IEEE_cliping & Filtering.rar_PAPR_signal processing

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Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">http://www.ieee.org/publications_standards/publications/rights/index.html for more information.</div><div class="t m0 x3 h2 y3 ff1 fs0 fc0 sc0 ls0 ws0">This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2017.2751620, IEEE Access</div><div class="t m0 x4 h3 y4 ff2 fs1 fc0 sc0 ls0 ws0">A new Approach to Iterative Clipping and Filtering</div><div class="t m0 x5 h3 y5 ff2 fs1 fc0 sc0 ls0 ws0">PAPR Reduction Scheme for OFDM Systems</div><div class="t m0 x6 h4 y6 ff2 fs2 fc0 sc0 ls0 ws0">Kelvin Anoh</div><div class="t m0 x7 h5 y7 ff3 fs3 fc0 sc0 ls0 ws0">∗</div><div class="t m0 x8 h4 y6 ff2 fs2 fc0 sc0 ls0 ws0">, Cagri Tanriover</div><div class="t m0 x9 h5 y7 ff3 fs3 fc0 sc0 ls0 ws0">†</div><div class="t m0 xa h4 y6 ff2 fs2 fc0 sc0 ls0 ws0">, Bamidele Adebisi</div><div class="t m0 xb h5 y7 ff3 fs3 fc0 sc0 ls0 ws0">∗</div><div class="t m0 xc h4 y6 ff2 fs2 fc0 sc0 ls0 ws0">and Mohammad Hammoudeh</div><div class="t m0 xd h5 y7 ff3 fs3 fc0 sc0 ls0 ws0">∗</div><div class="t m0 xe h5 y8 ff3 fs3 fc0 sc0 ls0 ws0">∗</div><div class="t m0 xf h4 y9 ff2 fs2 fc0 sc0 ls0 ws0">School of Engineering, Manchester Metropolitan University, UK</div><div class="t m0 x10 h5 ya ff3 fs3 fc0 sc0 ls0 ws0">†</div><div class="t m0 x11 h4 yb ff2 fs2 fc0 sc0 ls0 ws0">Intel Labs, USA</div><div class="t m0 x12 h4 yc ff2 fs2 fc0 sc0 ls0 ws0">[k.anoh, b.adebisi, m.hammoudeh]@mmu.ac.uk, cagri.tanriover@intel.com</div><div class="t m0 x13 h6 yd ff4 fs4 fc0 sc0 ls0 ws0">Abstract—While achieving reduced/good peak-to-average</div><div class="t m0 x14 h7 ye ff5 fs4 fc0 sc0 ls0 ws0">power (PAPR) in orthogonal frequency division multiplexing</div><div class="t m0 x14 h7 yf ff5 fs4 fc0 sc0 ls0 ws0">(OFDM) systems is attractive, this must not be performed at</div><div class="t m0 x14 h7 y10 ff5 fs4 fc0 sc0 ls0 ws0">the expense of the transmitted signal with over-reduced signal</div><div class="t m0 x14 h7 y11 ff5 fs4 fc0 sc0 ls0 ws0">power as it leads to degraded bit error ratio (BER). We introduce</div><div class="t m0 x14 h7 y12 ff5 fs4 fc0 sc0 ls0 ws0">a uniform distribution approach to solving the PAPR reduction</div><div class="t m0 x14 h7 y13 ff5 fs4 fc0 sc0 ls0 ws0">problem of OFDM signals and then use Lagrange multiplier</div><div class="t m0 x14 h7 y14 ff5 fs4 fc0 sc0 ls0 ws0">(LM) optimization to minimize the number of iterations involved</div><div class="t m0 x14 h7 y15 ff5 fs4 fc0 sc0 ls0 ws0">in an adaptive fashion. Due to the nonlinear attenuation of the</div><div class="t m0 x14 h7 y16 ff5 fs4 fc0 sc0 ls0 ws0">PAPR reduction scheme, we compensate the output signal using</div><div class="t m0 x14 h7 y17 ff5 fs4 fc0 sc0 ls0 ws0">a correlation factor that minimizes the error ﬂoor in the in-band</div><div class="t m0 x14 h7 y18 ff5 fs4 fc0 sc0 ls0 ws0">distortion of the clipped signal using minimum mean square error</div><div class="t m0 x14 h7 y19 ff5 fs4 fc0 sc0 ls0 ws0">(MMSE) method so as to improve the BER performance. Three</div><div class="t m0 x14 h7 y1a ff5 fs4 fc0 sc0 ls0 ws0">different methods are introduced each enabling PAPR reduction</div><div class="t m0 x14 h7 y1b ff5 fs4 fc0 sc0 ls0 ws0">by clipping followed by ﬁltering with no direct dependency on a</div><div class="t m0 x14 h7 y1c ff5 fs4 fc0 sc0 ls0 ws0">clipping ratio parameter. We ﬁnd that our approach signiﬁcantly</div><div class="t m0 x14 h7 y1d ff5 fs4 fc0 sc0 ls0 ws0">reduces the PAPR of the OFDM signals (especially with LM</div><div class="t m0 x14 h7 y1e ff5 fs4 fc0 sc0 ls0 ws0">optimization) better than the conventional adaptive iterative</div><div class="t m0 x14 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">clipping and ﬁltering operating without LM optimization. Based</div><div class="t m0 x14 h7 y20 ff5 fs4 fc0 sc0 ls0 ws0">on our proposed methods, we additionally outline two simple</div><div class="t m0 x14 h7 y21 ff5 fs4 fc0 sc0 ls0 ws0">steps for achieving perfect PAPR reduction (i.e. 0dB). We also</div><div class="t m0 x14 h7 y22 ff5 fs4 fc0 sc0 ls0 ws0">evaluate the performance of the three new models over high</div><div class="t m0 x14 h7 y23 ff5 fs4 fc0 sc0 ls0 ws0">power ampliﬁer (HPA) for completeness; the HPA is found to</div><div class="t m0 x14 h7 y24 ff5 fs4 fc0 sc0 ls0 ws0">induce negligible BER degradation effects on the processed signal</div><div class="t m0 x14 h7 y25 ff5 fs4 fc0 sc0 ls0 ws0">compared to the unprocessed signal.