1
An Iterative Geometric Mean Decomposition
Algorithm for MIMO Communications Systems
Chiao-En Chen, Member, IEEE, Yu-Cheng Tsai, and Chia-Hsiang Yang, Member, IEEE
Abstract—This paper presents an iterative geometric mean
decomposition (IGMD) algorithm for multiple-input-multipleoutput (MIMO) wireless communications. In contrast to the
conventional geometric mean decomposition (GMD) algorithm,
the proposed IGMD does not require the explicit Kth root
computation in the preprocessing stage but depends on a carefully
constructed iterative procedure that generates the GMD in
its limit. We prove analytically that the proposed IGMD is
guaranteed to converge to the exact GMD under certain sufficient
conditions, and propose three different constructions achieving
this condition. Both numerical simulations and complexity analysis of the proposed IGMD have been conducted and compared
with the conventional GMD. Simulation results show that our new
IGMD algorithm effectively reduces the complexity overhead and
hence is more advantageous for low-complexity implementations.
Index Terms—Geometric mean decomposition (GMD), MIMO,
QR, SVD, Tomlinson-Harashima precoding (THP)
I. I NTRODUCTION
M
ULTIPLE-input-multiple-output (MIMO) communications [1, 2] have continued to be one of the key
technologies of the next generation wireless systems because
of their potential to provide higher data rate and better
reliability compared to the conventional single-input-singleoutput (SISO) systems. When the channel state information
(CSI) is available at both the transmitter and receiver, it is
well known that the closed-loop gain can be further acquired
by jointly designing the precoder and the equalizer. Among
these closed-loop transceiver design schemes, singular-valuedecomposition (SVD)-based linear transceiver decomposes
the MIMO channel into multiple parallel subchannels and
is known to achieve the channel capacity if proper power
allocation [3] is applied. However, because of the variation
of the signal-to-noise-ratio (SNR) in each subchannel, the bit
error rate (BER) performance is dominated by the subchannel
with the worst SNR. Consequently, without sophisticated bitallocation schemes, fundamental trade-off between the BER
and capacity cannot be avoided in this type of design [4, 5].
In addition to the SVD-based linear design, geometricmean-decomposition (GMD)-based nonlinear transceiver design has also been proposed [4, 6]. With the help of GMD
[6, 7], the MIMO channel is decomposed into multiple subchannels with identical SNR, and hence the simple identical
This work was supported by the National Science Council (NSC), Taiwan,
R.O.C. under Grant Number NSC 102-2221-E-194-009-MY2.
Chiao-En Chen is with the Department of Electrical Engineering and
the Department of Communications Engineering, National Chung Cheng
University, Chiayi, Taiwan, R.O.C. (e-mail: [email protected]).
Yu-Cheng Tsai and Chia-Hsiang Yang are with the Electronics Engineering
Department, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
bit allocation can be used for all subchannels. It has also been
shown that the GMD-based transceiver under the zero-forcing
(ZF) constraint asymptotically achieves both the optimal BER
and capacity at sufficiently high SNR. Because of these good
properties, various extensions and generalizations of GMDbased transceivers have been proposed in the literature [5, 8]–
[13].
As the GMD is the core of many advanced MIMO
transceiver designs, the associated implementation issues began to draw researchers’ attention [14, 15]. In [14], a scaled
GMD algorithm was proposed to simplify the detection logic.
In [15], the authors presented a constant throughput GMD
implementation which also supports hardware sharing between
precoding and signal detection modules. In this paper, a new
implementation issue of the GMD algorithm is addressed.
It is noted that the existing GMD algorithms require the
computation of the geometric mean (GM) σ̄ of all the positive
singular values in the pre-processing stage.
√ This requires the
capability of computing the Kth root K A of some positive
real number A and hence results in additional complexity
overhead.
In this paper, we propose1 an iterative GMD (IGMD)
algorithm based on the successive approximation method. The
advantages of the proposed IGMD algorithm as well as our
main contributions are summarized as follows.
1) A new algorithm for computing the geometric mean decomposition is proposed. Unlike the conventional GMD
algorithm, the proposed algorithm has a regular structure
that simplifies the control logics and is easier to accommodate different signal dimensions from the hardware
implementation perspective. Another important feature of
the proposed IGMD algorithm is that it does not require
the explicit Kth root computation of the geometric mean
σ̄ but depends on a carefully constructed iterative procedure that generates the GMD in its limit. The proposed
algorithm substantially reduces the complexity overhead
and hence is more advantageous compared to the conventional GMD for applications with limited computing
capability.
2) We prove analytically that the proposed IGMD algorithm
always converges to the exact GMD in its limit under
certain sufficient conditions. From the sufficient condition, we propose three different constructions: IGMDAM (arithmetic mean), IGMD-GM (geometric mean),
1 Although we we have focused on the GMD problem of a point-to-point
MIMO channel in this paper, the proposed IGMD algorithm can be easily
extended to other GMD applications such as the BD (block diagonal)-GMD
[8] precoders design in a multi-user MIMO scenario.
2
and IGMD-HM (harmonic mean), and verify their convergence numerically. We also find that the convergence
behaviour of the proposed IGMD not only depends on
the topological property of the mapping (to be designed)
but also on how the algorithms are initialized. Meansquare-error (MSE) and error rate performance using
different initializations such as QR factorization, QR
factorization with V-BLAST (Vertical-Bell Laboratories
Layered Space-Time) [16, 17] sorting, singular value
decomposition (SVD), and interleaved-SVD have been
studied through numerical simulations.
3) The proposed IGMD algorithm under various constructions and initializations is implemented using CORDIC
(COordinate Rotation DIgital Computer) arithmetic [18]
and compared with the conventional GMD from the
complexity perspective. The complexity of the building
blocks in the conventional GMD and the proposed IGMD
algorithms have been analyzed and the overall performance versus complexity tradeoff has been simulated.
