and per transmit super-symbol X. X ∈ C, a codebook
of unitary T ×M matrices.
In the case M = N = 1, we use a channel model with
unknown phase only
(2)
y = x ejϕh + w/ ρT
to better discriminate among the performances of different receivers [9]. We suppose that the random phase ϕh is uniformly
distributed in [0, 2π).
At the receiver the decision is made according to the GLRT,
which does not require the knowledge of channel statistics
since it is defined as X̂ = arg maxX∈C supH p(Y|X, H)
[5]. In our case, that is i.i.d. fadings and unitary codebook,
the GLRT is equivalent to the Maximum Likelihood (ML)
criterion [5], and it takes the form
X̂ = arg max ||Y† X||2F = arg min ||Y† X⊥ ||2F
X∈C
X∈C
(3)
where X⊥ is the orthogonal complement of X, so that [X X⊥ ]
is square unitary, || · ||F is the Frobenius norm and (·)† is
the transpose conjugate. We assume N = M and 2M ≤ T ,
because it is the optimal choice of the transmit antenna number
from an information theoretical perspective [3] and it permits
to have full diversity [5].
III. C ODING P ROCEDURE
A. The exponential parameterization
Each codeword X of C can be seen as a point over GT,M ,
i.e. it represents the subspace generated by its columns. We
adopt the exponential parameterization of the Grassmannian
presented in [1]: {X = exp(αB)Xref , B ∈ B}. B is a full–
rate codebook designed for coherent systems, formed by M ×
(T − M ) complex matrices. Xref = exp(0T,M ) = IT,M =
0-7803-8939-5/05/$20.00 (C) 2005 IEEE
[IM 0T −M ]† is called the reference subspace. The factor α > 0
controls the distribution of the principal angles between X and
Xref . The distribution of the principal angles among different
codewords in C is imposed by the choice of the code B itself.
Let
0M
B
[X X⊥ ] = exp
.
(4)
−B† 0T −M
†
Let B = UΛV be the thin singular value decomposition
(SVD) [10] of B, where U is M × M and unitary, V is
(T − M ) × M and has orthonormal columns, Λ is M × M
and diagonal. The Cosine-Sine (CS) decomposition [10] of (4)
yields
USV†
UCU†
⊥
,
X =
(5)
X=
−VSU†
VCV† + V⊥ (V⊥ )†
where C = cos Λ and S = sin Λ. Hence Λ is the matrix of
the principal angles between X and Xref . Multiplying by α
modifies only the singular values matrix of αB which becomes
αΛ. It is possible to inverse the map. Let X† = [X†1 X†2 ],
where X1 is a positive defined Hermitian M × M matrix, X2
is (T − M ) × M , and apply the thin SVD to −X2 = V2 SU†2 .
If all the singular values in S are distinct, U2 = U D is equal
to U up to a right diagonal factor D = diag(d1 , . . . , dM ) with
|dm | = 1 for all m [10]. V2 is uniquely determined by U2
and D. As long as U2 and V2 comes from the same matrix
X2 , B = U2 sin−1 (S)V2† can be recovered, since the diagonal
factor annihilates. Since C2 + S2 = IM , the columns of X are
orthogonal. Condition 0 < αλ ≤ π/2 on the generic singular
value of αB prevents the folding up of the constellation over
the Grassmannian [1].
B. The One Transmit Antenna Case
where λ represents the principal angle between the subspace
spanned by x and the reference subspace xref = e1 =
[1 0 . . . 0]† . It is possible to find an explicit inverse map from
GT,1 to the set of vectors b ∈ CT −1 whose norm b = λ
satisfies λ < π/2 (see [11]). Let z = [z1∗ . . . zT∗ ]† ∈ GT,1 be
of the form (6), i.e. z1 > 0. Then the corresponding bz is
ρz
[z2 . . . zT ] ,
sin ρz
∆1 , ∆2 = tr{∆†1 ∆2 } = 2 Re(tr{B1 B†2 })
ρz = arccos z1 .
(7)
For codewords with small principal angle (λ → 0), map (6)
can be linearized (x† [1 − b]).
C. Metrics
The exponential map has a clear geometrical interpretation.
†
The T ×M matrix ∆ = −X⊥
ref B represents the direction of
X ∈ GT,M in the space tangent at Xref , denoted by Tπ [12].
