CLJL-PIC Method

Since the methods in [45], [46] suffer from the MUI, the method in [47], called the CLJL-PIC method, further suppress the MUI using the PIC technique. The CLJL-PIC method can be summarized in the following steps:

1. Using the CLJL method to obtain the initial estimate^{ @o} for all active users

 @o  µ @ —

where^{ y@} is the estimated data of user ^U at the^e th iteration.

2. Regenerate the MUI by ^—

Û @

and ^{ yWAB'@} obtained at the ^{êeb } th stage. Then cancel the

regenerated MUI from the received signal

—Ú @

y is the output signal after the PIC processing at the^e th stage.

3. Using the CLJL method to compensate for the CFO for each active user

 @

The main drawback of this method is that its performance is affected by the initial estimate

 @o

used for the PIC processing. When a large CFO occurs, the performance improvement by the PIC technique with the poor initial estimates (owing to the large MUI) will be limited. If

the modulation scheme has a large QAM-size such as 64-QAM, the performance will become sensitive to the residual MUI. Note that Step 2 can be implemented more efficiently as follows:

—Ú @

This formula will be used to evaluate its computational complexity in a comparison described later.

§ 5.3 ZF Method

§ 5.3.1 Proposed Newton-ZF Method

From (5.5), we can see that a straightforward method to compensate for the CFO effect is the ZF method given by [51]



Although the direct ZF method can completely suppress the CFO effect, it needs to invert the ICI matrix with dimension , the FFT size. When the size is large, the required computa-tional complexity can become prohibitively high. Unfortunately, in real-world applications, the symbol size is usually large. For example, for IEEE 802.16e, the size can be as large as 2048.

Here, we propose a low-complexity ZF method to solve the problem. The main idea is to use an iterative procedure such that the direct matrix inversion can be avoided. Specifically, we use Newton’s method as we did in the case of mobility-induced ICI.

Base on the previous discussion on the mobility-induced ICI, we also approximate the ma-trix inversion of^œ via the expanded form of Newton’s iteration as

¶·

Moreover, our final objective is to obtain the CFO-compensated result ^¶·{— not the matrix inversion^¶· itself. Multiplying both sides of (5.13) by the receive signal^—

Ú

. According to these definitions, we can rewrite (5.14) as As a result, ^{÷ ¾} can be recursively calculated. With this approach, we have transformed the matrix-to-matrix multiplications in (5.13) into the matrix-to-vector multiplications in (5.15) and (5.16).

To complete our low-complexity algorithm, we further let ^{¶ o} be a diagonal matrix and explore the special structure inherent in the CFO-induced ICI matrix, ^œ . Recall that ^{œ }

© Î

Note that operations in (5.17) only involve vector multiplications, IDFTs, and a DFT. It is well-known that DFT/IDFT can be implemented with FFT/IFFT and then the required computational complexity can be greatly reduced. Thus, evaluation of (5.15) only involves vector multiplica-tions, FFTs, and IFFTs. The required computational complexity is reduced from ^ to

 r €;

s  Ž!F

.

With the interleaved-OFDMA structure, the computational complexity can be reduced fur-ther. Let^{÷ ¾ Mx}



This is to say that^{à ƾ} corresponds to an upsampled sequence of the desired elements in^{÷ ¾} . The nonzero elements in^{à ƾ} , denoted by^{Œ ƾ Mx}

zE| , can be obtained by circularly shifting^{à ¾Æ} with^{‘ b } elements and downsampling the result with a factor of ^; . Let ^{Œ ¾Æ  „ ˜ ; ’“yñð Œ Æ¾} , where ^{’ yñð} is an ^ºpV
ºp DFT matrix, and construct an^{
^} vector by duplicating^{Œ ƾ} ,^; times shown as

´ ƾ Consequently, we can obtain^{’ “ µ Æ ÷ ¾} by the following method

’ “ µ Æ ÷ ¾  î Æ ´ ƾ ) (5.20)

Þ . Note that the operation ^{î Æ} results from the circlur shift of ^{à ƾ} . Equation (5.20) implies that we can implement^{’“)µ Æ ÷ ¾} by an IDFT with dimension^{»„ ;} instead of . Using this approach, we can reduce the computational complexity further by rewriting (5.16) as

÷ ¾

are complex diagonal matrices. As as-sumed,^{¶ o} is a diagonal matrix. Equation (5.21) only involves one DFT with size ,^; IDFTs with size^{»„ ;} , and some vector operations. As mentioned, DFT/IDFT can be efficiently imple-mented with FFT/IFFT. Finally, the required computational complexity is reduced from^{ 2*}

to^{ !}

 „ ;

r .

The final thing we have to deal with is how to determine the initial matrix ^{¶ o} . A well-designed initial values can reduce the number of iterations significantly and provide good

mit-igation performance. Let ^{¶ o a†Uˆ‡‰ xÒ o )šÒ\'*) JWJWJ )šÒ y AB'{zE|} . Again, we adopt the minimum-Frobenius-norm criterion to obtain optimum initial values. The criterion is given by

¶¬öÔy÷

with respect to^Ò_Õ· to zero. The first derivative of ^{® ” y b^¶ o œ  ®} For further complexity reduction, we can make an approximation to (5.25) as

ÒÄöÔy÷

where is a parameter controlling the number of ICI terms considered (^{3 º º »„ b } ). The approximation is based on the fact that the CFO-induced ICI on a subcarrier mainly comes from neighboring subcarriers. Moreover, this approximation is only for the initial matrix calculation.

The final result will be updated by Newton’s method. For easy reference, we denote this method as the N-ZF method.

For the direct ZF method, the matrix inversion is obtained by solving a set of linear equations as ^{œ   £¢  —}

Ú

, where ^{  £¢} is the ZF-compensated ^— . This can be implemented by triangular factorization (Gaussian elimination), and forward and backward substitution [69]. Finally, for signal detection, a conventional one-tap FEQ is applied to each subcarrier (compensating for the channel effect). We can express the result as^{1 }— AB' } , where^ is the estimate of^— while

 is the CFO-compensated^— .