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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007
A Tight Upper Bound on the PEP of a Space-Time Coded System
Ranjan K. Mallik, Senior Member, IEEE, and Q. T. Zhang, Senior Member, IEEE
Abstract— For a space-time coded system subject to flat Rayleigh fading, we formulate the pairwise error probability (PEP) in a unified expression, which accounts for the cases of perfect channel-state information, imperfect channel estimation, and unknown channel. The unified PEP expression is then bounded by an exponential-type inequality leading to a tight upper bound on the PEP. The bound indicates that to design a good code set, we not only need to maximize the geometric mean of the nonzero eigenvalues of the code difference matrix, but also minimize their harmonic mean. Index Terms— Code design criterion, pairwise error probability (PEP), Rayleigh fading, space-time coding, upper bound.
I. I NTRODUCTION
D
ETERMINATION of the pairwise error probability (PEP) is important to space-time coding. It provides not only an indication of the system performance, but also a necessary means for the design of the signal constellation. The expression of an exact PEP usually takes a complicated form thereby providing little insight into the influence of the signal constellation. In their seminal work [1], Tarokh et al. first derive a Chernoff upper bound (CUB) on the PEP, though quite accurate for large signal-to-noise ratio (SNR), offers little physical insight. They therefore use the inequality x 0, (1 + x)−1
× ln
G(ν 2 , q ) 2 G(ν 1 , q ) 1 1 MNR αbM NR 1 H −1 (ν 2 , q ) 1 F1 MNR + 2 ; MNR +1 ; − Γ
1 F1
(41) If G(ν 2 , q 2 ) > G(ν 1 , q 1 ) and H(ν 2 , q 2 ) 0, (62) where ∼ denotes “distributed as” and 0N ×1 the N × 1 vector of zeros. The c.f. of Y 2 is clearly 1 Ψ Y 2 (jω) = . det (IN − jωKY ) (63) Y ∼ CN (0N ×1 , KY ) ,
10
−8
10
−9
upper bound, ν=[4], q=[4] PEP, ν=[4], q=[4] upper bound, ν=[14.9282, 1.0718], q=[2, 2] PEP, ν=[14.9282, 1.0718], q=[2, 2] 5 5.5 6 6.5 7 7.5 8 average SNR per diversity branch (dB) 8.5 9 9.5 10
Fig. 3. Comparison of performance of the 4 × 4 orthogonal design code having (ν, q) = ([4], [4]) with that of the orthogonal code of [9] having (ν, q) = ([14.9282, 1.0718], [2, 2]) for BPSK constellation.
M=3, N R =6, geometric progression ratio ε=0.25 equal eigenvalues geometrically progressing unequal eigenvalues 10
−3
10
−2
10
−4
10
−5
10 PEP 10
−6
−7
10
−8
10
−9
10
−10
10
−11
0
1
2 3 4 average SNR per diversity branch (dB)
5
6
Fig. 4. PEP versus average SNR per diversity branch for the cases of equal and unequal geometrically progressing eigenvalues (geometric progression ratio = 0.25) when M = 3 and NR = 6 with a fixed determinant of unity.
of our upper bound in comparison with the other bounds is clear. The performance comparison of two code pairs having the same geometric mean of unity but different sets of nonzero eigenvalues is shown in Fig. 2. We find that the code pair with well separated eigenvalues performs better than the one with repeated eigenvalues. A similar comparison is made in Fig. 3 for BPSK constellation between the full-rank 4 × 4 orthogonal design code [3] which has (ν, q) = ([4], [4]) for code pairs with determinant of 256 and the full-rank 4×4 orthogonal code obtained in [9] which has (ν, q) = ([14.9282, 1.0718], [2, 2]) for pairs with the same determinant of 256. We observe the closeness of the bound to the actual PEP and as well as the superior performance of the latter code. Fig. 4 shows the PEP versus average SNR per diversity branch plots for the cases of equal and unequal geometrically progressing eigenvalues when M = 3 and NR = 6 with a
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 2007
Let D be a real-valued random variable given by D = = Y+Z − Z 2Re YH Z + Y
2 2 2
,
(64)
where · denotes the Euclidean norm. We wish to find the probability Pr(D < 0). This can be done by two methods. In the first method, we find the conditional distribution of 2Re YH Z , conditioned on Y, and from it we find the conditional probability Pr(D < 0|Y) in terms of the Gaussian Q-function. Since 2Re Y Z Y ∼ N 0, 2a Y , (65) we can conclude from (64) that the conditional probability is given by 1 Y , (66) Pr(D < 0|Y) = Q √ 2a where Q(·) denotes the Gaussian Q-function. Using Craig’s formula for the Gaussian Q-function [12] in (66) and averaging over the distribution of Y 2 , we get 1 Pr(D < 0) = π 1 Pr(D < 0) = π
π 2 π 2
the poles being determined by the nonzero eigenvalues of KY . Let λ1 , . . . , λK be the K distinct nonzero eigenvalues of KY , with λn having multiplicity pn for n = 1, . . . , K. Then the c.f. in (74) can be written as 1 . (75) ΨD (jω) = K 1 − jωλn + ω 2 aλn
n=1 pn
In addition, from (68), we get Pr(D < 0) = 1 π
π 2
K n=1
H
2
0
sin2 θ 2 λn 4a + sin θ
pn
dθ .
