MIT 6 441 - Capacity of Multi-antenna Gaussian Channels

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1 Capacity of Multi-antenna Gaussian Channels by Costas Pelekanakis 1. Abstract The demand for higher data rates in bandwidth-constraint fading channels (e.g. radio channels, underwater acoustic channels) has motivated research in multi-input multi-output (MIMO) systems. The goal is that signals emitted from Nt transmit antennas are combined at the Nr receive antennas in such way that the data rate and the reliability of the link is increased. Capacity formulas are derived for channels where the channel gains (fades) and noise are independent and Gaussian at different receiving antennas. The benefit of using MIMO systems relative to having a single transmit and receive antenna is demonstrated. 2. Introduction In this paper, the capacity of three different channels is computed. All the channels are described by the linear model: wxHy += where tNC∈xis the transmitted signal with an average power constrain []()P≤=′Qxx trE where Q is the covariance matrix of the input x and P is the total average power at the transmitter, rNC∈y is the received signal, the noise w is a zero mean circularly symmetric Gaussian random vector, spatially and temporally independent, i.e., )CN(0, ~rN2Iwσ and the H is the channel NtxNr complex matrix. Three cases are examined: I. H is deterministic. II. H is random and changes independently in every use of the channel (fast fading). Every element of the matrix is i.i.d. CN(0,1) (also called Rayleigh fading). III. H is random but stays fixed during transmission (slow fading). Every element of the matrix is i.i.d. CN(0,1).2 3. Determistic channel The capacity of this channel (time-invariant) is found to be: ()()HHQIQQ′+=≤rNP:C detlogmax)tr( where the maximization is over the input covariance matrix Q subject to the constraint: tr(Q) ≤ P. The capacity achieving input vector must be circularly symmetric Gaussian, i.e., ()Qx,0~CN. Applying the SVD on H, namely VUΣH′=, it is proved in the paper that the best Q is of the form VVΛQ′= where V is the matrix of the right singular vectors of H, ,0,...,0),...,diag(1**ΛrPP= and r = rank(H) ≤ min(Nt,Nr). The diagonal matrix Λ is computed by the waterfiling power allocation algorithm. Note that tr(Q) = tr(Λ). It is useful to express the capacity formula in terms of the singular values of H: ∑=+=r1i2*i2iσPλC 1log where λi is the i-th singular value of H. Hence, The SVD decomposes the MIMO channel into r parallel, independent scalar channels (or eigenchannels) with power gain 2iλ each and the optimal power allocation among the r eigenchannels is done via waterfiling. Waterfiling claims that one should put more power to higher gain eigenchannels. In order to achieve the above rate H must be known to the transmitter. Capacity is achieved by coding separately for each eigenchannel. The figure below illustrates the SVD-based architecture for reliable communication. Figure 1: SVD architecture for MIMO channels. The SVD architecture can be viewed as two coordinate transformations: the input is expressed in terms of the coordinate system defined by the columns of V and the output is expressed in terms of the coordinate system defined by the columns of U. ][mx1][mx2][mxtN][my1][my2][myrN1w2w][~mx1][~mxr0][mxr][~my1][~my2][~myrdecoder decoder decoder H V U’ AWGN Coder ),R(P*11 . . . rNwAWGN Coder ),R(Pr*r3 Question: The capacity formula is derived assuming independent coding for each eigenchannel. Does coding across the eigenchannels yield a higher capacity? Answer: The paper argues that this is true but it is not actually! Although using a finite length block code, by coding jointly across the parallel eigenchannels will yield a smaller probability of error than by coding independently in each of the eigenchannels for a given rate, the capacity of the parallel eigenchannels is the largest achievable rate using independent coding within each eigenchannel. We can justify that by considering a geometric point of view. Assume we use an NNr block length code, coding over all Nr channels with N symbols for each eigenchannel. In high dimensions, the NNr, received vector after passing through the parallel channel lies in an ellipsoid with different axes stretched and shrunk by the different channel gains λi. It can be proved that the volume of the ellipsoid is proportional to ( )∏=+rNiNiiP122σλ* while the volume of the noise sphere is proportional to ()rNN2σ. Thus, the maximum number of codewords that can be packed in the ellipsoid is: ∏=+rNiNiiP1221σλ* and the maximum reliable rate that can be achieved is: ∑∏==+=+rrNiiiNiNiiPPN1221221log1log1σλσλ** which is precisely the rate achieved using independent coding within each eigenchannel. Hence, coding across the eigenchannels cannot yield a higher capacity but only improve the probability of error for a given rate. Question: (not addressed in the paper) How does the rank of H affects the capacity in high SNR/low SNR regimes? Answer: At high SNR, it is asymptotically optimal to put equal power among the eigenchannels so the capacity is: ∑∑==λ+⋅≈⋅σλ+=riiriirσPrrPC12C2122loglog1logAWGN434214 Obviously, the capacity scales up linearly with the rank of H. Thus, the rank denotes the multiplexing capability of the MIMO channel: up to r parallel scalar AWGN channels can be spatially multiplexed by the MIMO channel. Note there is no multiplexing capability if one antenna is used either at the receiver or at the transmitter. Hence, a MIMO channel provides a higher data rate than a scalar channel given the transmit power and bandwidth. At low SNR, it is optimal to put all the power to the strongest eigenchannel. The capacity becomes: 43421AWGNC2elogmaxmax1log ⋅⋅≈σ⋅+=22ii22iiσPλPλC In this regime, the rank of H is not important and the MIMO channel merely provides a power gain of 2iiλmax . 4. Fast fading channel The key assumption is that the channel


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MIT 6 441 - Capacity of Multi-antenna Gaussian Channels

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