MIMO-OFDMA Signal Model

In the previous section, we have derived the SISO-OFDMA signal model and applied the low-complexity algorithms developed in Chapter 2 for ICI mitigation. Next we will further tackle the ICI problem in MIMO-OFDMA systems. In this section, we will develop a MIMO-OFDMA signal model facilitating the application of the proposed low-complexity algorithms. First, we will generalize the signal model derived in Section 4.1 to the scenario of MIMO-OFDMA sys-tems. Let ^{ Æk} be the transmit symbol vector of user ^‘ from its ^u th antenna and ^{ @Æqk} be the

cor-responding receive OFDMA symbol vector in the^Uth receive antenna (without CP and noise).

Based on these definitions, we have the signal model as follows:

 Æ

Æk , where ^{— Æk} is the corresponding frequency-domain symbol vector of ^{ Æk} . The matrix,^{} Æ@qk} , is the time-domain channel matrix for user^‘ consisting of the time-variant channel response between the^u th transmit antenna and the ^U th receive antenna. Note that only in the designated subcarrier positions are the elements of^{— Æk} non-zeros. We then have the receive time-domain OFDMA signal, for the channel between the^u th transmit antenna and the ^U th receive antenna, as

where^{ @qk} is the receive time-domain signal vector in the ^Uth receive antenna contributed from the^u th transmit antenna of all users. Transforming (4.11) with DFT, we have the corresponding frequency-domain signal as For user ^‘ , we define ^{—6 Ækƒq·} as the transmit signal at the ^Q th subcarrier from its^u th transmit antenna. Then we let^{— k Mx—6 kƒqo ) —6 kƒq'*) JWJWJ ) 6— kƒqy AB'{z|} , where^{6— kƒq·  6— Ækƒq·} if^Q¦„–Å_Æ . Thus, we have

—

Æk

 µ Æ — k

. Now we can express the receive signal in (4.12) as

— @qk  1 Î channel matrix between the^U th receive antenna and the^u th transmit antenna (for all users). With

(4.13), we can further express the receive frequency-domain signal in the ^U th receive antenna (from all transmit antennas of all users) as

— @  ,

where^{— @} is the receive frequency-domain noise vector in the^Uth receive antenna. By stacking all the receive signal vectors into a column vector, we finally obtain the following signal model

—

4œ 

— s — ) (4.15)

where ^{—  x— |' )r— | ) JWJWJ )š— |}

,

zE| is the overall receive frequency-domain signal vector, ^—

x— frequency-domain noise vector, and ^œ is the frequency-domain ICI channel matrix expressed as

Using the LTV channel model and (4.15), we can further derive a model facilitating ICI mit-igation in MIMO-OFDMA systems. With the LTV model, we again express the channel in a specific OFDMA symbol as

j Æ.q@qk

ml

n j

Æorq

.q@qk s l
j Æ' q.q@qk ) (4.17) where ^{j Æ.q@qk ml} is the ⁿth-tap channel response at time instant^l between the ^Uth receive antenna and the^u th transmit antenna for user^‘ ,^{j Æorq.q@qk} is its constant term, and^{j Æ' q.q@qk} is its variation rate. user’s time-domain channel matrix representing the channel between the ^Uth receive antenna and the^u th transmit antenna as

} Æ@qk  }

Transforming ^{} Æ@qk} from the time domain to the frequency domain and denoting the result as

. More importantly, we can see that each submatrix ^{œ @qk} is a combination of diagonal and DFT/IDFT matrices.

So far, we have derived the signal model for MIMO-OFDMA systems in (4.15), (4.16), and (4.21). Its signal structure is the same as that in MIMO-OFDM systems. Again, we can exploit this special structure to reduce the required computational complexity as we did in Chapter 3.

Thus, we can have low-complexity algorithms for MIMO-OFDMA systems. As for the required computational complexity, it is the same as that in MIMO-OFDM systems.

§ 4.2.1 Simulations

In this subsection, we provide simulation results to demonstrate the effectiveness of the pro-posed method. We consider a ^7
5 MIMO-OFDMA system with ^{ } , ^{¹Ð¤ R} ,

and ^{; ž:} . The simulated fading channel is generated by Jakes’ model [70]. In addi-tion, the power delay profile of the ^‘ th user is characterized by an exponential function, i.e.,

9 .qÆ

, and ^{i NX} . For the direct ZF and MMSE methods, we assume that the channel response is exactly known. For the N-ZF and N-MMSE methods, the parameters of the LTV channel model are obtained by LS fittings. The chosen modulation scheme is 16-QAM.

