Proposed Newton-MMSE Method

8704 (0.003, 0.85) 256 (0.016, 0.5) 9344 (0.003, 0.745)

N-ZF (^{%  3} ,

Q

5 )

19968 (0.007, 1.95) 256 (0.016, 0.5) 22656 (0.008, 1.806)

N-ZF (^{%  3} ,

Q

4R )

42496 (0.015, 4.15) 256 (0.016, 0.5) 49280 (0.017, 3.929)

N-ZF (^{% Ä} ,

Q

43 )

17007 (0.006, 1.661) 254 (0.015, 0.496) 12315 (0.004, 0.982)

N-ZF (^{% Ä} ,

Q

# )

23655 (0.008, 2.310) 254 (0.015, 0.496) 19987 (0.007, 1.593)

N-ZF (^{% Ä} ,

Q

5 )

36951 (0.013, 3.609) 254 (0.015, 0.496) 35331 (0.012, 2.817)

§ 2.3.1 Proposed Newton-MMSE Method

Now, we know that the MMSE method has a problem of high-complexity. Although existing iterative methods [58] can reduce the complexity of the matrix inversion, the matrix multipli-cation operation remains. Motivated by this issue, we develop a method to solve the problem.

First, we rewrite the direct MMSE solution with a new form as

The key idea to avoid the matrix multiplication and inversion is to apply an efficient iterative matrix inversion method twice in (2.50). In Chapter 2, we have shown that the ZF method with Newton’s iteration has good performance and its performance is almost as good as that of the direct ZF method. Here, we extend the idea to reduce the computational complexity of the direct MMSE method.

Let the estimated ^{x— AB'} at the^Q th iteration be^¶"· . Then from Newton’s iteration,¶"·DF' can be described as follows [61]– [66]:

¶·DF'

From (2.51), it is obvious that Newton’s iteration requires matrix-to-matrix multiplications.

Thus, direct application of Newton’s iteration for matrix inversion is not feasible.

As we did in the N-ZF method, we can iterate (2.51) to obtain a sequence of matrices

$<¶

where^{~ · q¾} is the coefficient of the^¿ th summation term in (2.52). It turns out that to obtain a low-complexity algorithm, we have to use the expanded form.

Note that the matrix inverse^¶"· is not the desired result, whereas^¶ò·— is the desired result.

Let^{ ·  ¶·—} and^{‚ ¾  Š¶ o x— ¾ ¶ o —} . Then, multiplying (2.52) by^— , we have

From the definition of^{‚ ¾} , it is simple to see that^{‚ ¾} can be recursively calculated as iteration be ^Å_y . We then have

Å_y As we can see, the number of the expansion terms in Newton’s iteration grows exponentially.

Now, we have two iterative processes, i.e., (2.53) and the summation terms in (2.56), the former for the approximation of ^{x— AB'} and the latter for the approximation of ^{œ A “} . The required computational complexity will be high if we fully iterate these two processes. From (2.50), we can see that ^{œ A “} is weighted by ^? which will be much less than one in high SNR scenarios.

As a result, the approximation of^{œ  A “} is less critical. For simplicity, we only use the first-order expansion for the approximation of ^{œ A “} in (2.56), i.e.,^{e #} . From (2.56), we then have rewrite (2.57) as follows:

‚ ¾

Note that ^{}— o} , ^}1'— , and ^' are all diagonal matrices. If we further let^{¶ o} and ^{Å o} be diago-nal matrices, (2.58) will only involve vector-to-vector and DFT/IDFT operations. It is well-known that DFTs/IDFTs can be efficiently implemented with FFTs/IFFTs, whose complexity

is ^{ ! }  . Thus, the computational complexity of the proposed MMSE algorithm is



!F

. The diagonal constraint on^{¶ o} and^{Å o} may not always yield satisfactory per-formance in all scenarios. To obtain higher perper-formance and at the same time to retain the low-complexity property, we can relax the constraint slightly. We may let^{¶ o} and^{Å o} be low-bandwidth circular band matrices. The structure of a circular band matrix is depicted in Fig.

2.12. Let the ^{U ){u} th entry of a circular band matrix ^Å be denoted as ^{ƒ@qk  Å U ){u} . Given a fixed index^U , ^ƒ@qk is non-zero only for ^{u2„#$ U b–%) U b¤% s  ) JWJWJ ) U s % +} . Here, ^% is the bandwidth of a circular band matrix. Note that the index,^u , is calculated with modulo- arith-metic. Thus, from Fig. 2.12, we can see that there are non-zero elements in the upper right and lower left corners of a circular band matrix. If ^{% 3} , the circular band matrix is reduced to a diagonal matrix. If ^{% 8} , the circular band matrix will have three non-zero diagonal vectors.

With this type of ^{¶ o} and ^{Å o} , the computational complexity in (2.58) will only be increased slightly. Thus, we only consider the case where ^{% } . For easy reference, we denote the proposed low-complexity MMSE method as the Newton-MMSE (N-MMSE) method.

§ 2.3.2 Derivation of the Initial Matrix

In the previous subsection, we have proposed the N-MMSE method to solve the problems of the matrix multiplication and inversion. However, we have to determine the initial matrices^{¶ o} and ^{Å o} . Good initial matrices can reduce the number of iterations significantly and provide good mitigation performance.