</div><div class="t m0 x13 h6 y26 ff4 fs4 fc0 sc0 ls0 ws0">Index Terms—OFDM, iterative clipping and ﬁltering (ICF),</div><div class="t m0 x14 h7 y27 ff5 fs4 fc0 sc0 ls0 ws0">adaptive ICF, PAPR, optimization, Lagrange Multiplier, uniform</div><div class="t m0 x14 h7 y28 ff5 fs4 fc0 sc0 ls0 ws0">distribution, high power ampliﬁer (HPA)</div><div class="t m0 x15 h8 y29 ff2 fs5 fc0 sc0 ls0 ws0">I. INTRODUCTION</div><div class="t m0 x13 h8 y2a ff2 fs5 fc0 sc0 ls0 ws0">Orthogonal frequency division multiplexing (OFDM) is an</div><div class="t m0 x14 h8 y2b ff2 fs5 fc0 sc0 ls0 ws0">efﬁcient multicarrier communication style over fading chan-</div><div class="t m0 x14 h8 y2c ff2 fs5 fc0 sc0 ls0 ws0">nels since the narrow-bandwidths subtended in the frequency</div><div class="t m0 x14 h8 y2d ff2 fs5 fc0 sc0 ls0 ws0">domain allow long symbol period. This property provides</div><div class="t m0 x14 h8 y2e ff2 fs5 fc0 sc0 ls0 ws0">protection against channel impulse response effects and makes</div><div class="t m0 x14 h8 y2f ff2 fs5 fc0 sc0 ls0 ws0">OFDM attractive in the design of modern communication</div><div class="t m0 x14 h8 y30 ff2 fs5 fc0 sc0 ls0 ws0">systems for efﬁcient management of scarce radio frequency</div><div class="t m0 x14 h8 y31 ff2 fs5 fc0 sc0 ls0 ws0">bandwidths. Further spectral efﬁciency can be achieved by</div><div class="t m0 x14 h8 y32 ff2 fs5 fc0 sc0 ls0 ws0">using wavelets which can operate OFDM without cyclic</div><div class="t m0 x14 h8 y33 ff2 fs5 fc0 sc0 ls0 ws0">preﬁx [1], [2]. Unfortunately, high peak-to-average power ratio</div><div class="t m0 x14 h8 y34 ff2 fs5 fc0 sc0 ls0 ws0">(PAPR) problem limits its wide adoption in some communica-</div><div class="t m0 x14 h8 y35 ff2 fs5 fc0 sc0 ls0 ws0">tion devices. For example, while OFDM is applied in downlink</div><div class="t m0 x14 h8 y36 ff2 fs5 fc0 sc0 ls0 ws0">transmissions of mobile communication standards, it is not</div><div class="t m0 x14 h8 y37 ff2 fs5 fc0 sc0 ls0 ws0">preferred in uplink transmissions due to PAPR limitation [3].</div><div class="t m0 x14 h8 y38 ff2 fs5 fc0 sc0 ls0 ws0">High PAPR leads power ampliﬁers to operate in the saturation</div><div class="t m0 x14 h8 y39 ff2 fs5 fc0 sc0 ls0 ws0">region expending large system power and induces bit error</div><div class="t m0 x14 h8 y3a ff2 fs5 fc0 sc0 ls0 ws0">ratio (BER) degradation due to smearing of signals.</div><div class="t m0 x13 h8 y3b ff2 fs5 fc0 sc0 ls0 ws0">In the literature, different PAPR reduction techniques exist</div><div class="t m0 x14 h8 y3c ff2 fs5 fc0 sc0 ls0 ws0">and may be applied before or after OFDM modulation [4].</div><div class="t m0 x16 h8 y3d ff2 fs5 fc0 sc0 ls0 ws0">Two families of post-modulation PAPR reduction schemes in-</div><div class="t m0 x16 h8 y3e ff2 fs5 fc0 sc0 ls0 ws0">clude companding and clipping [4]–[7]. Companding destroys</div><div class="t m0 x16 h8 y3f ff2 fs5 fc0 sc0 ls0 ws0">the orthogonality of OFDM subcarriers and may cause the</div><div class="t m0 x16 h8 y40 ff2 fs5 fc0 sc0 ls0 ws0">signals to be unrecoverable at the receiver. Clipping does not</div><div class="t m0 x16 h8 y41 ff2 fs5 fc0 sc0 ls0 ws0">require receiver-side processing, thus reduces the receiver-side</div><div class="t m0 x16 h8 y42 ff2 fs5 fc0 sc0 ls0 ws0">complexity unlike companding. In this study, we focus on</div><div class="t m0 x16 h8 y43 ff2 fs5 fc0 sc0 ls0 ws0">iterative clipping and ﬁltering (ICF) technique although the</div><div class="t m0 x16 h8 y44 ff2 fs5 fc0 sc0 ls0 ws0">conventional ICF PAPR reduction technique has been around</div><div class="t m0 x16 h8 y45 ff2 fs5 fc0 sc0 ls0 ws0">for some time [6]. ICF is attractive because it is simple to</div><div class="t m0 x16 h8 y46 ff2 fs5 fc0 sc0 ls0 ws0">implement, achieves better power ampliﬁer efﬁciency at the</div><div class="t m0 x16 h8 y19 ff2 fs5 fc0 sc0 ls0 ws0">cost of increasing in-band distortion while restricting out-</div><div class="t m0 x16 h8 y47 ff2 fs5 fc0 sc0 ls0 ws0">of-band power radiation due to power ampliﬁcation [8] and</div><div class="t m0 x16 h8 y48 ff2 fs5 fc0 sc0 ls0 ws0">can be designed to achieve good BER performances [9]–</div><div class="t m0 x16 h8 y49 ff2 fs5 fc0 sc0 ls0 ws0">[11]. After the introduction of clipping and ﬁltering by [6],</div><div class="t m0 x16 h8 y4a ff2 fs5 fc0 sc0 ls0 ws0">many studies [3], [4], [8]–[10], [12], [13] have been done to</div><div class="t m0 x16 h8 y1f ff2 fs5 fc0 sc0 ls0 ws0">perfect the technique as an ideal post-multicarrier modulation</div><div class="t m0 x16 h8 y4b ff2 fs5 fc0 sc0 ls0 ws0">PAPR reduction method for OFDM systems. However, all</div><div class="t m0 x16 h8 y4c ff2 fs5 fc0 sc0 ls0 ws0">these schemes are based on specifying a clipping ratio.