The rest of this article is organized as follows. Section
II reviews the geometric mean decomposition as well as its
conventional implementation algorithm. Section III presents
the proposed new IGMD algorithm. Analytical proof for the
sufficient condition such that the proposed IGMD converges to
the exact GMD is then provided. In Section IV, we explicitly
show three different constructions of the proposed IGMD that
can satisfy the required sufficient condition. Section V studies
the MSE and error rate performance of different constructions
of the proposed IGMD algorithm. Detailed computational
complexity IGMD algorithm is also provided in comparison
to the conventional GMD. Finally, Section VI concludes the
paper.
Notations: Throughout this paper, matrices and vectors are
set in boldface, with uppercase letters for matrices and lower
case letters for vectors. The superscripts T , H denote the
transpose and conjugate transpose of a matrix, respectively.
We use diag{x1 , · · · , xK } to represent the diagonal matrix
with diagonal elements {x1 , · · · , xK }, [X]p,q to represent the
(p, q)th component of X, and [X]m:n,p:q to represent the submatrix formed by the consecutive mth to nth rows and pth to
qth columns of X. We use the expression A := B to denote the
in-place update operation in which the value in A is updated
by the value of B.
II. R EVIEW ON THE S TANDARD G EOMETRIC M EAN
D ECOMPOSITION A LGORITHM
In this section, we review the main results of the GMD and
its implementation.
From [6, 7], it has been shown that given an N × M matrix
H of rank K, there exists semi-unitary matrices Q ∈ CN ×K
and S ∈ CM ×K , and an upper triangular matrix R ∈ RK×K
such that
H = QRSH ,
(1)
where the diagonal elements of R are all identical and equal
to the geometric mean σ̄ of the positive singular values of H,
that is,
√
σ̄ = A =
K
K
Y
i=1
σi
!1/K
, for all i = 1, · · · , K,
(2)
where σ1 ≥ σ2 ≥ · · · ≥ σK > 0. The decomposition of H
in (1) is referred as the QRS decomposition [6] or GMD [7]
in the literature. An efficient implementation procedure [19]
which computes Q, R, and S from H is described as follows.
Step 1. The algorithm starts with the SVD of H, given by
H = UΣVH ,
(3)
where U ∈ CN ×K and V ∈ CM ×K are both semiunitary and Σ = diag{σ1 , · · · , σK }. The algorithm
then sets Q := U, S := V, and R := Σ for
initialization, and computes the geometric mean σ̄
of the positive singular values of H via (2). The
algorithm then starts from k := 1.
Step 2. At stage k, where k ranges from 1 to K − 1,
the algorithm performs the following procedure. The
algorithm first checks the (k, k)th element of R,
denoted as Rk,k . If Rk,k ≥ σ̄, then the algorithm
chooses some p > k such that Rp,p ≤ σ̄; otherwise
the algorithm chooses some p > k such that Rp,p ≥ σ̄.
After p has been determined, the algorithm swaps the
Rp,p with Rk+1,k+1 , Q:,p with Q:,k+1 , and S:,p with
S:,k+1 . This can be achieved by setting
R := P(k)T RP(k) ,
(k)
Q := QP
(k)
S := SP
(4)
,
(5)
,
(6)
where P(k) is the associated permutation matrix.
(k)
(k)
Step 3. Construct 2 × 2 matrices ΘL and ΘR as
1
cRk,k
sRk+1,k+1
(k)
,
(7)
ΘL =
cRk,k
σ̄ −sRk+1,k+1
c −s
(k)
ΘR =
,
(8)
s c
such that
Rk,k
(k)
ΘL
0
0
Rk+1,k+1
(k)
ΘR
=
σ̄
0
⋆
Rk,k Rk+1,k+1
σ̄
(9)
Here ⋆ represents some number (generally nonzero)
that we don’t care. It is straightforward to verify that
(9) can always be achieved by choosing
s
2
p
σ̄ 2 − Rk+1,k+1
c=
, s = 1 − c2 .
(10)
2
2
Rk,k − Rk+1,k+1
(k)
(k)
Step 4. Construct GL , and GR from the identity ma(k)
trix IK with the submatrix [GL ]k:k+1,k:k+1 and
(k)
(k)
(k)
[GR ]k:k+1,k:k+1 replaced by ΘL and ΘR , respectively. Update R, Q, and S as
(k)
(k)
R := GL RGR ,
(11)
QGT
L,
(12)
(13)
Q :=
S := SGR .
.
3
Step 5. If k = K−1, then the algorithm terminates; otherwise,
the algorithm updates k := k + 1 and goes back to
step 2.
It follows that the matrices Q, R, and S generated by the
above-mentioned QRS algorithm can be explicitly expressed
as
(1)T
Q =UP(1) GL
S
R
(K−1)T
· · · P(K−1) GL
,
(1)
(K−1)
=VP GR · · · P(K−1) GR
,
(K−1) (K−1)T
(1) (1)T
=GL
P
· · · GL P
Σ
(1) (1)
(K−1) (K−1)
· P GR · · · P
GR
.
(1)
(14)
(15)
(16)
Note that the aforementioned GMD algorithm has to compute the geometric mean σ̄ of all the positive singular values
as in (2), and hence requires the capability of computing Kth
root of A. For special cases where K = 2L with L being some
positive integer, it is possible to decompose the computation
of σ̄ into successive geometric mean computations of two
numbers:
r q
q
√
√
√
√
σ1 σ2 σ3 σ4 · · ·
σK−3 σK−2 σK−1 σK ,
σ̄ = · · ·
where the square root operation can be carried out efficiently
using CORDIC-based computing
√ [18]. On the other hand, for
general cases where K 6= 2L , K A has to be performed
with
√
much more efforts. One way of computing K A is by first
transforming A into logarithmic domain and then converting
it back after dividing by K. This approach requires a large
look-up table and a piecewise polynomial (including linear)
approximation device to realize both the logarithmic and
exponential functions and hence calls for a mass of memory to
ensure the accuracy for such high dynamic-range computation.
