signal-to-noise ratio can preserve the diversity order of the original full-size MIMO ... Index Terms: Antenna selection, diversity order, diversity-multiplexing ...
On the Diversity Order of Spatial Multiplexing Systems with Transmit Antenna Selection: A Geometrical Approach Hongyuan Zhang, Student Member, IEEE, Huaiyu Dai*, Member, IEEE, Quan Zhou, Student Member, IEEE, and Brian L. Hughes, Senior Member, IEEE
Abstract In recent years, the remarkable ability of multiple-input multiple-output (MIMO) wireless communication systems to provide spatial diversity or multiplexing gains has been clearly demonstrated. For MIMO diversity schemes, it is well known that antenna selection methods that optimize the post-processing signal-to-noise ratio can preserve the diversity order of the original full-size MIMO system. On the other hand, the diversity order achieved by antenna selection in spatial multiplexing (SM) systems, especially those exploiting practical coding and decoding schemes, has not thus far been rigorously analyzed. In this paper, a geometrical framework is proposed to theoretically analyze the diversity order achieved by transmit antenna selection for separately encoded spatial multiplexing systems with linear and decisionfeedback receivers. When two antennas are selected from the transmitter, the exact achievable diversity order is rigorously derived, which previously only appears as conjectures based on numerical results in the literature. If more than two antennas are selected, we give lower and upper bounds on the achievable diversity order. Furthermore, the same geometrical approach is used to evaluate the diversity-multiplexing tradeoff curves for spatial multiplexing systems with transmit antenna selection.
Index Terms: Antenna selection, diversity order, diversity-multiplexing tradeoff, MIMO, spatial multiplexing
*
The authors are with the Department of Electrical and Computer Engineering, NC State University, Raleigh, NC 27695-7511. Phone: (919) 513-0299; Fax: (919) 515-7382; Email: {hzhang, hdai, qzhou, blhughes}@ncsu.edu. This work was supported in part by the National Science Foundation under Grant CCR 03-12696 and CCF-0515164.
I. Introduction Multiple-input multiple-output (MIMO) techniques are expected to be widely employed in future wireless communications to address the ever-increasing demand for capacity. A major potential problem with MIMO is increased hardware cost due to multiple analog/RF front-ends, which has recently motivated the investigation of antenna selection techniques for MIMO systems [2]. In many scenarios, judicious antenna selection may incur little or no loss in system performance, while significantly reducing system cost. MIMO systems can be exploited for spatial diversity (SD) or spatial multiplexing (SM) gains [7]. The majority of work on MIMO antenna selection focuses on the former, including selection combining, hybrid selection-maximum ratio combining (HS-MRC) [2][3], and antenna selection with space-time coding [4][5]. Essentially in these works, with independent and identically distributed (i.i.d.) Rayleigh fading, the system error performance or outage probability can be readily analyzed through order statistics [19]. It has been shown that the diversity order of the original full-size system can be maintained through the signal-to-noise ratio (SNR) maximization selection criterion. In contrast, antenna selection for MIMO systems with multiple data streams, i.e., spatial multiplexing systems, has received less attention. The few existing analytical results generally assume capacityachieving joint space-time coding and optimal decoding. Capacity-maximizing receive antenna selection is analyzed in [6] and shown to achieve the same diversity order as the full-size system. In [8], it is shown that the fundamental tradeoff between diversity and spatial multiplexing of the full-size system, obtained in [7], holds as well for MIMO systems with antenna selection. In practice, the multiple streams in a SM system may be uncoded or separately encoded and suboptimally decoded due to complexity and channel feedback overhead concerns. In [1], several transmit antenna selection algorithms for SM with linear receivers are proposed, and some conjectures on the achieved diversity orders are made from numerical results. To the best of our knowledge, the exact diversity order achieved by antenna selection for practical SM systems has not been rigorously obtained. In contrast to MIMO diversity schemes, the key challenge that hinders accurate performance analysis is that selection is performed among a list of inter-dependent random quantities, which are correlated in a complex manner. In this paper, we propose a new framework to analyze the diversity order achieved by transmit antenna selection for SM systems with separately (and independently) encoded data streams and linear or decision-feedback (DF) receivers (i.e., the V-BLAST structure [9]). In particular, we rigorously show that the optimal diversity order is ( N T − 1)( N R − 1) for an N R × N T separately encoded SM system employing
1
linear or DF receivers, when L = 2 antennas are selected from the transmit side. This should be compared with the diversity order of a two-stream SM system without antenna selection: N R − 1 . Such a diversity
gain can be tremendous for downlink high-data-rate communications, where there may be a large number of transmit antennas at the base stations but few receive antennas at the mobiles (e.g., N R = 2 ). For DF
receivers, it is also clarified that, the antenna selection rule that maximizes the performance of the first decoded data stream is not optimal, as the freedom to choose transmit antennas for subsequent data streams is restricted. Furthermore, following the same geometrical approach, we give an upper bound, ( N T − L + 1)( N R − 1) , and a lower bound, ( N T − L + 1)( N R − L + 1) , on the diversity order for general L , which coincide when L = 2 . The corresponding diversity-multiplexing tradeoff curves are also derived. Generally speaking, our results confirm and generalize some of the conjectures in [1], thus verifying that transmit antenna selection can achieve high data rates and robust error performance in practical SM systems without complex coding. Furthermore, the proposed geometrical approach may be used to solve other open problems related to MIMO communication systems. Among feasible sub-optimal receivers for separately encoded SM systems, we first investigate the linear zero-forcing (ZF) receiver [10]. The analysis for the zero-forcing decision-feedback (ZF-DF) receiver relies heavily on the former. Furthermore, their diversity order analyses, a study at high SNR regimes, hold for linear minimum mean-square-error (MMSE) and MMSE decision-feedback (MMSEDF) receivers as well. The rest of the paper is organized as follows. The system model and problem formulation are given in Section II. The main ideas of our approach are illustrated in Section III for L = 2 , where new notations and concepts can be more easily appreciated. Then extension to the general L scenario is discussed in Section IV, the analysis of which nonetheless becomes more involved. Finally Section V concludes the paper with future research directions.
II. System Model and Problem Formulation We consider a frequency non-selective block Rayleigh fading channel model, in which the N R × N T channel matrix is represented by H = [h1 , h 2 ,
, h NT ] , where {h k } are i.i.d. complex Gaussian column
vectors, whose entries are i.i.d. with zero mean and unit variance. With transmit antenna selection, there
⎛N ⎞ are a total of NU = ⎜ T ⎟ possible antenna subsets, defined as U1 ~ U NU 1: ⎝ L ⎠
1
Without loss of generality, it is assumed that in each subset the selected columns are ordered increasingly by index.
2
U1 = {h1 , h 2 ,
, hL}
U 2 = {h1 , h 2 ,
h L−1 , h L+1}
{
U NU = h NT − L+1 ,
(1)
}
, h NT .
If the subset U j is selected, of which the L column vectors compose the channel matrix H j , the SM system can then be expressed as: y=
ρ0 L
H js + n ,
(2)
where the N R × T matrix y is the received signal block with T the block length; n is the circularly symmetric complex Gaussian background noise with zero mean and identity covariance matrix; and the L × T matrix s represents the transmitted signal block, with independent rows each assuming unit
average energy per symbol. Therefore, ρ 0 is the average SNR per receive antenna. As in [7], we assume the fading channel keeps constant for the entire data frame. Each stream independently adopts a code of fixed data rate R0 2. As will be shown (see proof of Lemma I), channel coding in each data stream can provide coding gain but not diversity gain. Throughout the paper we assume N T ≥ L and N R ≥ L . Therefore given the assumption of i.i.d. Rayleigh fading, the selected channel matrix H j has full column rank with probability one [18]. −1
For a ZF receiver, a spatial equalizer G ZF = H†j = ⎡⎣ H Hj H j ⎤⎦ H Hj is applied to the received signal y to obtain an estimate of the transmitted signal:
sˆ = G ZF y = where
†
ρ0 L
s + G ZF n ,
(3)
denotes the pseudo-inverse of a matrix. Therefore L equivalent decoupled single-input single-
output (SISO) data links are formed as sˆk =
ρ0 L
sk + [G ZF ]k * n, 1 ≤ k ≤ L ,
(4)
where []k * denotes the kth row in matrix. The instantaneous channel capacity and error performance of these subchannels are determined by the corresponding post-processing SNRs, which can be directly derived from (4) as
2 We are mainly interested in the maximum diversity gain here, corresponding to a zero multiplexing gain scenario in [7] . An exception is in III.C, where a family of block codes with multiplexing gain r is considered for each stream.
