TY - JOUR

T1 - On the Maximum Achievable Partial Decode-and-Forward Rate for the Gaussian MIMO Relay Channel

AU - Gerdes, Lennart

AU - Hellings, Christoph

AU - Weiland, Lorenz

AU - Utschick, Wolfgang

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - This paper considers the so-called partial decode-and-forward (DF) strategy for the Gaussian multiple-input multiple-output (MIMO) relay channel. Unlike for the DF strategy or point-to-point (P2P) transmission from source to destination, for which Gaussian channel inputs are known to maximize the achievable rates, the input distribution that attains the maximum achievable partial DF rate for the Gaussian MIMO relay channel has remained unknown so far. For some special cases, e.g., for relay channels where the partial DF strategy reduces to the DF or P2P transmission, it could be deduced that Gaussian inputs maximize the rate that can be achieved with the partial DF strategy. For the general case, however, the problem has remained open until now. In this paper, we solve this problem by proving that the maximum achievable partial DF rate for the Gaussian MIMO relay channel is always attained by Gaussian channel inputs. Our proof relies on the channel enhancement technique, which was originally introduced by Weingarten et al. to derive the (private message) capacity region of the Gaussian MIMO broadcast channel. By combining this technique with a primal decomposition approach, we first establish that jointly Gaussian source and relay inputs maximize the achievable partial DF rate for the aligned Gaussian MIMO relay channel. Subsequently, we use a limiting argument to extend this result from the aligned to the general Gaussian MIMO relay channel.

AB - This paper considers the so-called partial decode-and-forward (DF) strategy for the Gaussian multiple-input multiple-output (MIMO) relay channel. Unlike for the DF strategy or point-to-point (P2P) transmission from source to destination, for which Gaussian channel inputs are known to maximize the achievable rates, the input distribution that attains the maximum achievable partial DF rate for the Gaussian MIMO relay channel has remained unknown so far. For some special cases, e.g., for relay channels where the partial DF strategy reduces to the DF or P2P transmission, it could be deduced that Gaussian inputs maximize the rate that can be achieved with the partial DF strategy. For the general case, however, the problem has remained open until now. In this paper, we solve this problem by proving that the maximum achievable partial DF rate for the Gaussian MIMO relay channel is always attained by Gaussian channel inputs. Our proof relies on the channel enhancement technique, which was originally introduced by Weingarten et al. to derive the (private message) capacity region of the Gaussian MIMO broadcast channel. By combining this technique with a primal decomposition approach, we first establish that jointly Gaussian source and relay inputs maximize the achievable partial DF rate for the aligned Gaussian MIMO relay channel. Subsequently, we use a limiting argument to extend this result from the aligned to the general Gaussian MIMO relay channel.

KW - Gaussian channel inputs

KW - Gaussian relay channel

KW - MIMO

KW - channel enhancement

KW - partial decodeand- forward

UR - http://www.scopus.com/inward/record.url?scp=84959421940&partnerID=8YFLogxK

U2 - 10.1109/TIT.2015.2485995

DO - 10.1109/TIT.2015.2485995

M3 - Article

AN - SCOPUS:84959421940

SN - 0018-9448

VL - 61

SP - 6751

EP - 6758

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 12

M1 - 7287756

ER -