Exact Symbol Error Rate Analysis of Orthogonalized Spatial ...
Exact Symbol Error Rate Analysis of Orthogonalized Spatial Multiplexing Systems with Optimal Precoding Jaesin Kim, Young-Tae Kim, Jihoon Kim, and Inkyu Lee School of Electrical Engineering, Korea University, Seoul, Korea Email: {cache, reftm}@korea.ac.kr, [email protected], and [email protected]
Abstract—Recently an orthogonalized spatial multiplexing scheme with optimal precoding, referred to as POSM, has been proposed. The POSM achieves optimal performance in terms of maximizing the received minimum Euclidean distance dmin,R in closed-loop multiple-input multiple-output (MIMO) systems. In this paper, we derive exact closed-form expressions of the symbol error rate (SER) for the POSM with 4-QAM and 16-QAM in Rayleigh fading channels. We first introduce a new system model for the given channel. Based on this system model and a unique feature of the POSM effective constellations, the final expression of the SER using polar coordinates can be evaluated with a single-integral form. Simulation results confirm the accuracy of our derived analysis.
I. I NTRODUCTION In wireless multiple-input multiple-output (MIMO) systems, channel knowledge at the transmitter can be utilized to obtain higher performance gains over open-loop MIMO systems. Many studies on such closed-loop MIMO systems have been carried out by computing singular value decomposition (SVD) of a channel matrix [1] [2]. The performance of such systems with linear receivers can be worse than that of systems with maximum likelihood (ML) receivers. To optimize the system with ML receivers in terms of uncoded symbol error rate (SER), the received minimum Euclidean distance dmin,R between the effective constellation points must be maximized [3]. Recently, orthogonalized spatial multiplexing (OSM) has been developed in [4] for closed-loop MIMO systems, which allows a simple symbol-by-symbol ML receiver. More recently, an optimal precoding scheme based on the OSM has been proposed, which maximizes dmin,R [5]. Surprisingly, it was shown that the performance of the OSM with this optimal precoding is identical to that of the optimal closedloop MIMO systems presented in [6]. Besides, this precoding scheme requires only a few real values for feedback. From now on, we refer to the OSM system with optimal precoding as POSM. Exact evaluation of an error rate is one of the fundamental problems for the system design. In this paper, we derive exact expressions of the SER for the POSM with 4-QAM and 16QAM in Rayleigh fading channels. Deriving an exact SER of the POSM offers many insights on the design of the optimal precoding matrix in terms of dmin,R for MIMO systems with This research was supported by the MKE (Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement).
ML receivers. Also, it helps us to understand the error performance of the POSM by showing the effective constellation structures dependent on the channel realization as well as the constellation. For example, while the effective constellation in general MIMO systems has the form of polygons with random inside angles, the effective constellation shape in the POSM is composed of a few parallelograms with relatively fixed inside angles, due to the selection of the optimal precoding matrix according to channel conditions. To derive an exact SER for the POSM, we first provide a new system model using singular values of a given channel. Based on this system model, we can compute the exact average SER by utilizing polar coordinates and the distributions of the singular values. Then, we investigate that the POSM has a unique feature that the effective constellations become relatively invariant with respect to the channel realization. Exploiting this characteristics, we show that the whole computation of the SER containing a triple-integral form is reduced to a single-integral form. The simulation results confirm the accuracy of our results. Throughout this paper, normal letters represent scalar quantities, boldface letters indicate vectors and boldface uppercase letters designate matrices. With a bar accounting for complex variables, for any complex notation c¯, we use [¯ c] and [¯ c] to denote the real and imaginary part of c¯, respectively. II. S YSTEM M ODEL We consider the baseband system model of the POSM with Mt transmit and Mr receive antennas in flat fading channels. ¯ ∈ CMr ×Mt and x Denoting H ¯ = [¯ x1 x ¯2 · · · x ¯Mt ]T ∈ CMt ×1 as the channel matrix and the transmitted symbol vector, respectively, the Mr -dimensional received signal vector is expressed as ¯x + n y ¯ = H¯ ¯
(1)
¯ are selected with an independent and where the elements of H identically distributed (i.i.d.) complex Gaussian distribution with zero mean and unit variance, and n ¯ = [¯ n1 n ¯2 · · · n ¯ Mr ]T is a circularly symmetric complex Gaussian noise with variance 2N0 . Although the POSM can be extended to a system with more than two transmit antennas using the method in [4] and [5], we focus on analyzing the POSM with Mt = 2 for simplicity. The POSM scheme encodes two transmitted symbol as F (¯ x, θo , θ1 , p, θ2 ) = R (θo , θ1 ) PRθ2 ¯s
978-1-4244-2515-0/09/$25.00 ©2009 IEEE
where R (θo , θ1 ) is the rotation matrix with phases θo and θ1 , described in [4] and [5] in detail, and P, Rθ2 and ¯s are defined as p 0 cos θ2 − sin θ2 P= , R θ2 = , sin θ2 cos θ2 2 − p2 0 x2 ] [¯ x1 ] + j[¯ and ¯s . x2 ] [¯ x1 ] + j[¯ Here the optimal values of p and θ2 are obtained in [5]. Applying the above transformation, the system model (1) is changed to ¯ (¯ ¯ e PRθ ¯s + n ¯ = HF ¯ ¯=H y x, θo , θ1 , p, θ2 ) + n 2
¯ e accounts for the effective channel matrix denoted where H ¯ ¯ (θo , θ1 ). Now, the real-valued representation of as He = HR the system model for the POSM can be rewritten as P 0 Rθ2 0 y = He s+n (2) 0 P 0 Rθ2 where s [[¯sT ][¯ sT ]]T , n [[¯ nT ][¯ nT ]]T , and He ¯ e ] −[H ¯ e] [H ¯ e ] . Here 0 indicates a 2-by-2 zero matrix. ¯ e ] [H [H Denoting He in (2) as He = [he,1 he,2 he,3 he,4 ], it is important to note that all columns of He achieve orthogonality among each other by the rotation angles θo and θ1 . Also, we have he,1 = he,3 ≥ he,2 = he,4 where he,1 and he,2 are represented by σ1 and σ2 , respectively [5]. Here σ1 and σ2 are equal to the first and second singular values of ˜ e,i = he,i /he,i ¯ respectively [7]. Defining h the matrix H, for i = 1, 2, 3, 4 as the normalized column, it follows ˜ e,1 σ2 h ˜ e,2 σ1 h ˜ e,3 σ2 h ˜ e,4 ] He = [σ1 h ˜ e,2 h ˜ e,3 h ˜ e,4 ]diag{σ1 , σ2 , σ1 , σ2 } ˜ e,1 h [h Σ 0 = U 0 Σ =