First, we discuss the determination of^{¶ o} . As we did in the N-ZF method, we also adopt the minimum-Frobenius-norm criterion to obtain optimal initial matrices. Let^¶ be a circular

…m†ˆ‡

…m†Š‰Œ‹Ž

…m†ˆ

Figure 2.12: The structure of a circular band initial matrix is depicted for ^=Z and ^{% >} . The elements in the shaded area are non-zeros, while the others are zeros.

band matrix with bandwidth^{% } . Then^{¶ o} can be obtained by the following criterion

¶ o 5ÌÍrnÎ6ÏÐ turns out that we only have to calculate ^{œ  A “} roughly. We will discuss this problem later. For the time being, we can simply assume that ^x— is known as a priori.

Define a vector consisting of non-zero elements in the^U th row of^{¶ o} as^à»@, i.e.,

à»@ (2.59) with respect to ^{Ò @Õqk} , and setting the corresponding result to zero, we can obtain the following equations:

where Note that the indices in^?(@, ^à»@, and ^{’ @} are calculated with modulo- arithmetic. Now, we can

A0

Figure 2.13: The structure of ^?(@ is presented for ^ and ^{% N} . Note that ^?(@ overlaps

?C@EDF' for modulo-^.U (i.e., the relationship is circular).

obtain the optimum solution of (2.59) by^{àC@  ? @AB' ’ @}. For easily understanding the structure of ^?C@, we show an example in Fig. 2.13 for ^P and ^{% >} . From Fig. 2.13, we can see that the lower right^0
1 submatrix of ^?(@ is exactly the same as the upper left^0
1 submatrix

of ^?C@EDF'. Note that this relationship is circular. The circular relationship means that the lower

right^0
7 submatrix of^?ô is exactly the same as the upper left^»
2 submatrix of^{? o} . Thus,

?C@ overlaps ^?C@EDF' for all modulo-^ÖU. Using this property, we can use the recursive algorithm

mentioned in Subsection 2.2.2 for fast computation of ^{? @AB'} . Consequently, we can obtain ^{? @AB'} recursively. Using this approach, we only have to actually calculate one matrix inversion, i.e.,

?

AB'

o , and its matrix dimension is ^{%  s  .
 %  s } . Note that as mentioned, ^?(@ has the circular relationship, we can start out from any^U .

To further reduce the computational complexity in the calculation of the initial matrix, we can use a circular band matrix derived from ^x— to calculate ^? . Define a matrix operation

~*Uh

and ^{îY U ){u} , respectively. The index,^u , is calculated with modulo-arithmetic. Using the operation, we define ^{x— a~*Uh ƒ& x— )* } and ^{?—  x— Õ x— |} . Moreover, let

œ . Here, ^ is the number of one-sided ICI terms that we want to take into calculation (^{3 º  º »„ b } ). Since ICI on a subcarrier mainly comes from a few neigh-boring subcarriers, we can make such an approximation safely. Note that this approximation is used only for the calculation of the initial matrix.

To obtain ^{Å o} , we can use the same approach, namely, ^{Å o} ÌÍrÁÎ6ÏЊ

® ” y

b5Å

œ “ ® ¢

. Recall that ^x— is needed in (2.59). A simple way to approximate ^x— is to use a zeroth-order expansion for ^x— , i.e., ^{x— ¨ œ s ?‹Å o} . The reason why we can use the zeroth-order expansion for ^x— is that the precision for initial values can be lower. Thus, its computational complexity can be reduced with limited performance loss. Note that^{Å o} is used twice; one is for the iterative step (2.57) and the other is for the initial matrix calculation.

§ 2.3.3 Complexity Analysis

In Subsections 2.3.1 and 2.3.2, we reformulate the MMSE solution and then obtain a low-complexity N-MMSE method. In this subsection, we analyze the required computational com-plexity of the proposed N-MMSE method and make a comparison between the N-MMSE and direct MMSE methods.

From Subsection 2.3.1, we know that the major computational load results from Eqs. (2.53), (2.54), and (2.56) shown as follows:

1. ^{‚ ¾} iteration, where^{‚ ¾  Š¶ o x— ‚ ¾ AB'} and^{‚ o  ¶ o —} ,

Due to the diagonal and DFT/IDFT structures in ^x— , ^{‚ ¾} can be obtained using (2.58). As a result, we require ^{x y G  s  yDF' %C' s  %  s  yD}

RDs for the case of ^{%  } . As for the direct MMSE method, the matrix inversion can be implemented by Gaussian elimination [69]. Finally, we summarize the required complexity of the N-MMSE method and the direct MMSE method for a SISO-OFDM system in Table 2.4.

We also summarize the complexity of calculating the initial matrix in the N-MMSE method in Table 2.5.

Table 2.4: Complexity comparison between the N-MMSE and direct MMSE methods in a SISO-OFDM system.

Methods Real multiplications Real divi-sions

Table 2.5: Complexity of the initial matrices calculation for the N-MMSE method in a SISO-OFDM system.

Methods Real multiplications Real divisions Real additions N-MMSE

In this subsection, we report simulation results to demonstrate the effectiveness of the proposed N-MMSE method. The simulated system parameters are the same as those in Subsection 2.2.5.

We compare the performance of the N-MMSE, one-tap FEQ, and direct MMSE methods. Here, we set^{HI 43KJ3X} , and use the BER as the performance measure. Fig. 2.14 shows the simulation results. From this figure, we see that the performance of the one-tap FEQ method suffers from an error floor phenomenon. However, the N-MMSE method effectively avoids this phenomenon