Pre-compensation Method - Previous Methods

5.2 Previous Methods

5.3.2 Pre-compensation Method

If CFOs are large, the performance of the proposed method may be affected. The larger the CFO, the worse the performance it will result. In OFDM systems, we can compensate for the CFO effect with a phase de-rotation operation. Although the same method cannot be used to compensate for all CFOs here, it can be used to reduce their magnitudes in some cases. Note that this is equivalent to a compensation (PC) approach. Let the receive signal be pre-compensated by a normalized CFO value ^ß . Thus, the resultant CFO for the th user, denoted as^ûÆ , is now changed to

ûÆ

ßLÆnb0ß

J (5.27)

Here, we propose a minimum square error criterion, shown below, to obtain optimum^ß , i.e.,

ßLöÔy÷

Setting the first derivative of ^© ^Î^Æ{¬F' ^û

Æ with respect to ^ß to zero, we can obtain the optimum^ß

From this result, it is simple to see that the optimum ^ß is just the mean of all CFOs. Compen-sating for a pre-determined CFO can be implemented by windowing the receive time-domain signal with the windowing vector^à ^Mx

o PC method will need extra complex multiplications which is minor compared to other oper-ations. In simulations, we will show that the PC method can greatly enhance the performance of the proposed algorithm in some cases.

§ 5.3.3 Complexity Analysis

In the previous subsections, we have proposed the N-ZF method for an interleaved-OFDMA uplink system. In this subsection, we will analyze the computational complexity of the proposed

method, and compare it with that of existing methods. From (5.15) and (5.16), we can clearly see that the computational complexity of the proposed method mainly consists of the following three parts: 3. ^¶ ^o calculation with (5.26).

Since the operations in ^÷ ^¾ iteration can be implemented by (5.21), we only need one FFT operation with size ,^; IFFT operations with size^» ^; , and a couple of vector operations. As a result, we require ^»
,x ^s ^» ^; ^s ^: ^;_z CMs and^{^x} ^s ^» ^; ^s ^;Cb ^z CAs. In addition, we need CMs for ^÷ ^o ^¶ ^o , and RMs for each ^~ ^·^¾ ^÷ ^¾ in (5.15). As to the construction of , we first have to obtain

Û Æ

by^à ^Æ ^¦à ^Æ for each user. This

will require ^; ^! CMs, and ^; ^! CAs. Despite the diagonal property of

¶ o

, we can use the special structure of to reduce the complexity further. To see this, we can rewrite as

where ^k ^´ denotes the circular shift of a column vector^´ downwards or a row vector ^´ left-wards by^u elements. From (5.30), it is straightforward to see that

U ) As mentioned, we can only take some neighboring ICI terms into account for^¶ ^o calculation.

Let^à ^Æ ^Sx^j ^Æo ⁾ ^j ^Æ^' ⁾ ^JWJWJ ⁾ ^j ^Æy ^AB' ^zE| . Thus, we only need the following values: ^$ ^j ^Æ^y ^A ^² ⁾ ^JWJWJ ⁾ ^j ^Æ^y ^AB' ⁾ ^j ^Æo ⁾

j Æ' ) JWJWJ

) j Æ² +

for ^# ⁾ ⁾ ^JWJWJ ^); . As a result, we require^;C ^s CMs,^;õ CAs, and ^; RDs

for ^¶ ^o calculation. The required computational complexity is summarized in Table 5.1. For the proposed method with PC, only extra complex multiplications are required. Table 5.1 also shows the required computational complexity of the CLJL-PIC method and the direct ZF method [69].

Table 5.1: Complexity comparison among proposed method, CLJL-PIC method, and direct ZF method.

Complexity Proposed method CLJL-PIC method Direct ZF method Real

§ 5.3.4 Performance Analysis

For an iterative algorithm, the convergence problem is usually a main concern. The proposed algorithm uses Newton’s iterative algorithm; however, the convergence is less critical here.

This is due to two facts described below. The first one is that for the application considered here, the proposed algorithm converges for most of the cases; only in few cases, will it diverge.

The second is that the required number of iterations is pre-determined. Thanks to the fast

convergence property of Newton’s iteration and good initial values we developed, only two or three iterations are necessary if it converges. Since the iteration number is finite, the proposed algorithm will never diverge. As shown below, even for divergence cases, SINR can still be increased for the designated iteration number. We will provide two analysis approaches mainly based on the analysis in Chapter 2. The first one is simpler, but the result is only approximated;

the other one is complicated, but the result is exact. The first approach can provide an intuitive understanding of the convergence behavior of Newton’s method. Here, we start with the simpler one. We first perform the eigenvalue decomposition for ^¹ ^o as follows:

is a diagonal matrix having the^Uth eigenvalue, ^@, as its ^Uth diagonal element.