Another possible way of finding σ̄ for the general case is
to compute it iteratively using Newton’s type of Kth-root
algorithm [20], which often requires a good starting point
to ensure reasonable rate of convergence. A computationally
efficient Kth-root algorithm that can be viewed as a slight
modification of the Newton’s algorithm using the technique
of binary approximation is described in [21].
It is worthwhile noting that the accuracy of σ̄ plays an
important role in the conventional GMD algorithm. This
is because the conventional GMD algorithm proceeds in a
sequential fashion, and hence any numerical error in σ̄ not only
causes numerical errors at each stage but also propagates and
accumulates to all the later stages. As a consequence, σ̄ has
to be computed with sufficient accuracy for the conventional
GMD algorithm to function properly, which results in nonnegligible complexity overhead.
III. P ROPOSED I TERATIVE G EOMETRIC M EAN
D ECOMPOSITION A LGORITHM
To mitigate the complexity overhead in the pre-processing
stage of the conventional GMD algorithm, a new iterative
GMD algorithm is proposed in this section. The main idea of
the proposed IGMD is to properly design the planar rotations
similar to those used in the conventional GMD so that the
spread of the diagonal elements of the updated R matrix can
be gradually reduced as the algorithm proceeds. With proper
design, it can be shown that the proposed IGMD can achieve
the exact GMD in the limit without the computation of σ̄, and
hence avoids the requirement of Kth-root computation. As it
will be elaborated in Section V, this feature brings in performance advantages for applications with limited computing
capability. The proposed iterative GMD algorithm is described
as follows.
Initialization: Given the matrix H ∈ CN ×M of rank K ≤
min(N, M ), the algorithm starts with some general orthogonal
decomposition of H
H = ŨR̃ṼH ,
(17)
where Ũ ∈ CN ×K and Ṽ ∈ CM ×K are both semi-unitary,
and R̃ ∈ CK×K is upper-triangular. For general N , M , and
K, one can always choose the SVD for initialization. For
special cases where H is full column rank with M = K, other
orthogonal decompositions such as the QR decomposition [22]
can also be used.
The algorithm initializes by setting Q := Ũ, S := Ṽ, R :=
R̃, and starts with iteration index ℓ := 1.
Iteration: In each iteration, the algorithm performs K − 1
stages of operations with the stage index k ranging from 1 to
K − 1.
At stage k, the algorithm first computes the SVD for the
2 × 2 submatrix of R
Rk,k
Rk,k+1
(k) (k)H
Rk:k+1,k:k+1 =
= U(k)
,
γ Σ γ Vγ
0
Rk+1,k+1
(18)
(k)
(k)
where the singular matrices Uγ ∈ C2×2
n and Voγ ∈
(k)
(k)
(k)
2×2
C
are both unitary, and Σγ = diag σγ,1 , σγ,2 is a
(k)
(k)
diagonal matrix with singular values σγ,1 and σγ,2 . With(k)
(k)
out loss of generality, we assume σγ,2 ≤ σγ,1 . After the
singular values are obtained, carefully designed planar rotations are then applied to obtain an upper triangular matrix with positive diagonal elements Ω (Rk,k , Rk+1,k+1 ) and
Rk,k Rk+1,k+1 /Ω (Rk,k , Rk+1,k+1 ), where Ω is a continuous
mapping from (0, ∞) × (0, ∞) to (0, ∞) with some desired
property to be discussed in details shortly. In matrix notations,
we then have
#
"
Ω (Rk,k , Rk+1,k+1 )
⋆
(k) (k) (k)
.
ΦL Σγ Φ R =
Rk,k Rk+1,k+1
0
Ω(Rk,k ,Rk+1,k+1 )
(19)
(k)
(k)
Note that the planar rotations ΦL and ΦR applied in (19) aliT
h
(k)
(k)
ways exist as long as σγ,1 , σγ,2 multiplicatively majorizes
T
[Ω (Rk,k , Rk+1,k+1 ) , Rk,k Rk+1,k+1 /Ω (Rk,k , Rk+1,k+1 )] ,
(k)
(k)
or equivalently when σγ,2 ≤ Ω (Rk,k , Rk+1,k+1 ) ≤ σγ,1
(k)
holds [23, 24]. It is easy to verify that the matrices ΦL and
(k)
ΦR can be constructed as
"
#
(k)
(k)
1
cσγ,1 sσγ,2
(k)
,
(20)
ΦL =
(k)
(k)
Ω (Rk,k , Rk+1,k+1 ) −sσγ,2
cσγ,1
c −s
(k)
ΦR =
,
(21)
s c
4
where
v
2
u
(k)
u
2
p
u Ω (Rk,k , Rk+1,k+1 ) − σγ,2
, s = 1 − c2 . (22)
c=u
2 2
t
(k)
(k)
σγ,1 − σγ,2
Combining the relations in (18) and (19), we then obtain
(k)
(k)
ΘL Rk:k+1,k:k+1 ΘR
"
Ω (Rk,k , Rk+1,k+1 )
=
0
(k)
(k)
⋆
Rk,k Rk+1,k+1
Ω(Rk,k ,R)k+1,k+1)
(k)H
(k)
(k)
#
,
(k)
k:k+1,k:k+1
and ΘR , respectively. The matrices R, Q, and S are finally
updated as
(k)
(k)
R := GL RGR ,
Q :=
S :=
(24)
(k)T
QGL ,
(k)
SGR .
