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On the Time-Frequency Localization of the Wavelet Signals, with Application to Orthogonal Modulations Marius Oltean, Alexandru Isar, Faculty of Electronics and Telecommunications, Timisoara, Romania
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ISSCS Iasi 2009 ETC Timisoara Contents Conclusions Results OFDM and WOFDM Time-frequency localization Orthogonal modulations concept
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ISSCS Iasi 2009 ETC Timisoara Objectives To prove that the time-frequency localization of the wavelet functions is better than the one of OFDM’s windowed complex exponentials To highlight the meaning of the above remark for an orthogonal modulation system
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ISSCS Iasi 2009 ETC Timisoara Orthogonal Modulations The transmitted symbol composed as a sum of orthogonal “carriers”: a k : data symbols, x k (t): orthogonal carriers Advantage: information distributed along low-rate carriers, less affected by ISI The orthogonality allows demodulation: (1)
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ISSCS Iasi 2009 ETC Timisoara Radio channels The radio channels are frequency - selective (multipath propagation) and time-variants (Doppler effect) A “time-frequency” localization of the channel can be introduced The carriers used in transmission should be localized as the channel itself Time-frequency localization
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ISSCS Iasi 2009 ETC Timisoara Effective bandwidth and duration Two measures are introduced: There isn’t “perfect” localization in time and frequency simultaneously: Time-frequency localization (4)
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ISSCS Iasi 2009 ETC Timisoara OFDM and WOFDM Properties & Representations OFDM WOFDM The signal The carriers p: rectangular window, m: subcarrier index The signal The carriers
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ISSCS Iasi 2009 ETC Timisoara OFDM Balyan-Low theorem: for all the time windows p(t) that gate complex exponential to generate orthonormal basis of L 2 (R), we have: Time-frequency localization
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ISSCS Iasi 2009 ETC Timisoara WOFDM ….When time meets frequency Cardinal sine Daub20 Daub4 Haar Time-frequency localization
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ISSCS Iasi 2009 ETC Timisoara Results The effective duration and bandwidth are normalized to unity The effective duration has a sharper evolution with N Numerically, the best time-frequency compromise is provided by Daubechies- 4 The choice of the wavelets mother must be dependent on the channel’s characteristics
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ISSCS Iasi 2009 ETC Timisoara Orthogonal modulation chain The channel is flat, and time-variant The variability in time is related to the maximum Doppler shift IFFT implements the OFDM modulator and IDWT implements the WOFDM modulator Orthogonal modulation in flat, time-variant channels [w est ] IDWT/ IFFT DWT/ FFT Decision s[n] ray[n] p[n] [w] Baseband implementation of an orthogonal modulation system. r[n]
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ISSCS Iasi 2009 ETC Timisoara BER results WOFDM has better results than OFDM For WOFDM, the time-localization of the carriers is the predominant factor which determines the BER performance Orthogonal modulation in flat, time-variant channel BER performance in various Doppler shift scenarios. Wavelets mother comparison in a WOFDM system.
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ISSCS Iasi 2009 ETC Timisoara BER Results Daubechies-12 has better results than Haar This time, the frequency-selectivity is predominant for the errors Orthogonal modulation in frequency-selective & time-variant channel It = the number of IDWT iterations Two ray channel model, with equal power of the two paths BER is computed independently at the third and the fourth scales
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ISSCS Iasi 2009 ETC Timisoara Conclusions Daubechies wavelets time-frequency localization is better than the time-frequency localization of OFDM’s windowed exponentials In flat, time-variant channels, WOFDM performs better than OFDM Wavelets with short compact time support are the best choice (e.g. Haar) In frequency-selective & time variant channels, wavelets with short compact frequency support provide better results The choice of the carrier family in an orthogonal modulation must be dependent on the channel characteristics
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Click to edit company slogan. Marius Oltean & Alexandru Isar
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