We assume that ^Ç ^@rÇ ^MÇ ^k ^Ç for^U ^º ^u . Since^¹ ^· ^¹ for Newton’s iteration to converge, the amplitudes of all eigenvalues of ^¹ ^o have to be smaller than one. As mentioned, this condition holds for most of the cases. In few cases, the condition dose not hold; however, the number of eigenvalues with amplitudes greater than one is small and their amplitudes do not deviate from one much. These results can be easily observed from simulations though difficult to be proved theoretically. In what follows, we will show that even for divergence cases, we may still benefit from Newton’s iteration.

Let ^ø ^AB' ^x ^o ⁾ ^') ^JWJWJ ⁾ ^y ^AB'zE| . By definition, ^¹ ^· ⁸ ^y ^bÉ¶· . We can represent the

CFO-compensated ICI matrix as Using (5.34), we can further express^¶ò· as

¶·

. With (5.36) and (5.38) , the CFO-compensated signal can be expressed as

where ^±· ^¶· . Since the eigenvectors ^$&Ã ^o ^)Ãt') ^JWJWJ ^)Ã ^y ^AB' ⁺ span the -dimensional space, respectively. Then, we can rewrite (5.39) as

· b ° the residual interference term and ^· ^q is the noise term. The average SINR for the^Q th iteration, denoted as ^@BACED^· , can be expressed as

@BACED ·

Assume that cross-correlations of ^Ã ^.’s are small, and can be ignored. Then, the powers of the desired signal, interference, noise can be approximated as

and Finally, the average SINR can be approximated as

@BACED · ¨ © y

. Now, we can examine the three terms in the denominator of (5.45). The first term^AÓ' , involving eigenvalues with amplitudes greater than one, is monoton-ically increased, and the second term, involving eigenvalues with amplitudes less than one, is monotonically decreased as ^Q is increased. Recall that only few eigenvalues’ amplitudes will be greater than one (i.e., is small), and their amplitudes often do not deviate from one too much. Also, from the definition of^C , it can be shown that the third term tends to be increased when^Q is increased and its variation is not large (see the definition of(

· ). Then, it is simple to see that in the first several iterations, the amount of decreasing in^A will be larger than that of increasing in^Aì' . We then conclude that for typical divergence cases, SINR will be increased and then decreased as the iteration is proceeded. Thus, if we can stop the iteration before SINR is degraded, we can still have the performance gain even though the iteration diverges eventu-ally. Due to the fast convergence property of Newton’s method, the number of iterations can be as small as two or three for convergent cases. For divergent cases, SINR is still increasing in the first two or three iterations. Note that the magnitudes of eigenvalues are related to those of CFOs. The proposed method with CFO pre-compensation can reduce the magnitudes of CFOs.

Thus, it can improve the performance of the proposed method. As an example, we let^"!9: ,

;

5: , ^G , and the normalized CFO (for each user) be randomly sampled from the interval

bB) z, where ^º ^3KJcX . After exhaustive simulations, we found that without PC, the largest

rendering the amplitudes of all eigenvalues smaller than one is^3KJ^R9 , and with PC, it can be as large as 0.5.

We now develop the second method to analyze the convergence behavior of the proposed algorithm. Recalling (5.5) and (5.14), we can rewrite the receive signal after CFO compensation as

DF' is the CFO-compensated ICI matrix. Define ^Å as a banded matrix with upper bandwidth and lower bandwidth , i.e.,^Å ^U ^){u ^ì3 whenever

Çu6b

Then we can define the average SINR for the proposed method with^Q iterations as follows:

@BACED

@CED . For comparison, the average SINR of the receive signal without

CFO-compensation is also calculated as

To obtain the ^@BACED^· in (5.48), we must calculate each element in ⁷ ^· and ^¶· . Since

Æ{¬F'

Û Æ µ Æ

, the CFO-compensated ICI matrix⁷ ^· can be expanded as

7 · ¼

expressed as

Next, we express ^?',^U '){u&'

By (5.56), we can expand^¿Y@ ^q^k as the desired signal can be formulated as

® o 7 · ® ¢ y

and ^® ^¶· ^® ^¢ can be calculated by

Thus, the average SINR of the proposed method with ^Q iterations can be explicitly calculated by (5.55), (5.59), (5.60), (5.61), and (5.62). As for (5.49), we can further express the result by

We also evaluate the SINR for each subcarrier. The SINR for the proposed algorithm with

Q iterations in the ^Uth subcarrier, denoted as ^@BACED ³^· ^q^@, is shown to be

@BACED (5.51), respectively. The^Uth subcarrier SINR of the received signal without CFO-compensation, denoted as ^@BACED ³^@, can be described as

@BACED

§ 5.3.5 Simulations

In this subsection, we report simulation results to demonstrate the effectiveness of the proposed method. We consider an interleaved-OFDMA uplink system with ^#9: , ^; ^!: , and the CP length ÇÐ0`<9

. The adopted modulation scheme is 16-QAM or 64-QAM. The length of the channel response,ⁱ , is set to 15 for all users, and the power delay profile of user is described with an exponential function, i.e.,⁹

a parameter of the function. For later simulations, we let ^$'?n'*)? ^)? ^)?

