Extension to the Multiple-Input Multiple-Output Case . 83

The transmission scheme can be extended to MIMO scenarios to benefit from additional gain from spatial diversity. As has been shown, BICM-OFDM can effectively capture time and frequency diversity; to further take advantage of spatial diversity in MIMO situations, a common way is to transform the spatial diversity to time or frequency diversity [30]. In the following we consider two techniques that achieve this goal with little modifications and overhead cost with respect to BICM-OFDM design for SISO cases. We choose a different approach from the obvious choice of combining BICM-OFDM with space time coding (STC), which potentially gives a even larger diversity order of N_T×min{L, d_free} [27], out of the considerations that the simple receivers often used for Alamouti-like schemes suffer greatly in double selective fading environments and the ML decoders for general STC usually carry prohibitively high cost [76–78].

4.3.1 Cyclic Delay Diversity

The first example is the CDD method used in OFDM systems [79] to transform spa-tial diversity to multipath diversity (or recognized as frequency diversity); it is done by inserting different cyclic delays to the signal at each transmit antenna. The

diver-sity order is analyzed via the framework established in Section 4.2.2; in essence, the asymptotic diversity order is determined by the rank of R_eq. For simplicity, the case of two transmit antennas and one receive antenna is considered and the results can be extended to more general cases. Starting from the SISO signal model of (2), inserting an intentional delay ∆ is equivalent to multiply a circular shift matrix P = [_I ^{0 I∆}

N −∆ 0 ] to the transmit signal F^Hx_p. The receive signal will be F H_pP F^Hx_p and we can combine P and H_p into an equivalent channel. Without loss of generality, assume that an intentional delay ∆ is inserted at the second antenna and (4.6) is modified for the MIMO case:

y = X(IP ⊗ FN ×2L⁰) (h1+ h2)

| {z }

h

+z = Xheq+ z (4.15)

where L⁰ = L + ∆ and h₁ and h₂ are respectively assembled by

h₁ = [˜h¹₁(0), . . . , ˜h¹₁(L − 1), 0_1×∆, ˜h¹₂(0), . . . , ˜h¹₂(L − 1), 0_1×∆, . . . , ˜h¹_P(0), . . . , ˜h¹_P(L − 1), 0_1×∆]^T_{P L}⁰_×1

h₂ = [0_1×∆, ˜h²₁(0), . . . , ˜h²₁(L − 1), 0_1×∆, ˜h²₂(0), . . . ,

˜h²₂(L − 1), . . . , 0_1×∆, ˜h²_P(0), . . . , ˜h²_P(L − 1)]^T_{P L}0×1

(4.16)

where superscript is used to denote the transmit antenna index.

Assume that the time and path ACF of h1 and h2 satisfy the Kronecker model in (9) individually and denote the spatial correlation matrix between two transmit antennas as Φ_S = [_ρ¹₂₁^ρ₁¹²]. Further assume that L ≤ ∆ ≤ N_cp− L where N_cp is CP length, then the taps of h₁ and h₂ will not overlap in the h in (15), and the length of h of each OFDM symbol is still within N_cp. As a result, the autocorrelation matrix of h will satisfy the MIMO Kronecker model:

R = E[hh^H] = ΦT ⊗ ΦS⊗ ΦL, (4.17)

and thus

rank(R_eq) = r_T × N_T × L. (4.18)

Following the approach in Section 4.2.2, it is straightforward to see from (4.18) that the maximum achievable diversity order is min{rT × NT × L, dfree}. Fig. 4.5 shows the BER performance of an MIMO BICM-OFDM system employing CDD with the normalized Doppler frequency set to 0.05 (thus r_T is 2). The slopes coincide with the analysis results, and the diversity order increases with the path and antenna counts.

