3.3 Proposed Algorithm with Conversion Vectors
3.3.2 Modified LWW Scheme with Proposed Stop Criterion
To reduce the computational complexity of our proposal further, LWW Scheme is considered with some modifications. First, the NL-point IDFT is replaced by our pro-posed algorithm which has been introduced in previous section. The second modification is about the length of base vectors which is length-NL in LWW Scheme. To adapt the conversion vectors to our schemes, the length of base vectors have to be changed to NL/r.
Figure 3.9 shows the architecture of convolution structure in which Ha1, Ha2, Hb1 and Hb2 denote the base vectors after length modification (i.e. length-(NL/r)). Therefore, the complete structure of our proposed scheme with conversion vectors is presented in Fig 3.10. Similarly, the computation of r × (NL/r)-point IDFTs are computed once for
Figure 3.8: Architecture of L&W Scheme I.
a given OFDM symbol, and a number of candidate sequences are generated by applying circular convolution of base vectors with the out samples of r × (NL/r)-point IDFTs.
Furthermore, since we can derive the length-(NL/r) output sequence which is a frac-tion of certain length-NL candidate sequence sequentially, the proposed stop criterion can also be considered in this modified scheme to lower the computational complexity.
3.3.3 Analysis of Computational Complexity
For all r × (NL/r)-point IDFTs, each of them only needs to be performed once for a given OFDM symbol when length-modified LWW Scheme is combined to our proposed algorithm. Therefore, our proposed schemes with length-modified conversion vectors result in lower computational complexity compares with conventional SLM scheme, con-ventional PTS scheme, and our proposed schemes in preceding section.
There are three LWW Schemes proposed in [12]. The scheme introduced in previous subsection is LWW Scheme I. LW Scheme II is constructed by combining LWW Scheme I and the conventional SLM scheme to enhance the PAPR reduction performance. Two
Figure 3.9: Architecture of modified convolution structure for generating candidate se-quences.
parallel IDFTs are required and a random phase sequence is adopted before the second IDFT operation. This random phase sequence can increase the diversity of candidate sequences, resulting in a better performance of PAPR reduction, but higher computa-tional complexity due to one more IDFT operation. LWW Scheme III has the same structure as LWW Scheme I while the third class conversion vectors proposed in [12] is used.
It can be noticed that for LWW Scheme I and LWW Scheme III, the corresponding schemes based on our proposed algorithm with length-modified conversion vectors can achieve the same number of complex multiplications. The corresponding Scheme II in our proposed algorithm requires less complexity than LWW Scheme II since the computation of the first stage only needs to compute once and reuses for two sets of r × (NL/r)-point IDFTs. Besides, our proposed corresponding schemes can achieve less number of
Figure 3.10: Architecture of our proposed scheme with conversion vectors.
complex additions due to the use of L-oversampling.
Table 3.6 and 3.7 present the computational complexity of various SLM schemes, including the conventional SLM scheme, LWW Schemes, and corresponding Schemes based on our proposed algorithm. However, only SLM scheme can be implemented in LWW Schemes so that Table 3.8 and 3.9, which are the comparison of various PTS schemes, just compare the conventional PTS scheme and corresponding scheme based on our proposed algorithm. It is worth noting that the worst case of our schemes is considered here while the computational complexity can be reduced further if the proposed stop criterion is considered.
In addition, computational complexity ratio for the conventional SLM and PTS schemes and our corresponding schemes is given in table 3.10 and 3.11.
Table 3.6: Computational complexity of various SLM schemes (I) Number of complex multiplications Conventional SLM Scheme U¡N L
2 · log2NL¢
LWW Scheme I N L2 · log2NL
LWW Scheme II NL · log2NL
LWW Scheme III N L2 · log2NL
Proposed modified Scheme I for SLM N L2 · log2NL Proposed modified Scheme II for SLM NL(1 + log2N) Proposed modified Scheme III for SLM N L2 · log2NL
Table 3.7: Computational complexity of various SLM schemes (II) Number of complex additions Conventional SLM Scheme U(NL · log2NL)
LWW Scheme I NL(log2NL + U + 7)
LWW Scheme II NL(2 · log2NL + U + 14)
LWW Scheme III NL(log2NL + 3U)
Proposed modified Scheme I for SLM NL(log2N + U + 8) Proposed modified Scheme II for SLM NL(2 · log2N + U + 15) Proposed modified Scheme III for SLM NL(log2N + 3U + 1)
3.3.4 Simulation Results
In this section, we investigate the PAPR performance of various PAPR reduction schemes when 16-QAM is employed with N = 256. Besides, L = 4 which is the factor of oversampling is considered.
