The system capacity is defined as the maximum possible transmission rate such that the probability of error is arbitrarily small.
Initially, we assume that the channel matrix is not known at the transmitter, while it is perfectly known at the receiver.
By the singular value decomposition (SVD) theorem [11] anynR×nT matrixHcan be written as
H=UDVH (1.11)
whereDis annR×nT non-negative and diagonal matrix,UandVarenR×nRandnT×nT unitary matrices, respectively. That is, UUH = InR andVVH =InT, where InR and InT
arenR×nR andnT ×nT identity matrices, respectively. The diagonal entries ofDare the non-negative square roots of the eigenvalues of matrix HHH. The eigenvalues of HHH, denoted byλ, are defined as
HHHy=λy, y=0 (1.12)
wherey is annR×1 vector associated withλ, called an eigenvector.
The non-negative square roots of the eigenvalues are also referred to as the singular values ofH. Furthermore, the columns ofUare the eigenvectors ofHHH and the columns of Vare the eigenvectors of HHH. By substituting (1.11) into (1.9) we can write for the received vectorr
r=UDVHx+n (1.13)
MIMO System Capacity Derivation 5 Let us introduce the following transformations
r=UHr x=VHx n=UHn
(1.14)
asUandVare invertible. Clearly, multiplication of vectorsr,xandnby the corresponding matrices as defined in (1.14) has only a scaling effect. Vectorn is a zero mean Gaussian random variable with i.i.d real and imaginary parts. Thus, the original channel is equivalent to the channel represented as
r =Dx+n (1.15)
The number of nonzero eigenvalues of matrix HHH is equal to the rank of matrix H, denoted by r. For the nR ×nT matrix H, the rank is at most m = min(nR, nT), which means that at mostmof its singular values are nonzero. Let us denote the singular values of H by √
λi, i = 1,2, . . . , r. By substituting the entries √
λi in (1.15), we get for the received signal components
ri=
λixi+ni, i=1,2, . . . , r ri=ni, i=r+1, r+2, . . . , nR
(1.16) As (1.16) indicates, received components,ri,i=r+1, r+2, . . . , nR, do not depend on the transmitted signal, i.e. the channel gain is zero. On the other hand, received components ri, fori=1,2, . . . , r depend only on the transmitted component xi. Thus the equivalent MIMO channel from (1.15) can be considered as consisting of r uncoupled parallel sub- channels. Each sub-channel is assigned to a singular value of matrixH, which corresponds to the amplitude channel gain. The channel power gain is thus equal to the eigenvalue of matrix HHH. For example, if nT > nR, as the rank of H cannot be higher than nR, Eq. (1.16) shows that there will be at mostnR nonzero gain sub-channels in the equivalent MIMO channel, as shown in Fig. 1.2.
On the other hand ifnR > nT, there will be at most nT nonzero gain sub-channels in the equivalent MIMO channel, as shown in Fig. 1.3. The eigenvalue spectrum is a MIMO channel representation, which is suitable for evaluation of the best transmission paths.
The covariance matrices and their traces for signals r, x and n can be derived from (1.14) as
Rrr =UHRrrU
Rxx =VHRxxV (1.17)
Rnn =UHRnnU tr(Rrr)=tr(Rrr)
tr(Rxx)=tr(Rxx) (1.18)
tr(Rnn)=tr(Rnn)
x
x
r
r
r
1
2 λ1
λ2
λn R
0
nR 1
2
xn
R
xn +1 R
xn T
0
Figure 1.2 Block diagram of an equivalent MIMO channel ifnT > nR
Figure 1.3 Block diagram of an equivalent MIMO channel ifnR> nT
MIMO System Capacity Derivation 7 The above relationships show that the covariance matrices ofr,x andn, have the same sum of the diagonal elements, and thus the same powers, as for the original signals,r,x andn, respectively.
Note that in the equivalent MIMO channel model described by (1.16), the sub-channels are uncoupled and thus their capacities add up. Assuming that the transmit power from each antenna in the equivalent MIMO channel model isP /nT, we can estimate the overall channel capacity, denoted byC, by using the Shannon capacity formula
C=W r i=1
log2
1+Pri σ2
(1.19) whereW is the bandwidth of each sub-channel andPri is the received signal power in the ith sub-channel. It is given by
Pri= λiP
nT (1.20)
where√
λi is the singular value of channel matrix H. Thus the channel capacity can be written as
C=W r i=1
log2
1+ λiP nTσ2
=Wlog2 r i=1
1+ λiP nTσ2
(1.21)
Now we will show how the channel capacity is related to the channel matrixH. Assuming thatm=min(nR, nT), Eq. (1.12), defining the eigenvalue-eigenvector relationship, can be rewritten as
(λIm−Q)y=0, y=0 (1.22)
whereQis the Wishart matrix defined as Q=
HHH, nR < nT
HHH, nR ≥nT (1.23)
That is, λ is an eigenvalue of Q, if and only if λIm−Q is a singular matrix. Thus the determinant ofλIm−Qmust be zero
det(λIm−Q)=0 (1.24)
The singular values λ of the channel matrix can be calculated by finding the roots of Eq. (1.24).
We consider the characteristic polynomialp(λ)from the left-hand side in Eq. (1.24)
p(λ)=det(λIm−Q) (1.25)
It has degree equal tom, as each row ofλIm−Qcontributes one and only one power of λ in the Laplace expansion of det(λIm−Q)by minors. As a polynomial of degree m
with complex coefficients has exactlymzeros, counting multiplicities, we can write for the characteristic polynomial
p(λ)=mi=1(λ−λi) (1.26)
whereλi are the roots of the characteristic polynomialp(λ), equal to the channel matrix singular values. We can now write Eq. (1.24) as
mi=1(λ−λi)=0 (1.27)
Further we can equate the left-hand sides of (1.24) and (1.27)
mi=1(λ−λi)=det(λIm−Q) (1.28) Substituting−nTPσ2 for λin (1.28) we get
mi=1
1+ λiP nTσ2
=det
Im+ P nTσ2Q
(1.29) Now the capacity formula from (1.21) can be written as
C=Wlog2det
Im+ P nTσ2Q
(1.30) As the nonzero eigenvalues ofHHH andHHHare the same, the capacities of the channels with matrices H and HH are the same. Note that if the channel coefficients are random variables, formulas (1.21) and (1.30), represent instantaneous capacities or mutual informa- tion. The mean channel capacity can be obtained by averaging over all realizations of the channel coefficients.