Design Criteria of SFC

Assuming the channel is constant over at least one OFDM symbol, and the perfect CSI is available at the receiver. The ML decoder decides the most likely transmitted sequence ˆc_k, k = 0, 1, . . . , N − 1, over all possible codewords according to space-frequency codeword, for a given channel realization, the pairwise error probability is given by where the squared Euclidean distance between the two codewords C and E is dented by

d²³

Using the Chernoff bound Q(x) ≤ e^−x2/2 into equation (3.17), we get P

Since the multipath channel were assumed to be i.i.d. complex Gaussian it follows that H³

e^j2πkN ´

are Gaussian as well and hence the vector Y is Gaussian. The average over all channel realizations of the right handside in (3.20) is equivalent to solving the characteristic function of Y and it is fully characterized by the eigenvalues of the covariance matrix of Y [28]. Therefore the pairwise error probability is given by

P (C → E) ≤

where r (C_Y) denotes the rank of C_Y , λ_i(C_Y) , i = 0, 1, . . . , r (C_Y) , are the nonzero eigenvalues of C_Y, the definition of C_Y is given by

CY = E£

where D = diag{e^−j2πkN }^{N −1}_k=0 , R_l denote the correlation matrix of MIMO channel between transmit and receive antennas at time delay l, A ⊗ B denotes the Kronecker product of the matrices A and B, and the superscript ∗ stands for complex conjugate operation. Assume the SFC systems are operated at high SNR, the PEP can be upper bounded by

P (C → E) ≤

The design criteria for space-frequency codes follow from equation (3.23) as the well-known rank and determinant criteria.

• Rank (diversity) criterion: The minimum rank of CY over all pairs of distinct codewords C and E should be as large as possible.

• Determinant criterion: The minimum product of nonzero eigenvalues^r(CQ^Y)

i=0

λ_i(C_Y) over all pairs of distinct codewords C and E should also be maximized.

Next, we shortly introduce the maximum achievable diversity of the space-frequency codes by the discussion of CY. Assume N > MTL, using the factorizations of Rl =

If the MIMO channels satisfy the condition of r (R_l) = M_R and by carefully design of space-frequency codeword to have r

³

D^l(C − E)^T

´

= M_T for all l over all distinct codeword pairs, the stacked matrix G (C, E) would be full rank. As above conditions are fulfilled, we thus have a full rank C_Y and the space-frequency codes can achieve the maximum diversity order of M_TM_RL.

Chapter 4

CP-reduced Coding Techniques with ICI Diversity Combining for OFDM Systems

Actually the ICI corrupted channel of the CP-reduced OFDM system can be viewed as a virtual MIMO channel when we regard the subcarriers as the virtual antennas. And the ICI induced by insufficient CP can be viewed as a sort of frequency diversity by the concepts of space-frequency coding (SFC). In this chapter, we study the space-frequency codes under the virtual MIMO channel of CP-reduced OFDM systems, and we term such codes as “virtual space-frequency codes”. The maximum achievable diversity order and the corresponding design criteria are also given by theoretical analysis of the derived PEP bound.

4.1 System Model of OFDM Systems

If we view the subcarriers as the virtual antennas, and combine the system blocks circled by dash lines in Figure 4.1(a) into a equivalent frequency-domain MIMO channels as shown in Figure 4.1(b) emphasized by dash lines, then conventional CP-reduced SISO-OFDM systems with perfect ISI canceller can be regarded as virtual MIMO-OFDM systems with their virtual MIMO channels of ith OFDM block index are specified by the frequency-domain ICI channels of

H⁽ⁱ⁾_eq ¡ e^jw¢

= F_N

³

H⁽ⁱ⁾− H⁽ⁱ⁾_ici

´

F^H_N, (4.1)

where FN is the N-point FFT matrix, H⁽ⁱ⁾ and H⁽ⁱ⁾_ici are the time-domain channel matrices in (2.3), and (2.5) which are the channel matrices corresponding to interference-free and ICI signal components. Under the aspects of subcarriers as virtual antennas and following the concepts of SFC, the ICI induced from reduced CP can equivalently be considered as the virtual MIMO channel gains from virtual transmit antennas to virtual received antennas.