4.3.2 Phase-roll Diversity

The second example is to generalize the phase-roll scheme [30], in which the correlation function of the equivalent channel h[k] = h₁+ h₂e^j2πkθ has zeros at certain delays, i.e., R_k[∆k] = ¹₂E{h[k]h^∗[k + ∆k]} = ¹₂(1 + e^j2π∆kθ) = 0 at ∆kθ = ¹₂,³₂,⁵₂, . . .. Zero correla-tion, as demonstrated in Section 4.2.2, in turn implies independent channel conditions and opportunities to exploit diversity. One interesting scenario happens when multi-ple CFOs exist among collaborating transmitters in cooperative communications. The scenario can be fitted into a PRD model with unintentional phase differences induced by CFOs. We have reported simulation results of similar schemes in [31] in which the effectiveness of PRD is clearly demonstrated.

Similarly as in the CDD case, by combining the phase rotation matrix E = diag(1, e^j2πεN , . . . , e^{j2πε(N −1)N} ) and H_p into an equivalent channel, (4.6) can be rewrit-ten as (when there are two transmit anrewrit-tennas)

y = X(I_P ⊗ F_{N ×L})(E₁h₁+ E₂h₂) + z = Xh_eq+ z (4.19)

where E_α = diag([1, 1, . . . , 1]⊗[e^j2πεα(N −1)/2

N , e^j2πεα2(N −1)/2

N , . . . , e^j2πεαP (N −1)/2

N ]), α ∈ {1, 2},

6 7 8 9 10 11 12 13 14

Figure 4.5: BER comparison of the MIMO BICM-OFDM employing CDD, PRD and STBC over doubly-selective fading channels. The DFT size is 64, P = 10, and a rate-1/2 convolutional code with the generator polynomial [133; 171] (dfree = 10) is adopted.

Notice that the considered diversity here is the effective diversity order based on the dominant eigenvalues. The channel is equal-gain two-path at l = 0 and l = 1 and the introduced cyclic delay ∆ is 5. The parameters of PRD are chosen as ε₁ = 0.05 and ε₂ = −0.05.

is an P L × P L diagonal matrix representing phase offsets, and h_α is defined as

hα = [˜h^α₁(0), . . . , ˜h^α₁(L − 1), ˜h^α₂(0), . . . , ˜h^α₂(L − 1), . . . , ˜h^α_P(0), . . . , ˜h^α_P(L − 1)]^T_{P L×1}.

(4.20)

Assume that we correct the phase offset from the first transmit antenna and thus, E₁

becomes an identity matrix. The rank of R_eq can be bounded by [75, Fact 2.10.7]

rank(Req) = rank(R + E2RE^H₂ )

≤ rank(R) + rank(E₂RE^H₂ ) = r_T × L × 2.

(4.21)

The last equation follows that rank(R) = rank(E₂RE^H₂ ) = r_T × L. The maximum diversity order is increased by N_T due to space diversity, if d_freeis not the limiting factor.

Though the analysis does not find the condition when the bound can be achieved, simulation results, such as those shown in Fig. 5, indicate that it is achievable.

For slow fading channels, it is worth noting that in [70] the diversity gain provided by multiple transmit antennas appears in the same form as our result. It is of no surprise since both methods transform spatial diversity into other types of diversity for easier harvest. Finally, the simulation results show the weakness of Alamouti scheme over doubly selective fading channels. Consider the example of space time Alamouti code where the signal matrix X in (4.6) becomes diag(X^ST₁ , . . . , X^ST_{P /2}) where X^ST_p = [_−X^X2(p−1)+1H ^X2(p−1)+2

2(p−1)+2 X^H_2(p−1)+1] with X_p = diag(X((p − 1)N ), X((p − 1)N + 1), . . . , X((p − 1)N + N − 1)). One would expect the diversity order to be P × min{L, d_free}; however, the performance degrades severely and an error floor occurs if a typical Alamouti receiver is deployed. The degradation is caused by channel variations within an Alamouti codeword and the induced destruction of the Alamouti structure.