3
−1 2 ⎛ρ ⎞ ⎛ρ ⎞ ) ρ k( ,jZF = ⎜ 0 ⎟ [G ZF ]k * = ⎜ 0 ⎟ ⎡⎣ H Hj H j ⎤⎦ , 1 ≤ k ≤ L , kk ⎝ L ⎠ ⎝ L ⎠
where
H
(5)
stands for conjugate transpose, and []kk represents the kth diagonal element of a matrix. Let h (k j )
be the kth column of H j , and H (k j ) the N R × ( L − 1) matrix resulting from leaving out h (k j ) from H j . It is −1
known that [10], 1/ ⎡⎣ H Hj H j ⎤⎦ is equal to Rk( j ) , the square of the projection height 3 from h(k j ) to the kk range of H (k j ) , or alternatively, the square of the norm of the projection of h(k j ) onto the null space of
(H )
( j) T k
. That is ) ρ k( ,jZF =
ρ0 L
Rk( j ) =
ρ0
2
h (k j ) sin 2 θ k( j ) ,
L
(6)
where h (k j ) is the norm of h(k j ) , while θ k( j ) is the angle between h (k j ) and its projection on the range of
H (k j ) , defined as
θ k( j ) = sin −1
Rk( j ) h(k j )
, 0 < θ k( j )
are similarly defined. Note that according to our notation, f ( x) ≤ g ( x) indicates f ( x ) ≥ g ( x ) for sufficiently small x . Lemma I: For separately encoded spatial multiplexing systems with linear ZF/MMSE receivers, the
antenna selection rule that chooses the antenna subset with the strongest weakest data link achieves the optimal diversity order among all antenna selection rules. Furthermore, the optimal diversity order can be evaluated as log Pr( RSL ≤ x) x →0 log( x)
d L = lim opt
(13)
for both receivers. Proof: See Appendix A. Remarks: Lemma I indicates that we only need to evaluate (13) for the optimal diversity order of separately encoded SM systems with transmit antenna selection and linear ZF/MMSE receivers. We also make no claims on the practicality of this selection algorithm, as the main focus of this paper is on theoretical analysis. Note that efficient antenna selection algorithms exist in literature (e.g., [2][15] and references therein). 4 By frame error probability, we mean an error is declared if any substream is decoded unsuccessfully. For each substream, assuming coding over a single block with constant fading, this is the true error probability of a code averaged over the transmitted codewords, channel fading, and additive noise.
5
III. Diversity Order and Diversity-Multiplexing Tradeoff when L = 2 In this section, we discuss the main idea of our geometrical approach for L = 2 , which also admits an exact result. Extension to the general L scenario is not trivial, as will be seen in Section IV. Theorem I: In an N R × NT spatial multiplexing system with linear ZF/MMSE or ZF/MMSE decision-
feedback receivers satisfying N T ≥ 2 and N R ≥ 2 , if separately encoded data streams are transmitted from two selected antennas, the optimal achievable diversity order is d opt = ( NT − 1)( N R − 1) .
(14)
The optimal diversity-multiplexing tradeoff curve is the piecewise function of r connecting the points (r , d opt ( r )) for 0 ≤ r ≤ 2 , with d opt ( r ) = ( N T − 1)( N R − 1)(1 − r / 2) + ,
(15)
+
where ( x) = max( x,0) . This theorem will be proved in three steps: we start with linear ZF/MMSE receivers by evaluating (13) in III.A. Then we extend the analysis to ZF/MMSE-DF receivers in III.B. Finally we analyze the corresponding diversity-multiplexing tradeoff curves in III.C. As L = 2 , each possible antenna subset in (1) contains two selected column vectors from the original channel matrix H . For convenience, we will use Rkj to denote the squared projection height from h k to h j . In another word, given that U s = {h k , h j } is selected, Rkj
R1( s ) and R jk
R2( s ) (c.f. (6) in Section II).
Similar notation transforms are adopted for θ k( j ) . Clearly Rkj = h k
2
sin 2 θ kj ,
where it can be shown that [12][18][26][27] h k
2
(16)
is χ 2 (2 N R ) distributed with probability density
function (PDF)5 f ||h ||2 ( x ) = k
1 1 x N R −1e − x = x N R −1e − x , Γ( N R ) ( N R − 1)!
(17)
and θ kj assumes a PDF of fθkj (θ ) = ( N R − 1)sin 2θ (sin θ ) 2 N R − 4 , Furthermore, h k
5
2
, hj
2
⎛ π⎞ θ ∈ ⎜ 0, ⎟ . 2 ⎝
(18)
⎠
and θ kj are mutually independent. Also, Rkj is a χ 2 (2( N R − 1)) distributed
χ 2 (2 N R ) denotes Chi-square distribution with 2 N R degrees of freedom and mean N R . Γ( x ) =
∫
∞
0
6
e − t t x −1dt is the Gamma function.
random variable, which leads to Pr( Rkj ≤ x)
x N R −1 .
(19)
From (19) we can see that without antenna selection ( NT = L ), any data stream with a ZF/MMSE receiver bears a diversity order of N R − 1 , so does the overall error probability. When NT > L , through the diversity-maximization transmit antenna selection (c.f. (7) and (10)) RSL =
max
k ≠ j∈{1, , NT }
{min {R , R }} , kj
(20)
jk
we will show that a product gain of N T − 1 on the diversity order can be achieved.
A. Linear ZF/MMSE Receivers L In the following, the optimal diversity order (13) for linear ZF/MMSE receivers d opt will be explicitly
explored. Note that neither the exact PDF of RSL nor its polynomial expansion near zero seems tractable, which motivates us to solve the problem through tight upper and lower bounds. By definition Pr( RSL ≤ x) = Pr(min( R12 , R21 ) ≤ x ,min( R13 , R31 ) ≤ x,
, min( R( NT −1) NT , RNT ( NT −1) ) ≤ x)
(21)
⎛ N ⎞ = Pr ⎜⎜ Ai ⎟⎟ , ⎝ i =1 ⎠
∪
where the N = 2 NU events { Ai } are defined as A1 = {R12 ≤ x, R13 ≤ x,
, R1NT ≤ x, R23 ≤ x,
, R( NT −1) NT ≤ x}
A2 = {R12 ≤ x, R13 ≤ x,
, R1NT ≤ x, R23 ≤ x,
, RNT ( NT −1) ≤ x}
AN = {R21 ≤ x, R31 ≤ x,
, RNT 1 ≤ x, R32 ≤ x,
, RNT ( NT −1) ≤ x},
(22)
each of which corresponds to the case that one data stream from each of the NU subsets is in outage (i.e., close to zero for sufficiently small x). For example, event A1 picks the first element from each possible
⎧⎪ subset U1 ~ U NU , i.e., A1 = ⎨ Rkj ≤ x ⎪⎩1≤ k < j ≤ NT
⎫
}⎪⎬ . Then from (21) we have:
∩ {
⎪⎭
Pr( A1 ) ≤ Pr( RSL ≤ x) ≤
N
∑ Pr( A ) , i
(23)
i =1
which indicates
lim x →0
log max Pr( Ai ) i
log( x )
L ≤ d opt ≤ lim x →0
7
log Pr( A1 ) . log( x )
(24)
Clearly the lower bound in (24) is actually tight. However, it is generally difficult to identify the dominant terms at high SNR from (22), since the cardinality grows exponentially with NT2 . Alternatively, we take the following approach. First, we find a common upper bound for Pr( Ai ), ∀i , i.e., obtaining such a PU that max Pr( Ai ) ≤ PU ,
(25)
i
L (c.f. the lower bound in (24))6. We then obtain a lower bound which determines a lower bound for d opt
for Pr( A1 ) :
Pr( A1 ) ≥ PL ,
(26)
L (c.f. the upper bound in (24)). It and evaluate its error exponential, which gives an upper bound for d opt
turns out that these two bounds coincide and represent the best achievable diversity order, given by
( N T − 1)( N R − 1) . Diversity Lower Bound L Proposition I: d opt ≥ ( NT − 1)( N R − 1) .
⎛N ⎞ Proof: Define Si as the set consisting of the NU = ⎜ T ⎟ random variables in Ai (see (22)). For example, ⎝ 2 ⎠
{
S1 = R12 , R13 ,
, R1NT , R23 ,
}
, R( NT −1) NT = {Rkj }
1≤ k < j ≤ NT
. First we claim that, in any Si we can always find
{
a subset of NT − 1 random variables in the form of Si _ indep = Rk1k2 , Rk2k3 ,
, Rk N
k T −1 NT
} , where k ~ k
{
some permutation of the integer array 1 ~ NT . For example, S1_ indep = R12 , R23 ,
1
NT
is
}
, R( NT −1) NT , i.e.,
ki = i, 1 ≤ i ≤ N T . By Lemma II given in Appendix B, we obtain the following common upper bound for
Pr( Ai ) : Pr( Ai ) ≤ PU = [Pr( Rkj ≤ x )]( NT −1) , k ≠ j , ∀Ai .