That is, ICI can be regarded as frequency diversity. After a clear understanding of the virtual spatial-domain, next we derive the PEP based on the virtual MIMO-OFDM systems shown in Figure 4.1(b).

(a) Conventional SISO-OFDM systems

(b) Virtual MIMO-OFDM systems with subcarriers as virtual antennas

Figure 4.1: (a) Conventional SISO-OFDM Systems, (b) Virtual MIMO-OFDM with sub-carriers as virtual antennas

4.2 Pairwise Error Probability Analysis

From the viewpoints of regarding subcarriers as virtual antennas, we investigate a coding technique across B OFDM blocks based on the concepts of SFC for the virtual MIMO-OFDM systems with perfect ISI canceller. Assume the channel responses are static at least for one OFDM duration (symbol-wise fading), and the channel gains for different delay are independent. The input bit stream is divided into b bit-long segments, forming 2^b-ary constellation symbols. These symbols are then mapped onto a frequency-domain

codeword to be transmitted over the N virtual antennas (subcarriers). Each frequency-domain codeword can be express as an NB × 1 matrix

X =£

X^T₁, X^T₂, . . . , X^T_B¤_T

(4.2) where Xi = [xi,0, xi,1, , · · · , xi,N −1] is an N × 1 column vector, representing the ith coded OFDM block. At the receiver, the received signal of ith block can be written in matrix form as

Ri =p

EsH⁽ⁱ⁾_eq ¡ e^jw¢

Xi+ Z^T_i , (4.3)

where the complex additive white Gaussian noise (AWGN) vector of ith block index is defined as Zi = [zi,0, zi,1, . . . , zi,N −1], and zi,j ∼ CN (0, N0) is the complex AWGN of ith OFDM block at j subcarrier. Thus the average coded symbol to noise ratio at each subcarrier is given by Es/N0. Collect B blocks of received OFDM signals and denote R =£

where the noise vector Z is formatted as Z = £

Z^T₁, Z^T₂, . . . , Z^T_B¤_T

, and H_eq(e^jw) is the virtual MIMO channel matrix in frequency-domain of consecutive B OFDM blocks, and can be formulated as

Heq

Assume perfect channel state information (CSI) is available at the receiver, and the receiver applies a maximum likelihood decoder to detect signals. The PEP of coding across consecutive B OFDM blocks conditioned on the virtual MIMO channels of H_eq(e^jw) is given by

where ˆX stands for the erroneous decoded codeword, Q (x) is the complementary error function, and d²_H(X, ˆX) is the modified Euclidean distance with the definition of

d²_H³

By using the Chernoff bound, the conditional PEP can be upper bounded by P

Next we rewrite d²_H(X, ˆX) into a more compact form for computational simplicity of channel averaging. By the equalities of tr¡

AA^H¢

= kAk²_F and tr (AB) = vec¡ A^H¢_H

vec (B) [29], we can reformulated (4.7) as

d²_H(X, ˆX) = is an operator that stacks all the columns of a matrix into a super column matrix orderly according to their column indexes. Apply the equality of vec (ABC) = ¡

C^T ⊗ A¢ distance is then given by

d²_H

Now we are about to average the conditional PEP over all channel realization, but we can’t assure if the random vector h_i has a full rank covariance matrix for each i (actually h_i has a rank-deficient covariance matrix for each i, and its proof is shown in the appendix).

