4.2 Proposed New SLM and PTS Schemes
4.3.1 Computational Complexity Analysis
In our proposed scheme, phase sequences are multiplied by the intermediate se-quences at the output of stages vD = [1, v2, · · · , vβ] with the corresponding numbers of
sequences given by PD = [P1, P2, · · · , Pβ], respectively. Hence, the total number of IFFT stages required to generate Pβ candidate sequences excluding the first stage is
λ = (v2− 1) · P1+ (v3− v2) · P2+ · · · + (m − vβ) · Pβ, (4.2)
where m is the total number of stages to generate a specific candidate sequence:
m = 1 + log2 NL
r . (4.3)
Since the computational complexity is fixed in each stage (NL/2 multiplications and NL additions), we have the complexity analysis of our scheme, stated in Table 4.1 – ??.
The G&G SLM scheme, which make use of the multistage structure of radix-2 IFFT, applies phase sequences differently from ours. Their operation of phase rotation is con-sidered at the output of stages vR = [v1, v2, · · · , vβ] and done to PR = [P1, P2, · · · , Pβ] intermediate sequences, where PR is defined as the number of intermediate sequences generated at stages vR. Thus, the total number of stages required (including the first few stages) is given as
λR = v1+ (v2− v1) · P1+ · · · + (mR− vβ) · Pβ, (4.4)
where mR = log2NL. For fair comparison, radix-2 is considered for the last m−1 stages of our proposed scheme, where it is done to all IFFT stages in the G&G SLM scheme.
Note that though the value of m and mR are different, the computational complexity of our proposal with r = 4 at the first stage and the that of the G&G SLM scheme is identical if same numbers of phase sequences are applied to the corresponding stages.
Table 4.1 – 4.2 present the computational complexity of our proposed schemes, the G&G SLM scheme, and the conventional SLM schemes. In addition, the computational complexity ratio of the proposed schemes over the conventional schemes are shown in Table 4.3 for N = 256. Obviously, the complexity is reduced.
Table 4.1: Computational complexity of various SLM schemes (I) Number of complex multiplications Conventional SLM scheme U¡N L
2 · log2NL¢ Proposed SLM scheme NL +N L2 · λ
G&G SLM scheme N L2 · λR
Table 4.2: Computational complexity of various SLM schemes (II) Number of complex additions Conventional SLM scheme U(NL · log2NL)
Proposed SLM scheme NL(1 + λ) G&G SLM scheme NL · λR
4.3.2 Simulation Results
The comparison of PAPR reduction performance of various schemes is presented in this section. The evaluation considers the CCDF of the PAPR of 16 QAM-modulated OFDM symbols with U = 8 and N = 256.
The PAPR reduction performance of various SLM schemes is depicted in Fig. 4.4, where vD and vRdenote the stages phase sequences are multiplied for our scheme and the G&G scheme, respectively. As can be seen, the earlier the phase sequences are applied to, the better the PAPR reduction performance. This observation coincides with the intuition that the multiplication of phase sequences of the earlier stages introduces more diversity to the candidates. However, the cost of improving PAPR reduction capability is the higher computational complexity; as shown in Table 4.3. This gives us a direct trade-off between the PAPR performance and computational complexity which is beneficial for system design. Notice that when vD = [1, 4] is applied in our proposed scheme, it can achieve almost the same performance as the conventional SLM scheme with a complexity ratio of 60% only. This shows the potential improvement of complexity reduction with negligible performance degradation by joint design. From Table 4.1 and 4.2, the computational complexity of vD = [1, 6] in our proposed scheme and vR= [2, 7]
Table 4.3: Computational complexity ratio of the proposed SLM schemes with N = 256.
vD PD Rmul (%) Radd (%) [1, 4] [2, 8] 60.00 58.75 [1, 6] [2, 8] 45.00 43.75 [1, 8] [2, 8] 30.00 28.75
in the G&G SLM scheme are identical while shown in Fig. 4.4 the PAPR reduction performance of the latter is poorer than the former with 0.12 dB for U = 8 at the CCDF of 10−4, which shows the advantage of our scheme over G&G SLM scheme. The case with vD = [1, 8] has the worst PAPR reduction performance among all three cases since the second part of the phase sequences are multiplied to a later stage to obtain the lowest complexity.