(27)
With (19) we have .
max Pr( Ai ) ≥ x ( NT −1)( N R −1) ,
(28)
i
and from (24) Proposition I follows. The claim is proved as follows. A graph is drawn in Figure 3 to visualize Si , where the nodes represent the transmit antenna elements and an arrow directed from node k1 to k2 appears in the graph if 6
Recall that an error probability with lower diversity order corresponds to an exponentially larger one, and vice versa.
8
Rk1k2 exists in Si . Based on the definition of Si , there is one and only one arrow between any two nodes. For such a graph of size n, a complete path is defined as one that goes through a series of n − 1 arrows, passing each of the n nodes one and only one time. Therefore we can find a subset Si _ indep in Si if at least one complete path exists in the graph of size N T , passing the nodes with the order
k1 → k2 →
→ k NT . It is easily shown that all graphs of size 3 contain one complete path. Now suppose
that all graphs of size n − 1 contain at least one complete path. Then in any graph of size n , we can always find a path passing n − 1 nodes one and only one time, assumed in the order k1 → k2 →
→ kn −1 .
From Figure 3, we see that if there is no complete path existing in the graph of size n, the arrow between
k1 and kn should be in the direction of k1 → kn , otherwise kn → k1 →
→ kn −1 forms a complete path.
For the same reason, the arrow between k2 and kn should be in the direction of k2 → kn , otherwise
k1 → kn → k2
→ kn −1 forms a complete path. By repeating the same reasoning, we finally reach
kn −1 → kn . However, in this case k1 → k2 →
→ kn is a complete path, and we conclude that a complete
path always exists for any graph of size n . Thus the claim is proved for any N T by induction.
□ Diversity Upper Bound L Proposition II: d opt ≤ ( N T − 1)( N R − 1) .
Proof: This proof is technically involved, so we divide it into three parts for ease of illustration: (1) by some geometrical analysis Pr( A1 ) is further lower bounded (see (31)); (2) by some lemmas in Appendix C, the probability lower bound is transformed into an exponentially equivalent form with known statistics (see (35)); (3) the diversity upper bound is explicitly evaluated (see (38)). Part (1):
Given
⎛ ⎞ ⎛ ⎞ 2 Pr( A1 ) = Pr ⎜ Rkj ≤ x ⎟ = Pr ⎜ h k sin 2 θ kj ≤ x ⎟ , ⎜ 1≤k < j ≤ N ⎟ ⎜ 1≤k < j ≤ N ⎟ T T ⎝ ⎠ ⎝ ⎠
∩
and by defining z =
NT −1
∑
hk
2
∩
, distributed as χ 2 (2( NT − 1) N R ) , and ψ 0 = (π / 2) /( NT − 1) , we have:
k =1
⎛ ⎞ ⎛ ⎞ Pr( A1 ) ≥ Pr ⎜ z sin 2 θ kj ≤ x ⎟ ≥ Pr ⎜ z sin 2 θ kj ≤ x, max θ1k < ψ 0 ⎟ , ⎜ 1≤k < j ≤ N ⎟ ⎜ 1≤ k < j ≤ N ⎟ 2≤k ≤ NT T T ⎝ ⎠ ⎝ ⎠
∩
∩
9
(29)
where for the second inequality we have further restricted the ranges of the N T − 1 i.i.d. random variables
{θ1k }kN=2 T
within (0,ψ 0 ) (c.f. Corollary I in Appendix B).
Based on the geometric structure involved, given θ1k and θ1 j , 2 ≤ k < j ≤ N T , θ kj is constrained as
θ kj ≤ θ1k + θ1 j , where the equality holds only when h1 , h k , h j are linearly dependent (located in the same subspace with a dimension less than 3) [11]. Then within the range (0,ψ 0 ) , we have sin 2 θ kj ≤ sin 2 (θ1k + θ1 j ) ≤ sin 2 (θ12 + θ13 + where θ Σ =
NT
∑θ
1k
+ θ1NT ) = sin 2 θ Σ ,
(30)
is still in the range of (0, π / 2) . Therefore (29) can be further lower bounded as:
k =2
Pr( A1 ) ≥ PL = Pr ⎛⎜ z sin 2 θ Σ ≤ x, max θ1k < ψ 0 ⎞⎟ 2≤ k ≤ NT ⎝ ⎠
Pr( z sin 2 θ 'Σ ≤ x ),
(31)
where we define a new set of i.i.d. random variables θ '12 ~ θ '1NT with PDF of fθ '1i ( x ) =
∫
ψ0
0
fθ1i ( x )
=
fθ1i ( x )
fθ1i ( x )dx
C
, 0 < x < ψ 0 , 2 ≤ i ≤ NT ,
i.e., the restriction of θ12 ~ θ1NT in the range of (0,ψ 0 ) , and θ 'Σ =
(32)
NT
∑θ '
1i
.
i =2
Part (2):
With Lemma III ~ VI on exponential equivalence given in Appendix C, the evaluation of (31) can be further simplified. Specifically, by defining θ '0 = max θ '1k and m( x ) = sin 2 ( x ) for x ∈ (0,ψ 0 ) , from k
Lemma IV (whose proof requires Lemma III) we have Pr(sin 2 θ 'Σ ≤ x)
Pr(sin 2 θ '0 ≤ x) ,
(33)
i.e., with respect to monotonic functions, replacing the sum of a set of independent random variables with the maximum of them does not change the associated exponential behavior. Further by Lemma V, we have Pr( z sin 2 θ 'Σ ≤ x )
Pr( z sin 2 θ '0 ≤ x ) ,
(34)
i.e., the exponential behavior is still not changed by multiplication of an independent random variable. Finally by Lemma VI, we have Pr( z sin 2 θ '0 ≤ x )
Pr( z sin 2 θ 0 ≤ x ) ,
where θ 0 = max θ1k , i.e. without the constraint of (0,ψ 0 ) on θ12 ~ θ1NT . k
10
(35)
Part (3):
We are then left to evaluate the smallest exponential in Pr( z sin 2 θ 0 ≤ x) in (35). Note that z is a χ 2 (2( NT − 1) N R ) distributed random variable with cumulative distribution function (CDF):
Fz ( x) = 1 − e
−x
M + NT − 2
∑ k =0
xk , k!
M = ( NT − 1)( N R − 1) ,
(36)
while from (18) the PDF of θ 0 can be easily derived as:
π
fθ 0 (θ ) = M [sin 2 θ ]M −1 sin 2θ , M = ( NT − 1)( N R − 1), θ ∈ (0, ) . 2
(37)
After some algebra presented in Appendix D, we can get the following equivalent polynomial form as
x→0: NT − 3 ⎛ 1 ⎞ M k! M Pr(z sin 2 θ 0 ≤ x) = ⎜ −M ⎟⎟ x + o( x ), M = ( NT − 1)( N R − 1) , ⎜ M! ( 1)! M k + + k =0 ⎝ ⎠
∑
(38)
where the coefficient of x M in (38) is always positive (also shown in Appendix D), which completes the proof. □
For example, when NT = 3, N R = 3 , and L = 2 , from (38) we get as x → 0
Pr(z sin 2 θ 0 ≤ x) =
x4 + o( x 5 ) . 120
To illustrate the soundness of our approach, Figure 4 presents by simulations the exponential behavior of Pr(z sin 2 θ 0 ≤ x) , together with the outage probability in (21) and the probabilities representing the diversity bounds as in (23)(24). The results show a diversity order of 4, verifying our derivations above. With Proposition I and II, (14) is proved for the linear receiver case.
B. Decision-Feedback Receivers In this subsection we continue to explore the optimal achievable diversity order for separately encoded DF SM systems with transmit antenna selection and DF receivers, denoted as d opt . As expected, the
performance analysis for DF receivers will rely heavily on that for their linear counterparts in III.A. For DF receivers with L = 2 , the frame error probability is given by
Pe = Pe1 + Pe 2 (1 − Pe1 ) ,
(39)
where Pe1 is the error probability of the first decoded data stream, and Pe 2 is that of the second stream assuming perfect feedback. Therefore Pe
max{Pe1 , Pe 2 } . For a fixed-order ZF/MMSE DF receiver
11
without antenna selection, Pe1 ≥ Pe 2 is always fulfilled, so that we can investigate its diversity order solely from the first decoded data stream, which is processed by just a linear ZF/MMSE receiver (see, e.g. [7][17]). This may entice one to consider an antenna selection rule that maximizes the performance of the first decoded data stream. However, as will be shown, this antenna selection rule is generally not optimal. Instead of seeking an optimal antenna selection rule for DF receivers like Lemma I, we take the DF following simpler approach to evaluate d opt , thanks to the close connection between DF receivers and
DF their linear counterparts. First, since for any antenna selection rule j , Pe( j ) ≤ Pe(1j ) , we have d (DF j ) ≤ d1,( j ) ,
where d1,(DFj ) is the diversity order of the corresponding first decoded data stream. Consequently, DF d opt ≤ d1,DF opt ,
(40)
where d1,DF opt is the best achievable diversity order of the first decoded data stream. Note that there may be multiple choices of antenna selection rules to achieve the optimum for both sides of (40), and a selection rule that is best for one may fail for the other. Nonetheless, following a procedure similar to what we have discussed above for linear receivers, we will explicitly evaluate d1,DF opt and derive a diversity upper bound DF for d opt . We then continue by constructing a specific antenna selection algorithm whose diversity order is DF easy to assess and serve as a lower bound for d opt .