(25)
(26)
It is clear that R remains upper-triangular, whereas Q and S
both remain unitary after (24)-(26) are performed at the end
of each stage. If the stage index k is smaller than K − 1, the
algorithm sets k := k+1 and performs the procedure (18)-(26).
Otherwise, the algorithm sets the iteration index ℓ := ℓ+1 and
starts a new iteration unless the prescribed number of iterations
is attained.
For convenience of subsequent discussion, we denote Q[ℓ] ,
[ℓ]
R , and S[ℓ] as the updated Q, R, S, respectively, at the
end of (K − 1)th stage in the ℓth iteration. It is then easy to
verify that the following relations hold for the proposed IGMD
algorithm:
(1)T
Q[ℓ+1] =Q[ℓ] GL
(K−1)T
· · · GL
,
(1)
(K−1)
S[ℓ+1] =S[ℓ] GR · · · GR
,
(K−1)
(1) [ℓ] (1)
[ℓ+1]
R
=GL
· · · GL R GR
(27)
(28)
(K−1)
.
· · · GR
(29)
for all ℓ = 0, 1, · · · . Here Q[0] , S[0] , and R[0] are defined
as Ũ, o
Ṽ, and R̃, respectively.
The planary rotation matrices
n
n
o
(k)
K−1
(k)
ℓ→∞
k=1
k,k
IV. D ESIGN OF MAPPING Ω
For the ease of following discussions, we introduce several
new notations. For the given τ > 0, we define a subset A(τ ) ⊂
RK :
)
(
K
Y
K
xk = τ ,
A(τ ) = x ∈ R x > 0,
k=1
where x = [x1 , · · · , xK ]T . We also define continuous mappings T (j) : A(τ ) → A(τ ), j = 1, · · · , K − 1, given by

 

x1:j−1
x1:j−1
  Ω (xj , xj+1 ) 
 xj
.

 
xj xj+1
(31)
T (j) 

 xj+1  =  Ω(x
j ,xj+1 )
xj+2:K
xj+2:K
If we denote the vector on the main diagonal of R[ℓ] as r[ℓ] ,
then the diagonal vectors of R[ℓ+1] and R[ℓ] can be related
from (24) and (29) using the new notations
r[ℓ+1] = T (K−1) · · · T (2) T (1) r[ℓ]
···
(32)
(33)
= T r[ℓ] = T ℓ+1 r[0] ,
where
T (x) = T (K−1) ◦ T (K−2) ◦ · · · ◦ T (2) ◦ T (1) (x),
(34)
n o
r[0] = diag R̃ , and T ℓ+1 (x) is the (ℓ + 1)-fold repeated
composition of T (x). With these new notations, the main
results for designing Ω is given by the following Proposition.
Proposition 1 If the mapping Ω : (0, ∞) × (0, ∞) → (0, ∞)
satisfies the following property
z1 z2
≤ z1 + z2 ,
(35)
Ω(z1 , z2 ) +
Ω(z1 , z2 )
for all z1 , z2 > 0, with equality holds when z1 = z2 , then
K−1
GL
and GR
clearly also depend on the
k=1
k=1
iteration index ℓ, but the dependency is not denoted explicitly
in (27)-(29) for simplicity as long as no confusion results.
Unlike the conventional GMD that requires proper swapping
of (4)-(6) that depends on the value σ̄, the proposed IGMD
algorithm does not require the computation of σ̄ and always
performs on the diagonal elements of R with consecutive
indices at each stage. The IGMD algorithm therefore has a
more regular structure that simplifies the control logics and
is easier to accommodate to problems of different dimensions
from the implementation perspective. These advantages rely
k,k
for all k = 1, · · · , K. The exact GMD is therefore achieved
when the algorithm converges.
(23)
where ΘL = ΦL Uγ , and ΘR = Vγ ΦR . Because
(k)
(k)
ΘL and ΘR are both products of unitary matrices, they are
unitary matrices as well.
(k)
(k)
(k)
(k)
After ΘL and ΘR are obtained, GL and GR are then
from the identity
hconstructed
i
h
imatrix IK with the submatrix
(k)
(k)
(k)
GL
and GR
replaced by ΘL
k:k+1,k:k+1
(k)
on the careful design of Ω. In the following section, we show
that it is possible to design mapping Ω such that
!1/K
K h i
i
h
Y
[ℓ]
= σ̄ =
lim R
R̃
,
(30)
lim r[ℓ] = σ̄1.
ℓ→∞
Proof
QK
(36)
h
i
[0]
[0]
Given r[0] = r1 , · · · , rK = diag{R̃}, we let τ =
[0]
, and consider
k=1 rkP
K
F (x) = k=1 xk . From
the function F : A(τ ) → (0, ∞),
the arithmetic mean-geometric mean
(AM-GM) inequality, we have
!1/K
K
K
X
Y
xk ≥ K
F (x) =
= Kτ 1/K = K σ̄, (37)
xk
k=1
k=1
5
in which the absolute minimum of F (x) is attained in A(τ )
at x = σ̄1. In addition, we have the following inequality if
Proposition 1 holds:
j−1
K
X
X
F T (j) (x) =
xk + Ω(xj , xj+1 )
xk +
k=1
V. S IMULATION R ESULTS AND C OMPLEXITY C OMPARISON
k=j+2
xj xj+1
Ω(xj , xj+1 )
K
X
xk = F (x),
≤
+
(38)
k=1
with equality achieved when xj = xj+1 . It follows readily that
T (x), which is a composite mapping of T (1) , · · · , T (K−1) ,
satisfies
F (T (x)) ≤ F (x),
(39)
with the equality holds when x1 = x2 = · · · = xK . Consequently, y [ℓ−1] = F (T (r[ℓ−1] )) is a monotonically decreasing
sequence in (0, ∞), and hence is guaranteed to converge to
the greatest lower bound K σ̄ [25]. As F is continuous, we
then have
= K σ̄, (40)
= F lim T r[ℓ−1]
lim F T r[ℓ−1]
ℓ→∞
ℓ→∞
which is attained when limℓ→∞ T r[ℓ−1] = σ̄1. As a result,
we have limℓ→∞ r[ℓ] = limℓ→∞ T r[ℓ−1] = σ̄1, which
completes the proof.
