Marius Oltean, Alexandru Isar, Faculty of Electronics and Telecommunications, Timisoara, Romania

1. 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

2. Contents Orthogonal modulations concept Time-frequency localization OFDM and WOFDM Results Conclusions ETC Timisoara

3. 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 ETC Timisoara

4. Orthogonal Modulations • The transmitted symbol composed as a sum of orthogonal “carriers”: • ak: data symbols, xk(t): orthogonal carriers • Advantage: information distributed along low-rate carriers, less affected by ISI • The orthogonality allows demodulation: (1) ETC Timisoara

5. Time-frequencylocalization 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 ETC Timisoara

6. Time-frequencylocalization Effective bandwidth and duration • Two measures are introduced: • There isn’t “perfect” localization in time and frequency simultaneously: (4) ETC Timisoara

7. OFDM and WOFDM Properties & Representations OFDM WOFDM The signal The signal The carriers The carriers p: rectangular window, m: subcarrier index ETC Timisoara

8. Time-frequency localization OFDM • Balyan-Low theorem: for all the time windows p(t) that gate complex exponential to generate orthonormal basis of L2(R), we have: ETC Timisoara

9. Daub20 Time-frequency localization WOFDM Haar Daub4 Cardinal sine ….When time meets frequency ETC Timisoara

10. 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 ETC Timisoara

11. ray[n] p[n] s[n] [w] IDWT/ IFFT [west] Decision DWT/ FFT r[n] Baseband implementation of an orthogonal modulation system. Orthogonal modulation in flat, time-variant channels 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 ETC Timisoara

12. Orthogonal modulation in flat, time-variant channel 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 Wavelets mother comparison in a WOFDM system. BER performance in various Doppler shift scenarios. ETC Timisoara

13. Orthogonal modulation in frequency-selective & time-variant channel BER Results • 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 • Daubechies-12 has better results than Haar • This time, the frequency-selectivity is predominant for the errors ETC Timisoara

14. 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 ETC Timisoara

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