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## Sum Capacity of a Gaussian Vector Broadcast Channel (2002)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 276 - 21 self |

### Citations

12169 |
Elements of information theory
- Cover, Thomas
- 2006
(Show Context)
Citation Context ...with more than two users. When a Gaussian broadcast channel has a scalar input and scalar outputs, it can be regarded as a degraded broadcast channel for which the capacity region is well established =-=[16]-=-. A broadcast channel is physically degraded if . Intuitively, this means that one user’s signal is a noisier version of the other user’s signal. Consider the Gaussian scalar broadcast channel where i... |

7513 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
Citation Context ...ation.) Because the right-hand side of the above is positive definite, if a square matrix C satisfies CT �√ΣV √ �−1 C = T HT QT Λ−1QHV Σ+I ,itmustbeoftheformC = UR where U is an 1 − o=-=rthogonal matrix [24]. In particu-=-lar, from (82), ∆ 2 G−T M T = UR. Then, the Cholesky factorization can be written as: G −1 ∆ −1 G −T = M T R T U T URM, (84) where URM is block-lower-triangular. For a fixed M, it is possi... |

5240 | Convex Analysis
- Rockafellar
- 1970
(Show Context)
Citation Context ...ct that the mutual information is a concave-convex function, and the two KKT conditions, corresponding to the two optimization problems are, collectively, sufficient and necessary at the saddle-point =-=[26]-=- [27]. Thus, the computation of the saddle-point is equivalent to solving the water-filling and the least favorable noise problems simultaneously. One might suspect that the following algorithm may be... |

1248 |
Noiseless coding of correlated information sources
- Slepian, Wolf
- 1973
(Show Context)
Citation Context ...egion for the general non-degraded broadcast channel is still unknown, superior coding schemes beyond that of superposition do exist. The key idea is a random binning argument which was first used in =-=[10] and-=- subsequently allowed Marton [11] [12] to derive an enlarged achievable rate region. For a two-user channel with independent information for each user, Marton’s region is as follows: R1 ≤ I(U1; Y1... |

1024 |
Writing on dirty paper
- Costa
- 1983
(Show Context)
Citation Context ...gnal whose entire non-causal realization is known to the transmitter but not to the receiver, and n is Gaussian noise independent of s. In a surprising result known as “writing-on-dirty-paper,” Co=-=sta [15]-=- showed that under a joint i.i.d. Gaussian condition on s and n, the capacity of the channel with interference s isthesameasifsdoes not exist. In addition, the optimal transmit signal x is statistical... |

579 |
On the achievable throughput of a multiantenna gaussian broadcast channel
- Caire, Shamai
- 2003
(Show Context)
Citation Context ...loss when receiver terminals lackcoordination? Second, what is the optimal encoding and decoding structure on such broadcast channels? These questions have been partially answered by Caire and Shamai =-=[6]-=- in the special case of a two-user broadcast channel with two transmit antennas and one receive antenna for each user, where they showed that a precoding strategy based on channels with transmitter si... |

562 | Broadcast channels
- Cover
- 1972
(Show Context)
Citation Context ...not among the receive terminals, the Gaussian vector channel becomes a broadcast channel. Unlike the multiple access channel, the capacity region for a broadcast channel is still not known in general =-=[5]-=-. The main difficulty is that a vector channel distributes information across several receive terminals, and without joint processing of the received signals, a data rate equal to I(X; Y) cannot be su... |

341 | Nested linear/lattice codes for structured multiterminal binning
- Zamir, Shamai, et al.
- 2002
(Show Context)
Citation Context ...ision-feedback equalizer viewpoint leads directly to a path for implementation. It also makes the capacity result amenable to practical coding schemes, such as the inflated-lattice precoding strategy =-=[12]-=- and the trellis shaping technique [13]. Further, the result in this paper is in fact more general than that of [10] and [11]. The result of this paper applies to broad0018-9448/04$20.00 © 2004 IEEE1... |

334 | Duality, Achievable Rates, and Sum-Rate Capacity of Gaussian MIMO Broadcast Channels
- Vishwanath, Jindal, et al.
(Show Context)
Citation Context ...rbitrary number of transmit antennas and an arbitrary number of users each equipped with multiple receive antennas. The sum capacity result has also been obtained in simultaneous and independent work =-=[10]-=- and [11]. These two separate pieces of work arrive at essentially the same result via a duality relation between the multiple-access channel capacity region and the dirty-paper precoding region for t... |