Therefore, to ease the calculation of integral during the step of channel averaging, we con-sider a linear transformation to transform h into a random vector with a full rank covariance

to simplify the computation of integral. Assume h_i is a complex Gaussian random vector with its mean denoted by µ_h and covariance denoted by R_h with rank(R_h) = r ≤ N². Consider the linear transformation of the ith channel vector

h_i = µ_h+ R_h¹²g_i, (4.12)

where g_i ∼ CN (0, I_r) and R_h¹² stands for the Cholesky decomposition of the covariance matrix R_h [30]. By this transformation, we can transform each h_i to another random vector g_i which has zero mean and an identity covariance matrix. Substitute (4.12) into (4.11), then the modified Euclidean distance can be finally formulated as

d²_H

Now we average the conditional PEP over all channel realizations P

K_i(X, ˆX)µ_h. Consider the multipath channels to be Rayleigh fading channels (i.e. µ_h= 0), and after some computations, the PEP can be upper bounded by

P

where K⁽ⁱ⁾_{ef f}(X, ˆX) is the effective codeword matrix of ith OFDM block, and is defined as K⁽ⁱ⁾_{ef f}

Here we use the phase “effective” to emphasize K⁽ⁱ⁾_{ef f}(X, ˆX) is a codeword distance matrix that not only depends on the codeword distance matrix A_i(X, ˆX), but also depends on the

covariance matrix of the multipath channels. Note that K⁽ⁱ⁾_{ef f}(X, ˆX) is nonnegative definite Hermitian, thus there exists a unitary matrix Ui and a real diagonal matrix Di such that

Ui

where the rows of U_i are the eigenvectors of K⁽ⁱ⁾_{ef f}(X, ˆX), forming a complete orthonormal basis of an N²-dimensional vector space. The diagonal matrix D_i can be represented as

D_i = the nonzero eigenvalues of K⁽ⁱ⁾_{ef f}(X, ˆX). At high SNR’s, the upper bound of PEP can be simplified as

here we further define a total effective codeword distance matrix with its definition is given by

Based on the upper bounds shown in (4.19) and (4.20), we have the insights on the factors that determine the diversity order and coding gain of system performance, and we can tell that both diversity and coding gains are closely related to the total effective codeword distance matrix. Thus we give a proof of the achievable diversity order along with detail discussions in next section.

4.3 Maximum Achievable Diversity and Design Crite-ria

We analyze the factors that affect the diversity order of a SF-coded virtual MIMO-OFDM system in this section. First, we derive an upper bound for the maximum achievable diversity for such a system. Second, we propose the design criteria according to the derivation results of the upper bound of maximum achievable diversity. From the PEP in (4.19) and (4.20), we can see that the diversity order is determined by the rank of the total effective distance matrix K_{ef f}(X, ˆX), and K_{ef f}(X, ˆX) depends on K⁽ⁱ⁾_{ef f}(X, ˆX), for i = 1, . . . , B. Therefore, in order to determine the upper bound of rank(K_{ef f}(X, ˆX)), we should firstly determine the upper bound of rank

³

K⁽ⁱ⁾_{ef f}(X, ˆX)

´

, and then applying the upper bound to determine the rank of K_{ef f}(X, ˆX). The upper bound of rank

³

K⁽ⁱ⁾_{ef f}(X, ˆX)

´

and rank(K_{ef f}(X, ˆX)) can be calculated in the following theorem.

Theorem 1. Consider a coding scheme based on CP-reduced OFDM systems under symbol-wise Rayleigh fading channels for a given subcarrier number N and channel order (L + 1), the maximum achievable diversity order is given by:

(1.a) If N ≥ (L+1) and K⁽ⁱ⁾_{ef f}(X, ˆX) 6= 0 for ∀X 6= ˆX in ith OFDM block, then rank

³

K⁽ⁱ⁾_{ef f}(X, ˆX)

´

≤ (L + 1) and therefore the rank of the coded B OFDM blocks are upper bounded by

rank(Kef f(X, ˆX)) ≤ N × (L + 1) .

(1.b) If N < (L+1) and K⁽ⁱ⁾_{ef f}(X, ˆX) 6= 0 for ∀X 6= ˆX in ith OFDM block, then rank³

K⁽ⁱ⁾_{ef f}(X, ˆX)´

≤ N × rank(Ai(X, ˆX)) and therefore the the rank of the coded B OFDM blocks are upper

bounded by rank(Kef f(X, ˆX)) ≤ N ×P^B

i=1

rank(Ai(X, ˆX)).