Diversity Upper Bound - Maximizing the SNR of the First Decoded Data Stream
We investigate an antenna selection algorithm that achieves d1,DF opt : selecting the antenna subset that maximizes the post-processing SNR of the first decoded data stream. We distinguish two scenarios with respect to detection order: arbitrary but fixed ordering and optimal ordering [9]. Our main result is summarized below. DF Proposition III: d opt ≤ d1,DFopt = ( NT − 1)( N R − 1) .
Proof: For fixed ordering, without loss of generality, we assume that decoding starts from the signal
transmitted from the antenna with the smallest index number in the selected antenna subset. The antenna selection rule indicates that the post-processing SNR of the first decoded data stream is given by (c.f. (20)) RSL1 = max
1≤ k < j ≤ NT
{R } . kj
(41)
Clearly we have (c.f. (22))
Pr( RSL1 ≤ x) = Pr( A1 ) ,
12
(42)
whose upper bound can be derived from (27), while its lower bound exponential behavior evaluation directly follows (29)~(38). Therefore for arbitrary but fixed ordering d1,DF opt = ( NT − 1)( N R − 1) .
(43)
For optimal ordering, our selection rule is reformulated with
RSL 2 =
max
k ≠ j∈{1, , NT }
{max {R , R }} , kj
(44)
jk
and we have:
(
⎛ ⎞ ⎛ 2 Pr( RSL 2 ≤ x ) = Pr ⎜ max( Rkj , R jk ) ≤ x ⎟ = Pr ⎜ max h k , h j ⎜ 1≤k < j ≤ N ⎟ ⎜ 1≤ k < j ≤ N T T ⎝ ⎠ ⎝
∩
∩
2
) sin θ 2
kj
⎞ ≤ x ⎟. ⎟ ⎠
(45)
It is straightforward to upper and lower bound (45) as (c.f. (27) and (31)): Pr( RSL 2 ≤ x) ≤ Pr( A1 ) ≤ ⎡⎣ Pr( Rkj ≤ x) ⎤⎦
NT −1
,
∀k ≠ j ,
(46)
and
(
2 Pr( RSL 2 ≤ x) ≥ Pr ⎜⎛ max h1 , ⎝
, h NT
2
)
⎛ ⎛ NT sin 2 θ Σ′ ≤ x ⎟⎞ ≥ Pr ⎜ ⎜ h ⎜ ⎜ i =1 i ⎠ ⎝⎝
∑
2
⎞ ⎞ 2 ⎟⎟ sin θ Σ′ ≤ x ⎟ . ⎟ ⎠ ⎠
The evaluation of the lower bound in (47) is similar as in (31), except that z is re-defined as z =
(47) NT
∑h
2 k
.
k =1
Following a similar approach as (31)~(38), we get NT + N R − 3 . ⎛ 1 ⎞ M k! M +1 Pr( RSL 2 ≤ x) ≤ ⎜ −M ⎟⎟ x + o( x ), M = ( NT − 1)( N R − 1) . ⎜ M! ( 1)! M k + + k =0 ⎝ ⎠
∑
(48)
So the same result as in (43) is obtained with optimal ordering. □
Remarks: Even though RSL 2 ≥ RSL1 ≥ RSL with probability 1, no advantage in diversity order can be achieved. Of course, some coding gain in SNR is naturally expected, which is beyond the scope of this DF L paper. So d opt ≤ d1,DF opt = d opt . On the other hand, we anticipate the performance of DF receivers is no
worse than their linear counterparts at high SNR, which will be verified below. In [7][12], the authors have shown that optimal ordering will not increase the diversity order in the first decoded data stream of a separately encoded SM system with DF receivers in the L = 2 case7. As a side product, here we reach the same conclusion in the antenna selection context. Finally we note that this antenna selection rule is in general sub-optimal with respect to the diversity order for the whole system. The reason is that although Pe1 in (39) is minimized, Pe 2 is not affected by 7
Extension to the general L scenario was done recently in [24][25].
13
the selection process. Rather, it behaves the same as in a non-selection scheme with N R -order diversity. Therefore Pe
max{Pe1 , Pe 2 } is mostly dominated by the second stream, and the diversity order is given by
d = min{( NT − 1)( N R − 1), N R } = N R , for NT ≥ 3, N R ≥ 2 . Diversity Upper Bound - QR Decomposition-Based Antenna Selection Algorithm Here we analyze a simple yet effective antenna selection algorithm, for which the first decoded data stream achieves a diversity order of ( NT − 1)( N R − 1) , while the second stream performs better than the first one. This antenna selection algorithm is based on QR decomposition, which was originally proposed for capacity maximization [14][15]. Compared with brute force methods, this algorithm greatly reduces the computational complexity while achieving a near optimal performance with respect to channel capacity. Here we apply it in SM systems with DF receivers with the goal of minimizing the error rate, and show that it also maximizes the diversity order, therefore verifying our observations in [15] and revealing its great potential. DF Proposition IV: d opt ≥ ( NT − 1)( N R − 1) .
Proof:
For simplicity we mainly focus on ZF-DF receivers. From a geometrical viewpoint, this
incremental antenna selection procedure starts by seeking the column vector with the largest norm (or the largest projection height to the null-space); and in each of the following steps, one column with the largest projection height to the space spanned by the selected column vectors is chosen until the Lth antenna is selected. The detection order is the reverse of the selection order, i.e., the stream decoded at the lth step is transmitted from the antenna selected at the ( L − l + 1)th step. The corresponding post-processing SNR for the stream decoded at the lth step is then proportional to the maximum value among NT − L + l independent random variables distributed as χ 2 (2( N R − L + l )) , resulting in a diversity order of ( NT − L + l )( N R − L + l ) . Therefore for L = 2 , the first decoded stream achieves a diversity order of
d1 = ( NT − 1)( N R − 1) and the second decoded stream achieves d 2 = NT N R , and the diversity order for the whole system equals ( NT − 1)( N R − 1) . For MMSE-DF receivers, the algorithm is slightly modified: at the lth step, a column h ( l ) is selected such that:
l = arg max l′
1 [[H( l −1) , hl ′ ] [H( l −1) , hl ′ ] + L / ρ 0I]ll−1 H
14
,
where H (l −1) is the matrix formed by the previous selected l − 1 column vectors, and [H (l −1) , hl ′ ] means appending hl ′ to the last column. Following the proof of Lemma I (in Appendix A), the same diversity lower bound is achieved. □
Simulation results are presented in Figure 5 for the NT = 3, N R = 3, L = 2 scenario, where the bit error rates (BER) of the ZF-DF receiver with the above two selection rules are shown, together with that for the DF system without antenna selection. The achieved diversity orders of different schemes are also provided for a clearer comparison. We see that although simpler, the QR based method (bearing a diversity of 4) outperforms the algorithm that optimizes only the first decoded data stream, which only achieves a diversity order of 3, due to the unaffected second data stream, as discussed above. With Proposition III and IV, (14) is also proved for ZF/MMSE-DF receivers.
C. The Diversity-Multiplexing Tradeoff Using the same geometrical approach, we can also obtain the diversity-multiplexing tradeoff curve introduced in [7] for the separately encoded SM systems with antenna selection. With quasi-static fading assumption, a family of codes {ς ( ρ0 )} over a block length shorter than fading coherence time is employed, one at each SNR level. We further assume that the rate of the code increases with SNR, so a scheme achieves a multiplexing gain r if the rate R( ρ0 ) = r log ρ0 . Based on the diversity order analysis in Section III.A and B (see especially proof of Lemma I in Appendix A), we get the following equation:
Pe
Pe _ max
Pout _ max ,
where Pout _ max can be viewed as the outage probability of the worst stream for linear receivers, and that of the first decoded data stream for DF receivers, respectively. The diversity-multiplexing tradeoff curve
d (r ) can be evaluated directly from Pout _ max as ρ0 → ∞ :8 ⎡ ⎛ ⎤ ρ ⎞ r Pout _ max = Pr ⎢log ⎜ 1 + 0 RSL ⎟ ≤ log ρ 0 ⎥ L L ⎠ ⎣ ⎝ ⎦ ⎡ − (1− r ) ⎤ ⎢ ρ0 L ⎥ ⎢⎣ ⎥⎦
M
⎛ r⎞ − M ⎜ 1− ⎟ ⎝ L⎠
= ρ0
r ⎡ − (1− ) ⎤ Pr ⎢ RSL ≤ Lρ 0 L ⎥ ⎣⎢ ⎦⎥
,
where M = ( NT − 1)( N R − 1) . Therefore (15) is proved and we finalize the proof of Theorem I.