In the following, with a slight abuse, we say that B “lies on
(8)
where ∆i is the direction of Xi and Re(·) is the real part.
From the canonical metric (8) a canonical distance is derived
dcan (X1 , X2 ) = ||∆1 − ∆2 ||2F = 2||B1 − B2 ||2F .
(9)
Different distances d(X1 , X2 ) can be defined over the manifold, all depending on the principal angles [12]. A simple link
between these distances and (9) seems hard to find.
IV. D ECODING
The continuous maximization problem
Yc = arg max ||Y† X||2F
(10)
X∈GT ,M
has a well known solution: Yc is the eigenspace spanned by
the first M greatest eigenvalues of YY † . Supposing that the
non–zero eigenvalues of YY† are all distinct, in our case, the
solution is unique when N ≥ M , since Y in (1) can not have
rank greater than N .
The maximization in (3) is made over a discrete set contained in GT,M . In this case the problem has always a solution,
since it is a maximum over a discrete set, but a reduced
complexity algorithm is hard to find.
ST
ST
w
xref
xref
(a)
If M = 1, matrices X and B become respectively the T ×1
vector x and the 1 × (T − 1) vector b. Map (4) simplifies in
cos λ
x=
,
λ = b < π/2 ,
(6)
− sinλ λ b†
b†z = −
(tangent space) Tπ ”, to refer to this interpretation. A canonical
inner product is defined in tangent space Tπ as
(b)
x
yc
λc
o
Lr
θr
Fig. 1. (a) The noise (dotted sphere) uniformly perturbs the direction of x.
(b) The level set Lr is given by s ∈ ST so that ψ(s) = cos θr .
A. Simplified decoding, case M = 1
For the case M = N = 1, a simplified decoder unable to
achieve GLRT performance was proposed in [11]: 1) calculate
yc = y/y, z = yc e−i arg(yc,1 ) , and bz via the inverse
map (7); 2) find the closest codeword to bz (e.g. via a
sphere decoder or threshold detector for uncoded QAMT −1
constellations). We propose another simplified decoder which
is suboptimal, more complex than the previous one, but able
to get closer to the GLRT performance.
A sent codeword x lies by construction on the complex
sphere ST = {s ∈ CT : s = 1} and it represents a
1–dimensional subspace of CT . The direction of x is not
modified by the channel phase ϕh , but it is perturbed by the
AWGN in a uniform way, since the noise process has spherical
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symmetry (see Fig. 1 (a)). The GLRT function in this case is
ψ(x) = |x† y|/y = |x† yc | = cos(λx,yc ), where λx,yc is the
principal angle between x and yc . The level set1 Lr of ψ(x)
over ST corresponding to the value cos θr is formed by the
vectors x ∈ ST with the same principal angle θr with yc . Lr
is a complex sphere2 ST −1 centered at o = cos θr yc and of
radius sin θr (see Fig. 1 (b)). The parameter θr can be linked
to the noise variance in order to have a certain probability to
find the sent x inside the part of ST bounded by Lr .
We would like to use an efficient closest point search
[13] in ST , but this is not possible, because the exponential
map does not transform a lattice in tangent space Tπ into a
lattice in ST . Conversely, we can use this algorithm in Tπ ,
since B is carved from a lattice. However, due to the map
non–linearity, the detection rule in Tπ , which corresponds
to the GLRT in the Grassmannian, does not coincide with
the smallest Euclidean distance except for points close to the
reference subspace (see Sec. III-B and the linearization of the
exponential map). Hence, our idea is to map back to Tπ the
level set Lr , and obtain its preimage L−1
r . As a matter of fact,
the codeword of B corresponding to the sent codeword will be
inside the subset of Tπ bounded by L−1
r with high probability.
Our proposition is to list all codewords inside this subset by
using the Pohst algorithm [13]. Then, GLRT metrics of these
“candidate” codewords are calculated and the codeword with
maximum metric is selected. This decoding strategy will be
called local GLRT, because the GLRT is performed only on a
list of codewords.
