Spectral Analysis of OFDM Signals and its Improvement by Polynomial

4. Windowing effect of PCCin the time domainIn the previous section it is shown that PCCimproves the rolloff of OFDM power spectrumand reduces the OOB power. This can also beexplained by the considering the effect of PCC inthe time domain. For weighted groups of twosubcarriers, we have a2k+ 1 = −a2k. The l-thoutput component of the transmitter inverse DFTdue to the pair of inputs a 2 k and a2 k+1 is givenby j2πkl j2πlbl= a2 k exp1− exp T T (11)This shows that PCC with data mapped ontoweighted subcarrier pairs is equivalent to thewindowing of the outputs of the inverse DFTwith nonzero inputs on the even numberedsubcarriers. The windowing function is complexand is given by ( 1− exp( j2πlT )). The plot ofthis window is shown in Figure 6. The plotshows that window is very smooth at the symbolboundaries. The real and imaginary part of thissignals are also smooth. This will reduce theOOB power in PCC-OFDM that might otherwisehave been caused by the sharp transitions atsymbol boundaries. Thus an equivalent systemcan be built using time domain windowing asshown in Figure 7. In the case of PCC with threeweighted subcarriers in a group, it can be easilyshown that the windowing function is given by( 1 − cos( 2πl T )). This is real and is a Hanningwindow.Power Envelope43.532.521.510.50-0.5 0 0.5t/TFigure 6. Variation in the power envelope across asymbol period for PCC OFDM. Solid line m = 2 ,dotted line m = 3 .5. ConclusionThe Fourier transform of an OFDM subcarrier isderived and is shown to be of the formk(− 1) sin( x)x , where k represents the subcarrierindex. Therefore polarity of a particularsubcarrier depends on k . The alternating sign onthe adjacent subcarriers is consistent with the ICIcancellation properties of PCC in OFDM. Thespectral analysis of OFDM is extended to PCC-OFDM. It is shown that PCC improves thespectral rolloff of OFDM signals. Theimprovement in the spectral rolloff of PCC-OFDM is discussed in the time domain. Theeffect of PCC in the time domain is windowing.This will result in most of the signal energybeing concentrated at the middle of symbolperiod and smoothing of signals at the symbolboundaries. This will reduce the OOB power inPCC-OFDM.a o, i0a 2 ,ia N−2,i0N-pointInverseDFTb o,iFigure 7. Time domain equivalent to weighting pairsof subcarriersAcknowledgement: This study was supportedby an Australian Research Council (ARC) grant.6. References:b N−1,iComplexWindow[1] D. Bhatoolaul and G. Wade, “SpectrumShaping in N -channel QPSK-OFDM systems”,IEE proc.-Vis. ImageSignal Processing, Vol.142, 1995, pp. 333-338.[2] R. Van Nee and R. Prasad. OFDM forWireless Multimedia Communications. ArtechHouse. 2000.[3] J. Armstrong, “Analysis of new and existingmethod of reducing intercarrier interference dueto carrier frequency offset in OFDM,” IEEETrans. Commun., vol 47, pp. 365-9, March 1999.[4] J. G. Proakis, and M. Salehi. CommunicationSystems Engineeing. Prentice Hall. 1994.[5] Yuping Z., “In-band and out-band spectrumanalysis of OFDM communication systems usingICI cancellation methods,” WCC 2000 - ICCT2000. 2000 ICCT Proc. IEEE. Part vol.1, 2000,pp.773-6 vol.1. Piscataway, NJ, USA.[6] K. Panta and J. Armstrong, “The performanceof Overlap PCC-OFDM with error-correctingcodes,” 6th International Symposium on DSP forCommunication Systems, Sydney, January 2002,pp.118-122.