Fig. 3.11 compares the PAPR reduction performance of our Proposed Modified
Table 3.8: Computational complexity of various PTS schemes (I) Number of complex multiplications Conventional PTS Scheme (N L)2r 2 · log2NL
Proposed modified Scheme I for PTS N L2 · log2NL Proposed modified Scheme II for PTS N L2 log2r + NL · log2 N Lr Proposed modified Scheme III for PTS N L2 · log2NL
Table 3.9: Computational complexity of various PTS schemes (II) Number of complex additions Conventional PTS Scheme (N L)r 2 · log2NL
Proposed modified Scheme I for PTS NL(log2NL + U + 6) Proposed modified Scheme II for PTS NL¡
log2r + 2 · log2 N Lr + U + 13¢ Proposed modified Scheme III for PTS NL(log2NL + 3U − 1)
Table 3.10: Computational complexity ratio for Proposed SLM Schemes over the con-ventional SLM scheme with N = 256.
Proposed Scheme I Proposed Scheme II Proposed Scheme III Rmul (%) Radd (%) Rmul (%) Radd (%) Rmul (%) Radd (%)
U = 8 12.50 30.00 22.50 48.75 12.50 41.25
U = 32 3.13 15.00 5.63 19.69 3.13 32.81
Schemes I and II, LWW Scheme I and II, and conventional SLM schemes. It is seen that for a given number of candidate sequences (U), the PAPR reduction performance of both Proposed Modified Scheme I and II is similar to that of LWW Scheme I and II. From a detailed inspection, the performance degradation of Proposed Modified Scheme I and II relative to that of conventional SLM scheme are 0.4 and 0.16 dB, respectively for U = 32 and CCDF of of 10−4. As expected, PAPR performance of Proposed Modified Scheme II is better that that of Proposed Modified scheme I for an extra IDFT operation.
Fig. 3.12 shows the PAPR reduction performance of our Proposed Modified Scheme III, LWW Scheme III and conventional SLM scheme. Note that our Proposed Modified Scheme III can achieve similar performance as LWW Scheme III, and the maximum performance loss of Proposed Modified Scheme III relative to the conventional SLM scheme is 0.12 dB for U = 32 and CCDF of 10−4.
As mentioned in previous subsection, LWW Schemes are just applied to SLM scheme, so comparison of PTS schemes just includes our Proposed Modified Schemes and the conventional PTS scheme. In Fig. 3.13, PAPR reduction performance of Proposed Modified Schemes I and II, and conventional PTS scheme is presented with U = 8. It
Table 3.11: Computational complexity ratio for Proposed PTS Schemes over the con-ventional PTS scheme with N = 256 and U = 8.
Proposed Scheme I Proposed Scheme II Proposed Scheme III Rmul (%) Radd (%) Rmul (%) Radd (%) Rmul (%) Radd (%)
r = 32(M = 32) 3.13 7.50 4.69 11.25 3.13 10.31
r = 128(M = 8) 12.50 30.00 16.25 42.50 12.50 41.25 r = 256(M = 4) 25.00 60.00 30.00 82.50 25.00 82.50
can be noted that for a given number of subblocks (M), PAPR performances of Proposed Modified Scheme I and II are poorer than that of conventional PTS scheme. However, lower computational complexity of Proposed Modified schemes is provided.
Figure 3.14 illustrates the PAPR performance of Proposed Modified Scheme III and conventional PTS scheme with U = 8. For M = 4, two schemes provide almost the same performance. Although there are slight degradation of PAPR reduction performance for M = 8 and M = 32, less computational complexity is achieved for Proposed Modified Scheme III as shown in preceding section. It is shown that the performance loss of Proposed Modified Scheme III relative to the conventional PTS scheme is 0.12 dB for M = 32 and CCDF of 10−4.