Due to lack of space, we rapidly sketch how to bound
with a family of spheres. To calculate L−1
L−1
r
r , we express
Lr in the orthonormal basis {z, f1 , . . . , fT −1 }, where z =
yc e−i arg(yc,1 ) , f1 = (e1 − cos λc z)/ sin λc , and the other
vectors complete the basis of CT . Then
Lr
= {eiξr (cos θr z + eiξ1 sin θr (sin θ1 f1 + cos θ1 v)) :
ξr , ξ1 ∈ [0, 2π), θ1 ∈ [0, π/2], v = 1}
where v belongs to a (T − 2)–dimensional sphere in the
space spanned by f2 , . . . , fT −1 . The parameter eiξr is used
to impose form (6) to the points of Lr , in order to invert
the map and to obtain L−1
r . It can be shown that, if 0 ≤
λc ≤ max(θr , π/2 − θr ) (condition verified with probability
one when noise variance and θr go to zero), L−1
r is contained
inside the closed surface bounded by the set of spheres in
CT −1 of constant radius R = (λc + θr ) sin θr / sin(λc + θr )
and centered at a set of points bo = bo ẑ with ẑ = −[z2∗ . . . zT∗ ],
and bo ∈ C depends on ξ1 , θ1 , θr and λc .
In order to simplify the decoder, the set of spheres is approximated by three spheres (hence our algorithm is suboptimal).
To summarize, once θr < π/4 is fixed, the proposed decoding
algorithm is as follows: 1) calculate z, bz and λc , 2) if λc <
π/5, the non–linearity effect can be neglected: find the closest
1 The level set of a function ψ : CT → R corresponding to the value r is
the subset of CT where ψ = r.
2 In fact, there are two spheres at the opposite side of ST , but one of them
is sufficient to locate the subspaces.
codeword to bz in Tπ and stop. 3) If π/5 ≤ λc ≤ π/2 − θr ,
perform three sphere searches of radius R and centers b1,2,3 =
oeiϕ1,2,3 ẑ, with ϕ1,2,3 = 0, ± arcsin(tan θr tan λc )/2, o =
ρ0 cos θr sin λc / sin ρ0 , where ρ0 = arccos(cos θr cos λc ). 4)
If π/2 − θr < λc < π/2, perform three sphere searches
with the same parameters as in case 3) except ϕ1,2,3 =
0, ± arccos(1 − (R/o)2 /2). 5) calculate GLRT metrics on
the list coming from step 3) or 4) and select the codeword
corresponding to the maximum.
The computational complexity of the proposed algorithm
is roughly O(8(T − 1)3 ), due to the Pohst algorithm. About
Lav T cells of memory are needed to store the candidate
codewords, where Lav is the average length of the list. Lav
can be controlled, since the lattice is known and θr can be
chosen to set properly the radius R of the sphere (if θr → 0,
then R → 0) according to the noise level.
In the case N > 1, the same decoder can be used, preceded
by an algorithm which exploits the different received vector
to provide a better estimation yc of x.
B. Simplified decoding, case M > 1
Codes over GT,M have no apparent structure, since the
exponential map is non–linear. Moreover, in this case the
analytical study of Sec. IV-A cannot be pursued. Again,
our idea is to take the decision on the tangent space at
Xref = IT,M , exploiting the properties of code B. The
detector chooses the codeword whose direction in Tπ is closest
†
to the direction ∆z = −X⊥
ref Bz , corresponding to the solution
of (10). To this aim, a maximization of the cosine between
pairs of directions ∆z , ∆/(||∆z ||F ||∆||F ) is performed.
From (9) it is equivalent to
Bz − B2F
.
(11)
B∈B Bz F BF
We give an algorithm to calculate Bz , which yields the exact
sent matrix when no additive white Gaussian noise is present
in the channel. The received codeword can always be written
as
min
Y = Z He ,
with
Z† Z = IM , He ∈ CM ×N
(12)
where Z has the form of Equation (5) and He is in general a
noisy estimation of the channel.
1) Let Yc1 be the upper M × M sub-matrix of Yc given
by (10). Calculate the SVD Yc1 = U1 C1 V1† .
2) Z in form of (5) is Z = Yc V1 U†1 . Inverse the exponential map to obtain Bz , as described in Sec. III-A.
Finally, He (a rough channel estimation) can be obtained from
(12), i.e. He = Z† Y.
