IEEE TRANSACTIONS ON COMMUNICATIONS. VOL 50. NO. 1 JANUARY 2002 tennas(or vice versa),(e.g, by fixing M= I and let where the minimization is over all possible Stc codewordS= N-o, the ergodic capacity converges to the capacity Eipp, kipk. Assuming equal transmitted power at all trans- of AwGn channels [26]) mitter antennas, using the Chernoff bound, the PEP of trans- In summary, we have seen the different impacts of two di- mitting r and deciding in favor of another codeword I at the versity resources-the spatial diversity and the selective-fading decoder is upper bounded by diversity--on the channel capacity of a multiple-antenna cor related fading OFDM system. Increasing the spatial diversity P(x→21)≤e(~F(x,) (10) order(i.e, N, M) can always bring capacity(outage capacity and or ergodic capacity increase at the expense of extra phys- where y is the total signal power transmitted from all N trans- ical costs. By contrast, the selective-fading diversity is a free re- mitted antennas(recall that the noise at each receiver antenna source, but its effect on improving the channel capacity becomes is assumed to have unit variance). Using(4) -(6), d2(a, r)is less as L becomes larger. Since both diversity resources can im- 13), shown at the bottom of the page. In(12), prove the capacity of a multiple-antenna OFDM system, It is (ep, ke[p, k] is a rank-one matrix, which equals to a zero crucial to have an efficient channel coding scheme, which can matrix if the entries of codewords r and T corresponding to the take advantage of all available diversity resources of the system. kth subcarrier and the pth time slot are the same. Let D denote the number of instances when ep, keh[p, A+0, vp, Vh, IV PAIRWISE ERROR PROBABILITY similarly, as in [101 Deff, which is the minimum D over every two possible codeword pair, is called the effective length In the previous section, the potential information rate of a of the code. Denoting T=rank(D), it is easily seen that multiple-antenna OFDM system in correlated fading channels minr, r< min( Deff, NL). Since wf (k:)and w(p)vary with is studied. In order to obtain more insights on coding design, in different multipath delay profiles and Doppler power spectrum this section, we analyze the pairwise error probability(PEP)of shapes, the matrix D is also variant with different channel this system with coded modulation. environments. However it is observed that D is a nonnegative With perfect CSI at the receiver, the maximum likelihood definite Hermitian matrix; by an eigendecomposition, it can be ML) decision rule of the signal model (1)is given by wrItten as argon 9-∑HJ where V is a unitary matrix and/dag{入1,…,,,0,……,0} with Aili= being the positive eigenvalues of D.Moreover, as (9)assumed in Section Ill, all the (NML)elements of . iJi j are d(x)=∑∑∑∑H小小小 i=1=1k=0=1 Wi(P)w/(kelp, kle"p, Aw(kw!(p) NL)×1 P=1k= LX(NL) ∑D lx,-, e4]千[y4],,N1 W八(k)=dig{w()…,/(k)(NL)xN W()三dieg{w()…,W(p)} W(p)W/ (k:)ep, k]e [p, k/()Wt(p) (NLX(NL 9全H…,gLx178 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002 tennas (or vice versa), (e.g., by fixing 1 and let , the ergodic capacity converges to the capacity of AWGN channels [26]). In summary, we have seen the different impacts of two di￾versity resources—the spatial diversity and the selective-fading diversity—on the channel capacity of a multiple-antenna cor￾related fading OFDM system. Increasing the spatial diversity order (i.e., ) can always bring capacity (outage capacity and/or ergodic capacity) increase at the expense of extra phys￾ical costs. By contrast, the selective-fading diversity is a free re￾source, but its effect on improving the channel capacity becomes less as becomes larger. Since both diversity resources can im￾prove the capacity of a multiple-antenna OFDM system, it is crucial to have an efficient channel coding scheme, which can take advantage of all available diversity resources of the system. IV. PAIRWISE ERROR PROBABILITY In the previous section, the potential information rate of a multiple-antenna OFDM system in correlated fading channels is studied. In order to obtain more insights on coding design, in this section, we analyze the pairwise error probability (PEP) of this system with coded modulation. With perfect CSI at the receiver, the maximum likelihood (ML) decision rule of the signal model (1) is given by (9) where the minimization is over all possible STC codeword . Assuming equal transmitted power at all trans￾mitter antennas, using the Chernoff bound, the PEP of trans￾mitting and deciding in favor of another codeword at the decoder is upper bounded by (10) where is the total signal power transmitted from all trans￾mitted antennas (recall that the noise at each receiver antenna is assumed to have unit variance). Using (4)–(6), is given by (11)–(13), shown at the bottom of the page. In (12), ( ) is a rank-one matrix, which equals to a zero matrix if the entries of codewords and corresponding to the th subcarrier and the th time slot are the same. Let denote the number of instances when ; similarly, as in [10], , which is the minimum over every two possible codeword pair, is called the effective length of the code. Denoting , it is easily seen that . Since and vary with different multipath delay profiles and Doppler power spectrum shapes, the matrix is also variant with different channel environments. However, it is observed that is a nonnegative definite Hermitian matrix; by an eigendecomposition, it can be written as (14) where is a unitary matrix and 0 0 , with being the positive eigenvalues of . Moreover, as assumed in Section III, all the ( ) elements of are (11) with (12) (13)