318 | Sum Capacity of the Vector Gaussian Broadcast Channel and Uplink-Downlink Duality
- Viswanath, Tse
- 2003
(Show Context)
Citation Context ...number of transmit antennas and an arbitrary number of users each equipped with multiple receive antennas. The sum capacity result has also been obtained in simultaneous and independent work [10] and =-=[11]-=-. These two separate pieces of work arrive at essentially the same result via a duality relation between the multiple-access channel capacity region and the dirty-paper precoding region for the broadc... |

233 |
A coding theorem for the discrete memoryless broadcast channel
- Marton
- 1979
(Show Context)
Citation Context ...oadcast channel is still unknown, superior coding schemes beyond that of superposition do exist. The key idea is a random binning argument which was first used in [10] and subsequently allowed Marton =-=[11] [12] to-=- derive an enlarged achievable rate region. For a two-user channel with independent information for each user, Marton’s region is as follows: R1 ≤ I(U1; Y1) (6) R2 ≤ I(U2; Y2) (7) R1 + R2 ≤ I(... |

223 |
Coding for channel with random parameters,” Prob
- Gelfand, Pinsker
- 1980
(Show Context)
Citation Context ... signal, y is the received signal, and s is the channel state information, whose entire sample sequence is known to the transmitter prior to transmission but not to the receiver. Gel’fand and Pinske=-=r [13] a-=-nd Heegard and El Gamal [14] characterized the capacity of such channels using an auxiliary random variable U: C = max {I(U; Y ) − I(U; S)}. (9) p(u,x|s) The achievability proof of this result uses ... |

154 |
Gamal, “On the capacity of computer memory with defects
- Heegard, El
- 1983
(Show Context)
Citation Context ...nal, and s is the channel state information, whose entire sample sequence is known to the transmitter prior to transmission but not to the receiver. Gel’fand and Pinsker [13] and Heegard and El Gama=-=l [14] c-=-haracterized the capacity of such channels using an auxiliary random variable U: C = max {I(U; Y ) − I(U; S)}. (9) p(u,x|s) The achievability proof of this result uses a random binning argument, and... |

116 |
Capacity and coding for degraded broadcast channels
- Gallager
- 1980
(Show Context)
Citation Context ...s first introduced by Cover [1], who also proposed an achievable coding strategy based on superposition. Superposition coding has been shown to be optimal for the class of degraded broadcast channels =-=[2]-=-, [3]. However, it is in general suboptimal for nondegraded broadcast channels. The largest achievable region for the nondegraded broadcast channel is due to Marton [4], [5], but no converse has been ... |

116 |
The worst additive noise under a covariance constraint
- Diggavi, Cover
- 2001
(Show Context)
Citation Context ...iance matrix . Let be a Gaussian random vector, and let be any other random vector with the same covariance matrix, but with possibly a different distribution. Then, . This fact is proved in [27] and =-=[28]-=-. Thus, to minimize ,itis without loss of generality to restrict attention to that are jointly Gaussian. In this case, the cooperative capacity is just . So, the least favorable noise is the solution ... |

103 |
Optimum decision feedback multiuser equalization with successive decoding achieves the total capacity of the Gaussian multiple-access channel
- Varanas, Guess
- 1997
(Show Context)
Citation Context ...e transmitter coordinated case. In addition, the lackof transmitter coordination makes the diagonalization of the vector channel impossible. Instead, the vector channel can only be triangularized [3] =-=[4]-=-. Such triangularization decomposes a vector channel into a series of single-user sub-channels each interfering with only subsequent sub-channels. The triangular structure enables a coding method base... |

103 |
Vectored transmission for digital subscriber line systems
- Ginis, Cioffi
- 2002
(Show Context)
Citation Context ...ntenna broadcast channel. Unfortunately, their procedure does not generalize to the -receiver case easily, and it does not reveal the structure of the optimal . In a separate effort, Ginis and Cioffi =-=[22]-=- demonstrated a precoding technique for an broadcast channel based on a QR decomposition of the channel matrix. The QR method transforms the matrix channel into a triangular structure, and by doing so... |