8
Replace RSL with RSL1 or RSL 2 for DF receivers.
15
IV. Extension to General L Analyses in Section III can be partially extended to the general scenario. In particular, the diversity lower bound can be obtained following a similar approach. In contrast, a tight diversity upper bound could not be achieved with straightforward extension. Evaluations become more involved as now the postprocessing SNR is proportional to the squared projection height from a column vector to a nondegenerated space. In the following, we present our results for general L , highlight the key challenges, and finally give our conjecture and future direction. Our main result for general L is given below, which should be compared with Theorem I. Theorem II: In an N R × NT spatial multiplexing system with linear ZF/MMSE or ZF/MMSE decision-
feedback receivers satisfying N T ≥ L and N R ≥ L , if separately encoded data streams are transmitted from L selected antennas, the optimal achievable diversity order is bounded as M L ≤ d opt ≤ M U ,
(49)
where M L = ( NT − L + 1)( N R − L + 1) , and M U = ( NT − L + 1)( N R − 1) . The optimal diversity-multiplexing tradeoff curve is bounded as: r r M L (1 − ) + ≤ d opt (r ) ≤ M U (1 − ) + . L L
(50)
We will mainly focus on the extension for linear receivers; those for diversity order of DF receivers and diversity-multiplexing tradeoff are relatively straightforward and will be briefly discussed. Employing the antenna selection method (10), we can then derive a similar outage probability expression as (21):
⎛ Pr( RSL ≤ x ) = Pr ⎜ min{Rk(1) } ≤ x, min{Rk(2) } ≤ x, U2 ⎝ U1 ⎛ ⎞ = P r ⎜ Ai ⎟ , ⎝ i =1 ⎠ N
⎞ , min{Rk( NU ) } ≤ x ⎟ UN ⎠ U
(51)
∪
where U1 ~ U NU are defined in (1),
Rk( j )
⎛ NT ⎞ ⎜ ⎟ ⎝ L ⎠
in (6), and the N = L
events { Ai } are similarly defined with
a generic form (c.f. (22)): ⎧⎪ Aj = ⎨ R (ji()i ) ≤ x ⎩⎪1≤i ≤ NU
∩{
⎫
}⎪⎬ , ⎭⎪
1≤ j ≤ N ,
where 1 ≤ j (i ) ≤ L denotes a pick from subset U i . Clear (23) still holds and we proceed with the evaluation of upper and lower bounds. 16
Proposition V: For general L , the optimal diversity order for linear receivers can be lower bounded as L dopt ≥ ( NT − L + 1)( NR − L + 1) .
Proof: From the definition of Ai and U1 ~ U NU , in any Si (the set of random variables in Ai ) we can
{
always find a subset bearing the form Si _ indep = Rk(1j1 )′ , Rk(2j2 )′ ,
}
, Rk NNT−L−+L1+1 ′ , with NT − L + 1 random (j
T
)
variables. In this form, Rk( j )′ is slightly modified from Rk( j ) with k indicating the index of column vector in the original channel matrix H , instead of H j , i.e., the squared projection height from h k (instead of
h(k j ) ) to the subspace spanned by the remaining column vectors in subset U j , with the implication that h k ∈ U j . Here k1 ~ k NT − L +1 are some NT − L + 1 distinct integers within [1, NT ] , and U j1 ~ U jN
T − L+1
is an
ordered list of NT − L + 1 different subsets from U1 ~ U NU . It is further required that h ki ∈ U ji , and
h ki ∉ U jl for l > i , i.e., h ki belongs to subset U ji but not those after. Therefore by Lemma VII in Appendix E we can get the following upper bound for any Pr( Ai ) : Pr( Ai ) ≤ ⎡ Pr( Rk(1j1 )′ ≤ x ) ⎤ ⎣ ⎦
NT − L +1
.
(52)
Furthermore, since Rk( j )′ is χ 2 (2( N R − L + 1)) distributed, we have Pr( Ai ) ≥ x ( NT − L +1)( N R − L+1) ,
(53)
and Proposition V is proved. □
On the other hand, for L > 2 scenarios, the derivation of a tight lower bound for Pr( A1 ) is much more involved as compared to the L = 2 case, because the angles {θ k(1) } are correlated in a complicated manner, and a general form of their joint PDF expressions is not accessible. Nonetheless, we have the following result. Proposition VI: For general L, the optimal diversity order for linear receivers can be upper bounded as L d opt ≤ ( NT − L + 1)( N R − 1) .
Proof: Since the projection height from a vector to a subspace represents the shortest distance from the vector to any point in the subspace, we have Rk( j ) ≤ Rkl( j ) , for any h (k j ) , hl( j ) ∈ U j , where Rkl( j ) denotes the squared projection height from h (k j ) to hl( j ) . It is then not difficult to build up the following lower bound:
17
⎛ ⎞ ⎛ ⎞ Pr( A1 ) = Pr ⎜ ∩ { R1( i ) ≤ x} ⎟ ≥ Pr ⎜ ∩ { R12( i ) ≤ x} ⎟ . ⎜ 1≤i ≤ N ⎟ ⎜ 1≤i ≤ N ⎟ U U ⎝ ⎠ ⎝ ⎠
(54)
Carefully examining the first two elements in all subsets (see (1)) reveals that the last term in (54) bears a similar form as the Pr( A1 ) in L = 2 case (see (22)), replacing NT − 1 with NT − L + 1 . Following the same lines as in Section III, we have for general L .
.
Pr( RSL ≤ x) ≤ Pr( A1 ) ≤ x ( NT − L +1)( N R −1) .
(55) □
Combining the above two propositions, (49) is proved for linear receivers. Also, as the L = 2 case it is straightforward to show that (49) applies for the first decoded data stream of DF receivers. Since the .
diversity order of the first decoded data stream is upper bounded by ( NT − L + 1)( N R − 1) and Pe ≤ Pe1 , the upper bound ( NT − L + 1)( N R − 1) also applies for Pe with DF receivers. On the other hand, by employing the QR based selection algorithm for DF receivers, a diversity order lower bound ( NT − L + 1)( N R − L + 1) is achieved. Therefore (49) also holds for DF receivers. Finally the derivation of (50) follows the same method as given in Section III.C, and Theorem II follows. Remarks: Note that when L = 2 , the two bounds in (49) and (50) coincide and conform to the results obtained in Section III. A conjecture on the diversity order of separately encoded SM systems with transmit antenna selection and linear ZF receivers was made in [1] based on numerical results, which actually has motivated our research: for linear ZF receivers, when N R = L , the achievable diversity order is NT − L + 1 . Our results prove its correctness and further extend it to general N R . In Figure 6 we investigate the exponential behavior of Pr( R SL ≤ x) and Pr( A1 ) for an exemplary scenario N T = N R = 4, L = 3 through simulations. By comparing with the reference curve x 4 , we find that both of them present a diversity of M L = 4 , verifying its achievability as we claim in Theorem II. On the other hand, it shows that in this scenario the probability lower bound Pr( A1 ) also achieves the diversity lower bound M L , revealing that the diversity orders between M L + 1 and M U may actually not be achievable. We thus have the following conjecture, which constitutes part of our future work. The key seems to lie on finding a better lower bound of Pr( A1 ) than (54). Conjecture: for general L , the optimal transmit antenna selection achieves a diversity order exactly
equal to M L .
18
V. Conclusions In this paper, we have analyzed the diversity order achieved by transmit antenna selection for separately encoded SM systems with linear and decision-feedback receivers. Using a geometric approach, we have rigorously derived their achievable diversity order for the L = 2 scenario. We have also used the same geometrical approach to obtain bounds on the achievable diversity order for general L . Our results prove and extend the previous conjectures in literature drawn from simulations, and verify the predicted potential of antenna selection for practical spatial multiplexing systems. Furthermore, the proposed geometrical approach may also be used to solve other open problems related to MIMO. Besides verifying the conjecture for general L , the analysis for maximum-likelihood receivers, joint transmitter and receiver selection, and multiuser MIMO scenarios also direct our future research.