˜ e,1 h ˜ e,2 h ˜ e,3 h ˜ e,4 ] represents an orthogonal where U = [h matrix and Σ is defined as Σ = diag{σ1 σ2 }. It was shown in [5] that there exist only two and four distinct cases for 4-QAM and 16-QAM, respectively, for representing the optimum precoder parameters p and θ2 , and that each case is determined by a single variable k, defined as σ12 /σ22 . By using these parameters, the POSM system attains the optimal performance in terms of maximizing dmin,R [5]. Finally, employing UT as the receive filter, the receive filter output r is given by T r1 ] [¯ r1 ] [¯ r2 ] [¯ r2 ] r = UT y = [¯ Σ 0 P 0 Rθ2 0 = s + w (3) 0 Σ 0 P 0 Rθ2 T x1 ] [¯ x1 ] [¯ x2 ] [¯ x2 ] where s is represented as s = [¯ and w is defined as w = UT n. Thus, denoting [[¯ zl ] [¯ zl ]]T = ΣPRθ2 [[¯ xl ] [¯ xl ]]T as the transmitted symbol, we obtain the following complex-valued ML detection metric as 2 zˆ ¯l = arg min |¯ rl − z¯l | , z¯l ∈χ
for l = 1, 2
(4)
where χ is the effective constellation formed by z¯l . As shown above, the ML detection at the receiver can be done by searching for a single symbol (called single-symbol decodable). This new formation is convenient for geometric consideration in order to evaluate the error probability in the subsequent section. III. E XACT E RROR P ROBABILITY A NALYSIS In this section, we will derive an exact SER of the POSM. First, we start with a conditional SER of the POSM for a given channel gain with σ1 and σ2 for 4-QAM. In [5], the parameters for the optimum precoding for 4-QAM are divided into two cases of 1 ≤ k < 7 and k ≥ 7. We can depict decision boundaries of the effective constellations for these two cases based on (4), as shown in Fig. 1. Denoting Es as the average transmit energy per symbol, the transmitted symbol 1 , S2 , S3 , S4 } for 4-QAM constellation becomes set {S
{[± E2s ± E2s ]T }. the 1 ≤ k < 7 case with the symbol [[¯ xl ] [¯ xl ]]T = For Es T [ E2s 2 ] , we have ⎡ E ⎤ s p 0 σ1 0 cos θ2 − sin θ2 [¯ zl ] ⎣ 2 ⎦ = 2 sin θ2 cos θ2 0 σ2 0 2 − p Es [¯ zl ] =
0 √
2
σ2 Es
where p = popt = 1 and θ2 = θ2opt = π/4 are used as in [5]. By applying similarly to the remaining symbols, we can see that the effective constellation of the 1 ≤ k < 7 case forms a rhombus shape, while the k ≥ 7 case has a straight line-shaped effective constellation as shown in Fig. 1. Note that the decision regions of the 1 ≤ k < 7 case are no longer perpendicular. For this reason, we evaluate an exact error probability employing polar coordinates [8]–[10]. To utilize polar coordinates, we denote r and φ as the magnitude and the angle of w1 + jw2 , respectively, where wi is an ith element of w in (3). Then, since unitary filter UT at the receiver does not have an effect on the property ¯ in (1), r has a Rayleigh distribution with of the noise n E[|w1 + jw2 |2 ] = 2N0 , and φ is uniformly distributed. Also, the joint probability density function (pdf) of r and φ is given as 1 r r2 f (r, φ) = exp − . (5) 2π N0 2N0 Thus, error decisions are made if the received signal corrupted by the noise with the above distribution falls outside the decision region. Let us define D(φ) as the distance from a symbol to the decision boundary for a certain angle φ ∈ [0, φ0 ]. Then we can obtain the error decision probability by integrating the pdf (5) as φ0 ∞ 1 r r2 Pr(error decision) = exp − drdφ 2N0 D(φ) 2π N0 0 φ0 D(φ)2 1 exp − dφ. = 2π 0 2N0 (6) Φ(φ0 , D(φ)).
σ 2 εs
s1 θt
π − 2θt 2
s4
s2
θt
σ 1 εs
√ √ 1 Es 1 Es + Q κ2 σ√ where Pe,2|S1 (σ12 , σ22 ) = Q κ1 σ√ N N0 √ 0 σ√ 2 2 1 Es and Pe,2|S2 (σ1 , σ2 ) = Q κ2 N . Here κ1 and κ2 are 0 defined as κ1 = cos 0.464 − sin 0.464 and κ2 = sin 0.464, respectively. Note that the Q-function can be evaluated using the Craig’s form with a definite integral term [11]. Utilizing the Craig’s form instead of the Q-function, it can be shown that (8) is represented as π π Pe,2 (σ12 , σ22 ) = 2Φ( , D3 (φ)) + Φ( , D4 (φ)) (9) 2 2 √
s3 a) 1 ≤ k < 7 case
s4
s3
s2
s1
2(cos 0.464 − sin 0.464)σ 1 εs
2sin 0.464σ 1 εs
b) k ≥ 7 case Fig. 1.
POSM decision regions for 4-QAM
For simple representation, let us denote Pe,1 and Pe,2 as the SER of the 1 ≤ k < 7 case and the k ≥ 7 case, respectively. For a given channel with σ1 and σ2 , we first evaluate the conditional error probability for 1 ≤ k < 7 with a rhombic effective constellation. In this case, the conditional probability of error decision for the symbols S1 and S2 can be obtained as π π Pe,1|S1 (σ12 , σ22 ) = 2Φ( + θt , D1 (φ))+2Φ( − 2θt , D2 (φ)), 2 2 π Pe,1|S2 (σ12 , σ22 ) = 2Φ( + θt , D1 (φ)) 2 √ 2 2 √ (σ1 +σ2 )Es 2 Es where D1 (φ) = 2 cos(θt −φ) and D2 (φ) = σcos φ . Here √ θt is defined as tan−1 (σ2 /σ1 ) = tan−1 (1/ k). Since the symbols S1 and S2 are symmetric to S3 and S4 with respect to the origin, we can simply check that Pe,1|S3 = Pe,1|S1 and Pe,1|S4 = Pe,1|S2 . Assuming that the transmitted symbols are equally likely, we can obtain the probability of error given σ1 and σ2 as 1 2Pe,1|S1 (σ12 , σ22 ) + 2Pe,1|S2 (σ12 , σ22 ) Pe,1 (σ12 , σ22 ) = 4 π π = 2Φ( + θt , D1 (φ)) + Φ( − 2θt , D2 (φ)). (7) 2 2 Next, for the k ≥ 7 case, we can easily express the conditional error probability as the combination of Q-function similar to pulse amplitude modulation (PAM). Since the effective constellation points are symmetric with respect to the origin, we have 1 Pe,2 (σ12 , σ22 ) = 2Pe,2|S1 (σ12 , σ22 ) + 2Pe,2|S2 (σ12 , σ22 ) (8) 4
√
σ1 Es 1 Es where D3 (φ) = κ2 σsin φ and D4 (φ) = κ1 sin φ . Now we have the probability of error given the channel realization as in (7) and (9). Since the channel gain σ1 and σ2 are random, we need to average (7) and (9) in terms of σ12 and σ22 . For compact representation, we replace σ12 and σ22 by λ1 and λ2 , respectively. Then we can obtain the total probability of symbol error for 4-QAM as ∞ 7λ2 Pe = Pe,1 (λ1 , λ2 )f (λ1 , λ2 )dλ1 0 λ2∞ + Pe,2 (λ1 , λ2 )f (λ1 , λ2 )dλ1 dλ2 (10) 7λ2
where f (λ1 , λ2 ) represents the joint pdf of the square of singular values of a 2-by-2 matrix, which is derived as [12] f (λ1 , λ2 ) = (λ1 − λ2 )2 e−(λ1 +λ2 )
for λ1 ≥ λ2 .
