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## Finite-SNR Diversity-Multiplexing Tradeoff for Rayleigh MIMO Channels

### Citations

2913 | Capacity of multi-antenna Gaussian channels
- Telatar
- 1995
(Show Context)
Citation Context ... represented by the multiplexing gain. In the sequel, motivated by the work in [13], we derive the exact finite-SNR DMT for systems with dual transmit and/or receive antennas over uncorrelated Rayleigh fading channels. III. COMPUTATION OF FINITE-SNR DMT A. Mutual information pdf In this section, we derive an analytical expression for the pdf of the mutual information between received and transmitted signals for Nt×2 and 2×Nr MIMO systems. By assuming that X is a zero-mean white complex Gaussian random variable, the MIMO mutual information I conditioned on the channel realization H is given by [15]: I = log det ( INr + ρ Nt HHH ) = log det ( INt + ρ Nt HHH ) (8) where the superscript H stands for conjugate transpose. It is known that when H is a complex normally distributed matrix, matrices HHH and HHH are central complex Wishart distributed [16]. We define m Δ = min (Nt, Nr) and n Δ = max (Nt, Nr). The joint pdf of the m nonzero ordered eigenvalues λ1, ..., λm of the complex Wishart matrix is given in [16]. For m = 2, the joint pdf of the two nonzero ordered eigenvalues λ1 and λ2 (λ1 ≥ λ2 ) is given by: f(λ1, λ2) = (λ1λ2) n−2 Γ (n) Γ (n− 1) (λ1 − λ2) 2 e−(λ1+λ2) (9) where Γ(x) is the g... |

1778 | Space-time codes for high data rate wireless communication: Performance criterion and code construction
- Tarokh, Seshadri, et al.
- 1998
(Show Context)
Citation Context ...ly designed to achieve the asymptotic DMT frontier, this finite-SNR DMT could provide a new insight to design STCs for practical MIMO systems optimized at operational SNRs. Index Terms—MIMO systems, diversity multiplexing tradeoff. I. INTRODUCTION MULTIPLE-input multiple-output (MIMO) systems havebeen adopted in most of the recently developed wireless communication systems, e.g., WiMAX, Wi-Fi, LTE, etc. Indeed, their potential benefits are data rate increase and better reliability. Spatial multiplexing techniques [1] have been incorporated for data rate increase, while space-time codes (STCs) [2] have been designed to improve the channel reliability through spatial diversity. Zheng and Tse in [3] have shown that both gains can be simultaneously delivered with a fundamental tradeoff between them. The diversitymultiplexing tradeoff (DMT) defines the optimal tradeoff between achievable diversity and multiplexing gains of any transmission over Nt ×Nr MIMO channels. The DMT formulated in [3], for uncorrelated Rayleigh MIMO channels, is an asymptotic framework as the signal to noise ratio (SNR) tends to infinity. In [2], asymptotic analysis was also used to obtain the well-known rank-determ... |

1164 | Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels
- Zheng, Tse
- 2003
(Show Context)
Citation Context ... design STCs for practical MIMO systems optimized at operational SNRs. Index Terms—MIMO systems, diversity multiplexing tradeoff. I. INTRODUCTION MULTIPLE-input multiple-output (MIMO) systems havebeen adopted in most of the recently developed wireless communication systems, e.g., WiMAX, Wi-Fi, LTE, etc. Indeed, their potential benefits are data rate increase and better reliability. Spatial multiplexing techniques [1] have been incorporated for data rate increase, while space-time codes (STCs) [2] have been designed to improve the channel reliability through spatial diversity. Zheng and Tse in [3] have shown that both gains can be simultaneously delivered with a fundamental tradeoff between them. The diversitymultiplexing tradeoff (DMT) defines the optimal tradeoff between achievable diversity and multiplexing gains of any transmission over Nt ×Nr MIMO channels. The DMT formulated in [3], for uncorrelated Rayleigh MIMO channels, is an asymptotic framework as the signal to noise ratio (SNR) tends to infinity. In [2], asymptotic analysis was also used to obtain the well-known rank-determinant criteria for space-time block codes (STBCs) design. STBCs for MIMO systems are commonly designed... |