As mentioned in previous subsection, to guarantee the BER performance, our pro-posed schemes have to follow some rules to ensure the factors of equivalent phase se-quences all locate on the unit circle while the G&G SLM scheme does not take into account. Figure 4.5 and 4.6 show the magnitudes of equivalent phase sequences of our scheme and the G&G SLM scheme, respectively. These results conform to what we discussed before. To investigate the effect of destroying the unimodularity, the BER performance of our SLM scheme and the G&G SLM scheme is also given in Fig. 4.7 where a 256-subcarrier QPSK-modulated OFDM system is considered. As our expecta-tion, the BER performance of the G&G SLM scheme is much poorer than our proposal.
4 5 6 7 8 9 10 11 12 13 1E-4
1E-3 0.01 0.1 1
CCDF
PAPR (dB)
Conventional SLM scheme Proposed SLM: v D = [1,4]
Proposed SLM: v D = [1,6]
Proposed SLM: v D = [1,8]
Original OFDM
G&G SLM scheme: v R - [2,7]
Figure 4.4: Comparison of the PAPR reduction performance of the proposed SLM Scheme, G&G SLM scheme and conventional SLM scheme.
50 100 150 200 250 0
0.2 0.4 0.6 0.8 1 1.2 1.4
subcarrier
magnitude
Figure 4.5: Magnitude of an equivalent phase sequence of our scheme.
50 100 150 200 250 0
0.2 0.4 0.6 0.8 1 1.2 1.4
subcarrier
magnitude
Figure 4.6: Magnitude of an equivalent phase sequence of the G&G SLM scheme.
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 1E-7
1E-6 1E-5 1E-4 1E-3 0.01 0.1 1
BER
SNR (dB)
Proposed multi-stage SLM scheme G&G SLM scheme
Theoretical
Figure 4.7: Comparison of the BER performance of the proposed SLM Scheme and the G&G SLM scheme.
Chapter 5 Conclusion
Several novel low complexity SLM-based and PTS-based schemes for PAPR reduction are proposed. These schemes make use of some special properties of the IFFT structures which are known as DIT IFFT or DIF IFFT. We also suggest a stop criterion to further reduce the computational complexity for the DIT-IFFT implementation.
The new schemes achieve the same PAPR reduction performance as the conventional schemes with much reduced computational complexity. For instance, when the number of subcarriers (N) is 256 and the number of mapping sequences (U) is 8, the multipli-cation complexity ratios (normalized with respect to that required by the conventional SLM/PTS schemes) are 82.5% and 60% for the SLM and PTS (which uses four sub-blocks) schemes, respectively. If the stop criterion is applied, the reduction ratio can be further improved. Besides variants of the conventional SLM and PTS schemes, we can apply the proposed stop criterion to reduce the complexity of the LWW schemes [12] as well.
We extend our study to the multistage DIT IFFT structure. A similar approach, which we referred to as the G&G SLM scheme [16], was presented before. A shortcoming of this scheme is that the equivalent mapping sequence does not have constant modulus entries and, as a result, the receiver’s de-mapping process brings about noise enhance-ment and BER performance degradation. Our new design overcomes this disadvantage
by generating multiple constant modulus candidate frequency domain sequences.
The concept of taking advantage of the special IFFT structures in designing new PAPR reduction schemes can be extended to modify other existing PAPR reduction schemes and develop new low-complexity schemes. One such example is the bit-flipping based PAPR reduction schemes. Moreover, as there are several recent proposals for simplifying the implementation of IDFTs, it is worthwhile to consider these alternative structures, e.g., split-radix FFT algorithm, and investigate the feasibility of further com-putational complexity reduction when used in conjunction with some PAPR reduction schemes.
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