91 |
An outer bound to the capacity region of broadcast channels
- Sato
- 1978
(Show Context)
Citation Context ...erate. Fix an input distribution p(x). The sum capacity of the broadcast channel is clearly bounded by the capacity of the vector channel I(X; Y1, Y2) where y1 and y2 cooperate. As recognized by Sato =-=[21]-=-, this bound can be further tightened. Because y1 and y2 cannot coordinate in a broadcast channel, the broadcast channel capacity does not depend on the joint distribution p(n1, n2) but only on the ma... |

73 |
der Meulen, “A proof of marton’s coding theorem for the discrete memoryless broadcast channel
- Gamal, van
- 1981
(Show Context)
Citation Context ...st channel is still unknown, superior coding schemes beyond that of superposition do exist. The key idea is a random binning argument which was first used in [10] and subsequently allowed Marton [11] =-=[12] to deri-=-ve an enlarged achievable rate region. For a two-user channel with independent information for each user, Marton’s region is as follows: R1 ≤ I(U1; Y1) (6) R2 ≤ I(U2; Y2) (7) R1 + R2 ≤ I(U1; Y... |

59 | Trellis precoding for the broadcast channel
- Yu, Cioffi
- 2001
(Show Context)
Citation Context ...s directly to a path for implementation. It also makes the capacity result amenable to practical coding schemes, such as the inflated-lattice precoding strategy [12] and the trellis shaping technique =-=[13]-=-. Further, the result in this paper is in fact more general than that of [10] and [11]. The result of this paper applies to broad0018-9448/04$20.00 © 2004 IEEE1876 IEEE TRANSACTIONS ON INFORMATION TH... |

55 |
Vector coding for partial response channels
- Kasturia, Cioffi
(Show Context)
Citation Context ...mple, under a total power constraint P , the maximization is over all Sxx such that trace(Sxx) ≤ P . This leads to the well-known water-filling solution based on the singular-value decomposition of =-=H [2]-=-. Assuming that Snn is an identity matrix, the optimum Sxx must have its eigenvectors equal to the right singular-vectors of H, and its eigenvalues obeying the water-filling power allocation on the si... |

52 | On the capacity of multiple input multiple output broadcast channels - Vishwanath, Jindal, et al. - 2002 |

49 | Sum capacity of the multiple antenna Gaussian broadcast channel and uplink-downlink duality”, submitted to - Viswanath, Tse - 2002 |

37 |
On the capacity of channels with additive non-Gaussian noise
- Ihara
- 1978
(Show Context)
Citation Context ...ussian random vector, and let n ′ be any other random vector with the same covariance matrix, but with possibly a different distribution. Then, I(X; HX + N) ≤ I(X; HX + N ′ ). This fact was prov=-=ed in [22] an-=-d [23]. So, for a Gaussian input, Gaussian noise is the least favorable distribution among all distributions. Thus, to minimize I(X; Y1, ···, YK), it is without loss of generality to restrict atten... |

37 |
On the capacity of the multiple antenna broadcast channel
- Tse, Viswanath
- 2003
(Show Context)
Citation Context ... fully address the capacity region for the vector broadcast channel. The difficulty appears to be in proving that Gaussian inputs are optimal for non-rate-sum points. In fact, as is shown in [14] and =-=[15]-=-, the dirty-paper precoding region is the capacity region if an additional Gaussianity assumption is made. The capacity region of Gaussian vector broadcast channels is still an open problem. The remai... |

36 | Writing on colored paper
- Yu, Sutivong, et al.
- 2001
(Show Context)
Citation Context ...omposed by a discrete Fourier transform [16].) Nevertheless, the “dirty-paper” result may be generalized to the vector case to implement interference pre-subtraction at the transmitter. Lemma 1 ([=-=17] [18]-=-) Consider a channel y = x + s + n, wheresand n are independent i.i.d. vector Gaussian signals. Suppose that non-causal knowledge of s is available at the transmitter but not at the receiver. The capa... |