Observing the development of (10), we can see that the error probability for the α ≤ k < β case is generally comprised as the following triple-integral form ∞ βλ2 Φ(φ0 , D(φ))f (λ1 , λ2 )dλ1 dλ2 0
1 = 2π
αλ2 βλ2 φ0
∞ 0
αλ2
0
D(φ)2 exp − f (λ1 , λ2 )dφdλ1 dλ2 .(11) 2N0
This triple integral may require high complexity in numerical computation. Now, we will show that the above triple-integral form can be computed with a single-integral form. To obtain a final SER expression with reduced complexity, we investigate the effect of the optimal precoding on the effective constellation shapes in terms of dmin,R . It can be concluded that in MIMO systems, the effective constellation generally has the form of polygons with random inside angles, whereas that in the POSM is composed of a few parallelograms with almost invariant inside angles. For example, θt for 1 ≤ k < 7 shown in Fig. 1 varies from 0.361 to 0.785 radian. Based on this observation, we make an assumption that θt is independent of random channel gains σ1 and σ2 in order to simplify the computation in (11), and then we substitute θt by a fixed value. To determine this fixed value, we first compute the median value of k. Utilizing the result in [13], we can derive the pdf of k as f (k) = 6(k + 1)−4 (k − 1)2 , k ≥ 1. Then, based on this distribution, the median value of k is√obtained as 3.94 for 1 ≤ k < 7. Since θt equals √ tan−1 (1/ k), −1 we replace the angle θt by 0.467 = tan (1/ 3.94). In the
simulation section, we will show that this approximation does not sacrifice the accuracy of the SER analysis, since the basic structure of the effective constellation does not vary much. With this substitution in θt , equation (11) is transformed into φ0 ∞ βλ2 1 (λ1 − λ2 )2 e−A(φ)λ1 e−B(φ)λ2 dλ1 dλ2 dφ. (12) 2π 0 0 αλ2
d1
θc
d2
yX
d3
yY
a) 7.59 ≤ k < 43.1 case
Then the integrand in the triple integrals now becomes integrable with respect to λ1 and λ2 . Thus it can be shown that equation (12) equals φ0 2(α − 1)2 1 2(α − 1) 2 + 2 + 3 2π 0 A(αA + B)3 A (αA + B)2 A (αA + B) 2(β − 1) 2 2(β − 1)2 − 2 − 3 − dφ A(βA + B)3 A (βA + B)2 A (βA + B) (13) Ψ(α, β, φ0 , A(φ), B(φ)). In this expression, we omit the variable φ in A(φ) and B(φ) for simplicity. Hence, representing Pe in (10) with the Ψ-function in (13), the final SER for 4-QAM is expressed by π + θt , C1 (φ), C1 (φ)) 2 π +Ψ(1, 7, − 2θt , 1, C2 (φ)) 2 π π +2Ψ(7, ∞, , C3 (φ), 1) + Ψ(7, ∞, , C4 (φ), 1) (14) 2 2
b) 43.1 ≤ k < 101 case
Pe = 2Ψ(1, 7,
where C1 (φ) = κ2 γ
γ 8 cos2 (0.467−φ)
+ 1, C2 (φ) = κ2 γ
γ 2 cos2 φ
+ 1,
C3 (φ) = 2 sin2 2 φ + 1 and C4 (φ) = 2 sin1 2 φ + 1. Here γ denotes the signal-to-noise ratio (SNR) defined as Es /N0 . Note that when β is equal to infinity in the Ψ-function, the last three terms of the integrand go to zero in (13). Next we consider a SER for 16-QAM. Unlike 4-QAM, 16-QAM is classified into four cases with boundary points of k = 7.59, 43.1 and 101 as shown in [5]. Considering the effective constellation, the 1 ≤ k < 7.59 case forms a rhombus-shaped effective constellation as in 4-QAM and the effective constellation for k ≥ 101 becomes 16-PAM. Therefore, it is straightforward to derive the error probabilities of these two cases for 16-QAM by using the solution for 4QAM. On the other hand, the other two cases of 7.59 ≤ k < 43.1 and 43.1 ≤ k < 101 are more complicated since there are many asymmetric decision regions. In Fig. 2, we illustrate the decision boundaries for these two cases. Note that the decision boundaries are quite different from conventional 16-QAM constellations. As shown in Fig. 2, any vertex of decision regions becomes an outer center of a triangle formed by neighboring three constellation points. Therefore we define all inside angles of decision region as an inverse cosine function for convenience. For example, θc in Fig. 2 is defined as d1 θc = cos−1 ( 2R ). Here the radius of a circumscribed circle 1 R1 is calculated by d1 d2 d3 R1 = 4 s(s − d1 )(s − d2 )(s − d3 )
Fig. 2.