290 |
Distributions of matrix variates and latent roots derived from normal
- James
- 1964
(Show Context)
Citation Context ... A. Mutual information pdf In this section, we derive an analytical expression for the pdf of the mutual information between received and transmitted signals for Nt×2 and 2×Nr MIMO systems. By assuming that X is a zero-mean white complex Gaussian random variable, the MIMO mutual information I conditioned on the channel realization H is given by [15]: I = log det ( INr + ρ Nt HHH ) = log det ( INt + ρ Nt HHH ) (8) where the superscript H stands for conjugate transpose. It is known that when H is a complex normally distributed matrix, matrices HHH and HHH are central complex Wishart distributed [16]. We define m Δ = min (Nt, Nr) and n Δ = max (Nt, Nr). The joint pdf of the m nonzero ordered eigenvalues λ1, ..., λm of the complex Wishart matrix is given in [16]. For m = 2, the joint pdf of the two nonzero ordered eigenvalues λ1 and λ2 (λ1 ≥ λ2 ) is given by: f(λ1, λ2) = (λ1λ2) n−2 Γ (n) Γ (n− 1) (λ1 − λ2) 2 e−(λ1+λ2) (9) where Γ(x) is the gamma function defined by ∫∞ 0 t x−1e−tdt. Therefore, the mutual information is expressed by: I = log ( 1 + λ1 ρ Nt ) + log ( 1 + λ2 ρ Nt ) . (10) Let us define new variables x and y as: x Δ = log ( 1 + λ1 ρ Nt ) and y Δ = log ( 1 + λ2 ρ Nt ) . (11) Usin... |

59 | Explicit space-time codes achieving the diversity-multiplexing gain tradeoff,”
- Elia, Kumar, et al.
- 2006
(Show Context)
Citation Context ... Brest Iroise CS 83818, 29238 Brest Cedex 3, Universite europeenne de Bretagne, France (e-mail: {ammar.elfalou, charlotte.langlais, charbel.abdelnour, catherine.douillard}@telecom-bretagne.eu). A. El Falou was a visiting Researcher at Concordia University from 01 April to 30 June 2012. W. Hamouda is with Concordia University, Montreal, QC, Canada H3G 1M8 (e-mail: hamouda@ece.concordia.ca). This work was partly funded by the French ANR Mobile MultiMedia project, ANR-10-VERS-0010. Digital Object Identifier 10.1109/LCOMM.2013.022213.130007 sufficient to reach the frontier of the asymptotic DMT [5]. Recently, several papers have noted that STBCs designed to perform better at high (asymptotic) SNRs are not effective at low to medium SNRs [6] (and references therein) where practical communication systems operate. Moreover, we have shown in [7] that STBCs design parameters are SNRdependent and that adaptive STBCs are more effective for practical communication systems. A novel framework has been proposed in [8–10] to characterize the tradeoff between diversity and multiplexing gains at finite-SNRs. This finite-SNR DMT could provide new insights to design MIMO systems optimized at operationa... |

24 | Transmit diversity vs. spatial multiplexing in modern MIMO systems,”
- Lozano, Jindal
- 2010
(Show Context)
Citation Context ...elnour, catherine.douillard}@telecom-bretagne.eu). A. El Falou was a visiting Researcher at Concordia University from 01 April to 30 June 2012. W. Hamouda is with Concordia University, Montreal, QC, Canada H3G 1M8 (e-mail: hamouda@ece.concordia.ca). This work was partly funded by the French ANR Mobile MultiMedia project, ANR-10-VERS-0010. Digital Object Identifier 10.1109/LCOMM.2013.022213.130007 sufficient to reach the frontier of the asymptotic DMT [5]. Recently, several papers have noted that STBCs designed to perform better at high (asymptotic) SNRs are not effective at low to medium SNRs [6] (and references therein) where practical communication systems operate. Moreover, we have shown in [7] that STBCs design parameters are SNRdependent and that adaptive STBCs are more effective for practical communication systems. A novel framework has been proposed in [8–10] to characterize the tradeoff between diversity and multiplexing gains at finite-SNRs. This finite-SNR DMT could provide new insights to design MIMO systems optimized at operational SNRs. In [9, 10], Narasimhan has pointed out that exact form of the outage probability and therefore the finite-SNR DMT for any Nt × Nr MIMO ch... |