Then, an efficient method to implement (11) is the Schnorr–
Euchner algorithm [13] as long as ||B||F is constant for all the
codewords in B. If this is not the case, the same algorithm can
be used for each code subset with constant ||B||F . To improve
the performance, the GLRT metric can be computed on the list
of codewords (local GLRT). Since the code lattice is known
a priori, the radius of the sphere can be controlled. Then, the
Pohst algorithm efficiently calculates lists with stable lengths,
roughly independent of the SNR [14].
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0
10
C. Mismatched Decoding
Another decoder for the case M > 1 is introduced. This
heuristic mismatched distance takes into account the effect of
the channel:
−1
10
(13)
Rule (13) can be justified as follows. If all the codewords
have small principal angles with respect to Xref , in a first
order
approximation,
the exponential map (5) becomes X† =
IM −BX , since cos(Λ) IM and sin(Λ) Λ. Then,
the system approximatively reduces to the training–based
systems as in [7]. The derivation of (13) is trivial. For our
codebooks, the approximation is good only for codewords
close to Xref (in the sense of small principal angles). On the
other hand, when codewords are far away from the reference
space, the mismatched metric does not take into account the
non–linearity of the exponential map. Thus, a performance
degradation is expected. The computational complexity of this
decoder can be roughly estimated as O(8[(T − M )M ]3 ) +
O(Lav T N M ), where Lav is the average length of candidate
codewords lists. The first term is due to the Pohst algorithm,
while the second one is due to the GLRT metric calculations
of the codewords in the list.
Pe supersymbol
dheu (Z, X) = ||H†e (Bz − B)||F .
−2
10
exhaustive, T=3
new algoritm, T = 3
new algorithm, T = 6
inversion + threshold detector ,T = 3
inversion + threshold detector, T = 6
Sweldens’ algorithm, T = 3
Sweldens’ algorithm, T = 6
−3
10
10
11
12
13
14
15
SNR [dB]
16
17
18
19
20
Fig. 2.
Performances of Grassmann codes obtained from B = (16–
QAM)T −1 . Comparison with codes from (16–PSK)T −1 proposed in [9].
Case G
, spectral efficiency η = 2 bits/s/Hz, M = N = 2
4,2
0
10
code 2 exhaustive
code 1 exhaustive
code 2 mismatched list 10−30
code 1 mismatched list 10−30
code 1 threshold
code 1 Pohst
training + TAST sphere
222
In the GT,1 case, with N = 1, we use a uncoded unit–
with T = 3 and T = 6.
energy (16–QAM)T√−1 constellation,
The factor α = π 10/(2(3 2(T − 1) + 1)) is justified in
[11]. Spectral efficiency η is 4(1 − 1/T ) bits/s/Hz.
In the G4,2 case, let M = N = 2 and T = 4, s =
[s∗1 s∗2 s∗3 s∗4 ]† . Symbols si are drawn from a 4–QAM and an
8–QAM, so that the spectral efficiency η is respectively 2
bits/s/Hz and 3 bits/s/Hz. Two codes are compared. Code B1
is proposed in [15], gives the non–coherent code C1 (see also
[1]), and its codewords are given by
1
ϑ(s3 + φs4 )
s1 + φs2
√
B=
s1 − φs2
2 ϑ(s3 − φs4 )
with ϑ2 = φ = eiπ/4 . It holds ||B||F = ||vec(B)|| =
||Φ1 s|| = ||s||, because Φ1 is unitary [15], and where vec (B)
is a vector built with the columns of B.
Code B2 , presented in [16], gives the non–coherent code
C2 , and its codewords are given by
1 φr (s1 + rs2 ) φr (s3 + rs4 )
B= √
5 iφ̄r (s3 − r̄s4 ) φ̄r (s1 + r̄s2 )
√
√
where r = (1+ 5)/2, r̄ = (1− 5)/2, φr = 1+i(1−r) and
φ̄r = 1 + i(1 − r̄). As in the previous case ||B||F = ||Φ2 s|| =
||s|| is constant for M–PSK alphabets, because Φ2 is unitary.
For the 4–QAM case, α has been chosen to minimize the
super–symbol error probability at 15 dB SNR. For C1 we chose
8–QAM case,
α = 0.566 and for C2 we set α = 0.6. In the M
we maximized the minimum product distance m=1 sin2 θm
where θ1 , . . . , θM are the principal angles between any two
codewords [1]. For C1 we chose α = 1.267, while for C2
α = 1.400.