</div><div class="t m0 x17 h8 y4d ff2 fs5 fc0 sc0 ls0 ws0">In this study, a different approach that does not require</div><div class="t m0 x16 h8 y4e ff2 fs5 fc0 sc0 ls0 ws0">a predeﬁned clipping ratio is taken. This approach is based</div><div class="t m0 x16 h8 y25 ff2 fs5 fc0 sc0 ls0 ws0">on the fact that transforming the amplitude distribution to a</div><div class="t m0 x16 h8 y4f ff2 fs5 fc0 sc0 ls0 ws0">uniform distribution can lead to perfect PAPR reduction; thus</div><div class="t m0 x16 h8 y50 ff2 fs5 fc0 sc0 ls0 ws0">we establish the steps for achieving such distribution through</div><div class="t m0 x16 h8 y51 ff2 fs5 fc0 sc0 ls0 ws0">clipping. At this stage, let us ﬁrst recapitulate that conven-</div><div class="t m0 x16 h8 y52 ff2 fs5 fc0 sc0 ls0 ws0">tional OFDM signal has characteristic amplitude distribution</div><div class="t m0 x16 h8 y53 ff2 fs5 fc0 sc0 ls0 ws0">that follows Rayleigh distribution. The amplitudes distributed</div><div class="t m0 x16 h8 y54 ff2 fs5 fc0 sc0 ls0 ws0">above the mean are fundamentally responsible for high PAPR</div><div class="t m0 x16 h8 y55 ff2 fs5 fc0 sc0 ls0 ws0">problem and drive the high power ampliﬁer (HPA) towards</div><div class="t m0 x16 h8 y56 ff2 fs5 fc0 sc0 ls0 ws0">saturation region where it consumes large amount of power</div><div class="t m0 x16 h8 y57 ff2 fs5 fc0 sc0 ls0 ws0">and smears the signals thereby degrading the BER. We explore</div><div class="t m0 x16 h8 y58 ff2 fs5 fc0 sc0 ls0 ws0">the realization of a uniform distribution by ICF without using</div><div class="t m0 x16 h8 y59 ff2 fs5 fc0 sc0 ls0 ws0">the conventional predetermined clipping ratio.</div><div class="t m0 x17 h8 y5a ff2 fs5 fc0 sc0 ls0 ws0">Companding is the foremost PAPR reduction scheme that</div><div class="t m0 x16 h8 y5b ff2 fs5 fc0 sc0 ls0 ws0">explicitly imposes uniform distribution probability density</div><div class="t m0 x16 h8 y5c ff2 fs5 fc0 sc0 ls0 ws0">function (PDF) constraint unto the Rayleigh PDF of the con-</div><div class="t m0 x16 h8 y5d ff2 fs5 fc0 sc0 ls0 ws0">ventional OFDM signal amplitudes (e.g. [7], [14], [15]). How-</div><div class="t m0 x16 h8 y5e ff2 fs5 fc0 sc0 ls0 ws0">ever, companding destroys the orthogonality of the subcarriers,</div><div class="t m0 x16 h8 y5f ff2 fs5 fc0 sc0 ls0 ws0">unfairly expands the low amplitude signals or compresses the</div><div class="t m0 x16 h8 y60 ff2 fs5 fc0 sc0 ls0 ws0">larger amplitude signals - due to amplitude distortion, these</div><div class="t m0 x16 h8 y61 ff2 fs5 fc0 sc0 ls0 ws0">increase noise overhead and lead to poor BER performance.</div><div class="t m0 x17 h8 y62 ff2 fs5 fc0 sc0 ls0 ws0">We propose the possibility of addressing PAPR problem</div><div class="t m0 x16 h8 y63 ff2 fs5 fc0 sc0 ls0 ws0">by clipping without setting thresholds. For example, if the</div><div class="t m0 x16 h8 y64 ff2 fs5 fc0 sc0 ls0 ws0">signal amplitude peaks can be made to approach a uniform</div><div class="t m0 x16 h8 y3b ff2 fs5 fc0 sc0 ls0 ws0">distribution, then the PAPR problem can also be eliminated.</div><div class="t m0 x16 h8 y3c ff2 fs5 fc0 sc0 ls0 ws0">Unlike converting the PDFs by using different companding</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6260eefd090cbf2c4ef55cf2/bg2.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">2169-3536 (c) 2017 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">http://www.ieee.org/publications_standards/publications/rights/index.html for more information.</div><div class="t m0 x3 h2 y3 ff1 fs0 fc0 sc0 ls0 ws0">This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2017.2751620, IEEE Access</div><div class="t m0 x18 h9 y65 ff2 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m0 x14 ha y66 ff2 fs3 fc0 sc0 ls0 ws0">Figure 1. OFDM system model showing ICF technique for PAPR reduction; ICF block envelopes the adaptive ICF solution implemented with Lagrange</div><div class="t m0 x14 ha y67 ff2 fs3 fc0 sc0 ls0 ws0">multiplier optimization to reduce clipping and ﬁltering iterations</div><div class="t m0 x14 h8 y68 ff2 fs5 fc0 sc0 ls0 ws0">transforms, we restrict our design to the mean amplitude</div><div class="t m0 x14 h8 y69 ff2 fs5 fc0 sc0 ls0 ws0">distribution. For example, we estimate the mean amplitude of</div><div class="t m0 x14 h8 y6a ff2 fs5 fc0 sc0 ls0 ws0">OFDM signals, then clip all other amplitudes higher than the</div><div class="t m0 x14 h8 y6b ff2 fs5 fc0 sc0 ls0 ws0">mean amplitude; this we called Method 1. We found that this</div><div class="t m0 x14 h8 y6c ff2 fs5 fc0 sc0 ls0 ws0">greatly reduced the PAPR of the system by 10dB and 11dB in</div><div class="t m0 x14 h8 y6d ff2 fs5 fc0 sc0 ls0 ws0">1 and 3 iterations, respectively. As there are many subcarriers</div><div class="t m0 x14 h8 y6e ff2 fs5 fc0 sc0 ls0 ws0">exhibiting the characteristic amplitudes higher than the mean</div><div class="t m0 x14 h8 y6f ff2 fs5 fc0 sc0 ls0 ws0">amplitude, this led to high in-band distortion which adversely</div><div class="t m0 x14 h8 y70 ff2 fs5 fc0 sc0 ls0 ws0">impacted the BER performance.</div><div class="t m0 x13 h8 y71 ff2 fs5 fc0 sc0 ls0 ws0">Since the BER is greatly degraded, we further introduced</div><div class="t m0 x14 h8 y72 ff2 fs5 fc0 sc0 ls0 ws0">another method that can improve the BER at the expense</div><div class="t m0 x14 h8 y73 ff2 fs5 fc0 sc0 ls0 ws0">of the PAPR. This was achieved by scaling up the mean</div><div class="t m0 x14 h8 y74 ff2 fs5 fc0 sc0 ls0 ws0">amplitude so that the number of clipped signals is reduced</div><div class="t m0 x14 h8 y75 ff2 fs5 fc0 sc0 ls0 ws0">thus reducing the in-band distortion; this we called Method 2.