Appendix A. Proof of Lemma I Proof: Suppose subset U j is selected following an arbitrary antenna selection rule, which may be channel dependent. Given any channel realization H , let the corresponding selected antenna subset be H j (as in ( j) (2)). Let random variable Rmin denote the minimum squared projection height for this antenna selection
rule (c.f. (7)). We claim that the maximum diversity order achieved for this antenna selection rule is given by ( j) log Pr( Rmin ≤ x) . x →0 log( x)
d (Lj ) = lim
(56)
( j) Since our antenna selection rule (10) dictates RSL ≥ Rmin with probability 1, the lemma readily follows.
The claim of (56) is proved as follows. Let us consider a linear ZF receiver first. For block fading channels, given any channel realization H , let Pe( j ) (H) and Pel( j ) (H) denote the conditional error probability of the whole frame and the lth substream, respectively, assuming optimal one-dimensional ) (H ) = max Pel( j ) (H ) , coding for each substream. The worst link, with conditional error probability Pe(_jmax l
has instantaneous post-detection SNR
ρ0 L
( j) Rmin (H ) 9 . The corresponding average error probabilities are
) denoted as Pe( j ) , Pel( j ) and Pe(_j max . Clearly we have
9
( j) ( j) Rmin ( H) refers to the realization of Rmin with respect to H .
19
L
) Pe(_jmax (H) ≤ Pe( j ) (H) ≤ ∑ Pel( j ) (H) ,
(57)
l =1
which leads to L
) max Pel( j ) ≤ Pe(_jmax ≤ Pe( j ) ≤ ∑ Pel( j ) . l
(58)
l =1
Since the upper and lower bounds above bear the same diversity order [7][10], we have
Pe( j )
) Pe(_j max
) E {Pe(_j max ( H)} .
(59)
) is (uniformly) lower bounded by the It is known from Lemma 5 of [7] that the error probability Pe(_jmax
⎛ ⎛ ρ ( j) ⎞ ⎞ corresponding outage probability, given by Pr ⎜ log ⎜ 1 + 0 Rmin ⎟ ≤ R0 ⎟ , where R0 is the fixed data rate L ⎠ ⎝ ⎝ ⎠ (see Section II). Therefore the diversity order is upper bounded by
d (ZFj)
⎛ ( j ) L R0 ⎞ ≤ (2 − 1) ⎟ log Pr ⎜ Rmin ( j) ρ0 ⎝ ⎠ = lim log Pr( Rmin ≤ x) . ≤ − lim ρ0 →∞ x →0 log( ρ 0 ) log( x)
(60)
The above error probability lower bound is obtained without specifying any coding scheme. On the other hand, the error probability with uncoded square M-QAM signaling can serve as an upper bound [7][21]: ) ) Pe(_jmax ≤ Pe(_jmax_ QAM ≤ 4 P ( s0 → s1 ) ,
(61)
where P( s0 → s1 ) denotes the pairwise error probability between two closest QAM constellation points. By Eq. (21) in [7], we have P ( s0 → s1 )
3 ⎛ ρ ( j) ⎞ P ⎜ 0 Rmin ⋅ < 1⎟ . M −1 ⎠ ⎝ L
(62)
We therefore obtain ( j) log Pr( Rmin ≤ x) , x →0 log( x)
d (ZFj) ≥ lim
(63)
and thus have proved the claim of (56) for a linear ZF receiver. An immediate conclusion from analysis above is that, one-dimensional channel coding has no impact on system diversity order. Note that the above analysis (see (58)) also indicates that ) ( j) log Pr( min { ρ k , ZF } ≤ 1) ≤ 1) log Pr( ρ k( ,jZF ≤ x) log Pr( Rmin k =1 L = lim = lim = min lim . x →0 k =1 L ρ0 →∞ ρ 0 →∞ log( x) log(1/ ρ 0 ) log(1/ ρ 0 ) ( j)
d
ZF ( j)
As for a linear MMSE receiver, following a similar approach, we have
20
(64)
d
MMSE ( j)
= lim
) log Pr( min { ρ k( ,jMMSE } ≤ 1)
k =1 L
log(1/ ρ 0 )
ρ0 →∞
= min lim
) log Pr( ρ k( ,jMMSE ≤ 1)
log(1/ ρ 0 )
k =1 L ρ 0 →∞
.
(65)
In the following we complete the proof by showing that ) Pr( ρ k( ,jZF ≤ 1)
) Pr( ρ k( ,jMMSE ≤ 1) , ∀k .
(66)
We alternatively have expressions of post-processing SNRs for ZF and MMSE receivers as (see, e.g., [10])
ρ k( ,jZF) = ) = ρ k( ,jMMSE
ρ0
(h ) L
( j) H k
(
⎡ ( j) ( j) H ( j) ⎢I − Hk ( Hk ) Hk ⎣
ρ
−ρ
( j) H k
⎤ ( j) ⎥ hk , ⎦
(67)
−1 ⎡ ⎤ L ⎞ ( j) H ( j) ⎛ ( j) H ( j) ( j) H ( j) ⎢ − + h I H H H I ( ⎟ ( Hk ) ⎥ hk . k ) k ⎜( k ) k L ρ0 ⎠ ⎢⎣ ⎥⎦ ⎝
(
( j) k , ZF
(H )
ρ0
By matrix inversion Lemma [20], with A (k j ) = ( H (k j ) ) H (k j ) ( j) k , MMSE
)
−1
= (h
)
H
)
(68)
−1
, we have −1
( j) H k
H A ( j) k
( j) k
⎛ L ( j) ⎞ ( j) ( j) H ( j) ⎜ A k + I ⎟ A k ( Hk ) hk ⎝ ρ0 ⎠
ηk( j ) ( ρ 0 ) .
(69)
Clearly η k( j ) ( ρ 0 ) ≥ 0 and thus ) ) Pr( ρ k( ,jMMSE ≤ 1) ≥ Pr( ρ k( ,jZF ≤ 1) .
(70)
) , as the latter is Another key observation is that, η k( j ) ( ρ 0 ) is statistically independent with ρ k( ,jZF
(
proportional to the squared norm of the projection of h (k j ) onto the null space of H (k j )
)
T
, while the
former is the correlation of h (k j ) with a vector in the range of H (k j ) [18][26]. Furthermore, as ρ0 → ∞ , it can be shown that a.s.
ηk( j ) ( ρ0 ) →ηk( j ) ,
(71)
i.e., it converges almost surely to a positive random variable with finite mean. This observation was also made in [24] where the PDF of η k( j ) was explicitly given. For our purpose, it is sufficient to know that Pr(η k( j ) ≤ 1/ 2) is a positive probability of O(1) (the same order as 1). We thus have ) log Pr( ρ k( ,jMMSE ≤ 1)
log(1/ ρ 0 )
=
) log Pr( ρ k( ,jZF + η k( j ) ( ρ 0 ) ≤ 1)
log(1/ ρ0 )
≥
) log Pr( ρ k( ,jZF ≤ 1/ 2)
log(1/ ρ 0 )
+
log Pr(ηk( j ) ( ρ0 ) ≤ 1/ 2) , (72) log(1/ ρ 0 )
with log Pr(ηk( j ) ( ρ0 ) ≤ 1/ 2) log Pr(η k( j ) ≤ 1/ 2) = lim = 0, ρ0 →∞ ρ 0 →∞ log(1/ ρ0 ) log(1/ ρ0 ) lim
(73)
where (73) is due to the fact that almost sure convergence always leads to convergence in distribution. (72) and (73) together gives rise to
21
) ) Pr( ρ k( ,jMMSE ≤ 1) ≤ Pr( ρ k( ,jZF ≤ 1) ,
(74)
and (66) follows with (70) and (74).
□
B. Lemma II and Corollary I Lemma II: For any permutation of the integer array 1 ~ NT , denoted as k1 ~ k NT , we have
(
Pr Rk1k2 ≤ x, Rk2k3 ≤ x, i.e. random variables Rk1k2 , Rk2 k3 ,
, Rk( N
, Rk( N
k T −1) NT
k T −1) NT
)
≤ x = [Pr( Rk1k2 ≤ x )]( NT −1) ,
(75)
are jointly independent.
Proof: Essentially Rki ki+1 is only a function of h ki and h ki+1 , denoted as Rki ki+1 = g (h ki , h ki+1 ) , therefore the conditional PDF of Rki ki+1 given those variables appearing earlier in the sequence admits: f ( Rki ki+1 | Rki−1ki ,
, Rk1k2 ) = f ( g (h ki , h ki+1 ) | g (h ki−1 , h ki ),
, g (h k1 , h k2 ))
= f ( g (h ki , h ki+1 ) | g (h ki−1 , h ki )) = f ( Rki ki+1 | Rki−1ki ), where the second equality holds because the states of
h k1 ~ h ki−1 do not affect h ki and h ki+1 .