In the beginning, we discuss the ZF method. We consider two cases here; in the first case,

HI ’s are set to^{$ 3KJ3 ) 3KJ3X ) 3KJ3R ) 3KJ3: +} and in the second case, they are set to^{$ 3KJ3: ) 3KJL ) 3KJ3 ) 3KJ3 +} . Figure 4.5 shows the BER performance of case 1. In this figure, the performance of a two-tap FEQ method is also compared. The ^HI ’s in Fig. 4.1 and Fig. 4.5 are the same. From both figures, we find that the behavior of the BER curves is similar except that the BER is higher in Fig. 4.5. This is because the inter-antenna interference is introduced in a MIMO-OFDMA system. Figure 4.6 shows the BER performance in case 2. In this case, ^HI ’s are larger. As a result, we find that the N-ZF method needs three iterations to approach the direct ZF method.

Table 4.3 summarizes the required computational complexity for the direct ZF method and the N-ZF method in a ^1
– MIMO-OFDMA system. For case 1, two iterations are suffi-cient. The complexity ratio for multiplication/division/addition is 0.003/0.002/0.003. For case 2, three iterations are needed. The complexity ratio for multiplication/division/addition be-comes 0.005/0.002/0.006. As we can see from these figures, significant complexity reduction can still be obtained even though the iteration number is three.

We now compare the required computational complexity for ICI mitigation in MIMO-OFDMA systems and in SISO-MIMO-OFDMA systems. From Tables 4.1 and 4.3, we see that the multiplication complexity of the direct ZF method is increased up to almost eight times from SISO-OFDMA to^.
» MIMO-OFDMA. For the N-ZF method, it is only increased about three times. Thus, the complexity reduction achieved by the N-ZF method is greater in a^‹
\ MIMO-OFDMA system. In a SISO-MIMO-OFDMA system, the ratio of multiplication is 0.007 (^{Q a} and

HÊ

+ ). This is because the complexity of the direct ZF method is proportional to^{ hg *} , whereas that of the N-ZF method is proportional to^{ hg} GfF

. Even with three iterations, the complexity reduction achieved by the N-ZF method is still larger in MIMO-OFDMA systems. It can be inferred that the N-ZF method can save more computa-tions when^g or becomes larger.

Now, we report simulation results for the MMSE method. We also consider the above two cases; case 1 is that ^H&I ’s are set to ^{$ 3KJ3 ) 3KJ3X ) 3KJ3R ) 3KJ3: +} and in case 2, they are set to

+ . Figure 4.7 shows the BER performance of case 1. We find that the N-MMSE method with two iterations can avoid the error floor phenomenon of the two-tap FEQ method and in the meanwhile the N-MMSE method approaches the direct MMSE method.

Figure 4.8 shows the result of case 2. In this figure, the behavior of the N-MMSE method is similar to that in Fig. 4.7 except the required number of iterations is three.

Table 4.4 summarizes the required computational complexity for the direct MMSE method and the N-MMSE method in a ^Y
 MIMO-OFDMA system. For case 1, two iterations are enough. The complexity ratio of multiplication/division/addition is 0.0029/0.0389/0.0031. For case 2, three iterations are needed. The complexity ratio for multiplication/division/addition becomes 0.0056/0.0389/0.0062. From the above reports, it is clear that significant complexity reduction can be obtained by the N-MMSE method.

0 5 10 15 20 25 30 35 10⁻³

10⁻² 10⁻¹ 10⁰

SNR (dB)

BER

Two−tap FEQ Direct ZF N−ZF (k=0) N−ZF (k=1) N−ZF (k=2)

Figure 4.5: BER comparison among one-tap FEQ, direct ZF, and N-ZF (^{% 43} , ^! ) methods

in a ^0
1 MIMO-OFDMA system;^H&I = {0.02, 0.05, 0.03, 0.04} and 16-QAM modulation.

0 5 10 15 20 25 30 35

10⁻³ 10⁻² 10⁻¹ 10⁰

SNR (dB)

BER

Two−tap FEQ Direct ZF N−ZF (k=0) N−ZF (k=1) N−ZF (k=2) N−ZF (k=3)

Figure 4.6: BER comparison among one-tap FEQ, direct ZF, and N-ZF (^{% 43} , ^! ) methods

in a ^0
1 MIMO-OFDMA system;^H&I = {0.04, 0.1, 0.08, 0.07} and 16-QAM modulation.

0 5 10 15 20 25 30 35 10⁻³

10⁻² 10⁻¹ 10⁰

SNR (dB)

BER

Two−tap FEQ Direct MMSE N−MMSE (k=0) N−MMSE (k=1) N−MMSE (k=2)

Figure 4.7: BER comparison among two-tap FEQ, direct MMSE, and N-MMSE (^{%'  %  53} ,

-'

 

P ) methods in a ^{C
^} MIMO-OFDMA system; ^H&I = {0.02, 0.05, 0.03, 0.04} and 16-QAM modulation.

0 5 10 15 20 25 30 35

10⁻³ 10⁻² 10⁻¹ 10⁰

SNR (dB)

BER

Two−tap FEQ Direct MMSE N−MMSE (k=0) N−MMSE (k=1) N−MMSE (k=2) N−MMSE (k=3)

Figure 4.8: BER comparison among two-tap FEQ, direct MMSE, and N-MMSE (^{%'  %  53} ,

-'

 

>

) methods in a ^Y
 MIMO-OFDMA system; ^H&I = {0.04, 0.1, 0.08, 0.07} and 16-QAM modulation.