</div><div class="t m0 x14 h8 y76 ff2 fs5 fc0 sc0 ls0 ws0">We compared both the PAPR performances of Method 2 with</div><div class="t m0 x14 h8 y77 ff2 fs5 fc0 sc0 ls0 ws0">the original signal and observed that Method 2 reduced the</div><div class="t m0 x14 h8 y78 ff2 fs5 fc0 sc0 ls0 ws0">PAPR of unclipped OFDM signal by 9.5dB. In terms of the</div><div class="t m0 x14 h8 y79 ff2 fs5 fc0 sc0 ls0 ws0">BER, Method 2 achieved 3.4dB gain over Method 1. Method</div><div class="t m0 x14 h8 y7a ff2 fs5 fc0 sc0 ls0 ws0">2 is attractive since convergence after 1 iteration achieves</div><div class="t m0 x14 h8 y7b ff2 fs5 fc0 sc0 ls0 ws0">good (PAPR and BER) level, which offers a good trade-off</div><div class="t m0 x14 h8 y7c ff2 fs5 fc0 sc0 ls0 ws0">in terms of processing time and power consumption required</div><div class="t m0 x14 hb y7d ff2 fs5 fc0 sc0 ls0 ws0">for running 2M + 1 IFFT/FFT operations (where ’M’ is the</div><div class="t m0 x14 h8 y7e ff2 fs5 fc0 sc0 ls0 ws0">number of iterations).</div><div class="t m0 x13 h8 y7f ff2 fs5 fc0 sc0 ls0 ws0">We emphasize that the two approaches do not require any</div><div class="t m0 x14 h8 y80 ff2 fs5 fc0 sc0 ls0 ws0">predetermination of clipping ratio as it is the custom of the</div><div class="t m0 x14 h8 y81 ff2 fs5 fc0 sc0 ls0 ws0">conventional ICF. Based on these two methods, it follows that</div><div class="t m0 x14 h8 y82 ff2 fs5 fc0 sc0 ls0 ws0">the PAPR problem can be completely eliminated by two simple</div><div class="t m0 x14 h8 y83 ff2 fs5 fc0 sc0 ls0 ws0">steps, which leads us to a third method, namely Method 3; 1)</div><div class="t m0 x14 h8 y84 ff2 fs5 fc0 sc0 ls0 ws0">determine the signal amplitudes below the mean and scale</div><div class="t m0 x14 h8 y85 ff2 fs5 fc0 sc0 ls0 ws0">them up using the approach in Method 2, to transform these</div><div class="t m0 x14 h8 y86 ff2 fs5 fc0 sc0 ls0 ws0">lower energy signals to equal or higher amplitudes as the</div><div class="t m0 x14 h8 y87 ff2 fs5 fc0 sc0 ls0 ws0">mean signals; 2) determine the amplitudes distributed above</div><div class="t m0 x14 h8 y88 ff2 fs5 fc0 sc0 ls0 ws0">the mean, then clip the excess - this achieves the complete</div><div class="t m0 x14 hb y89 ff2 fs5 fc0 sc0 ls0 ws0">PAPR reduction to 0dB.</div><div class="t m0 x13 h8 y5e ff2 fs5 fc0 sc0 ls0 ws0">However, to reduce the number of iterations involved, we</div><div class="t m0 x14 h8 y5f ff2 fs5 fc0 sc0 ls0 ws0">apply the Lagrange multiplier (LM) optimization technique</div><div class="t m0 x14 h8 y60 ff2 fs5 fc0 sc0 ls0 ws0">to reduce the distortion noise which further reduced the PAPR</div><div class="t m0 x14 hb y61 ff2 fs5 fc0 sc0 ls0 ws0">by another 3.5dB so that the unclipped OFDM signal PAPR is</div><div class="t m0 x14 hb y8a ff2 fs5 fc0 sc0 ls0 ws0">reduced by 11.5dB in 3 iterations only using Method 1 and to</div><div class="t m0 x14 hb y63 ff6 fs5 fc0 sc0 ls0 ws0">0dB in Method 3. We also measured the amount of out of band</div><div class="t m0 x14 hb y64 ff2 fs5 fc0 sc0 ls0 ws0">emissions generated by our approach and realized 4.47dB,</div><div class="t m0 x14 hb y3b ff6 fs5 fc0 sc0 ls0 ws0">1.68dB and 3.23dB gains for Methods 1, 2 and 3 respectively</div><div class="t m0 x14 h8 y3c ff2 fs5 fc0 sc0 ls0 ws0">when compared with the results of unclipped signals.</div><div class="t m0 x19 h8 y8b ff2 fs5 fc0 sc0 ls0 ws0">II. PROPOSED ICF MODEL</div><div class="t m0 x17 h8 y8c ff2 fs5 fc0 sc0 ls0 ws0">At baseband, the discretely sampled OFDM symbols at</div><div class="t m0 x16 h8 y8d ff2 fs5 fc0 sc0 ls0 ws0">the Nyquist rate do not exhibit equivalent PAPR as the</div><div class="t m0 x16 h8 y8e ff2 fs5 fc0 sc0 ls0 ws0">continuous symbols, thus oversampling is usually required [5],</div><div class="t m0 x16 h8 y8f ff2 fs5 fc0 sc0 ls0 ws0">[16]. Given an oversampled frequency domain OFDM signal,</div><div class="t m0 x16 hb y90 ff7 fs5 fc0 sc0 ls0 ws0">X =</div><div class="t m0 x1a hc y91 ff8 fs5 fc0 sc0 ls0 ws0"></div><div class="t m0 x1a hc y92 ff8 fs5 fc0 sc0 ls0 ws0"></div><div class="t m0 x1a hc y93 ff8 fs5 fc0 sc0 ls0 ws0"></div><div class="t m0 xb hb y90 ff7 fs5 fc0 sc0 ls0 ws0">d</div><div class="t m0 x1b hd y94 ff9 fs6 fc0 sc0 ls0 ws0">0</div><div class="t m0 x1c hb y90 ff7 fs5 fc0 sc0 ls0 ws0">, d</div><div class="t m0 x1d hd y94 ff9 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x1e he y90 ff7 fs5 fc0 sc0 ls0 ws0">, ··· , d</div><div class="t m0 x1f hd y94 ffb fs6 fc0 sc0 ls0 ws0">N−1</div><div class="t m0 xb hc y95 ff8 fs5 fc0 sc0 ls0 ws0">| {z }</div><div class="t m0 x20 h9 y96 ff2 fs6 fc0 sc0 ls0 ws0">input data</div><div class="t m0 x21 he y90 ff7 fs5 fc0 sc0 ls0 ws0">, 0, 0, ··· , 0</div><div class="t m0 x22 hd y97 ffb fs6 fc0 sc0 ls0 ws0">N(`−1)</div><div class="t m0 x23 hc y98 ff8 fs5 fc0 sc0 ls0 ws0">| {z }</div><div class="t m0 x24 h9 y99 ff2 fs6 fc0 sc0 ls0 ws0">oversampling zeros</div><div class="t m0 x25 hc y91 ff8 fs5 fc0 sc0 ls0 ws0"></div><div class="t m0 x25 hc y92 ff8 fs5 fc0 sc0 ls0 ws0"></div><div class="t