34 | The capacity region of broadcast channels with intersymbol interference and colored Gaussian noise
- Goldsmith, Effros
- 2001
(Show Context)
Citation Context ...s typically not possible to decompose the vector channel into independent scalar broadcast channels. (An important exception is the ISI channel which can be decomposed by a discrete Fourier transform =-=[16].) N-=-evertheless, the “dirty-paper” result may be generalized to the vector case to implement interference pre-subtraction at the transmitter. Lemma 1 ([17] [18]) Consider a channel y = x + s + n, wher... |

29 |
Generalized decision-feedback equalization for packet transmission with ISI and Gaussian noise
- Cioffi, Forney
- 1997
(Show Context)
Citation Context ... The derivation is largely tutorial in nature. It is useful in fixing the notations used in the development and for setting the stage for a subsequent generalization of GDFE. This section is based on =-=[23]-=-. Decision-feedback equalization (DFE) is widely used to mitigate ISI in linear dispersive channels. To untangle the ISI, a decision-feedback equalizer decodes each input symbol sequentially, based on... |

26 | Capacity Bounds for Gaussian Vector Broadcast Channels
- Vishwanath, Kramer, et al.
- 2003
(Show Context)
Citation Context ...10], [11] fully address the capacity region for the vector broadcast channel. The difficulty appears to be in proving that Gaussian inputs are optimal for non-rate-sum points. In fact, as is shown in =-=[14]-=- and [15], the dirty-paper precoding region is the capacity region if an additional Gaussianity assumption is made. The capacity region of Gaussian vector broadcast channels is still an open problem. ... |

20 |
Estimation
- Kailath, Hassibi
- 2000
(Show Context)
Citation Context ...itly, W = SxyS −1 yy (23) = SxxH T (HSxxH T + I) −1 (24) = (H T H + S −1 xx ) −1 H T , (25) where (23) follows from standard linear estimation theory, and (25) follows from the matrix inversio=-=n lemma [20], which will be -=-used repeatedly in subsequent development, (A + BCD) −1 = A −1 − A −1 B(C −1 + DA −1 B) −1 DA −1 . (26) Now, it is clear that W may be split into a matched filter H T , and an estimati... |

17 |
Minimax theorems,” Proc
- Fan
- 1953
(Show Context)
Citation Context ...a concave function over the set of positive definite matrices, 1 2 log |HSxxH T + Snn|/|Snn| is convex in Snn and concave in Sxx. The constraints are convex. So, from a minimax theorem in game theory =-=[25], there exists-=- a saddle-point (Sxx,Snn) such that 1 2 log |HS′ xxHT + Snn| ≤ |Snn| 1 2 log |HSxxH T + Snn| ≤ |Snn| 1 2 log |HSxxH T + S ′ nn | |S ′ nn| for all (S ′ xx ,S′ nn ) in the constraint sets.... |

10 |
Saddle-points and convex analysis. Differential Games Related Topics
- Rockafellar
- 1971
(Show Context)
Citation Context ...at the mutual information is a concave-convex function, and the two KKT conditions, corresponding to the two optimization problems are, collectively, sufficient and necessary at the saddle-point [26] =-=[27]-=-. Thus, the computation of the saddle-point is equivalent to solving the water-filling and the least favorable noise problems simultaneously. One might suspect that the following algorithm may be able... |

8 |
Pantelides: An Interior Point Algorithm for Computing Saddle
- Zakovic, C
- 2000
(Show Context)
Citation Context ...m also suggests that general-purpose numerical convex programming algorithms can be used to solve the least favorable noise problem, or to solve for a saddle-point directly with polynomial complexity =-=[28] [29].-=- 4.4 Example The following numerical example illustrates the computation of the saddle-point (Sxx,Snn) and the construction of a precoder. Consider the following broadcast channel: ⎡ ⎤ ⎡ y1 1.0 ... |

8 |
MMSE Decision Feedback Equalizers and Coding
- Cioffi, Dudevoir, et al.
- 1995
(Show Context)
Citation Context ...rity condition (that rarely occurs by accident), a generalization of decision-feedback equalizer (that often consists of several DFEs) can achieve the capacity of a Gaussian linear dispersive channel =-=[24]-=-. The study of the decision-feedback equalizer is related to the study of multiple-access channels. If each transmitted symbol in an ISI channel is regarded as a data stream from a separate user, the ... |