Proof. By equation (4.16) and the inequality of rank(AB) ≤ min {rank(A), rank(B)}, then the upper bound of rank

³

Therefore we analyze the rank of R_h¹² and K_i(X, ˆX) individually to determine which matrix has the minimum rank that upper bounds the diversity order. We deal with K_i(X, ˆX) first.

Using the fact that each eigenvalue of the N × N matrix A_i(X, ˆX) is an eigenvalue of the N²× N² matrix I_N ⊗ A_i(X, ˆX) with multiplicity N, thus the rank of K_i(X, ˆX) is given by From (4.23) we can conclude that only if the ith codeword distance matrix Ai(X, ˆX) is nonzero for all pairs of ith distinct OFDM symbols, then rank(Ki(X, ˆX)) will be the mul-tiplication of rank(A_i(X, ˆX)) and subcarrier number N.

The rank of R_h¹² is equivalent to the rank of rank (R_h), therefore we calculate rank (R_h) in the following. By the equality of vec (ABC) = ¡

C^T ⊗ A¢ in which E[·] stands for taking expectation and “∗” stands for the complex conjugate op-erator. Using the inequality rank(AB) ≤ min {rank(A), rank(B)} again, then rank (R_h) can be upper bounded by

rank (R_h) ≤ min¡ rank¡

F^∗_N ⊗ F^H_N¢

, rank (Φ)¢

. (4.27)

Note that the FFT matrix F_N is a full rank matrix so we have rank¡

F^∗_N ⊗ F^H_N¢

= N² by the rank property of kronecker product. Thus rank (R_h) is always upper bounded by

rank (R_h) ≤ rank (Φ) . (4.28)

Now we shall determine rank (Φ) to determine the exact value of the upper bound of rank (R_h). Because the channel taps are assumed to be independent, the equality

rank (Φ) = L + 1 (4.29)

always holds^†. Therefore rank (R_h) is always upper bounded by the channel order of (L + 1), that is,

rank (R_h) ≤ (L + 1) . (4.30)

The rank of K⁽ⁱ⁾_{ef f}(X, ˆX) is upper bounded by the minor value of rank(Ki(X, ˆX)) and rank (Rh) which means the upper bound of rank

³

K⁽ⁱ⁾_{ef f}(X, ˆX)

´

for any one-to-one map-ping coding scheme is either N × rank(Ai(X, ˆX)) or the channel order (L + 1). There-fore we can conclude that rank

³

K⁽ⁱ⁾_{ef f}(X, ˆX)

´

≤ (L + 1) holds for N ≥ (L + 1), and rank

³

K⁽ⁱ⁾_{ef f}(X, ˆX)

´

≤ N × rank(Ai(X, ˆX)) holds for N < (L + 1). Note that the total effective codeword distance matrix Kef f(X, ˆX) is formed by placing 1th to Bth effective distance codeword matrix orderly at its diagonal and by the aforementioned maximum achievable diversity order of any one-to-one mapping coding schemes within one OFDM block, thus the maximum achievable diversity order of any one-to-one mapping coding schemes for consecutive B OFDM blocks is the cumulation of maximum achievable diver-sity order of each block index, i.e. rank(K_{ef f}(X, ˆX)) ≤ N × (L + 1) for N ≥ (L + 1) and rank(K_{ef f}(X, ˆX)) ≤ N ×P^B

i=1

rank(A_i(X, ˆX)) for N < (L + 1).

As shown in Theorem (1.a), when N ≥ (L + 1), a one-to-one mapping coding scheme within one OFDM block can always achieves (L + 1)-fold diversity, which means the vir-tual space-frequency code design based on CP-reduced OFDM systems under symbol-wise Rayleigh fading channels is no longer focusing on maximizing the diversity order since the full diversity can be always achieved only if the codes is designed to be one-to-one mapping.

