On the Time-Frequency Localization of the Wavelet Signals, with Application to Orthogonal Modulations

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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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Contents
Orthogonal modulations concept
Time-frequency localization
OFDM and WOFDM
Results
Conclusions
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Objectives

• To prove that the time-frequency localization of
  the wavelet functions is better than the one of
  OFDMs windowed complex exponentials
• To highlight the meaning of the above remark for
  an orthogonal modulation system

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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)
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Radio channels
Time-frequency localization

• 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

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Effective bandwidth and duration
Time-frequency localization

• Two measures are introduced
• There isnt perfect localization in time and
  frequency simultaneously

(4)
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OFDM and WOFDM
Properties Representations
OFDM
WOFDM
The signal
The signal
The carriers
The carriers
p rectangular window, m subcarrier index
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OFDM
Time-frequency localization

• Balyan-Low theorem for all the time windows
  p(t) that gate complex exponential to generate
  orthonormal basis of L2(R), we have

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WOFDM
Time-frequency localization
Haar
Daub4
Cardinal sine
.When time meets frequency
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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 channels characteristics

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Orthogonal modulation chain
Orthogonal modulation in flat, time-variant
channels

• 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

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

• WOFDM has better results than OFDM
• For WOFDM, the time-localization of the carriers
  is the predominant factor which determines the
  BER performance

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BER Results
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

• Daubechies-12 has better results than Haar
• This time, the frequency-selectivity is
  predominant for the errors

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Conclusions

• Daubechies wavelets time-frequency localization
  is better than the time-frequency localization of
  OFDMs 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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Thank You !
Marius Oltean Alexandru Isar

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