2 |
Diversity-multiplexing tradeoff in MIMO system with finite SNR,” in Proc.
- Ebrahimzad, Mohammadi
- 2007
(Show Context)
Citation Context ...de new insights to design MIMO systems optimized at operational SNRs. In [9, 10], Narasimhan has pointed out that exact form of the outage probability and therefore the finite-SNR DMT for any Nt × Nr MIMO channels are not tractable. Therefore, estimates of the finite-SNR DMT are given in [9–12] for uncorrelated, correlated Rayleigh and Rician fading Nt ×Nr MIMO channels where Nr ≥ Nt. In some cases, the exact expression of the outage probability and the finite-SNR DMT can be derived. To the best of our knowledge, exact expression of the finite-SNR DMT is derived only for 2 × 2 MIMO systems in [13] with uncorrelated Rayleigh fading and for both multiple-input single-output (MISO) and single-input multiple-output (SIMO) systems with uncorrelated [14] and correlated [12] Rayleigh fading. Inspired by the method in [13], we derive in this paper the exact finite-SNR DMT for any uncorrelated Rayleigh fading MIMO channels with dual antennas i.e., Nt × 2 and 2×Nr MIMO systems. II. SYSTEM MODEL AND DEFINITIONS We consider a MIMO system with Nt transmit antennas, Nr receive antennas operating over a flat Rayleigh fading channel. A perfect channel state information (CSI) is assumed at the receiver... |

1 |
diversity-multiplexing tradeoff for correlated Rayleigh and Rician MIMO channels,”
- “Finite-SNR
- 2006
(Show Context)
Citation Context ... several papers have noted that STBCs designed to perform better at high (asymptotic) SNRs are not effective at low to medium SNRs [6] (and references therein) where practical communication systems operate. Moreover, we have shown in [7] that STBCs design parameters are SNRdependent and that adaptive STBCs are more effective for practical communication systems. A novel framework has been proposed in [8–10] to characterize the tradeoff between diversity and multiplexing gains at finite-SNRs. This finite-SNR DMT could provide new insights to design MIMO systems optimized at operational SNRs. In [9, 10], Narasimhan has pointed out that exact form of the outage probability and therefore the finite-SNR DMT for any Nt × Nr MIMO channels are not tractable. Therefore, estimates of the finite-SNR DMT are given in [9–12] for uncorrelated, correlated Rayleigh and Rician fading Nt ×Nr MIMO channels where Nr ≥ Nt. In some cases, the exact expression of the outage probability and the finite-SNR DMT can be derived. To the best of our knowledge, exact expression of the finite-SNR DMT is derived only for 2 × 2 MIMO systems in [13] with uncorrelated Rayleigh fading and for both multiple-input single-output... |

1 | Impact of spatial correlation on the finite-SNR diversity-multiplexing tradeoff,”
- Rezki, Haccoun, et al.
- 2008
(Show Context)
Citation Context ...1) × ∫ r log(1+Gρ) 0 eI (∫ I/2 0 ∂g(y,I,ρ) ∂ρ dy ) dI (20) where ∂g(y,I,ρ) ∂ρ = ( eI−y − ey)2 (eI−y − 1)n−2 (ey − 1)n−2 ×e−Ntρ (eI−y+ey−2)Ntρ2 ( eI−y + ey − 2) . (21) The above expressions can also be calculated using numerical integration. After deriving Pout (r, ρ) and ∂Pout(r,ρ) ∂ρ , the finiteSNR DMT can be easily computed using (7). IV. NUMERICAL RESULTS In order to examine the derived finite-SNR DMT, in Fig. 1 we plot the derived exact finite-SNR DMT curves for n = 2 at SNR = 5, 10 dB and the asymptotic DMT [3]. We noted that the obtained DMT curves are identical to the ones obtained in [9, 11] by Monte Carlo simulations, which validates the derived exact DMT. Furthermore, the finite-SNR DMT derived in [9, 11] overestimate the achievable DMT. In order to discuss the achievability of our exact finite-SNR DMT, we have plotted the exact finite-SNR DMT for Alamouti code derived in [8]. It is noted that the Alamouti code always benefits from all the available diversity of the system (when the multiplexing gain r approaches zero), not only at asymptotic SNRs, but also at realistic SNRs. 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Multiplexing gain D iv er si ty g ai n SNR = 5 dB SNR = 10 dB S... |