And the maximum achievable diversity order in this setup is consistent with the maximum achievable diversity of conventional SFCs under the case of single transmit and receive an-tenna. On the other hand, if N < (L + 1), the diversity order of a one-to-one mapping

† The proof of rank (Φ) = (L + 1) in (4.29) is detailed in Appendix A.

coding scheme within one OFDM block can achieve N × rank(A_i(X, ˆX))-fold diversity as shown in Theorem (1.b). From Theorem 1, we propose the design criteria of any one-to-one mapping coding schemes within one OFDM block (B = 1, so K_{ef f}(X, ˆX) = K⁽¹⁾_{ef f}(X, ˆX) ) based on CP-reduced OFDM systems under symbol-wise Rayleigh fading channels.

Design Criteria 1: For virtual space-frequency codes within one OFDM symbol based on CP-reduced OFDM systems under symbol-wise Rayleigh fading channels, the design criteria are given as following.

I Rank (diversity) Criterion:

• For N ≥ (L + 1): Design a one-to-one mapping coding scheme, and then such a code can always achieve full diversity of (L + 1).

• For N < (L + 1): Maximize the minimum rank of codeword distance matrix A1(X, ˆX) over all pairs of distinct OFDM symbols.

I Determinant Criterion: Maximize th minimum determinant of total effective dis-tance matrix K_{ef f}(X, ˆX) over all pairs of distinct OFDM symbols.

Unlike the diversity criterion of conventional SFCs, Design Criterion 1 raises a different diversity criterion in the case of N ≥ (L + 1). Since the codes designed to be one-to-one mapping are able to achieved full diversity for N ≥ (L + 1), the coding redundancy shall be used to enlarge the determinant of K_{ef f}(X, ˆX), rather than the rank of K_{ef f}(X, ˆX), which means the diversity order in this setting is no longer determined by the minimum rank of the codes over all pairs of the distinct OFDM symbols, but determined by the channel order. On the other case of N < (L + 1), the rank of K_{ef f}(X, ˆX) is upper bounded by N × rank(A₁(X, ˆX)), thus maximize the rank of the codeword distance matrix is still be the main concern of code design which is coincided with the diversity criterion of conventional SFCs.

The determinant criterion is consistent in both circumstances, in order to minimize the error probability, the minimum determinant of K_{ef f}(X, ˆX) should be maximized. For a code set of the same achievable diversity order, the code with largest minimum det(K_{ef f}(X, ˆX))

outperforms other codes of the code set, thus we define the effective codeword distance to be the minimum determinant of K_{ef f}(X, ˆX) as a measure to evaluate of the superiority of different coding schemes when they have equivalent achievable diversity. Based on Design Criterion 1, we propose the design critera of any one-to-one mapping virtual space-frequency codes for consecutive B OFDM blocks based on CP-reduced OFDM systems under symbol-wise Rayleigh fading channels.

Design Criteria 2: For virtual space-frequency codes for consecutive B OFDM blocks based on CP-reduced OFDM systems under symbol-wise Rayleigh fading channels, the design criteria are given as following.

I Rank (diversity) Criterion:

• For N ≥ (L + 1): Maximize the number of nonzero ith effective codeword distance matrix K⁽ⁱ⁾_{ef f}(X, ˆX) for each block index over all pairs of distinct con-secutive B OFDM blocks.

• For N < (L + 1): Maximize the minimum rank of total effective distance matrix Kef f(X, ˆX) over all pairs of distinct consecutive B OFDM symbols.

I Determinant Criterion: Maximize the minimum determinant of total effective dis-tance matrix K_{ef f}(X, ˆX) over all pairs of distinct consecutive B OFDM symbols.

Because only if the codes have its K⁽ⁱ⁾_{ef f}(X, ˆX) 6= 0 for ∀X 6= ˆX, the codes achieve full diversity in ith block index. Therefore, we should maximize the number of nonzero effective codeword distance matrix for individual block index in the case of N ≥ (L+1), and then the maximum achievable diversity order of the codes is the multiplication of channel order and the number of nonzero ith effective codeword distance matrix out of B OFDM block indexes.