There exists potentially many functions that satisfy condi√
tion (35). The geometric mean ΩGM (z1 , z2 ) = z1 z2 clearly
satisfies Proposition 1 as
√
z1 z2
ΩGM (z1 , z2 ) +
= 2 z1 z2 ≤ z1 + z2 , (41)
ΩGM (z1 , z2 )
because of the AM-GM inequality. In addition to ΩGM (z1 , z2 ),
the arithmetic mean ΩAM (z1 , z2 ) = (z1 + z2 )/2 is another
choice that also satisfies Proposition 1. This can be observed
by squaring both sides of the AM-GM inequality
4z1 z2 ≤ (z1 + z2 )2
⇔ (z1 + z2 )2 + 4z1 z2 ≤ 2(z1 + z2 )2
z1 + z2
2z1 z2
⇔
+
≤ z1 + z2
2
z1 + z2
constructions not only depends on the topological property
of the mapping but also depends on how the algorithms are
initialized.
(42)
As a result, ΩAM (z1 , z2 )+ ΩAMz1(zz12,z2 ) ≤ z1 +z2 , with equality
holds if and only if z1 = z2 . Note that ΩAMz1(zz12,z2 ) is simply the
harmonic mean (HM) function ΩHM (z1 , z2 ) = 2z1 z2 /(z1 +z2 )
satisfying ΩAM (z1 , z2 ) = z1 z2 /ΩHM (z1 , z2 ). It is then clear
that ΩHM (z1 , z2 ) also satisfies Proposition 1 from the same
relation we obtained in (42).
Based on ΩAM , ΩGM , and ΩHM , we can then construct
three different types of IGMD algorithms: IGMD-AM, IGMDGM, and IGMD-HM, respectively. As these mappings are
highly nonlinear, it is very difficult to compare the convergence
speed of the proposed algorithm analytically in these three
constructions. Hence we resort to computer simulations as
shown in Section V and leave the more challenging theoretical
analysis to our future work. In fact, as it will be observed from
the simulation results, the convergence behaviour of different
In this section, we present simulation results of three different types of the proposed IGMD algorithms. Throughout the
simulation, we assume standard K ×K i.i.d. (independent and
identically distributed) Rayleigh fading channel in which every
element in the channel matrix H is modelled as a zero-mean
circularly symmetric complex Gaussian random variable with
unit variance. To highlight the applicability of the proposed
algorithm in the challenging K 6= 2L case, we choose K = 7
in most of the simulation. Each simulation point in the figure
is averaged over 104 channel realizations.
A. Convergence of the proposed IGMD
Figures 1(a) and 1(b) show the MSE of the diagonal elements of R using SVD and QR factorization as initialization,
respectively. For SVD initialization, it is possible to exploit
the degrees-of-freedom from the ordering of singular values
to enable more efficient averaging in each stage. Based on this
idea, we propose to use an interleaved-SVD (intrlv-SVD) so
that a large Rk,k tends to be averaged with a small Rk+1,k+1
and vice versa. To be more specific, we use the following
factorization:
˜ Σ̃
˜ Ṽ
˜ H ∈ C7×7 ,
H = Ũ
(43)
˜ and
˜ = diag {[σ , σ , σ , σ , σ , σ , σ ]}, and Ũ
where Σ̃
1
7
2
6
3
5
4
˜ are the corresponding left and right singular matrices,
Ṽ
respectively. For QR initialization, we also propose to use
VBLAST ordering (VBQR) [16, 17] to speed up convergence.
This idea is motivated by the fact that the diagonal elements
of R̃ in VBQR generally has less spread [26] than those in
QR. Hence, using VBQR as initialization generally requires
fewer iterations to achieve the same MSE compared to that
of using standard QR initialization as shown in Fig. 1(b).
From Figs. 1(a) and 1(b), the simulation results show that
the IGMD-HM achieves the same MSE with smallest number
of iterations, followed by the IGMD-GM, and the IGMD-AM
when QR, VBQR, and SVD are used as initialization. On the
other hand, when intrlv-SVD is used, the IGMD-AM achieves
the same MSE with smallest number of iterations, followed by
the IGMD-GM, and finally the IGMD-HM.
In the second simulation setting, we investigate the error rate
performance of the proposed IGMD applied to a 7 × 7 GMDbased ZF Tomlinson-Harashima precoded (ZFTHP) MIMO
system [4] with 16-quadrature amplitude modulation. Figs.
2(a) and 2(b) show the error rate of the proposed IGMD using
QR and VBQR, respectively. By comparing Figs. 2(a) and
2(b), it is observed that VBQR provides a better initialization
for the proposed IGMD, and results in faster convergence. At
the first iteration, the error rate of IGMD-AM and IGMD-HM
appears to be similar. For iteration number greater than 1, the
IGMD-GM and IGMD-HM both outperform the IGMD-AM
and perform very close to the exact GMD after four iterations.