Consequently, the above sequence forms a Markov chain. We are left to prove the independence between Rki ki+1 and Rki−1ki . Given Rki ki+1 = h ki 2
Rki−1ki = h ki−1 sin 2 θ ki−1ki , together with the independence between h ki
2
and h ki−1
2
2
sin 2 θ ki+1ki and
, and between vector
norms and directions (angles) [18], we only need to show that θ ki+1ki and θ ki−1ki are independent. Following a similar rotation approach as in [12], we define e1 ~ e N R as a fixed orthonormal basis (e.g., Cartesian coordinates) of the vector space
NR
. We rotate [h ki−1 , h ki , h ki+1 ] as a whole so that h ki is parallel
to e1 , denoted as [h ki−1 , h ki+1 ] = Q(ψ ki )[h ki−1 , h ki+1 ] , where ψ ki is the angle between h ki and e1 , and Q(ψ ki ) is the corresponding unitary rotation matrix. Since [h ki−1 , h ki+1 ] is an i.i.d. Gaussian matrix (therefore the joint distribution is rotationally invariant) and is independent with ψ ki , [h ki−1 , h ki+1 ] is still i.i.d. Gaussian. Because θ ki+1ki and θ ki−1ki are unchanged after the rotation, and equal to the angles between h ki+1 and e1 , and between h ki−1 and e1 , respectively (see Figure 2), given the fact that h ki+1 and h ki−1 are independent, it is
22
straightforward to show that θ ki+1ki and θ ki−1ki are independent, so are Rki ki+1 and Rki−1ki , and Lemma II follows.
□ Corollary I: θ12 ,θ13 ,
,θ1NT are jointly independent.
Proof: Using the same rotation approach as above, if we rotate [h1 , h 2 , parallel to e1 , h 2 ,
, h NT ] as a whole such that h1 is
, h NT are jointly independent vectors; their angles with e1 , which are equal to
θ12 ,θ13 , ,θ1N , respectively, are also jointly independent. T
□
C. Some Lemmas used in the Derivation of Proposition II Lemma III: If θ1 ,
,θ K are independent positive random variables, whose cumulative distribution
functions admit Fθ k ( x)
x nk , we have ⎡K ⎤ Pr ⎢ ∑θ k ≤ x ⎥ ⎣ k =1 ⎦
K
∑ nk
x k =1 .
(76)
Proof: At first we evaluate the exponential behavior of Pr(θ1 + θ 2 ≤ x) . Note that the following expression assumes x → 0 : x x −θ1 ⎡ ⎤ Pr(θ1 + θ 2 ≤ x ) = ∫ ⎢ ∫ fθ 2 (θ 2 )dθ 2 ⎥ fθ1 (θ1 )dθ1 ⎢ 0 0⎣ ⎦⎥ x
=
∫ fθ (θ ) Fθ ( x − θ )dθ 1
1
2
1
1
0 x
∫ ⎡⎣θ
n1 −1 1
+ o(θ1n1 −1 ) ⎤⎦ . ⎡⎣( x − θ1 ) n2 + o(( x − θ1 ) n2 ) ⎤⎦ dθ1
0 x
x n2 ∫ ⎡⎣θ1n1 −1 + o(θ1n1 −1 ) ⎤⎦ dθ1
,
0
x n1 + n2 , where o( x ) denotes a higher-order function of x such that lim o( x ) / x = 0 . Next, we can treat θ1 + θ 2 as x →0
a new random variable exponentially equivalent to x n1 + n2 and evaluate (θ1 + θ 2 ) + θ 2 following the same approach, whose CDF asymptotically behaves as x ( n1 + n2 ) + n3 . Lemma IV follows after such repeated operations. □
23
Lemma IV: Let m(θ ) be a positive function of θ , which monotonically increases with θ , satisfying
x n0 . If θ1 ,
m(0) = 0 and m −1 ( x)
,θ K are independent positive random variables, whose cumulative x nk , we have
distribution functions admit Fθ k ( x)
⎡ ⎛ K ⎞ ⎤ Pr ⎢ m ⎜ ∑θ k ⎟ ≤ x ⎥ ⎣ ⎝ k =1 ⎠ ⎦ Proof: Since θ1 ,
(
K
)
Pr ⎡ m max θ k ≤ x ⎤ ⎢⎣ ⎥⎦ k
n0
x
∑ nk k =1
.
(77)
,θ K are independent, we get: K
∑ nk
K
Pr[max θ k ≤ x] = ∏ Fθ k ( x) k
x k =1 .
(78)
k =1
By Lemma III we have K
∑ nk
⎡K ⎤ Pr ⎢ ∑θ k ≤ x ⎥ ⎣ k =1 ⎦
x k =1 .
(79)
Because both θ and m(θ ) are positive, and m(θ ) monotonically increases with θ , m −1 ( x) is also a positive function of x, which monotonically increases with x. Therefore, K
∑ nk Pr[m(max θ k ) ≤ x] = Pr[max θ k ≤ m ( x)] [m ( x)] k =1 −1
k
−1
k
K
n0
x
∑ nk k =1
.
(80)
Similarly, ⎡ ⎛ K ⎞ ⎤ Pr ⎢ m ⎜ ∑θ k ⎟ ≤ x ⎥ ⎣ ⎝ k =1 ⎠ ⎦
K
n0
x
∑ nk k =1
(81)
and Lemma V follows. □
For independent continuous random variables a , b1 and b2 satisfying a ≥ 0 with
Lemma V:
Pr(a ≤ x)
x na , and 0 ≤ b1 , b2 ≤ 1 with Pr(b1 ≤ x) Pr(ab1 ≤ x)
Pr(b2 ≤ x)
x nb , we have
.
Pr(ab2 ≤ x) ≤ x na .
(82)
Proof: We have 1
x Pr(ab1 ≤ x) = Pr(a ≤ ) fb1 ( y ).dy y 0
∫
ε
=
1
x x Pr( a ≤ ) f b1 ( y ).dy + Pr( a ≤ ) f b1 ( y ).dy y y ε 0
∫
∫
= t11 ( x) + t12 ( x), and similarly,
24
(83)
ε
1
x x Pr(a ≤ ) f b2 ( y ).dy + Pr(a ≤ ) fb2 ( y ).dy y y 0 ε
∫
Pr(ab2 ≤ x) =
∫
(84)
= t21 ( x) + t22 ( x), where ε > 0 is a fixed small positive number, small enough to make the PDF approximations for b1 and
b2 : fb1 ( x) = c1 x nb −1 + o( x nb −1 ),
f b2 ( x) = c2 x nb −1 + o( x nb −1 )
(85)
hold true. x Since Pr(a ≤ ) is positive, and is a decreasing function of y, when x → 0 we have y x
1
x t12 ( x) ≤ Pr(a ≤ ) fb1 ( y ).dy ≤ Pr(a ≤ )
ε
∫ε
ε
x na .
(86)
.
In another word, t12 ( x) ≥ x na . On the other hand, since 0 ≤ b1 ≤ 1 , we have Pr(ab1 ≤ x) ≥ Pr(a ≤ x)
x na ,
(87)
.
which means that Pr(ab1 ≤ x) ≤ x na . The same inequalities as in (86)(87) also hold true for t22 ( x) and Pr( ab2 ≤ x) , respectively. Meanwhile, since ε is small enough, by applying (85), we get that t11 ( x) t21 ( x) If t12 ( x ) > x the case t12 ( x )
na
∫
ε
0
x Pr(a ≤ ) y nb −1.dy . y
or t22 ( x ) > x a , t11 ( x )
(88)
t21 ( x ) ≤ x na dominate and (82) follows. We are left to check
n
x na , which together with (88) leads to (82) as well.
t22 ( x )
□ Lemma VI: With z , θ '0 , and θ 0 as defined in Section III A, we have
Pr( z sin 2 θ '0 ≤ x)
Pr( z sin 2 θ 0 ≤ x) .