For N > (L + 1), the diversity gain is maximized by enlarging the rank of K_{ef f}(X, ˆX), to be more exact, it is done by maximizing the rank of A_i(X, ˆX) for each block index. Moreover, the determinant criterion claims that for the codes own larger effective codeword distance have the larger coding gain under the same diversity order, so the minimum determinant of K_{ef f}(X, ˆX) should be maximized.

Chapter 5

Simulation Results

In addition to theoretical analysis, we also carry out simulations to investigate the perfor-mance of virtual space-frequency codes based on CP-reduced OFDM systems by choosing bit-error rate (BER) as our figure of merit, and the BER plots are also used to demonstrate the derived maximum diversity order and design criteria in Chapter 4. The global setting of the simulations and the assumptions are given as following.

Global setting and assumptions:

• Assume the CSI is available at the receiver side, and the ISI component can be canceled perfectly. Timing and frequency synchronizations are assumed to be perfect as well.

• We employ QPSK modulation for all OFDM systems, and use the ML decoder to detect signals.

• There are two types of Rayleigh fading channels used in simulations corresponding to different delay spread, and their power profile are specified as following.

. For L = 1, the channel power profile is [ 0.8, 0.2 ].

. For L = 2, the channel power profile is [ 0.642, 0.256, 0.102 ] (SUI-4)[31].

A. Demonstrate the maximum achievable diversity order for N ≥ (L + 1)

From the derivation of achievable diversity bound in Chapter 4, we asserts that the maximum achievable diversity order of virtual space-frequency coding within one OFDM

symbol based on CP-reduced OFDM systems for N ≥ (L + 1) is upper bounded by channel order rather than the rank of effective codeword distance matrix, and this fact disagrees with the common sense of the rank criterion of conventional SFCs. Therefore we apply orthogonal designed SFBCs as the virtual space-frequency codes with their ranks are smaller or larger than the channel order to examine whether the slope of the BER curves is consistent with the channel order or the rank of the SFBCs. There are three orthogonal designed SFBCs applied in the simulations [32][33], and their codeword matrices are given by

SFBC_4×2 = (5.1) can achieve full diversity, so the rank of SFBC_4×2, SFBC_4×4, and SFBC_8×8are given accordingly as 2, 4, and 8. And the SFBCs used here have one-to-one correspondence, thus applying the SFBCs to the CP-reduced OFDM systems when N ≥ (L + 1) shall achieve the full diversity of (L + 1). Moreover, note that the subcarriers are playing the role of virtual spatial domain, and we assume the channel is static during each SFBC transmission which is quasi-static fading.

For N = 4 and L = 1, we intentionally apply the SFBCs with rank 2 and 4 based on CP-free OFDM systems with perfect ISI canceler to examine if the achievable diversity

order is channel order. As shown in Figure 5.1, the slope of BER curve corresponding to SFBC_4×2 is consistent to the slope of SFBC_4×4, and it is about the order of 2. As matter of fact, we can classify the same fact by computing the minimum rank of K⁽¹⁾_{ef f}(X, ˆX) for all distinct pair of OFDM symbols, and the minimum value of K⁽¹⁾_{ef f}(X, ˆX) is truly 2 which coincides with the simulation results. Furthermore, the effective codeword distance of SFBC_4×4 (the minimum det³

K⁽¹⁾_{ef f}(X, ˆX)´

of SFBC_4×4) is 0.1584 which is larger than the effective codeword distance of SFBC_4×2 (0.048), consequently, SFBC_4×4 outperforms SFBC_4×2 as shown in Figure 5.1. And we note that the uncoded CP-free OFDM systems have better performance than the CP-sufficient (which means the CP length is equal to

of SFBC_4×4) is 0.1584 which is larger than the effective codeword distance of SFBC_4×2 (0.048), consequently, SFBC_4×4 outperforms SFBC_4×2 as shown in Figure 5.1. And we note that the uncoded CP-free OFDM systems have better performance than the CP-sufficient (which means the CP length is equal to

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