POSM decision regions for 16-QAM
where d1 , d2 and d3 are the length of three sides of a triangle, respectively, and s is defined as s = (d1 + d2 + d3 )/2. There are two distinct radii R1 and R2 for the 7.59 ≤ k < 43.1 case, while three distinct radii exist for the 43.1 ≤ k < 101 case as shown in Fig. 2. Now, we can calculate the error probabilities of each case with the same approach in 4-QAM. The detailed derivation is omitted to simplify the presentation. We denote Pe,i for i ∈ {1, 2, 3, 4} as the SER of the 1 ≤ k < 7.59, 7.59 ≤ k < 43.1, 43.1 ≤ k < 101 and k ≥ 101 cases, respectively. The fixed angle values for 22 different terms are listed in Table I. Presenting with the Ψ-function in (13), an exact expression of the SER for 16-QAM can be finally expressed as 22
Pe =
1 ci Ψ(αi , βi , φ0i , Ai (φ), Bi (φ)) 16 i=1
(15)
where ci represents the coefficient related to the number of Ψ(αi , βi , φ0i , Ai (φ), Bi (φ)). IV. S IMULATION R ESULTS In this section, we provide the results for our analysis and compare them with numerical simulation. Fig. 3 depicts the average SER of the POSM for 4-QAM and 16-QAM over Rayleigh fading channels with Mt = Mr = 2 versus the averaged received SNR γ. Here, we plot our final analysis results in (14) and (15) as the solid line and numerical simulations as the square point. Also, we include the SER results with a triple-integral form for 4-QAM in (10) and 16QAM, respectively. As shown in Fig. 3, it can be observed
TABLE I PARAMETERS OF Pe FOR 16-QAM i
ci
φ0i
Ai (φ)
0
10
Bi (φ)
Analysis w/ single−integral Analysis w/ triple−integral Simulation
−1
10
Pe,1 : αi = 1, βi = 7.59 48
0.458
2
24
2.029
3
36
0.655
γ +1 40 cos2 φ γ + 40 cos2 (0.458−φ)
1
1
γ +1 40 cos2 φ γ + 40 cos2 (0.458−φ) γ +1 10 cos2 φ
−2
10
1
SER
1
−3
10
Pe,2 : αi = 7.59, βi = 43.1 (mc = cos 0.488, ms = sin 0.488) 4
8
1.617
5
8
2.187
6
8
0.616
7
4
2.572
8
24
0.403
9
12
1.525
10
12
0.552
11
12
1.168
+1
m2 cγ 20 cos2 (0.616−φ)
8
1.993
13
4
2.213
14
4
2.922
15
12
0.407
16
12
1.485
17
12
0.775
18
12
0.608
19
8
1.015
20
4
2.586
−5
10 m2 sγ +1 20 cos2 (1.001−φ) (mc −2ms )2 γ +1 20 cos2 φ
m2 cγ +1 20 cos2 (1.001−φ) (2mc +ms )2 γ +1 20 cos2 φ
(mc −ms )2 γ 20 cos2 (0.552−φ)
+1
(mc +ms )2 γ 20 cos2 (0.552−φ)
+1
m2 cγ 20 cos2 (0.403+φ)
+1
m2 sγ 20 cos2 (0.403+φ)
+1
m2 sγ 20 cos2 (0.642−φ)
+1
m2 sγ +1 20 cos2 (1.351−φ) (mc −3ms )2 γ +1 20 cos2 φ (mc −2ms )2 γ +1 20 cos2 (0.522−φ) (mc −ms )2 γ +1 20 cos2 (0.407+φ) m2 cγ +1 20 cos2 (0.963+φ) m2 sγ +1 20 cos2 φ m2 sγ +1 20 cos2 (1.015−φ)
m2 cγ 20 cos2 (0.642−φ)
+1
m2 cγ +1 20 cos2 (1.351−φ) (3mc +ms )2 γ +1 20 cos2 φ (2mc +ms )2 γ +1 20 cos2 (0.522−φ) (mc +ms )2 γ +1 20 cos2 (0.407+φ) m2 sγ +1 20 cos2 (0.963+φ) m2 cγ +1 20 cos2 φ m2 cγ +1 20 cos2 (1.015−φ)
Pe,4 : αi = 101, βi = ∞ (mc = cos 0.245, ms = sin 0.245) 21 22
48
π 2
12
π 2
16−QAM
+1
Pe,3 : αi = 43.1, βi = 101 (mc = cos 0.345, ms = sin 0.345) 12
4−QAM
−4
10 m2 sγ 20 cos2 (0.616−φ)
m2 sγ +1 10 sin2 φ (mc −3ms )2 γ + 10 sin2 φ
1 1
1
that our analysis exactly matches numerical simulation results, even though we substitute angles by fixed values in order to form a single integral. Also, SER curves in Fig. 3 exhibit a slope of 4 and these clearly show a full diversity advantage of the POSM. V. C ONCLUSION In this paper, we have addressed analytical characteristics for the optimal precoding in terms of dmin,R by deriving a simple and exact SER expression for the POSM in Rayleigh fading channels. Also, we have shown the accuracy of our analysis for 4-QAM and 16-QAM through simulations. This analysis method of exploiting the channel gain distributions and polar coordinates can be applied to general MIMO systems adopting the optimal precoding algorithm of the POSM. For future work, it will be interesting to extend our analysis to a Rician fading channel. In addition, by observing regular
0
5
10
15 20 SNR [dB]
25
30
35
Fig. 3. SER comparison of Monte Carlo simulation and our analysis for 4-QAM and 16-QAM
patterns in the bit error event, we can also develop the exact bit error rate expression of the POSM. R EFERENCES [1] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion,” IEEE Transactions on Communications, vol. 49, pp. 2198– 2206, December 2001. [2] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization,” IEEE Transactions on Signal Processing, vol. 51, pp. 2381–2401, September 2003. [3] D. J. Love and R. W. Heath, “Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Transactions on Information Theory, vol. 51, pp. 2967–2976, August 2005. [4] H. Lee, S. Park, and I. Lee, “Orthogonalized spatial multiplexing for closed-loop MIMO systems,” IEEE Transactions on Communications, vol. 55, pp. 1044–1052, May 2007. [5] Y. T. Kim, H. Lee, S. Park, and I. Lee, “Optimal precoding for orthogonalized spatial multiplexing in closed-loop MIMO systems,” IEEE Journal on Selected Areas in Communications, vol. 26, pp. 1556– 1566, October 2008. [6] L. Collin, O. Berder, P. Rostaing, and G. Burel, “Optimal minimum distance-based precoder for MIMO spatial multiplexing systems,” IEEE Transactions on Signal Processing, vol. 52, pp. 617–627, March 2004. [7] H. Lee, S. Park, and I. Lee, “Transmit beamforming method based on maximum-norm combining for MIMO systems,” IEEE Transactions on Wireless Communications, vol. 8, pp. 2067–2075, April 2009. [8] R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Transactions on Communications, vol. 30, pp. 1828–1841, August 1982. [9] F. S. Weinstein, “Simplified relationship for the probability distribution of the phase of a sine wave in narrow-band normal noise,” IEEE Transactions on Information Theory, vol. IT-20, pp. 658–661, September 1974. [10] J. Kim, W. Lee, J.-K. Kim, and I. Lee, “On the symbol error rates for signal space diversity schemes over a Rician fading channel,” IEEE Transactions on Communications, vol. 57, July 2009. [11] J. W. Craig, “A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations,” in Proc. IEEE MILCOM Conf. Rec., vol. 2, pp. 571–575, November 1991. [12] R. J. Muirhead, Aspects of multivariate statistical theory. New York: Wiley, 1982. [13] A. Edelman, “On the distribution of a scaled condition number,” Mathematics of Computation, vol. 58, pp. 185–190, January 1992.
