1m. &帝子ae2 hanee A.Discrete-Time Input-Output Relation er in this sec 1=s(+后HRH門)e and the capacity of the MIMO channel follows as[7] y=√+n C=爱如(n+原Rr）能可 M MIMO channel matrix,n is additive te porally white and B Acquiring channel knowledge at the transmitter is in symbol period.We constrain the total average transmitted implies that B.Capacity ofa Deterministic MIMO Channel In the following.we assume that the channel H is by [7]and [20] a=s(+点m） (10 Although H is random.we shall first study the capacityo which may be decomposed as with Gaussian code books,i.e.s symmetric (11) PAULRAJL:AN OVERVIEW OF MIMO COMMUNICATIONS-A KEY TO GIGABIT WIRELESS 20m uso limitod to:ETH BIBLIOTHEK ZURICH ry 10,2010 at 11:50 Xplore.Rostn
Fig. 4. Measured time-frequency response of an , MIMO channel. denotes the scalar subchannel between the th transmit and the th receive antenna. A. Discrete-Time Input–Output Relation For the sake of simplicity we assume that the channel is frequency-flat fading (the capacity of frequency-selective fading MIMO channels will be discussed later in this section). The input–output relation over a symbol period assuming single-carrier (SC) modulation is given by (8) where is the 1 received signal vector, with is the 1 transmitted signal vector, is the MIMO channel matrix, is additive temporally white complex Gaussian noise with , and is the total average energy available at the transmitter over a symbol period. We constrain the total average transmitted power over a symbol period by assuming that the covariance matrix of , , satisfies Tr . B. Capacity of a Deterministic MIMO Channel In the following, we assume that the channel is perfectly known to the receiver (channel knowledge at the receiver can be maintained via training and tracking). Although is random, we shall first study the capacity of a sample realization of the channel, i.e., we consider to be deterministic. It is well known that capacity is achieved with Gaussian code books, i.e., is a circularly symmetric complex Gaussian vector [7]. The corresponding mutual information for having covariance matrix is given by b/s Hz and the capacity of the MIMO channel follows as [7] det b/s Hz (9) where the maximization is performed over all possible input covariance matrices satisfying Tr . Furthermore, given a bandwidth of Hz, the maximum asymptotically (in the block-length) error-free data rate supported by the MIMO channel is simply b/s. Acquiring channel knowledge at the transmitter is in general very difficult in practical systems. In the absence of channel state information at the transmitter, it is reasonable to choose to be spatially white, i.e., . This implies that the signals transmitted from the individual antennas are independent and equi-powered. The mutual information achieved with this covariance matrix is given by [7] and [20] (10) which may be decomposed as (11) PAULRAJ et al.: AN OVERVIEW OF MIMO COMMUNICATIONS—A KEY TO GIGABIT WIRELESS 203 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply
12 1 18 20 nfigurations.Note that the SIMO channel efollow.The sial at the iv have a C.E ation (11)ex 张=hs十n (12) where the 1 xMfr vector h;represents the ith row of H follows that multiple scala ent of n.S spatial data pipe I an known ase.For example,Ic increases by rb/s/Hz for every 3-dB E./N see below that in a fadin tially two notions of cap hewere known to individual spaial apaciy722.231. ich relate to the mean and tail be If th transmitted codewords span ling blocks.t can be allocated c ss the chieved by be ymmetric complex Gaussian with Rss=resulting in [7).[24] capacity C C=efa (13) C.Capacity of Fading MIMO Channels C=min(MR:Mr)logap+0(1) 14 owledge at the receiver and no channel state inform the taorewe水ch. which clearly shows the linear increase in capacity in the tan 三 ergod PROCEEDINGS OF THE IEEE VOL.92.NO.2.FEBRUARY 2004 Authorized liceneod use limited to:ETH BIBLIOTHEK ZURICH.Downlosdod on February 10,2010 at 11:50 trm IEEE Xplore.apply
Fig. 5. Ergodic capacity for different MIMO antenna configurations. Note that the SIMO channel has a higher ergodic capacity than the MISO channel. where is the rank of and denotes the positive eigenvalues of . Clearly, we have . Equation (11) expresses the spectral efficiency of the MIMO channel as the sum of the capacities of SISO channels with corresponding channel gains and transmit energy . It follows that multiple scalar spatial data pipes (also known as spatial modes) open up between transmitter and receiver resulting in significant performance gains over the SISO case. For example, increases by b/s/Hz for every 3-dB increase in transmit power (for high transmit power), as opposed to 1 b/s/Hz in conventional SISO channels. If the channel were known to the transmitter, the individual spatial channel modes can be accessed through linear processing at transmitter and receiver (modal decomposition), following which transmit energy can be allocated optimally across the different modes via the “waterfilling algorithm” [21], [7] so as to maximize the mutual information and achieve the capacity . C. Capacity of Fading MIMO Channels We now consider the capacity of fading MIMO channels. In particular, we shall assume with perfect channel knowledge at the receiver and no channel state information at the transmitter. Furthermore, we assume an ergodic block fading channel model where the channel remains constant over a block of consecutive symbols, and changes in an independent fashion across blocks. The average SNR at each of the receive antennas is given by , which can be demonstrated as follows. The signal at the th receive antenna is obtained as (12) where the 1 vector represents the th row of and is the th element of . Since and Tr , it follows that and, hence, the average SNR at the th receive antenna is given by . We shall see below that in a fading channel there are essentially two notions of capacity—ergodic capacity and outage capacity [7], [22], [23], which relate to the mean and tail behavior of , respectively. Ergodic Capacity: If the transmitted codewords span an infinite number of independently fading blocks, the Shannon capacity also known as ergodic capacity is achieved by choosing to be circularly symmetric complex Gaussian with resulting in [7], [24] (13) where the expectation is with respect to the random channel. It has been established that at high SNR [7], [25] (14) which clearly shows the linear increase in capacity in the minimum of the number of transmit and receive antennas. Fig. 5 depicts the ergodic capacity of several MIMO configurations as a function of SNR. As expected, the ergodic capacity increases with increasing and also with and . We note that the ergodic capacity of a SIMO ( 1) channel 204 PROCEEDINGS OF THE IEEE, VOL. 92, NO. 2, FEBRUARY 2004 Authorized licensed use limited to: ETH BIBLIOTHEK ZURICH. Downloaded on February 10, 2010 at 11:50 from IEEE Xplore. Restrictions apply