m0 x25 hc y93 ff8 fs5 fc0 sc0 ls0 ws0"></div><div class="t m0 x26 hb y90 ff7 fs5 fc0 sc0 ls0 ws0">, as shown in</div><div class="t m0 x16 h8 y9a ff2 fs5 fc0 sc0 ls0 ws0">Fig. 1, it can be converted to a time-domain after oversampling</div><div class="t m0 x16 h8 y9b ff2 fs5 fc0 sc0 ls0 ws0">as follows</div><div class="t m0 x27 hb y9c ffd fs5 fc0 sc0 ls0 ws0">x(n) =</div><div class="t m0 x28 hb y9d ff6 fs5 fc0 sc0 ls0 ws0">1</div><div class="t m0 x29 he y9e ffa fs5 fc0 sc0 ls0 ws0">√</div><div class="t m0 x28 hb y9f ff7 fs5 fc0 sc0 ls0 ws0">`N</div><div class="t m0 x2a hd ya0 ffb fs6 fc0 sc0 ls0 ws0">`N−1</div><div class="t m0 x2b hc ya1 ff8 fs5 fc0 sc0 ls0 ws0">X</div><div class="t m0 x2b hd ya2 ffb fs6 fc0 sc0 ls0 ws0">k=0</div><div class="t m0 x1f hb y9c ff7 fs5 fc0 sc0 ls0 ws0">X(k)e</div><div class="t m0 x2c hd ya3 ffb fs6 fc0 sc0 ls0 ws0">j2π</div><div class="t m0 x2d hf ya4 ffe fs7 fc0 sc0 ls0 ws0">kn</div><div class="t m0 x2d hf ya5 ffe fs7 fc0 sc0 ls0 ws0">lN</div><div class="t m0 x2e he y9c ffa fs5 fc0 sc0 ls0 ws0">∀n = 0, 1, ··· , `N −1 (1)</div><div class="t m0 x16 he ya6 ff2 fs5 fc0 sc0 ls0 ws0">where ` is an oversampling factor, usually ` ≥ 4, j =</div><div class="t m0 x2f he ya7 ffa fs5 fc0 sc0 ls0 ws0">√</div><div class="t m0 x30 he ya6 ffa fs5 fc0 sc0 ls0 ws0">−1</div><div class="t m0 x16 hb ya8 ff2 fs5 fc0 sc0 ls0 ws0">and N is the number of data symbols and `N is the number</div><div class="t m0 x16 hb ya9 ff2 fs5 fc0 sc0 ls0 ws0">over-sized subcarriers after oversampling. Since the x(n) is</div><div class="t m0 x16 h10 yaa ff2 fs5 fc0 sc0 ls0 ws0">characteristically complex with real x</div><div class="t m0 x31 h11 yab ffb fs6 fc0 sc0 ls0 ws0">r</div><div class="t m0 x32 hb yaa ff6 fs5 fc0 sc0 ls0 ws0">(n) and imaginary</div><div class="t m0 x16 h10 yac ffd fs5 fc0 sc0 ls0 ws0">x</div><div class="t m0 x33 h11 yad ffb fs6 fc0 sc0 ls0 ws0">i</div><div class="t m0 x17 hb yac ff6 fs5 fc0 sc0 ls0 ws0">(n) components, the amplitude of the signal can be cal-</div><div class="t m0 x16 h8 yae ff2 fs5 fc0 sc0 ls0 ws0">culated as</div><div class="t m0 x19 he yaf ffa fs5 fc0 sc0 ls0 ws0">|x(n)| =</div><div class="t m0 x34 hc yb0 ff8 fs5 fc0 sc0 ls0 ws0">p</div><div class="t m0 x2c h10 yaf ffd fs5 fc0 sc0 ls0 ws0">x</div><div class="t m0 x35 h11 yb1 ffb fs6 fc0 sc0 ls0 ws0">r</div><div class="t m0 x36 hb yaf ff6 fs5 fc0 sc0 ls0 ws0">(n)</div><div class="t m0 x37 hd yb2 ff9 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m0 x38 hb yaf ff6 fs5 fc0 sc0 ls0 ws0">+ x</div><div class="t m0 x39 h11 yb1 ffb fs6 fc0 sc0 ls0 ws0">i</div><div class="t m0 x3a hb yaf ff6 fs5 fc0 sc0 ls0 ws0">(n)</div><div class="t m0 x3b hd yb2 ff9 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3c h8 yaf ff2 fs5 fc0 sc0 ls0 ws0">(2)</div><div class="t m0 x16 h8 yb3 ff2 fs5 fc0 sc0 ls0 ws0">From (2), it can be shown as illustrated in Fig. 2 that</div><div class="t m0 x16 h8 yb4 ff2 fs5 fc0 sc0 ls0 ws0">OFDM signal has amplitude distribution that follows Rayleigh</div><div class="t m0 x16 h10 yb5 ff2 fs5 fc0 sc0 ls0 ws0">distribution since x</div><div class="t m0 x3d h11 yb6 ffb fs6 fc0 sc0 ls0 ws0">r</div><div class="t m0 x3e hb yb5 ff6 fs5 fc0 sc0 ls0 ws0">(n) and x</div><div class="t m0 x3f h11 yb6 ffb fs6 fc0 sc0 ls0 ws0">i</div><div class="t m0 x40 hb yb5 ff6 fs5 fc0 sc0 ls0 ws0">(n) are independently and</div><div class="t m0 x16 h8 yb7 ff2 fs5 fc0 sc0 ls0 ws0">identically distributed Gaussian random variables according</div><div class="t m0 x16 h8 yb8 ff2 fs5 fc0 sc0 ls0 ws0">to central limit theorem. Being Rayleigh distributed as in Fig.</div><div class="t m0 x16 h8 yb9 ff2 fs5 fc0 sc0 ls0 ws0">2 suggests that the small fraction of amplitudes distributed</div><div class="t m0 x16 h8 yba ff2 fs5 fc0 sc0 ls0 ws0">above the mean amplitude lead to high PAPR problem.</div><div class="t m0 x17 h8 y5b ff2 fs5 fc0 sc0 ls0 ws0">Our goal is to convert the amplitude distribution, by clip-</div><div class="t m0 x16 h8 ybb ff2 fs5 fc0 sc0 ls0 ws0">ping, to achieve a uniform distribution. However, the con-</div><div class="t m0 x16 h8 y5d ff2 fs5 fc0 sc0 ls0 ws0">ventional ICF scheme is usually limited to a preset clipping</div><div class="t m0 x16 h8 y5e ff2 fs5 fc0 sc0 ls0 ws0">threshold and clipping ratio; this limits how much PAPR that</div><div class="t m0 x16 h8 y5f ff2 fs5 fc0 sc0 ls0 ws0">be can reduced. It also requires many iterations [10], [17]</div><div class="t m0 x16 h8 y60 ff2 fs5 fc0 sc0 ls0 ws0">which expends the system power and expands the processing</div><div class="t m0 x16 h8 y61 ff2 fs5 fc0 sc0 ls0 ws0">time. A way of overcoming these limitations is by making the</div><div class="t m0 x16 h8 y8a ff2 fs5 fc0 sc0 ls0 ws0">clipping threshold adaptive as described in [13] while another</div><div class="t m0 x16 h8 y63 ff2 fs5 fc0 sc0 ls0 ws0">is by constructing a PAPR reduction vector [9]. Motivated</div><div class="t m0 x16 h8 y64 ff2 fs5 fc0 sc0 ls0 ws0">by the fact that OFDM symbols are dynamic with varying</div><div class="t m0 x16 h8 y3b ff2 fs5 fc0 sc0 ls0 ws0">amplitude distribution and the studies presented in [13] and</div><div class="t m0 x16 h8 y3c ff2 fs5 fc0 sc0 ls0 ws0">[9], we propose a technique that does not require the clipping</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6260eefd090cbf2c4ef55cf2/bg3.