6
0
10
−1
10
−1
10
−2
10
10
BER
MSE
−2
ZFTHP−QR
IGMD−ZFTHP−QR−AM (iter=1)
IGMD−ZFTHP−QR−AM (iter=2)
IGMD−ZFTHP−QR−AM (iter=4)
IGMD−ZFTHP−QR−GM (iter=1)
IGMD−ZFTHP−QR−GM (iter=2)
IGMD−ZFTHP−QR−GM (iter=4)
IGMD−ZFTHP−QR−HM (iter=1)
IGMD−ZFTHP−QR−HM (iter=2)
IGMD−ZFTHP−QR−HM (iter=4)
GMD−ZFTHP
−3
−3
10
10
IGMD−SVD−AM
IGMD−SVD−GM
IGMD−SVD−HM
IGMD−intrlv−SVD−AM
IGMD−intrlv−SVD−GM
IGMD−intrlv−SVD−HM
−4
10
0
−4
10
5
10
15
0
5
10
Iterations
Eb/N0 (dB)
(a)
(a)
15
20
25
15
20
25
0
10
−1
10
−1
10
−2
10
BER
MSE
−2
10
−3
ZFTHP−VBQR
IGMD−ZFTHP−VBQR−AM (iter=1)
IGMD−ZFTHP−VBQR−AM (iter=2)
IGMD−ZFTHP−VBQR−AM (iter=4)
IGMD−ZFTHP−VBQR−GM (iter=1)
IGMD−ZFTHP−VBQR−GM (iter=2)
IGMD−ZFTHP−VBQR−GM (iter=4)
IGMD−ZFTHP−VBQR−HM (iter=1)
IGMD−ZFTHP−VBQR−HM (iter=2)
IGMD−ZFTHP−VBQR−HM (iter=4)
GMD−ZFTHP
−3
10
10
IGMD−QR−AM
IGMD−QR−GM
IGMD−QR−HM
IGMD−VBQR−AM
IGMD−VBQR−GM
IGMD−VBQR−HM
−4
10
0
−4
10
5
10
15
0
5
10
Iterations
Eb/N0 (dB)
(b)
(b)
Fig. 1. MSE comparison of the diagonal elements of R under proposed IGMD
using (a) SVD and interleaved-SVD and (b) QR and VB-QR as initialization.
Fig. 2. BER performance of the proposed Iterative GMD algorithm in a 7×7
MIMO ZFTHP system using (a) QR (b) VB-QR as initialization.
Figures 3(a) and 3(b) show the error rate of the proposed
IGMD using standard SVD and interleaved SVD, respectively.
When standard SVD is used, the IGMD-GM performs the
best, followed by the IGMD-HM, and the IGMD-AM for
sufficiently high SNR. On the contrary, when the interleaved
SVD is used, the IGMD-HM performs much worse compared
to the IGMD-AM and IGMD-GM. For most SNR region of
practical interests in this setting, the IGMD-AM is comparable
to the IGMD-GM for iteration number greater than 1. From
Fig. 2(b) and Fig. 3(b), it is also observed that the proposed
IGMD-intrlv-SVD-GM and IGMD-intrlv-SVD-AM achieve
even better error rates than the IGMD-VBQR-GM and IGMDVBQR-HM after four iterations.
B. Complexity Comparison
To highlight the complexity advantages, we compare our
proposed IGMD algorithm with the conventional GMD algorithm. The required Kth root algorithm in the conventional
GMD is implemented as in Algorithm 1 [21], in which
only elementary functions including comparison, addition, bit
shifting, and multiplication are required. For general cases
√
(A 6= 1), the output y in Algorithm 1 converges to K A
through iteratively narrowing the search range bounded by M .
The number of iterations n determines the number of binary
digits of accuracy.
To make a quantitative comparison in computational complexity, multiplications involved are taken into account. A typical 32-bit fixed-point representation ({sign, integer, fraction}
= {1, 4, 27}) for channel matrix H is adopted. For hardware
implementation, the dynamic range of the datapaths needs
to be taken into consideration. As a first-order estimate, an
N × M -bit multiplier can be regarded as M N -bit adders or
as N M -bit adders, and an N -bit adder can be treated as N/16
16-bit adder(s) [27, 28]. Hence, the number of the atomic 16bit equivalent additions is used as our complexity metric for
fair comparison.
Table I shows the required complexity of the building blocks
in the conventional GMD and the proposed IGMD algorithms.
In the Kth root algorithm as shown in Algorithm 1, the
dynamic range ofQthe multiplication increases drastically for
K
calculating A = i=1 σi and (y + M )K . For example, multiplications with output word length ranging from 64 to 224
7
−1
10
−2
BER
10
SVD
IGMD−ZFTHP−SVD−AM (iter=1)
IGMD−ZFTHP−SVD−AM (iter=2)
IGMD−ZFTHP−SVD−AM (iter=4)
IGMD−ZFTHP−SVD−GM (iter=1)
IGMD−ZFTHP−SVD−GM (iter=2)
IGMD−ZFTHP−SVD−GM (iter=4)
IGMD−ZFTHP−SVD−HM (iter=1)
IGMD−ZFTHP−SVD−HM (iter=2)
IGMD−ZFTHP−SVD−HM (iter=4)
GMD−ZFTHP
−3
10
−4
10
0
5
10
15
20
25
Eb/N0 (dB)
(a)
−1
10
√
Algorithm 1: K A Algorithm using Binary Approximation [21]
Input: A, K, n
Output: y
M = 1;
if A < 1 then
while A ≤ M K do
M = M/2;
end
y = M;
else if A > 1 then
while A ≥ M K do
M = M × 2;
end
y = M/2;
for i = 1 : n do
M = M/2;
if (y + M )K ≤ A then
y =y+M ;
end
end
−2
BER
10
TABLE I
C OMPLEXITY IN T ERMS OF N UMBER OF 16- BIT E QUIVALENT A DDITIONS
SVD
IGMD−ZFTHP−intrlv−SVD−AM (iter=1)
IGMD−ZFTHP−intrlv−SVD−AM (iter=2)
IGMD−ZFTHP−intrlv−SVD−AM (iter=4)
IGMD−ZFTHP−intrlv−SVD−GM (iter=1)
IGMD−ZFTHP−intrlv−SVD−GM (iter=2)
IGMD−ZFTHP−intrlv−SVD−GM (iter=4)
IGMD−ZFTHP−intrlv−SVD−HM (iter=1)
IGMD−ZFTHP−intrlv−SVD−HM (iter=2)
IGMD−ZFTHP−intrlv−SVD−HM (iter=4)
GMD−ZFTHP
−3
10
−4
10
0
5
10
15
20
25
Eb/N0 (dB)
Arithmetic function
QK
√A = i=1 σi
K
A Algorithm [21]
2×2 SVDs/planar rotations
diagonal swap operations
CORDIC based sqrt
Conventional
1674
1674n
2016
5124
−
AM
−
−
2808l
−
−
HM
−
−
2808l
−
−
GM
−
−
2808l
−
198l
n : Number of iterations performed in the Kth root algorithm.