Abstract—Recently an orthogonalized spatial multiplexing scheme with optimal precoding, referred to as POSM, has been proposed. The POSM achieves optimal performance in terms of maximizing the received minimum Euclidean distance dmin,R in closed-loop multiple-input multiple-output (MIMO) systems. In this paper, we derive exact closed-form expressions of the symbol error rate (SER) for the POSM with 4-QAM and 16-QAM in Rayleigh fading channels. We first introduce a new system model for the given channel. Based on this system model and a unique feature of the POSM effective constellations, the final expression of the SER using polar coordinates can be evaluated with a single-integral form. Simulation results confirm the accuracy of our derived analysis.
I. I NTRODUCTION In wireless multiple-input multiple-output (MIMO) systems, channel knowledge at the transmitter can be utilized to obtain higher performance gains over open-loop MIMO systems. Many studies on such closed-loop MIMO systems have been carried out by computing singular value decomposition (SVD) of a channel matrix [1] [2]. The performance of such systems with linear receivers can be worse than that of systems with maximum likelihood (ML) receivers. To optimize the system with ML receivers in terms of uncoded symbol error rate (SER), the received minimum Euclidean distance dmin,R between the effective constellation points must be maximized [3]. Recently, orthogonalized spatial multiplexing (OSM) has been developed in [4] for closed-loop MIMO systems, which allows a simple symbol-by-symbol ML receiver. More recently, an optimal precoding scheme based on the OSM has been proposed, which maximizes dmin,R [5]. Surprisingly, it was shown that the performance of the OSM with this optimal precoding is identical to that of the optimal closedloop MIMO systems presented in [6]. Besides, this precoding scheme requires only a few real values for feedback. From now on, we refer to the OSM system with optimal precoding as POSM. Exact evaluation of an error rate is one of the fundamental problems for the system design. In this paper, we derive exact expressions of the SER for the POSM with 4-QAM and 16QAM in Rayleigh fading channels. Deriving an exact SER of the POSM offers many insights on the design of the optimal precoding matrix in terms of dmin,R for MIMO systems with This research was supported by the MKE (Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement).
ML receivers. Also, it helps us to understand the error performance of the POSM by showing the effective constellation structures dependent on the channel realization as well as the constellation. For example, while the effective constellation in general MIMO systems has the form of polygons with random inside angles, the effective constellation shape in the POSM is composed of a few parallelograms with relatively fixed inside angles, due to the selection of the optimal precoding matrix according to channel conditions. To derive an exact SER for the POSM, we first provide a new system model using singular values of a given channel. Based on this system model, we can compute the exact average SER by utilizing polar coordinates and the distributions of the singular values. Then, we investigate that the POSM has a unique feature that the effective constellations become relatively invariant with respect to the channel realization. Exploiting this characteristics, we show that the whole computation of the SER containing a triple-integral form is reduced to a single-integral form. The simulation results confirm the accuracy of our results. Throughout this paper, normal letters represent scalar quantities, boldface letters indicate vectors and boldface uppercase letters designate matrices. With a bar accounting for complex variables, for any complex notation c¯, we use [¯ c] and [¯ c] to denote the real and imaginary part of c¯, respectively. II. S YSTEM M ODEL We consider the baseband system model of the POSM with Mt transmit and Mr receive antennas in flat fading channels. ¯ ∈ CMr ×Mt and x Denoting H ¯ = [¯ x1 x ¯2 · · · x ¯Mt ]T ∈ CMt ×1 as the channel matrix and the transmitted symbol vector, respectively, the Mr -dimensional received signal vector is expressed as ¯x + n y ¯ = H¯ ¯
(1)
¯ are selected with an independent and where the elements of H identically distributed (i.i.d.) complex Gaussian distribution with zero mean and unit variance, and n ¯ = [¯ n1 n ¯2 · · · n ¯ Mr ]T is a circularly symmetric complex Gaussian noise with variance 2N0 . Although the POSM can be extended to a system with more than two transmit antennas using the method in [4] and [5], we focus on analyzing the POSM with Mt = 2 for simplicity. The POSM scheme encodes two transmitted symbol as F (¯ x, θo , θ1 , p, θ2 ) = R (θo , θ1 ) PRθ2 ¯s
978-1-4244-2515-0/09/$25.00 ©2009 IEEE
where R (θo , θ1 ) is the rotation matrix with phases θo and θ1 , described in [4] and [5] in detail, and P, Rθ2 and ¯s are defined as p 0 cos θ2 − sin θ2 P= , R θ2 = , sin θ2 cos θ2 2 − p2 0 x2 ] [¯ x1 ] + j[¯ and ¯s . x2 ] [¯ x1 ] + j[¯ Here the optimal values of p and θ2 are obtained in [5]. Applying the above transformation, the system model (1) is changed to ¯ (¯ ¯ e PRθ ¯s + n ¯ = HF ¯ ¯=H y x, θo , θ1 , p, θ2 ) + n 2
¯ e accounts for the effective channel matrix denoted where H ¯ ¯ (θo , θ1 ). Now, the real-valued representation of as He = HR the system model for the POSM can be rewritten as P 0 Rθ2 0 y = He s+n (2) 0 P 0 Rθ2 where s [[¯sT ][¯ sT ]]T , n [[¯ nT ][¯ nT ]]T , and He ¯ e ] −[H ¯ e] [H ¯ e ] . Here 0 indicates a 2-by-2 zero matrix. ¯ e ] [H [H Denoting He in (2) as He = [he,1 he,2 he,3 he,4 ], it is important to note that all columns of He achieve orthogonality among each other by the rotation angles θo and θ1 . Also, we have he,1 = he,3 ≥ he,2 = he,4 where he,1 and he,2 are represented by σ1 and σ2 , respectively [5]. Here σ1 and σ2 are equal to the first and second singular values of ˜ e,i = he,i /he,i ¯ respectively [7]. Defining h the matrix H, for i = 1, 2, 3, 4 as the normalized column, it follows ˜ e,1 σ2 h ˜ e,2 σ1 h ˜ e,3 σ2 h ˜ e,4 ] He = [σ1 h ˜ e,2 h ˜ e,3 h ˜ e,4 ]diag{σ1 , σ2 , σ1 , σ2 } ˜ e,1 h [h Σ 0 = U 0 Σ =