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">2169-3536 (c) 2017 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">http://www.ieee.org/publications_standards/publications/rights/index.html for more information.</div><div class="t m0 x3 h2 y3 ff1 fs0 fc0 sc0 ls0 ws0">This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2017.2751620, IEEE Access</div><div class="t m0 x18 h9 y65 ff2 fs6 fc0 sc0 ls0 ws0">3</div><div class="c x14 ybc w2 h12"><div class="t m1 x41 h13 ybd fff fs8 fc1 sc0 ls0 ws0">0 1 2 3 4 5 6</div><div class="t m1 x42 h14 ybe ff10 fs9 fc1 sc0 ls0 ws0">OFDM signal amplitudes </div><div class="t m1 x43 h13 ybf fff fs8 fc1 sc0 ls0 ws0">0</div><div class="t m1 x44 h13 yc0 fff fs8 fc1 sc0 ls0 ws0">0.1</div><div class="t m1 x44 h13 yc1 fff fs8 fc1 sc0 ls0 ws0">0.2</div><div class="t m1 x44 h13 yc2 fff fs8 fc1 sc0 ls0 ws0">0.3</div><div class="t m1 x44 h13 yc3 fff fs8 fc1 sc0 ls0 ws0">0.4</div><div class="t m1 x44 h13 yc4 fff fs8 fc1 sc0 ls0 ws0">0.5</div><div class="t m1 x44 h13 yc5 fff fs8 fc1 sc0 ls0 ws0">0.6</div><div class="t m2 x45 h15 yc6 ff10 fsa fc1 sc0 ls0 ws0">Probability</div><div class="t m1 x46 h13 yc7 fff fs8 fc1 sc0 ls0 ws0">0 1 2 3 4 5</div><div class="t m1 x47 h14 yc8 ff10 fs9 fc1 sc0 ls0 ws0">OFDM signal amplitudes</div><div class="t m1 x48 h13 yc9 fff fs8 fc1 sc0 ls0 ws0">0</div><div class="t m1 x49 h13 yca fff fs8 fc1 sc0 ls0 ws0">0.1</div><div class="t m1 x49 h13 ycb fff fs8 fc1 sc0 ls0 ws0">0.2</div><div class="t m1 x49 h13 ycc fff fs8 fc1 sc0 ls0 ws0">0.3</div><div class="t m1 x49 h13 ycd fff fs8 fc1 sc0 ls0 ws0">0.4</div><div class="t m1 x49 h13 yce fff fs8 fc1 sc0 ls0 ws0">0.5</div><div class="t m1 x49 h13 ycf fff fs8 fc1 sc0 ls0 ws0">0.6</div><div class="t m2 x4a h15 yd0 ff10 fsa fc1 sc0 ls0 ws0">Probability</div></div><div class="t m0 x14 ha yd1 ff2 fs3 fc0 sc0 ls0 ws0">Figure 2. PDF distribution of OFDM signal demonstrating the amplitude</div><div class="t m0 x14 ha yd2 ff2 fs3 fc0 sc0 ls0 ws0">distribution which inﬂuences the PAPR behaviour of the system</div><div class="t m0 x14 h8 yd3 ff2 fs5 fc0 sc0 ls0 ws0">ratio and threshold limitations in this study. For example, recall</div><div class="t m0 x14 h8 yd4 ff2 fs5 fc0 sc0 ls0 ws0">the conventional ICF method, usually deﬁned as [8]</div><div class="t m0 x4b hb yd5 ff6 fs5 fc0 sc0 ls0 ws0">ˆ</div><div class="t m0 x4c hb yd6 ffd fs5 fc0 sc0 ls0 ws0">x(n) =</div><div class="t m0 x4a hc yd7 ff8 fs5 fc0 sc0 ls0 ws0">(</div><div class="t m0 x4d he yd8 ff7 fs5 fc0 sc0 ls0 ws0">T × exp (j × θ</div><div class="t m0 x4e h11 yd9 ffb fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x4f he yd8 ff6 fs5 fc0 sc0 ls0 ws0">) , |x(n)| > T</div><div class="t m0 x4d he yda ffd fs5 fc0 sc0 ls0 ws0">x(n), |x(n)| ≤ T</div><div class="t m0 x50 h8 yd6 ff2 fs5 fc0 sc0 ls0 ws0">(3)</div><div class="t m0 x14 hb ydb ff2 fs5 fc0 sc0 ls0 ws0">where T is the desired amplitude derived from T = γ</div><div class="t m0 x10 h11 ydc ffb fs6 fc0 sc0 ls0 ws0">o</div><div class="t m0 x11 he ydd ffa fs5 fc0 sc0 ls0 ws0">√</div><div class="t m0 x51 hb ydb ff7 fs5 fc0 sc0 ls0 ws0">P</div><div class="t m0 x50 h11 ydc ffb fs6 fc0 sc0 ls0 ws0">av</div><div class="t m0 x52 h8 ydb ff2 fs5 fc0 sc0 ls0 ws0">,</div><div class="t m0 x14 hb yde ff7 fs5 fc0 sc0 ls0 ws0">γ</div><div class="t m0 x53 h11 ydf ffb fs6 fc0 sc0 ls0 ws0">o</div><div class="t m0 x54 hb yde ff2 fs5 fc0 sc0 ls0 ws0">is the clipping ratio and P</div><div class="t m0 x55 h11 ydf ffb fs6 fc0 sc0 ls0 ws0">av</div><div class="t m0 x56 hb yde ff6 fs5 fc0 sc0 ls0 ws0">=</div><div class="t m0 x57 hd ye0 ff9 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m0 x58 h11 ye1 ffb fs6 fc0 sc0 ls0 ws0">`N</div><div class="t m0 x59 hc ye2 ff8 fs5 fc0 sc0 ls0 ws0">P</div><div class="t m0 x5a hd ye3 ffb fs6 fc0 sc0 ls0 ws0">`N−1</div><div class="t m0 x5a hd ye4 ffb fs6 fc0 sc0 ls0 ws0">n=0</div><div class="t m0 x5b he yde ffa fs5 fc0 sc0 ls0 ws0">|x(n)|</div><div class="t m0 x5c hd ye3 ff9 fs6 fc0 sc0 ls0 ws0">2</div><div class="t m0 x52 h8 yde ff2 fs5 fc0 sc0 ls0 ws0">,</div><div class="t m0 x14 hb ye5 ff7 fs5 fc0 sc0 ls0 ws0">θ</div><div class="t m0 x53 h11 ye6 ffb fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x5d he ye5 ff6 fs5 fc0 sc0 ls0 ws0">= arg {x(n)} is the phase of x(n) and</div><div class="t m0 x5e hb ye7 ff6 fs5 fc0 sc0 ls0 ws0">ˆ</div><div class="t m0 x57 hb ye5 ffd fs5 fc0 sc0 ls0 ws0">x(n) is the output</div><div class="t m0 x14 h8 ye8 ff2 fs5 fc0 sc0 ls0 ws0">clipped signal. From (3), we emphasize that clipping PAPR</div><div class="t m0 x14 h8 ye9 ff2 fs5 fc0 sc0 ls0 ws0">reduction technique is amplitude-based PAPR scheme and</div><div class="t m0 x14 h8 yea ff2 fs5 fc0 sc0 ls0 ws0">does not impact the phase of the signal. Besides, to remove the</div><div class="t m0 x14 h8 yeb ff2 fs5 fc0 sc0 ls0 ws0">in-band distortion noise arising from excess clipped signals,</div><div class="t m0 x14 h8 yec ff2 fs5 fc0 sc0 ls0 ws0">a frequency domain ﬁltering is applied. This leads to peak</div><div class="t m0 x14 h8 yed ff2 fs5 fc0 sc0 ls0 ws0">regrowth and increases the PAPR. To cushion this effect, the</div><div class="t m0 x14 h8 yee ff2 fs5 fc0 sc0 ls0 ws0">clipping is repeated a few times until the desired PAPR is</div><div class="t m0 x14 h8 yef ff2 fs5 fc0 sc0 ls0 ws0">achieved.