(b)
l : Number of iterations performed in the IGMD algorithm.
Fig. 3. BER performance of the proposed Iterative GMD algorithm in a 7×7
MIMO ZFTHP system using (a) SVD (b) interleaved-SVD as initialization.
bits are required to retain full precision for K = 7. After some
proper truncation in word lengths, it follows that 1674×(n+1)
QK
16-bit additions are required for calculating A =
i=1 σi
and (y + M )K with n iterations. Additional diagonal swap
computations because of irregular data-dependent control flow
are also required in the conventional GMD. For the 2 × 2
SVDs and planar rotations required in all GMD algorithms, we
efficiently implemented them by using CORDICs where only
constant multiplications are necessary for scaling operations.
Through canonic signed digit (CSD) [29] coding, the scaling
operation by 0.60725 (000000.101001̄001̄0) for CORDIC can
be efficiently realized by shift-and-add operations. It follows
that 2808 16-bit equivalent additions are required for the 2 × 2
SVDs and planar rotations in each iteration. For the GMbased IGMD algorithm, additional computations for calculating square root are necessary. It can be shown that a CORDICbased square-root requires 198 16-bit equivalent additions per
iteration. Consequently, under our proposed implementation,
IGMD-GM has slightly higher complexity comparing to the
other IGMDs, whereas the IGMD-AM and the IGMD-HM
essentially have the same complexity.
Fixed-point simulations have been conducted to evaluate the
MSE performance of the GMD algorithms with respect to
computational complexity. Instead of directly implementing
IGMD-HM by constructing the planar rotation so that the
upper-left element is updated by ΩHM , we implement it by
constructing the planar rotations so that the lower-right element is updated by ΩAM . Through this novel implementation,
only shift-and-add operations are required in computing ΩAM
which is computationally more efficient than direct implementation which requires square root and division operations in
computing ΩHM .
In Fig. 4(a) and 4(b), the MSE performance versus complexity of both QR-based (QR and VB-QR) and SVD-based
(SVD and interleaved SVD) GMD algorithms has been simulated under a 7 × 7 i.i.d. Rayleigh fading channel. For
fair comparison, we also implement the conventional GMD
algorithm with two different initializations, namely the GMDQR and the GMD-SVD respectively. It is observed that in
this simulation scenario both VB-QR and interleaved SVD
provide substantial performance improvement when compared
with their counterparts. The proposed IGMD algorithms are
also observed to provide considerable performance advantages
when compared with the conventional GMD algorithms when
8
0
10
0
10
−1
10
−1
MSE
MSE
10
−2
10
GMD−QR
IGMD−QR−AM
IGMD−QR−GM
IGMD−QR−HM
IGMD−VBQR−AM
IGMD−VBQR−GM
IGMD−VBQR−HM
GMD−QR
IGMD−QR−AM
IGMD−QR−GM
IGMD−QR−HM
IGMD−VBQR−AM
IGMD−VBQR−GM
IGMD−VBQR−HM
−3
10
−3
10
−2
10
−4
0.4
0.6
0.8
1
1.2
Complexity
1.4
1.6
1.8
10
2
1500
2000
2500
4
x 10
(a)
3000
3500
Complexity
4000
4500
5000
(a)
0
10
0
10
−1
10
−1
MSE
MSE
10
−2
GMD−SVD
IGMD−SVD−AM
IGMD−SVD−GM
IGMD−SVD−HM
IGMD−SVD−Intrlv−AM
IGMD−SVD−Intrlv−GM
IGMD−SVD−Intrlv−HM
10
−3
10
4000
6000
8000
−2
10
−3
10
−4
10000
Complexity
12000
14000
16000
10
GMD−SVD
IGMD−SVD−AM
IGMD−SVD−GM
IGMD−SVD−HM
IGMD−SVD−Intrlv−AM
IGMD−SVD−Intrlv−GM
IGMD−SVD−Intrlv−HM
1500
2000
2500
Complexity
3000
3500
4000
(b)
(b)
Fig. 4. Complexity comparison under 7 × 7 i.i.d. Rayleigh fading channel:
(a) QR-based and (b) SVD-based.
Fig. 5. Complexity comparison under 4 × 4 i.i.d. Rayleigh fading channel:
(a) QR-based and (b) SVD-based.
operated in the low complexity region.
In Figs. 5(a) and 5(b), the same complexity comparison
is performed under a 4 × 4 i.i.d. Rayleigh fading channel.
Similar to the case in the 7 × 7 i.i.d. Rayleigh fading channel,
the proposed IGMD algorithms outperform the conventional
GMDs in the low complexity region. In addition, the proposed
IGMDs appear to have larger performance gain when the
problem dimension is smaller. This is because the spread on
the diagonal of R is generally smaller in lower dimension
problems and hence the proposed IGMDs can be very efficient
in this case as they only require a small number of iterations
to achieve the desired precision.