˜ e,1 h ˜ e,2 h ˜ e,3 h ˜ e,4 ] represents an orthogonal where U = [h matrix and Σ is defined as Σ = diag{σ1 σ2 }. It was shown in [5] that there exist only two and four distinct cases for 4-QAM and 16-QAM, respectively, for representing the optimum precoder parameters p and θ2 , and that each case is determined by a single variable k, defined as σ12 /σ22 . By using these parameters, the POSM system attains the optimal performance in terms of maximizing dmin,R [5]. Finally, employing UT as the receive filter, the receive filter output r is given by T r1 ] [¯ r1 ] [¯ r2 ] [¯ r2 ] r = UT y = [¯ Σ 0 P 0 Rθ2 0 = s + w (3) 0 Σ 0 P 0 Rθ2 T x1 ] [¯ x1 ] [¯ x2 ] [¯ x2 ] where s is represented as s = [¯ and w is defined as w = UT n. Thus, denoting [[¯ zl ] [¯ zl ]]T = ΣPRθ2 [[¯ xl ] [¯ xl ]]T as the transmitted symbol, we obtain the following complex-valued ML detection metric as 2 zˆ ¯l = arg min |¯ rl − z¯l | , z¯l ∈χ
for l = 1, 2
(4)
where χ is the effective constellation formed by z¯l . As shown above, the ML detection at the receiver can be done by searching for a single symbol (called single-symbol decodable). This new formation is convenient for geometric consideration in order to evaluate the error probability in the subsequent section. III. E XACT E RROR P ROBABILITY A NALYSIS In this section, we will derive an exact SER of the POSM. First, we start with a conditional SER of the POSM for a given channel gain with σ1 and σ2 for 4-QAM. In [5], the parameters for the optimum precoding for 4-QAM are divided into two cases of 1 ≤ k < 7 and k ≥ 7. We can depict decision boundaries of the effective constellations for these two cases based on (4), as shown in Fig. 1. Denoting Es as the average transmit energy per symbol, the transmitted symbol 1 , S2 , S3 , S4 } for 4-QAM constellation becomes set {S
{[± E2s ± E2s ]T }. the 1 ≤ k < 7 case with the symbol [[¯ xl ] [¯ xl ]]T = For Es T [ E2s 2 ] , we have ⎡ E ⎤ s p 0 σ1 0 cos θ2 − sin θ2 [¯ zl ] ⎣ 2 ⎦ = 2 sin θ2 cos θ2 0 σ2 0 2 − p Es [¯ zl ] =
0 √
2
σ2 Es
where p = popt = 1 and θ2 = θ2opt = π/4 are used as in [5]. By applying similarly to the remaining symbols, we can see that the effective constellation of the 1 ≤ k < 7 case forms a rhombus shape, while the k ≥ 7 case has a straight line-shaped effective constellation as shown in Fig. 1. Note that the decision regions of the 1 ≤ k < 7 case are no longer perpendicular. For this reason, we evaluate an exact error probability employing polar coordinates [8]–[10]. To utilize polar coordinates, we denote r and φ as the magnitude and the angle of w1 + jw2 , respectively, where wi is an ith element of w in (3). Then, since unitary filter UT at the receiver does not have an effect on the property ¯ in (1), r has a Rayleigh distribution with of the noise n E[|w1 + jw2 |2 ] = 2N0 , and φ is uniformly distributed. Also, the joint probability density function (pdf) of r and φ is given as 1 r r2 f (r, φ) = exp − . (5) 2π N0 2N0 Thus, error decisions are made if the received signal corrupted by the noise with the above distribution falls outside the decision region. Let us define D(φ) as the distance from a symbol to the decision boundary for a certain angle φ ∈ [0, φ0 ]. Then we can obtain the error decision probability by integrating the pdf (5) as φ0 ∞ 1 r r2 Pr(error decision) = exp − drdφ 2N0 D(φ) 2π N0 0 φ0 D(φ)2 1 exp − dφ. = 2π 0 2N0 (6) Φ(φ0 , D(φ)).
σ 2 εs
s1 θt
π − 2θt 2
s4
s2
θt
σ 1 εs
√ √ 1 Es 1 Es + Q κ2 σ√ where Pe,2|S1 (σ12 , σ22 ) = Q κ1 σ√ N N0 √ 0 σ√ 2 2 1 Es and Pe,2|S2 (σ1 , σ2 ) = Q κ2 N . Here κ1 and κ2 are 0 defined as κ1 = cos 0.464 − sin 0.464 and κ2 = sin 0.464, respectively. Note that the Q-function can be evaluated using the Craig’s form with a definite integral term [11]. Utilizing the Craig’s form instead of the Q-function, it can be shown that (8) is represented as π π Pe,2 (σ12 , σ22 ) = 2Φ( , D3 (φ)) + Φ( , D4 (φ)) (9) 2 2 √
s3 a) 1 ≤ k < 7 case
s4
s3
s2
s1
2(cos 0.464 − sin 0.464)σ 1 εs
2sin 0.464σ 1 εs
b) k ≥ 7 case Fig. 1.
POSM decision regions for 4-QAM
For simple representation, let us denote Pe,1 and Pe,2 as the SER of the 1 ≤ k < 7 case and the k ≥ 7 case, respectively. For a given channel with σ1 and σ2 , we first evaluate the conditional error probability for 1 ≤ k < 7 with a rhombic effective constellation. In this case, the conditional probability of error decision for the symbols S1 and S2 can be obtained as π π Pe,1|S1 (σ12 , σ22 ) = 2Φ( + θt , D1 (φ))+2Φ( − 2θt , D2 (φ)), 2 2 π Pe,1|S2 (σ12 , σ22 ) = 2Φ( + θt , D1 (φ)) 2 √ 2 2 √ (σ1 +σ2 )Es 2 Es where D1 (φ) = 2 cos(θt −φ) and D2 (φ) = σcos φ . Here √ θt is defined as tan−1 (σ2 /σ1 ) = tan−1 (1/ k). Since the symbols S1 and S2 are symmetric to S3 and S4 with respect to the origin, we can simply check that Pe,1|S3 = Pe,1|S1 and Pe,1|S4 = Pe,1|S2 . Assuming that the transmitted symbols are equally likely, we can obtain the probability of error given σ1 and σ2 as 1 2Pe,1|S1 (σ12 , σ22 ) + 2Pe,1|S2 (σ12 , σ22 ) Pe,1 (σ12 , σ22 ) = 4 π π = 2Φ( + θt , D1 (φ)) + Φ( − 2θt , D2 (φ)). (7) 2 2 Next, for the k ≥ 7 case, we can easily express the conditional error probability as the combination of Q-function similar to pulse amplitude modulation (PAM). Since the effective constellation points are symmetric with respect to the origin, we have 1 Pe,2 (σ12 , σ22 ) = 2Pe,2|S1 (σ12 , σ22 ) + 2Pe,2|S2 (σ12 , σ22 ) (8) 4
√
σ1 Es 1 Es where D3 (φ) = κ2 σsin φ and D4 (φ) = κ1 sin φ . Now we have the probability of error given the channel realization as in (7) and (9). Since the channel gain σ1 and σ2 are random, we need to average (7) and (9) in terms of σ12 and σ22 . For compact representation, we replace σ12 and σ22 by λ1 and λ2 , respectively. Then we can obtain the total probability of symbol error for 4-QAM as ∞ 7λ2 Pe = Pe,1 (λ1 , λ2 )f (λ1 , λ2 )dλ1 0 λ2∞ + Pe,2 (λ1 , λ2 )f (λ1 , λ2 )dλ1 dλ2 (10) 7λ2
where f (λ1 , λ2 ) represents the joint pdf of the square of singular values of a 2-by-2 matrix, which is derived as [12] f (λ1 , λ2 ) = (λ1 − λ2 )2 e−(λ1 +λ2 )
for λ1 ≥ λ2 .