+ $

for each user. Each channel tap is generated according to Rayleigh distribution. Also, we have found that the performance of the proposed algorithm with ^! is similar to that with ^5RK . Thus, in the following experiments, we will only consider the setting of ^! .

First, we evaluate the validity of our output SINR analytic results. Two cases are consid-ered. Case 1 corresponds to the case that the amplitudes of all eigenvalues of ^¹ ^o are smaller than one, and in case 2 some eigenvalues’ amplitudes are larger than one. In case 1, CFOs

$ and SNR = 15 dB. Note that most of the CFO values in case 2 are quite large and positive.

Figure 5.1 shows the average SINRs calculated for the proposed method with the approaches in Subsection 5.3.4. From this figure, we find that the simulated output SINRs are identical to the results of the exact analysis, which verifies the correctness of the exact analysis. We also see that in both cases, the average SINR obtained by the approximated analysis is close to that by the exact analysis, especially when the iteration number gets larger. Just as mentioned in Subsection 5.3.4, even in the divergent case, the SINR increases for first two iterations. In case 1, the SINR is saturated at the second iteration.

Next, we investigate the effect of input SNR on output SINR. We assume that CFOs are set

to ^$ ^3KJL ^)Wb ^3KJc ^)Wb ^3KJ^3X ⁾ ^3KJc ⁺ , and use the result of exact analysis. Figure 5.2 shows the average

SINR for the proposed method. In this figure, we also plot the theoretical average SINR without CFO compensation using (5.63). From this figure, we can see that the average output SINR is

improved when the number of iterations increases. For the proposed method with two iterations, the theoretical average output SINR is almost the same as the input SNR. This result indicates that the proposed method almost cancels the CFO-induced ICI thoroughly. The theoretical subcarrier SINR analysis for the proposed method is also shown in Fig. 5.3. Here, the input SNR is set to 30 dB. From this figure, we can see that different subcarriers have different SINRs and the SINRs are improved when the number of iterations is increased. When the number of iterations is two, all subcarriers almost have the same output SINR which is close to the input SNR (30 dB). We have also tried other scenarios and obtained the similar result. We then conclude that when amplitudes of CFOs are moderate, a suitable choice for the iteration number is two.

Now, we present simulation results to evaluate the BER performance of the proposed method.

We consider a 16-QAM modulation scheme and set CFOs to ^$ ^3KJL ^)Wb ^3KJc ^)Wb ^3KJ^3X ⁾ ^3KJc ⁺ . Figure 5.4 shows the simulation result. From the figure, it is obvious that the conventional method and the CLJL method both have a serious error floor phenomenon. This is because when the

th user’s CFO is compensated, other users’ CFOs may be enlarged, magnifying MUI. We can also see that the CLJL method performs slightly better than the conventional method. Since the CLJL-PIC method further processes the MUI, it improves the performance of the CLJL method.

The CLJL-PIC method with a 2-stage PIC can perform similarly as that with a 3-stage PIC. The performance bound, in which no CFOs are added, is also shown in the figure. Even with a 3-stage PIC, the CLJL-PIC method still cannot approach the performance bound. In higher SNR regions, the performance loss is larger. With only two iterations, the proposed method performs as well as the direct ZF method. Also, its performance closely approaches the performance bound.

We also consider a 64-QAM modulation scheme for the same CFO setting in Fig. 5.4 (CFOs

= ^$ ^3KJL ^)Wb ^3KJc ^)Wb ^3KJ^3X ⁾ ^3KJc ⁺ ). Figure 5.5 shows the BER performance comparison. We can see

that the performance of the CLJL-PIC method degrades. This is because in high QAM-size modulation, the performance of an OFDMA system is more sensitive to the residual MUI. The

performance of the proposed method can still approach that of the direct ZF method.

We further consider a worse scenario, where the CFO of some user is large. Specifically, we set CFOs to ^$ ^3KJL ^)Wb ^3KJ^: ^)Wb ^3KJ^3X ⁾ ^3KJc ⁺ and use a 16-QAM modulation scheme. Figure 5.6 shows the BER performance comparison. In addition to the conventional method and the CLJL method, the CLJL-PIC method (even with a 3-stage PIC) performs poorly. In this case, the PIC method fails to cancel MUI. This may be due to an error propagation effect inherent in the PIC scheme. The performance of the proposed method is slightly affected. This performance loss results from the insufficient iteration number. We have shown the result for the proposed method with three iterations in the figure. It can be seen that the performance can be further enhanced at the expense of the increased computational complexity.

