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

ISSCS Iasi 2009 ETC Timisoara Contents Conclusions Results OFDM and WOFDM Time-frequency localization Orthogonal modulations concept

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

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)

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

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)

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

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

ISSCS Iasi 2009 ETC Timisoara WOFDM ….When time meets frequency Cardinal sine Daub20 Daub4 Haar Time-frequency localization

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

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]

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.

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

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

Click to edit company slogan. Marius Oltean & Alexandru Isar