Observing the development of (10), we can see that the error probability for the α ≤ k < β case is generally comprised as the following triple-integral form ∞ βλ2 Φ(φ0 , D(φ))f (λ1 , λ2 )dλ1 dλ2 0
1 = 2π
αλ2 βλ2 φ0
∞ 0
αλ2
0
D(φ)2 exp − f (λ1 , λ2 )dφdλ1 dλ2 .(11) 2N0
This triple integral may require high complexity in numerical computation. Now, we will show that the above triple-integral form can be computed with a single-integral form. To obtain a final SER expression with reduced complexity, we investigate the effect of the optimal precoding on the effective constellation shapes in terms of dmin,R . It can be concluded that in MIMO systems, the effective constellation generally has the form of polygons with random inside angles, whereas that in the POSM is composed of a few parallelograms with almost invariant inside angles. For example, θt for 1 ≤ k < 7 shown in Fig. 1 varies from 0.361 to 0.785 radian. Based on this observation, we make an assumption that θt is independent of random channel gains σ1 and σ2 in order to simplify the computation in (11), and then we substitute θt by a fixed value. To determine this fixed value, we first compute the median value of k. Utilizing the result in [13], we can derive the pdf of k as f (k) = 6(k + 1)−4 (k − 1)2 , k ≥ 1. Then, based on this distribution, the median value of k is√obtained as 3.94 for 1 ≤ k < 7. Since θt equals √ tan−1 (1/ k), −1 we replace the angle θt by 0.467 = tan (1/ 3.94). In the
simulation section, we will show that this approximation does not sacrifice the accuracy of the SER analysis, since the basic structure of the effective constellation does not vary much. With this substitution in θt , equation (11) is transformed into φ0 ∞ βλ2 1 (λ1 − λ2 )2 e−A(φ)λ1 e−B(φ)λ2 dλ1 dλ2 dφ. (12) 2π 0 0 αλ2
d1
θc
d2
yX
d3
yY
a) 7.59 ≤ k < 43.1 case
Then the integrand in the triple integrals now becomes integrable with respect to λ1 and λ2 . Thus it can be shown that equation (12) equals φ0 2(α − 1)2 1 2(α − 1) 2 + 2 + 3 2π 0 A(αA + B)3 A (αA + B)2 A (αA + B) 2(β − 1) 2 2(β − 1)2 − 2 − 3 − dφ A(βA + B)3 A (βA + B)2 A (βA + B) (13) Ψ(α, β, φ0 , A(φ), B(φ)). In this expression, we omit the variable φ in A(φ) and B(φ) for simplicity. Hence, representing Pe in (10) with the Ψ-function in (13), the final SER for 4-QAM is expressed by π + θt , C1 (φ), C1 (φ)) 2 π +Ψ(1, 7, − 2θt , 1, C2 (φ)) 2 π π +2Ψ(7, ∞, , C3 (φ), 1) + Ψ(7, ∞, , C4 (φ), 1) (14) 2 2
b) 43.1 ≤ k < 101 case
Pe = 2Ψ(1, 7,
where C1 (φ) = κ2 γ
γ 8 cos2 (0.467−φ)
+ 1, C2 (φ) = κ2 γ
γ 2 cos2 φ
+ 1,
C3 (φ) = 2 sin2 2 φ + 1 and C4 (φ) = 2 sin1 2 φ + 1. Here γ denotes the signal-to-noise ratio (SNR) defined as Es /N0 . Note that when β is equal to infinity in the Ψ-function, the last three terms of the integrand go to zero in (13). Next we consider a SER for 16-QAM. Unlike 4-QAM, 16-QAM is classified into four cases with boundary points of k = 7.59, 43.1 and 101 as shown in [5]. Considering the effective constellation, the 1 ≤ k < 7.59 case forms a rhombus-shaped effective constellation as in 4-QAM and the effective constellation for k ≥ 101 becomes 16-PAM. Therefore, it is straightforward to derive the error probabilities of these two cases for 16-QAM by using the solution for 4QAM. On the other hand, the other two cases of 7.59 ≤ k < 43.1 and 43.1 ≤ k < 101 are more complicated since there are many asymmetric decision regions. In Fig. 2, we illustrate the decision boundaries for these two cases. Note that the decision boundaries are quite different from conventional 16-QAM constellations. As shown in Fig. 2, any vertex of decision regions becomes an outer center of a triangle formed by neighboring three constellation points. Therefore we define all inside angles of decision region as an inverse cosine function for convenience. For example, θc in Fig. 2 is defined as d1 θc = cos−1 ( 2R ). Here the radius of a circumscribed circle 1 R1 is calculated by d1 d2 d3 R1 = 4 s(s − d1 )(s − d2 )(s − d3 )
Fig. 2.