Table 4.3: Complexity comparison between direct ZF and N-ZF methods in a ^‘ MIMO-OFDMA system ("#

,^{% 43} , and ^! ).

Methods Real multiplications (ratio)

Real divisions (ratio) Real additions (ra-tio)

Direct ZF 22828288 65792 22728576

N-ZF (^{Q #} ) 35712(0.002) 128(0.002) 34560(0.002) N-ZF (^{Q 5} ) 64384(0.003) 128(0.002) 65280(0.003) N-ZF (^{Q 4R} ) 121728(0.005) 128(0.002) 126720(0.006)

Table 4.4: Complexity comparison between the N-MMSE and direct MMSE methods in a^ì
0

MIMO-OFDMA system (^ , $&%('*)%

,+

 $ 3 ) 3 +

, and^{$ -'*)* ,+  $  )  +} ).

Methods Real multiplications (ratio)

Real divisions (ratio) Real additions (ra-tio)

Direct MMSE 45086976 65792 45027072

N-MMSE (^{Q #} ) 69377 (0.0015) 2560 (0.0389) 68352 (0.0015) N-MMSE (^{Q 5} ) 130817 (0.0029) 2560 (0.0389) 139008 (0.0031) N-MMSE (^{Q 4R} ) 253697 (0.0056) 2560 (0.0389) 280320 (0.0062)

Chapter 5

CFO-induced ICI Mitigation for OFDMA Uplink Systems

§ 5.1 Signal Model

In the previous chapters, we discuss the mobility-induced ICI in SISO/MIMO-OFDM(A). Ex-cept for mobility, CFO also induces ICI in an OFDM-based system. In this chapter, we will focus on the CFO-induced ICI in an OFDMA uplink system. In an OFDMA uplink system with ^; active users, the available bandwidth is divided into equally spaced subbands. Each subcarrier uses a subband with bandwidth ^{„ orpr} , where^oqp is the sampling period. In such a system, ^; users share the subcarriers. Without loss of generality, we assume that each user

uses ^{ºp_P»„ ;} subcarriers. For the ^‘ th user, the transmitted frequency-domain signal at the

Q th subcarrier is denoted by ^{—6 Æ·} , where ^QׄØÅ_Æ and ^Å_Æ is the set of the subcarrier indices for the ^‘ th user. It is assumed that ^{ŵ@ÉËÅ k ÙÌ} for ^{UCÆ u} and ^{ÍÏÎÆ{¬F' Å_Æ  $ 3 )  )  ) JWJWJ ) b  +} . Usually OFDMA adopts the interleaved subcarrier allocation scheme. In other words, ^{ÅºÆ }

$ ‘ b  ) ‘ b  s

;») JWJWJ

) ‘ b  s

»„

;/b

 t

; +

. Since the subcarriers assigned to the different users are interleaved in the whole bandwidth, this scheme can achieve the maximum frequency diversity. For each user, we assume that the CP length is long enough to prevent the ISI effect.

Note that the channel we consider here is quasi-static, i.e., it is time-invariant in one OFDMA symbol period.

Consider a specific OFDMA symbol for the ^‘ th user. The channel output signal, after CP removal, can be expressed as

 Æ  } Æ  Æ ) (5.1)

where^{ Æ} is the vector representation of the^‘ th user’s time-domain OFDMA symbol, i.e., ^{ Æ }

„ ˜ 1

’ “ — Æ

. Here^{— Æ} is the corresponding frequency-domain signal vector. The matrix,^{} Æ} , is a circulant channel matrix with the first column vector being^{w Æ} which is the channel response

 Æ

experiences. Zeros are padded in^{w Æ} since the channel length is assumed to be smaller than the CP length. Note that^{} Æ} can be decoupled as ^{’“ }— Æ ’} , where ^{}— Æ} is a diagonal matrix with the diagonal vector of ^{—w Æ  ˜ ’w Æ} and elements of ^{— Æ} are nonzeros only in the designated subcarrier positions. The receive time-domain OFDMA symbol from ^; active users can be

expressed as ^Ú

Þ and ^ßLÆ is the normalized CFO (with respect to the subcarrier spacing) for user^‘ . Also,^‚ denotes the noise vector. After the FFT operation, we have the corresponding frequency-domain signal as

is a circulant matrix.

Denote its first column as^{—à Æ} . Then,^{—à Æ  „˜  ’¦à Æ} , where ^{à Æ Zx­}

—

Thus we can express the receive signal in (5.3) as [50], [51]

—Ú  1 Î

Æ is the CFO-induced ICI matrix. From the above formulation, we can see that the ICI matrix is composed of diagonal and DFT/IDFT matrices. This ICI matrix structure will be exploited in the developed low-complexity method.