</div><div class="t m0 x13 hb yf0 ff2 fs5 fc0 sc0 ls0 ws0">In the adaptive case [13], the authors argue that T must be</div><div class="t m0 x14 h8 yf1 ff2 fs5 fc0 sc0 ls0 ws0">recalculated based on the output amplitude of</div><div class="t m0 x5f hb yf2 ff6 fs5 fc0 sc0 ls0 ws0">ˆ</div><div class="t m0 x60 hb yf1 ffd fs5 fc0 sc0 ls0 ws0">x(n) instead</div><div class="t m0 x14 h8 yf3 ff2 fs5 fc0 sc0 ls0 ws0">of the hard-ﬁxed threshold which improved the PAPR perfor-</div><div class="t m0 x14 h8 yf4 ff2 fs5 fc0 sc0 ls0 ws0">mance. However, an OFDM signal frame is characterized by</div><div class="t m0 x14 h8 yf5 ff2 fs5 fc0 sc0 ls0 ws0">three different amplitudes [18], namely</div><div class="t m0 x48 hb yf6 ff7 fs5 fc0 sc0 ls0 ws0">x</div><div class="t m0 x15 hd yf7 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x61 hb yf6 ff6 fs5 fc0 sc0 ls0 ws0">=</div><div class="t m0 x62 hb yf8 ff6 fs5 fc0 sc0 ls0 ws0">1</div><div class="t m0 x63 hb yf9 ff7 fs5 fc0 sc0 ls0 ws0">`N</div><div class="t m0 x64 hd yfa ffb fs6 fc0 sc0 ls0 ws0">`N−1</div><div class="t m0 x65 hc yfb ff8 fs5 fc0 sc0 ls0 ws0">X</div><div class="t m0 x65 hd yfc ffb fs6 fc0 sc0 ls0 ws0">n=0</div><div class="t m0 x66 he yf6 ffa fs5 fc0 sc0 ls0 ws0">|x(n)| (4a)</div><div class="t m0 x67 hb yfd ff7 fs5 fc0 sc0 ls0 ws0">x</div><div class="t m0 x68 h11 yfe ffb fs6 fc0 sc0 ls0 ws0">min</div><div class="t m0 x48 hb yfd ff6 fs5 fc0 sc0 ls0 ws0">= arg</div><div class="t m0 x12 h11 yff ffb fs6 fc0 sc0 ls0 ws0">x</div><div class="t m0 x69 hf y100 ffe fs7 fc0 sc0 ls0 ws0">n=0,··· ,`N −1</div><div class="t m0 x4e he yfd ff6 fs5 fc0 sc0 ls0 ws0">min {|x(n)|} (4b)</div><div class="t m0 x6a hb y101 ff7 fs5 fc0 sc0 ls0 ws0">x</div><div class="t m0 x6b h11 y102 ffb fs6 fc0 sc0 ls0 ws0">max</div><div class="t m0 x4d hb y101 ff6 fs5 fc0 sc0 ls0 ws0">= arg</div><div class="t m0 x12 h11 y103 ffb fs6 fc0 sc0 ls0 ws0">x</div><div class="t m0 x69 hf y104 ffe fs7 fc0 sc0 ls0 ws0">n=0,··· ,`N −1</div><div class="t m0 x6c he y101 ff6 fs5 fc0 sc0 ls0 ws0">max {|x(n)|} (4c)</div><div class="t m0 x14 h8 y105 ff2 fs5 fc0 sc0 ls0 ws0">In general, we can also summarize (4) into a vector of the</div><div class="t m0 x14 h8 y106 ff2 fs5 fc0 sc0 ls0 ws0">form</div><div class="t m0 x6d he y107 ffa fs5 fc0 sc0 ls0 ws0">|x(n)| = [|x(n)| < x</div><div class="t m0 x6e hd y108 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x6f hb y107 ff7 fs5 fc0 sc0 ls0 ws0">, x</div><div class="t m0 x70 hd y108 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x71 he y107 ff7 fs5 fc0 sc0 ls0 ws0">, |x(n)| > x</div><div class="t m0 x72 hd y108 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x73 hb y107 ff6 fs5 fc0 sc0 ls0 ws0">] ,</div><div class="t m0 x74 he y109 ffa fs5 fc0 sc0 ls0 ws0">∀n = 0, 1, ··· , `N − 1. (5)</div><div class="c x16 y10a w2 h16"><div class="t m0 x41 h17 y10b ff13 fsb fc1 sc0 ls0 ws0">0 500 1000 1500 2000 2500 3000 3500 4000 4500</div><div class="t m0 x6b h15 y10c ff14 fsa fc1 sc0 ls0 ws0">Subcarrier indices</div><div class="t m0 x75 h17 y10d ff13 fsb fc1 sc0 ls0 ws0">-2</div><div class="t m0 x76 h17 y10e ff13 fsb fc1 sc0 ls0 ws0">-1.5</div><div class="t m0 x75 h17 y10f ff13 fsb fc1 sc0 ls0 ws0">-1</div><div class="t m0 x76 h17 y110 ff13 fsb fc1 sc0 ls0 ws0">-0.5</div><div class="t m0 x43 h17 y111 ff13 fsb fc1 sc0 ls0 ws0">0</div><div class="t m0 x44 h17 y112 ff13 fsb fc1 sc0 ls0 ws0">0.5</div><div class="t m0 x43 h17 y113 ff13 fsb fc1 sc0 ls0 ws0">1</div><div class="t m0 x44 h17 y114 ff13 fsb fc1 sc0 ls0 ws0">1.5</div><div class="t m0 x43 h17 y115 ff13 fsb fc1 sc0 ls0 ws0">2</div><div class="t m3 x77 h15 y116 ff14 fsa fc1 sc0 ls0 ws0">Output signal amplitude</div><div class="t m0 x78 h18 y117 ff13 fsc fc0 sc0 ls0 ws0">original</div><div class="t m0 x78 h18 y118 ff13 fsc fc0 sc0 ls0 ws0">distorted</div></div><div class="t m0 x16 ha y119 ff2 fs3 fc0 sc0 ls0 ws0">Figure 3. Amplitude distribution of conventional OFDM signals compared</div><div class="t m0 x16 ha y11a ff2 fs3 fc0 sc0 ls0 ws0">to the clipped signals using the proposed scheme</div><div class="t m0 x16 h8 y11b ff2 fs5 fc0 sc0 ls0 ws0">If (4a) is seen as the average power signals, then (4c) and</div><div class="t m0 x16 h8 y11c ff2 fs5 fc0 sc0 ls0 ws0">(4b) are the high and low power signals, respectively. A sister</div><div