POSM decision regions for 16-QAM
where d1 , d2 and d3 are the length of three sides of a triangle, respectively, and s is defined as s = (d1 + d2 + d3 )/2. There are two distinct radii R1 and R2 for the 7.59 ≤ k < 43.1 case, while three distinct radii exist for the 43.1 ≤ k < 101 case as shown in Fig. 2. Now, we can calculate the error probabilities of each case with the same approach in 4-QAM. The detailed derivation is omitted to simplify the presentation. We denote Pe,i for i ∈ {1, 2, 3, 4} as the SER of the 1 ≤ k < 7.59, 7.59 ≤ k < 43.1, 43.1 ≤ k < 101 and k ≥ 101 cases, respectively. The fixed angle values for 22 different terms are listed in Table I. Presenting with the Ψ-function in (13), an exact expression of the SER for 16-QAM can be finally expressed as 22
Pe =
1 ci Ψ(αi , βi , φ0i , Ai (φ), Bi (φ)) 16 i=1
(15)
where ci represents the coefficient related to the number of Ψ(αi , βi , φ0i , Ai (φ), Bi (φ)). IV. S IMULATION R ESULTS In this section, we provide the results for our analysis and compare them with numerical simulation. Fig. 3 depicts the average SER of the POSM for 4-QAM and 16-QAM over Rayleigh fading channels with Mt = Mr = 2 versus the averaged received SNR γ. Here, we plot our final analysis results in (14) and (15) as the solid line and numerical simulations as the square point. Also, we include the SER results with a triple-integral form for 4-QAM in (10) and 16QAM, respectively. As shown in Fig. 3, it can be observed
TABLE I PARAMETERS OF Pe FOR 16-QAM i
ci
φ0i
Ai (φ)
0
10
Bi (φ)
Analysis w/ single−integral Analysis w/ triple−integral Simulation
−1
10
Pe,1 : αi = 1, βi = 7.59 48
0.458
2
24
2.029
3
36
0.655
γ +1 40 cos2 φ γ + 40 cos2 (0.458−φ)
1
1
γ +1 40 cos2 φ γ + 40 cos2 (0.458−φ) γ +1 10 cos2 φ
−2
10
1
SER
1
−3
10
Pe,2 : αi = 7.59, βi = 43.1 (mc = cos 0.488, ms = sin 0.488) 4
8
1.617
5
8
2.187
6
8
0.616
7
4
2.572
8
24
0.403
9
12
1.525
10
12
0.552
11
12
1.168
+1
m2 cγ 20 cos2 (0.616−φ)
8
1.993
13
4
2.213
14
4
2.922
15
12
0.407
16
12
1.485
17
12
0.775
18
12
0.608
19
8
1.015
20
4
2.586
−5
10 m2 sγ +1 20 cos2 (1.001−φ) (mc −2ms )2 γ +1 20 cos2 φ
m2 cγ +1 20 cos2 (1.001−φ) (2mc +ms )2 γ +1 20 cos2 φ
(mc −ms )2 γ 20 cos2 (0.552−φ)
+1
(mc +ms )2 γ 20 cos2 (0.552−φ)
+1
m2 cγ 20 cos2 (0.403+φ)
+1
m2 sγ 20 cos2 (0.403+φ)
+1
m2 sγ 20 cos2 (0.642−φ)
+1
m2 sγ +1 20 cos2 (1.351−φ) (mc −3ms )2 γ +1 20 cos2 φ (mc −2ms )2 γ +1 20 cos2 (0.522−φ) (mc −ms )2 γ +1 20 cos2 (0.407+φ) m2 cγ +1 20 cos2 (0.963+φ) m2 sγ +1 20 cos2 φ m2 sγ +1 20 cos2 (1.015−φ)
m2 cγ 20 cos2 (0.642−φ)
+1
m2 cγ +1 20 cos2 (1.351−φ) (3mc +ms )2 γ +1 20 cos2 φ (2mc +ms )2 γ +1 20 cos2 (0.522−φ) (mc +ms )2 γ +1 20 cos2 (0.407+φ) m2 sγ +1 20 cos2 (0.963+φ) m2 cγ +1 20 cos2 φ m2 cγ +1 20 cos2 (1.015−φ)
Pe,4 : αi = 101, βi = ∞ (mc = cos 0.245, ms = sin 0.245) 21 22
48
π 2
12
π 2
16−QAM
+1
Pe,3 : αi = 43.1, βi = 101 (mc = cos 0.345, ms = sin 0.345) 12
4−QAM
−4
10 m2 sγ 20 cos2 (0.616−φ)
m2 sγ +1 10 sin2 φ (mc −3ms )2 γ + 10 sin2 φ
1 1
1
that our analysis exactly matches numerical simulation results, even though we substitute angles by fixed values in order to form a single integral. Also, SER curves in Fig. 3 exhibit a slope of 4 and these clearly show a full diversity advantage of the POSM. V. C ONCLUSION In this paper, we have addressed analytical characteristics for the optimal precoding in terms of dmin,R by deriving a simple and exact SER expression for the POSM in Rayleigh fading channels. Also, we have shown the accuracy of our analysis for 4-QAM and 16-QAM through simulations. This analysis method of exploiting the channel gain distributions and polar coordinates can be applied to general MIMO systems adopting the optimal precoding algorithm of the POSM. For future work, it will be interesting to extend our analysis to a Rician fading channel. In addition, by observing regular
0
5
10
15 20 SNR [dB]
25
30
35
Fig. 3. SER comparison of Monte Carlo simulation and our analysis for 4-QAM and 16-QAM
patterns in the bit error event, we can also develop the exact bit error rate expression of the POSM. R EFERENCES [1] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion,” IEEE Transactions on Communications, vol. 49, pp. 2198– 2206, December 2001. [2] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization,” IEEE Transactions on Signal Processing, vol. 51, pp. 2381–2401, September 2003. [3] D. J. Love and R. W. Heath, “Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Transactions on Information Theory, vol. 51, pp. 2967–2976, August 2005. [4] H. Lee, S. Park, and I. Lee, “Orthogonalized spatial multiplexing for closed-loop MIMO systems,” IEEE Transactions on Communications, vol. 55, pp. 1044–1052, May 2007. [5] Y. T. Kim, H. Lee, S. Park, and I. Lee, “Optimal precoding for orthogonalized spatial multiplexing in closed-loop MIMO systems,” IEEE Journal on Selected Areas in Communications, vol. 26, pp. 1556– 1566, October 2008. [6] L. Collin, O. Berder, P. Rostaing, and G. Burel, “Optimal minimum distance-based precoder for MIMO spatial multiplexing systems,” IEEE Transactions on Signal Processing, vol. 52, pp. 617–627, March 2004. [7] H. Lee, S. Park, and I. Lee, “Transmit beamforming method based on maximum-norm combining for MIMO systems,” IEEE Transactions on Wireless Communications, vol. 8, pp. 2067–2075, April 2009. [8] R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Transactions on Communications, vol. 30, pp. 1828–1841, August 1982. [9] F. S. Weinstein, “Simplified relationship for the probability distribution of the phase of a sine wave in narrow-band normal noise,” IEEE Transactions on Information Theory, vol. IT-20, pp. 658–661, September 1974. [10] J. Kim, W. Lee, J.-K. Kim, and I. Lee, “On the symbol error rates for signal space diversity schemes over a Rician fading channel,” IEEE Transactions on Communications, vol. 57, July 2009. [11] J. W. Craig, “A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations,” in Proc. IEEE MILCOM Conf. Rec., vol. 2, pp. 571–575, November 1991. [12] R. J. Muirhead, Aspects of multivariate statistical theory. New York: Wiley, 1982. [13] A. Edelman, “On the distribution of a scaled condition number,” Mathematics of Computation, vol. 58, pp. 185–190, January 1992.