Proof: From (18)(32) the PDF of θ '0 = max θ '1k is derived through results in order statistics: k
fθ '0 (θ ) =
M C NT −1
[sin 2 θ ]M −1 sin 2θ ,
M = ( NT − 1)( N R − 1), θ ∈ (0,ψ 0 ) ,
25
(89)
where C =
∫
ψ0
0
fθ1i ( x)dx, 2 ≤ i ≤ NT . Therefore, we have
Pr( z sin 2 θ '0 ≤ x) =
ψ0
⎛
⎞ ⎟ fθ '0 (θ ).dθ
x
∫ F ⎜⎝ sin z
θ⎠
2
0
=
ψ0
M C
NT −1
⎛
x
∫ F ⎜⎝ sin z
2
0
t =sin θ 2
a0
M
=
∫
C NT −1
t t1 = a0
0
⎞ M −1 2 ⎟ [sin θ ] sin 2θ .dθ θ⎠
⎛ x⎞ Fz ⎜ ⎟ t M −1.dt ⎝t⎠
(90)
⎛ ( x / a0 ) ⎞ M −1 Ma M = N 0−1 Fz ⎜ ⎟ t1 .dt1 , T C ⎝ t1 ⎠ 0 1
∫
where a0 = sin 2 ψ 0 is a positive real number. On the other hand, by applying (37) we have π /2
Pr( z sin 2 θ 0 ≤ x) =
⎛
z
0
π /2
=M
⎞ ⎟ fθ0 (θ ).dθ
x
∫ F ⎜⎝ sin ⎛
2
θ⎠ x
∫ F ⎜⎝ sin z
2
0
⎞ M −1 2 ⎟ [sin θ ] sin 2θ .dθ θ⎠
(91)
1
t =sin 2 θ
⎛ x⎞ = M Fz ⎜ ⎟ t M −1.dt. ⎝t⎠ 0
∫
By comparing (90) and (91), we get Pr( z sin 2 θ '0 ≤ x)
Pr( z sin 2 θ 0 ≤ x) . □
D. The Derivation of (38). We continue the evaluation of (91) to derive the polynomial expansion of Pr( z sin 2 θ 0 ≤ x) : 1
⎛ x⎞ Pr( z sin θ 0 ≤ x) = M Fz ⎜ ⎟ t M −1.dt ⎝t⎠ 0
∫
2
∞
m =1/ t
= M Fz ( mx )
∫ 1
1 m
M +1
.dm
∞ M + NT − 2 ⎡ mk x k ⎤ 1 .dm = M ⎢1 − e − mx ⎥ , k ! ⎦⎥ m M +1 ⎢ k =0 1⎣
∫
∑
= 1 − P1 ( x) − P2 ( x),
26
(92)
M
xk where P1 ( x) = M EM +1− k ( x) , and P2 ( x) = M k =0 k !
∑
M + NT − 2
∑
k = M +1
∞
xk e − xm E− ( k − M −1) ( x) , with Ek ( x) = dm the k! mk 1
∫
integral exponential function. From [13], we have the following recursive rules: Ek +1 ( x) =
1 −x (e − xEk ( x)) k
Ek +1 ( x) =
e− x k!
k −1
∑
(−1)i (k − i − 1)! xi +
i =0
(−1) k k x E1 ( x). k!
Therefore, after some involved mathematical manipulations, we have P1 ( x) = Me − x
M −1
∑c x
k
k
,
k =0
where ck =
k
∑ i =0
(−1) k −i ( M − k − 1)! , for k ≤ M − 1 . We can expand e− x by its Taylor’s series and get the ( M − i )!i !
polynomial equivalent form: P1 ( x) =
∞
∑a x n
n
. For n ≤ M − 1 ,
n =0
an = M =M
(−1) n − k k (−1) k −i ( M − k − 1)! ( M − i )!i ! k = 0 ( n − k )! i = 0 n
∑
∑
1 n (−1) n n!( M − k − 1)! k (−1)i M ! n! k =0 (n − k )!M ! i = 0 ( M − i )!i !
∑
∑
⎛n⎞ (−1) n ⎜ ⎟ k 1 ⎝ k ⎠ ( −1)i ⎛ M ⎞ = ⎜ ⎟ n ! k =0 ⎛ M − 1⎞ i =0 ⎝ i ⎠ ⎜ ⎟ ⎝ k ⎠ n
∑
∑
⎛n⎞ (−1) n ⎜ ⎟ 1 ⎝ k ⎠ (−1) k ⎛ M − 1⎞ = ⎜ ⎟ n ! k =0 ⎛ M − 1⎞ ⎝ k ⎠ ⎜ ⎟ ⎝ k ⎠ n
∑
=
( −1) n n!
n
∑ (−1) k =0
k
where we use the equality [13]:
k
n=0 ⎛ n ⎞ ⎧1, ⎜ ⎟=⎨ ⎝ k ⎠ ⎩0, 0 < n ≤ M − 1,
⎛M ⎞
∑ (−1) ⎜⎝ i ⎟⎠ = (−1) i
k
i =0
aM = −
⎛ M − 1⎞ ⎜ ⎟ . With a similar approach, we can obtain ⎝ k ⎠
1 , therefore M!
P1 ( x) = 1 −
1 M x + o( x M ) . M!
27
(93)
On the other hand, P2 ( x) can be represented as P2 ( x) = Me
−x
M + NT − 2
∑
k = M +1
⎧⎪ (k − M − 1)! ⎡ k − M −1 xi + M ⎨ ⎢ k! ⎣ i =0 i ! ⎩⎪
∑
⎤ ⎫⎪ ⎥⎬ , ⎦ ⎭⎪
and after some manipulations we obtain P2 ( x) = bM x M + o( x M ) , where bM = M
NT − 3
(94)
∑ (M + k + 1)! . By combining (92), (93) and (94), we can derive (38). Furthermore, k!
k =0
from [13], we have ∞
∑ (M + k + 1)! = M (M + 1)! , k!
1
k =1
therefore NT − 3
∑ k =0
∞ k! k! < ( M + k + 1)! k =0 ( M + k + 1)!
∑
,
∞
∑
k! 1 1 1 1 = + = + = ( M + 1)! M ( M + 1)! ( M + 1)! M .M ! k =1 ( M + k + 1)! so the coefficient of x M in (38) is always positive.
E. Lemma VII Lemma VII: For any NT − L + 1 distinct integers within [1, NT ] , denoted as an ordered list k1 ~ k NT − L +1 ,
and any NT − L + 1 different subsets from U1 ~ U NU , denoted as an ordered list U j1 ~ U jN
T − L +1
, suppose
h ki ∈U ji , and h ki ∉U jl for l > i , i.e., h ki belongs to subset U ji but not those after. We have:
)
(
(j ) , RkNNT−L−+L1+1 ′ ≤ x = ⎡⎢ Pr( Rk(1j1 )′ ≤ x) ⎤⎥ T ⎣ ⎦
Pr Rk(1j1 )′ ≤ x, Rk(2j2 )′ ≤ x,
( NT − L +1)
,
(95)
where Rk( j )′ is modified from Rk( j ) with k indicating the index of the column vector in the original channel matrix H , instead of H j 10. Proof: From the definition, Rk(1j1 )′ = Ph k1
{
2
, where P = I − BB† is the projection matrix to the null space
}
of range U j1 \ h k1 , the subspace spanned by the remaining column vectors in subset U j1 except h k1 [20], and B is composed of any basis of this subspace. Since any projection matrix is idempotent, i.e. P 2 = P , 10
Lemma II is a special case of Lemma VII. However, the proof of Lemma II bears some interesting geometric elements, and is needed for Corollary I and Proposition II.
28
its eigenvalues are either 1 or 0. Noting that P is Hermitian, we can write the eigenvalue decomposition of P as P = VΛV H , where V is unitary, and Λ = diag (1N R − L +1 ,0 L −1 ) . Therefore Rk(1j1 )′ = VΛV H h k1
2
2
= ΛV H h k1 ,
(96)
where the second equality follows by the fact that a unitary transformation preserves length. From the definition of P , the unitary matrix V H is independent of h k1 . Therefore the conditional PDF
(
{
f Rk(1j1 )′ | U j1 \ h k1
}) = f ( ΛV
H
0
h k1
2
) = f ( Λh ) = f ( R ′ ) , 2
k1
( j1 ) k1
where V0 H is a fixed unitary matrix dependent on the given realizations of
{U
(97)
j1
}
\ h k1 , and the second
equality comes from the rotationally invariant property of the i.i.d. Gaussian vector h k1 [18][26]. That is,
(
)
Rk(1j1 )′ is independent of any vector in range U j1 \ h k1 . It is therefore straightforward to show that Rk(1j1 )′ ~ RkNNT−L−+L1+1 ′ are jointly independent and identically distributed, and Lemma VII holds. (j
)
T
□
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30
Figure 1. Illustration of (6)
Figure 2. Independence of θ ki−1ki and θ ki+1ki
31
Figure 3. Illustration of the Proof for Proposition I: A Graph with Size n
Figure 4. Exponential Behaviors of the Outage Probabilities for the NT = N R = 3, L = 2 Scenario
32
Figure 5. BER Performance of the ZF-DF Receiver with NT = 3, N R = 3, L = 2
Figure 6. Exponential Behaviors of the Outage Probabilities for the N T = N R = 4, L = 3 Scenario
33