(76)
Therefore, if the c.f. of a real-valued random variable D is given by (75), then the probability Pr(D < 0) can be expressed as (76). A PPENDIX B We can rewrite (32) as P (ν, q) ≤ b Γ
MNR
0
Ψ Y
2
1 − 4a sin2 θ
1
K n=1 q νnn M
dθ .
(67)
I,
Substitution of (63) in (67) yields
0
(68) In the second method, we treat the decision variable as a quadratic form in complex Gaussian random variables, by denoting the composite vector X as X= and D in (64) as D = XH IN 0N 0N −IN X, (70) Y+Z Z , (69)
1 KY det IN + 4a sin2 θ
−1
dθ .
Γ 1 ≥ , b min(ν)f (α) where I denotes the integral given by I = 1 π
π 2
(77)
0
sin2 θ
MNR K n=1
× exp −
αbM Γ
−1 qn νn sin2 θ dθ . (78)
Using the series expansion of the exponential term in (78), we get
∞
where 0N denotes the N × N matrix of zeros. Since E XXH = KY + aIN aIN aIN aIN , (71)
π 2
I
=
k=0
(−1)k k! 1 × π
π 2
αbM Γ
0
K n=1
k −1 qn νn
sin2 θ
MNR +k
dθ .
(79)
the c.f. of D can be expressed as [13] ΨD (jω) = ⎛ ⎜ det ⎜ ⎝ IN 0N 0N IN × 1 − jω 0N IN 0N −IN KY + aIN aIN aIN aIN ⎞ , (72) ⎟ ⎟ ⎠
Substituting the result [8] Γf M N R + 1 + k Γf 1 2 2 , 2πΓf (M NR + 1 + k) 0 (80) where Γf (·) denotes the gamma function, in (79) and noting √ that Γf (1/2) = π, we further obtain 1 π sin2 θ
MNR +k
dθ =
which, on simplification yields ΨD (jω) = 1 . det (IN − jωKY + ω 2 aKY ) (73)
I
=
1 Γf M N R + 1 2 √ 2 π Γf (M NR + 1) Γf (M NR + 1) × Γf M N R + 1 2
∞ k=0
(−1)k k!
The probability Pr(D < 0) can be obtained using the inversion theorem [14] as Pr(D < 0) = − sum of residues of ΨD (z) z (74)
αbM Γ
K n=1
k −1 qn νn
Γf M N R + 1 + k 2 . × Γf (M NR + 1 + k)
(81)
From the definition of the confluent hypergeometric function 1 F1 (·; ·; ·) [8] and the fact that 1 Γf M N R + 1 1 2 √ = 2 2 π Γf (M NR + 1) 2M NR M NR 1 , 4MNR
at poles on left-half z-plane ,
MALLIK and ZHANG: A TIGHT UPPER BOUND ON THE PEP OF A SPACE-TIME CODED SYSTEM
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we can rewrite (81) as I = 1 2 2M NR M NR M NR + 1 4MNR 1 αbM ; M NR + 1 ; − 2 Γ
K n=1 −1 qn νn .(82)
[14] J. Gil-Pelaez, “Note on the inversion theorem,” Biometrika, vol. 38, pp. 481–482, 1951. Ranjan K. Mallik (S’88–M’93–SM’02) received the B.Tech. degree from the Indian Institute of Technology, Kanpur, in 1987 and the M.S. and Ph.D. degrees from the University of Southern California, Los Angeles, in 1988 and 1992, respectively, all in Electrical Engineering. From August 1992 to November 1994, he was a scientist at the Defence Electronics Research Laboratory, Hyderabad, India, working on missile and EW projects. From November 1994 to January 1996, he was a faculty member of the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur. In January 1996, he joined the faculty of the Department of Electronics and Communication Engineering, Indian Institute of Technology, Guwahati, where he worked till December 1998. Since December 1998, he has been with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, where he is a Professor. His research interests are in communication theory and systems, difference equations, and linear algebra. Dr. Mallik is a member of Eta Kappa Nu. He is also a member of the IEEE Communications and Information Theory Societies, the American Mathematical Society, the International Linear Algebra Society, and The Institution of Engineering and Technology, a fellow of the Indian National Academy of Engineering and The Institution of Electronics and Telecommunication Engineers, a life member of the Indian Society for Technical Education, and an associate member of The Institution of Engineers (India). He is an Editor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS . Q. T. Zhang (S’84–M’85–SM’95) received the B.Eng. degree from Tsinghua University, Beijing, and the M.Eng. degree from South China University of Technology, Guangzhou, China, both in wireless communications, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, Ontario, Canada. After graduation from McMaster in 1986, he held a research position and Adjunct Assistant Professorship at the same institution. In January 1992, he joined the Spar Aerospace Ltd., Satellite and Communication Systems Division, Montreal, as a Senior Member of Technical Staff. At Spar Aerospace, he participated in the development and manufacturing of the Radar Satellite (Radarsat). He was subsequently involved in the development of the advanced satellite communication systems for the next generation. He joined Ryerson Polytechnic University, Toronto, in 1993 and became a Full Professor in 1999. In 1999, he took one-year sabbatical leave at the National University of Singapore, and is now a Professor with the City University of Hong Kong. His research interest is on transmission and reception over fading channels with current focus on wireless MIMO and UWB systems. He is an Associate Editor for the IEEE C OMMUNICATIONS L ETTERS .
×1 F1
Substitution of (82) in (77) yields (35). R EFERENCES
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