In Fig. 6(a) and 6(b), the complexity of proposed algorithms
are compared in a 7 × 7 correlated Rayleigh fading channel.
Uniform linear arrays at both transmit and receive sides
with antenna spacing of 0.3λ have been considered, and the
correlated channel is generated using the typical Kronecker
model. It can be observed from the figures that the proposed
IGMDs become less efficient in the correlated channel when
compared with the conventional GMDs. This is because the
diagonal of R generally has larger spread when the channel is
more correlated, and hence it generally takes more iterations
for the proposed IGMDs to achieve the desired precision.
From the complexity analysis, it is also observed that one
common characteristic for the SVD, QR, and QR-based IGMDs is that the HM usually has better performance, followed
by the GM, and then the AM. This characteristic is related
to the fact that the SVD, QR, and VB-QR initializations all
tend to have larger elements on the upper left corner when
compared with the lower right corner on the diagonal of
R. As the AM mapping also generates larger element on
the upper-left corner and smaller element on the lower-right
corner because of the AM-HM inequality, when applying
these initializations to IGMD-AM, larger elements are then
discouraged from being averaged with smaller elements as the
algorithm proceeds and hence AM construction is expected to
take more iterations to achieve the same performance. This
characteristic also explains why IGMD-HM tends to have
better performance when SVD, QR, and VB-QR initializations
are used. For interleaved-SVD initialization, the performance
of IGMDs is very difficult to characterize as it depends
on the mapping and also the interleave pattern. In addition,
9
ACKNOWLEDGMENT
0
10
The authors would like to thank Prof. Chiu-Chu Melissa
Liu for the helpful discussion and the anonymous reviewers
for their valuable suggestions.
−1
10
MSE
R EFERENCES
−2
10
GMD−QR
IGMD−QR−AM
IGMD−QR−GM
IGMD−QR−HM
IGMD−VBQR−AM
IGMD−VBQR−GM
IGMD−VBQR−HM
−3
10
0.4
0.6
0.8
1
Complexity
1.2
1.4
1.6
1.8
4
x 10
(a)
0
MSE
10
−1
10
GMD−SVD
IGMD−SVD−AM
IGMD−SVD−GM
IGMD−SVD−HM
IGMD−SVD−Intrlv−AM
IGMD−SVD−Intrlv−GM
IGMD−SVD−Intrlv−HM
−2
10
0.4
0.6
0.8
1
1.2
Complexity
1.4
1.6
1.8
2
4
x 10
(b)
Fig. 6. Complexity comparison under 7 × 7 correlated Rayleigh fading
channel: (a) QR-based and (b) SVD-based.
it is observed that although the proposed interleave pattern
provides significant gain in the 7 × 7 i.i.d. Rayleigh fading
channel, it does not always outperform the plain SVD in some
other scenarios. This suggests a good interleave pattern not
only depends on the types of algorithm but also depends on
the data (channel matrix). A more rigorous treating on the
performance characterization is beyond the scope of this article
and will be left for our future research.
VI. C ONCLUSION
A new algorithm for computing the geometric mean decomposition is proposed. We prove analytically that the proposed
IGMD is guaranteed to converge to the exact GMD under
certain sufficient conditions and present three different constructions. The proposed IGMD algorithm does not require
computing Kth root at the pre-processing stage and has a
regular structure that is easily scalable for different problem
dimensions. These advantages lead to a more efficient hardware design when operated in the low complexity regime and
have been verified from extensive numerical simulations.
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Chiao-En Chen (M’ 05) was born in Kaohsiung,
Taiwan in 1976. He received the B.Sc. and M.Sc.
degrees in Electrical Engineering from National Taiwan University, Taipei, Taiwan in 1998 and 2000
respectively. From 2003 to 2008, he was with the
Electrical Engineering Department at University of
California, Los Angeles, where he received his Ph.D.
degree. Since 2008, he joined both the Department of Electrical Engineering and the Department
of Communications Engineering at National Chung
Cheng University, Chiayi, Taiwan, and is currently
an associate professor. His research interests include statistical signal processing and multiple-input-multiple-output (MIMO) communications.
Dr. Chen was a co-recipient of the Best Paper Award in IEEE WCNC 2012,
and a co-author of the book “Detection and Estimation for Communication
and Radar Systems,” published by Cambridge University Press, 2013.
Yu-Cheng Tsai received the B.S. degree from
the Department of Electrical Engineering, National
Chung Hsing University, Taichung, Taiwan. He is
currently pursuing the M.S. degree in Electronics
Engineering form National Chiao Tung University,
Hsinchu, Taiwan.
His research interests include algorithms development of signal processing and VLSI design for
wireless baseband processing.
Chia-Hsiang Yang (S’07-M’10) received his B.S.
and M.S. degrees from the National Taiwan University, Taiwan, in 2002 and 2004, respectively, all in
Electrical Engineering. He received his Ph.D. degree
from the Department of Electrical Engineering of
the University of California, Los Angeles in 2010.
He then joined the faculty of the Electronics Engineering Department at the National Chiao Tung
University, Taiwan, as an Assistant Professor. His
current research interests include energy-efficient
integrated circuits and architectures for biomedical
and communication signal processing.
Dr. Yang was a winner of the DAC/ISSCC Student Design Contest in 2010.
He received the 2010-2011 Distinguished Ph.D. Dissertation in Circuits &
Embedded Systems Award from the Department of Electrical Engineering,
University of California, Los Angeles. In 2013, he was a co-recipient of the
ISSCC Distinguished-Technical-Paper Award.