Zhao et al. [17]
−1
10
Pe supersymbol
V. E XAMPLES : GT,1 AND G4,2
−2
10
−3
10
6
8
10
12
14
16
18
20
SNR per supersymbol and rx antenna [dB]
22
24
26
Fig. 3. Performances of code C1 and C2 with symbols si drawn from 4–
QAM constellation, with GLRT and mismatched metric. Comparison with the
training–based systems.
For this two codes, by letting bz = vec(Bz ) and b =
vec(B), since Φ1 and Φ2 are unitary, minimizing the metric
(11) is equivalent to minimize bz − b = sz − s, k =
1, 2. Hence the decoder (11) can be implemented by a simple
threshold decoder.
VI. S IMULATIONS AND D ISCUSSION
In the M = 1 case, Fig. 2, shows that our simplified
decoder closely follows the GLRT performance for T = 3.
For T = 6, the size of the codebook (220 ) prevented us from
simulating the GLRT performance. However, in both cases
the new decoder works better than the previous simplified
decoder. Moreover, since the codes are drawn from uncoded
QAM constellations, when λc < π/5, we do not use a
Pohst algorithm but just a threshold detector. We make a
comparison for the same spectral efficiency as Sweldens’ fast
0-7803-8939-5/05/$20.00 (C) 2005 IEEE
spectral efficiency η = 3 bits/s/Hz, M = N = 2
0
by the GLRT in the tangent space. Hence, we succeeded in
coping with the strong non–linearity introduced by the non–
linear map and a sub–optimal decoder based on the Pohst
algorithm has been proposed. In the multiple–antenna case,
the analytical approach was not feasible. We proposed other
simplified decodings which, nevertheless, do not achieve full
diversity. Finally, we made comparisons with other codes
proposed in the literature.
10
−1
Pe supersymbol
10
−2
10
ACKNOWLEDGEMENT
The first author wishes to thank Slim Chabbouh for many
helpful discussions.
R EFERENCES
G code 2 exhaustive
4,2
G code 1 exhaustive
4,2
G code 2 mismatched list 10−30
4,2
G code 1 mismatched list 10−30
4,2
G training + TAST exhaustive
4,2
222
G training + TAST sphere
4,2
222
−3
10
10
12
14
16
18
20
22
SNR per supersymbol and rx antenna [dB]
24
26
Fig. 4. Performances of code C1 and C2 with symbols si drawn from rectangular 8–QAM constellation, with GLRT and mismatched metric. Comparison
with training–based systems.
block decoding algorithm [9], which works only with PSKT −1
constellations.
In the case M = 2, with ML decoding (Fig. 3), code
C2 slightly outperforms code C1 . This is due to a better
distances distribution of the principal angles between pairs of
codewords. The change of the curve slope (loss of diversity)
for the threshold detector is due to the fact that the metric (11)
does not take into account the channel effect, and it does not
correspond to the GLRT rule in GT,M .
The performance of the proposed threshold decoder slightly
outperforms the behavior of the decoders in [8, fig. 2] and [17]
below 18 dB SNR, while the trend changes at higher SNR.
Our detector is very simple, it inverts the map, it rotates the
symbol and makes two comparisons. The super–symbol error
probability can be decreased performing a local GLRT over a
list of points around bz . In Fig. 3, by using lists of average
length 12 an improvement of about 4 dB can be appreciated,
but the diversity is not recovered.
At low SNR, the training–based codes of [6], [7], [18] with
sphere decoder have comparable performance to the proposed
local GLRT decoder, where the list of points is calculated
using the mismatched metric (13) (see Fig. 3 and Fig. 4). Our
proposed decoder does not maintain full diversity, as explained
in Sec. IV–B. However, even if training–based non–unitary
codes are far better than unitary ones [18], our proposed
unitary codes, which do not follow a genuine training–based
approach, are still comparable with the best training–based
non–unitary codes at spectral efficiencies of 2 and 3 bits/s/Hz
(with ML decoding for every SNR, with mismatched decoding
only at low SNR).
VII. C ONCLUSIONS
In this paper, we presented some simplified decoding strategies for unitary space–time codes obtained via the exponential
map from coherent codes. In the one transmit antenna case, we
were able to analyze the shape of the decision regions induced
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