class="t m0 x16 h8 y11d ff2 fs5 fc0 sc0 ls0 ws0">approach to ICF, namely companding, expands the energy of</div><div class="t m0 x16 h8 y11e ff2 fs5 fc0 sc0 ls0 ws0">the low amplitude signals [19] or simultaneously compresses</div><div class="t m0 x16 h8 y11f ff2 fs5 fc0 sc0 ls0 ws0">and expands the amplitude of (4c) and (4b) to achieve PAPR</div><div class="t m0 x16 h8 y120 ff2 fs5 fc0 sc0 ls0 ws0">reduction [7]. One of the notable companding examples [14],</div><div class="t m0 x16 h8 y121 ff2 fs5 fc0 sc0 ls0 ws0">[20] can be said to have derived from (4), segmenting the char-</div><div class="t m0 x16 h8 y122 ff2 fs5 fc0 sc0 ls0 ws0">acteristic amplitudes of (1) into (4a), (4b) and (4c) to construct</div><div class="t m0 x16 h8 y123 ff2 fs5 fc0 sc0 ls0 ws0">an amplitude transforming PDF model that converts Rayleigh</div><div class="t m0 x16 h8 y124 ff2 fs5 fc0 sc0 ls0 ws0">distribution into a near-uniform distribution. The trapezoidal</div><div class="t m0 x16 h8 y125 ff2 fs5 fc0 sc0 ls0 ws0">distribution [14], [20] followed in that discussion leads to the</div><div class="t m0 x16 h8 y126 ff2 fs5 fc0 sc0 ls0 ws0">unsolved PAPR problem as the realized distribution is non-</div><div class="t m0 x16 h8 y127 ff2 fs5 fc0 sc0 ls0 ws0">uniform due to the central peak. These methods present some</div><div class="t m0 x16 h8 y128 ff2 fs5 fc0 sc0 ls0 ws0">exploitative insights yet unexplored with the use of ICF PAPR</div><div class="t m0 x16 h8 y129 ff2 fs5 fc0 sc0 ls0 ws0">reduction style.</div><div class="t m0 x17 h8 y12a ff2 fs5 fc0 sc0 ls0 ws0">Now, we summarize the PAPR problem in OFDM systems</div><div class="t m0 x16 h8 y12b ff2 fs5 fc0 sc0 ls0 ws0">to avoid non-uniformly distribution of signal amplitudes. In</div><div class="t m0 x16 h8 y12c ff2 fs5 fc0 sc0 ls0 ws0">order to make these peaks uniformly distributed, we clip</div><div class="t m0 x16 he y12d ff2 fs5 fc0 sc0 ls0 ws0">the amplitudes (|x(n)| > x</div><div class="t m0 x23 hd y12e ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x2c hb y12d ff6 fs5 fc0 sc0 ls0 ws0">) within the upper bound of (5).</div><div class="t m0 x16 h8 y12f ff2 fs5 fc0 sc0 ls0 ws0">Consequently, we restate the ICF solution in (3) as</div><div class="t m0 xb hb y130 ff6 fs5 fc0 sc0 ls0 ws0">ˆ</div><div class="t m0 xb hb y131 ffd fs5 fc0 sc0 ls0 ws0">x(n) =</div><div class="t m0 x79 hc y132 ff8 fs5 fc0 sc0 ls0 ws0">(</div><div class="t m0 x7a hb y133 ff7 fs5 fc0 sc0 ls0 ws0">x</div><div class="t m0 x7b hd y134 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x1f he y133 ffa fs5 fc0 sc0 ls0 ws0">× exp (j × θ</div><div class="t m0 x2e h11 y134 ffb fs6 fc0 sc0 ls0 ws0">n</div><div class="t m0 x38 he y133 ff6 fs5 fc0 sc0 ls0 ws0">) , |x(n)| > x</div><div class="t m0 x7c hd y134 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x7a he y135 ffd fs5 fc0 sc0 ls0 ws0">x(n), |x(n)| ≤ x</div><div class="t m0 x7c hd y136 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x3c h8 y131 ff2 fs5 fc0 sc0 ls0 ws0">(6)</div><div class="t m0 x16 h8 y137 ff2 fs5 fc0 sc0 ls0 ws0">Considering the other two characteristic amplitudes of an</div><div class="t m0 x16 hb y138 ff2 fs5 fc0 sc0 ls0 ws0">OFDM system in (4), choosing x</div><div class="t m0 x7d h11 y139 ffb fs6 fc0 sc0 ls0 ws0">min</div><div class="t m0 x7e h8 y138 ff2 fs5 fc0 sc0 ls0 ws0">in (4b) to determine</div><div class="t m0 x16 h8 y13a ff2 fs5 fc0 sc0 ls0 ws0">the clipped signal will reduce the energy in the signals to</div><div class="t m0 x16 h8 y13b ff2 fs5 fc0 sc0 ls0 ws0">mere noise and will be severely attenuated/convolved with</div><div class="t m0 x16 h8 y13c ff2 fs5 fc0 sc0 ls0 ws0">the in-band distortion from the excess amplitudes to become</div><div class="t m0 x16 hb y13d ff2 fs5 fc0 sc0 ls0 ws0">irrecoverable at the receiver. On the other hand, choosing x</div><div class="t m0 x7f h11 y13e ffb fs6 fc0 sc0 ls0 ws0">max</div><div class="t m0 x16 h8 y13f ff2 fs5 fc0 sc0 ls0 ws0">in (4c) will be too large that nothing will be clipped and will</div><div class="t m0 x16 h8 y140 ff2 fs5 fc0 sc0 ls0 ws0">also lead to smearing at HPA degrading BER and causing</div><div class="t m0 x16 h8 y141 ff2 fs5 fc0 sc0 ls0 ws0">the HPA to expend more power. In Fig. 3, we exemplify the</div><div class="t m0 x16 h8 y142 ff2 fs5 fc0 sc0 ls0 ws0">amplitude distribution of the clipped signal for large number</div><div class="t m0 x16 hb y143 ff2 fs5 fc0 sc0 ls0 ws0">of subcarrier N = 1024 with ` = 4.</div><div class="t m0 x17 h8 y3b ff2 fs5 fc0 sc0 ls0 ws0">From Fig. 3, the clipped signals are nicely uniformly</div><div class="t m0 x16 he y3c ff2 fs5 fc0 sc0 ls0 ws0">distributed, U (−x</div><div class="t m0 x7a hd y144 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x80 hb y3c ff7 fs5 fc0 sc0 ls0 ws0">, x</div><div class="t m0 x81 hd y144 ffc fs6 fc0 sc0 ls0 ws0">†</div><div class="t m0 x82 hb y3c ff6 fs5 fc0 sc0 ls0 ws0">). This will signiﬁcantly reduce the</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>