6 |
A simple converse for broadcast channels with additive white Gaussian noise
- Bergman
- 1974
(Show Context)
Citation Context ...word intended for y2 can be decoded by y1. Thus, y1 can subtract the codeword due to P2, and in effect get a cleaner channel with noise σ 2 1 instead of σ2 1 + P2. In fact, as it was shown by Bergma=-=n [9]-=-, this superposition and interference subtraction scheme is optimum for the degraded Gaussian broadcast channel. Unfortunately, when a Gaussian broadcast channel has multiple transmit terminals, it is... |

4 |
The Gaussian watermarking game: Part I,” submitted to
- Cohen, Lapidoth
- 2001
(Show Context)
Citation Context ...e decomposed by a discrete Fourier transform [16].) Nevertheless, the “dirty-paper” result may be generalized to the vector case to implement interference pre-subtraction at the transmitter. Lemma=-= 1 ([17]-=- [18]) Consider a channel y = x + s + n, wheresand n are independent i.i.d. vector Gaussian signals. Suppose that non-causal knowledge of s is available at the transmitter but not at the receiver. The... |

3 |
The Gaussian watermarking game - parts I
- Cohen, Lapidoth
- 2001
(Show Context)
Citation Context ...nsmitter but not at the receiver, has the same capacity as if did not exist, i.e., (13) Further, the capacity-achieving is statistically independent of . This result has been noted by several authors =-=[20]-=-, [21] under different conditions. Lemma 1 suggests a coding scheme for the broadcast channel as shown in Fig. 5. The following theorem formalizes this idea. Theorem 1: Consider the Gaussian vector br... |

2 |
convex analysis,” in Differential Games and Related
- “Saddle-points
- 1971
(Show Context)
Citation Context ...e the mutual information is a concave–convex function, and the two KKT conditions, corresponding to the two optimization problems are, collectively, sufficient and necessary at the saddle-point [31], =-=[32]-=-. Thus, the computation of the saddle-point is equivalent to simultaneously solving the water-filling problem and the least favorable noise problem. One might suspect that the following algorithm can ... |

2 |
The structure of lease-favorable noise in the Gaussian vector broadcast channels
- Yu
(Show Context)
Citation Context ...peration. C. GDFE With Singular Noise To complete the argument, it remains to show that Lemma 3 holds even when the least favorable noise is singular. Part of the following proof has also appeared in =-=[36]-=- Lemma 4: Consider the Gaussian vector channel , where . There exists a GDFE structure for the Gaussian vector channel with a block-diagonal feedforward matrix if and only if is the minimizing solutio... |

1 |
Generalized decision-feedbackequalization for packet transmission with ISI and Gaussian noise
- Cioffi, Forney
- 1997
(Show Context)
Citation Context ...o the transmitter coordinated case. In addition, the lackof transmitter coordination makes the diagonalization of the vector channel impossible. Instead, the vector channel can only be triangularized =-=[3]-=- [4]. Such triangularization decomposes a vector channel into a series of single-user sub-channels each interfering with only subsequent sub-channels. The triangular structure enables a coding method ... |

1 |
Vector DMT: a FEXT cancellation scheme
- Ginis, Cioffi
- 2000
(Show Context)
Citation Context ...n’s region indeed coincides with the outer bound. Unfortunately, this numerical procedure does not generalize easily, and it does not reveal the structure of the optimal Si. In an independent effort=-=, [7] -=-demonstrated a precoding technique for a broadcast channel with a transmitter having N terminals, and N receivers each having a single terminal. The channel is modeled as y = Hx + n, wherey is an N ×... |

1 |
MMSE decision feedbackequalizers and coding
- Cioffi, Dudevoir, et al.
- 1995
(Show Context)
Citation Context ...ion does not occur. Under this assumption, it can be shown that a minimum mean-square error decision-feedbackequalizer (MMSE-DFE) achieves the channel capacity of a Gaussian linear dispersive channel =-=[19]-=-. The study of the decision-feedbackequalizer is related to the study of the multiple access channel. Each transmitted symbol in an ISI channel can be regarded as a separate user. Suppose that the tra... |