To clearly see the impact of the CFO magnitude, we consider a scenario that the fourth user’s CFO is increased from 0 to 0.5. If the CFO of the fourth user is increased, the MUI from the fourth user will be increased. The CFOs of other three users are set to^3KJL , ^b ^3KJc , and ^b ^3KJ^3X , respectively. The adopted modulation scheme is 16-QAM and the simulated SNRs are 25 dB and 35 dB. We simulate the average BER of the first three users. Figure 5.7 shows the simulation result. From this figure, we can see that the CLJL-PIC method (with^e ^SR ) is sensitive to the CFO variation. For SNR = 35 dB, we find that the BER for the CLJL-PIC method begins to increase when the fourth user’s CFO is 0.1. When the CFO is increased further, its performance is degraded rapidly. For the proposed method, only little performance loss is observed. If the iteration number is three, the proposed method almost does not have performance degradation compared to the direct ZF method. As the results observed above, the proposed method can have the same performance as the direct ZF method.

Since OFDMA is a multiuser system, the near-far phenomenon may occur. In such a case, some users may have stronger receive power than others. To realize the impact of the near-far effect, we report simulations with a scenario that the powers of the first three users are equal and fixed and that of the fourth user is varied. The power ratio of the fourth user to anyone of the first three users is defined as the near-far power ratio ^E ranging from ^b ^X dB to ^X

dB. Let CFOs be ^$ ^3KJL ^)Wb ^3KJc ^)Wb ^3KJ^3X ⁾ ^3KJc ⁺ and modulation be 16-QAM. Similar to the previous simulation setting, we calculate the average BER of the first three users. Figure 5.8 shows the result. From the figure, we find that both the CJLJ-PIC and proposed algorithms are affect by the near-far problem. However, the CJLJ-PIC is more sensitive in the near-far environment.

We consider an extreme case, in which most CFOs are very large and positive. Here, we set CFOs to ^$ ^3KJ^:d ⁾ ^3KJ^:d ⁾ ^3KJL ⁾ ^3KJ^: ⁺ and use a 16-QAM modulation scheme. In this case, ^ßêöÔy÷

is calculated as ^3KJ^R , which is large. We compare the performance of all methods mentioned above. For the proposed method, we also try the PC technique. Figure 5.9 shows the simulation results. In this case, the conventional method, the CLJL method, and the CLJL-PIC method all have bad performance. Note that the conventional method can even have better performance than the CLJL-PIC method. The proposed method without PC does not perform well either.

Only does the proposed method with PC perform well. Its performance is almost identical to that of the direct ZF method. The result shows the effectiveness of the proposed PC method.

We also consider another case that CFOs are not all positive. Specifically, CFOs are set to

$fb

+ . Figure 5.10 shows the BER result. We again see that the proposed method with PC has similar performance with the direct ZF method. These results show that the PC method can always be applied to improve the performance of the proposed method.

Finally, we present simulation results in an interleaved-OFDMA uplink system with the large number of subcarriers and more users. Specifically, ^M3: , ^; ^P<9 , and ^¹ÐÖP . The modulation scheme is 16-QAM. The channel length is set to 127 for all users and let

$'?t'*)? -0.3, 0, -0.1, 0.4, -0.3, 0.05, 0, -0.1, 0.05, -0.1, 0.3, 0.15}. The performances of five methods, namely, the conventional, CLJL, direct ZF, banded ZF, and proposed methods, are compared in our simulations. The banded ZF method indicates that it modifies the ICI matrix into a banded matrix with bandwidth ^F . Figure 5.11 shows the simulation results. From this figure, we find that the conventional and CLJL methods both have a serious error floor phenomenon. The performance of the proposed method with three iterations can approach that of the direct ZF

method. The complexity of the banded ZF method depends greatly on its matrix bandwidth.

For a fair comparison, we let^F be 16 for the banded ZF method. In this case, the complexity of the banded ZF method and the proposed method (^Q ^!R ) are roughly equal. Figure 5.11 shows that the proposed method performs much better than the banded ZF method (^F ^<9 ).

From the above simulations, we can see that the proposed method is more robust to the large modulation QAM-size and CFOs compared to the CLJL-PIC method. In what follows, we will compare the computational complexity of the direct ZF method, the CLJL-PIC method, and the proposed method in an OFDMA system shown above (^M9: , ^; ^S: ). Substituting the required parameters into Table 5.1, we then derive the computational complexity of each method, and show the result in Table 5.2. The number of iterations for the proposed method

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