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

Contents
Orthogonal modulations concept
Time-frequency localization
OFDM and WOFDM
Results
Conclusions
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”:
  k a x k
 
 a k
: data symbols, x k
( t ): orthogonal carriers
 Advantage: information distributed along low-rate carriers, less affected by ISI
 The orthogonality allows demodulation: a k
    
(1)
(2)
ETC Timisoara ISSCS Iasi 2009

Time-frequency localization
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
ISSCS Iasi 2009 ETC Timisoara

Time-frequency localization
Effective bandwidth and duration
 Two measures are introduced:
  t





2
( )
2 t x t dt
( )
2 x t dt
, and
 





 2
X ( )
2 d

X ( )
2 d

(3)
 There isn’t “perfect” localization in time and frequency simultaneously:
   t


2
(4)
ETC Timisoara ISSCS Iasi 2009

OFDM and WOFDM
Properties &
Representations
OFDM
The signal
  m n a
, w m

0
, nt
0
The carriers w m

0
, nt
0
     nt
0
  e jm

0 t p: rectangular window, m: subcarrier index
WOFDM
The signal

0   j

J k d

, j
( t
 k )

The
  k a



J
( t
 k ) carriers

 j
 t
 k


2 j / 2 

2 j t
 k
 
ETC Timisoara ISSCS Iasi 2009

Time-frequency localization
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:  t
2   2

 
 2

 
ISSCS Iasi 2009 ETC Timisoara

Time-frequency localization
Daub4
Haar
H
 t
1 3
H

  
N

Dau t
N
M
3


N m
3
  max

N
 2
Dau
3
 sc
 t sc

   3
14 / 3
Daub20 Cardinal sine
N lim

N

Dau
   
N lim

N   
ISSCS Iasi 2009
2
 
.
sc
 
….When time meets frequency
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Results
ISSCS Iasi 2009
 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

Orthogonal modulation in flat, time-variant channels
Orthogonal modulation chain ray[n] p[n]
[w]
IDWT/
IFFT s[n]
[w est
]
Decision
DWT/
FFT r[n]
Baseband implementation of an orthogonal modulation system.
 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
ISSCS Iasi 2009 ETC Timisoara

10
0
Orthogonal modulation in flat, time-variant channel
BER results
10
0
: Daub10 WOFDM,fm=0.001,1 level
: Daub10 WOFDM,fm=0.05,1 level
: Haar WOFDM,fm=0.001,1 level
: Haar WOFDM,fm=0.05,1 level
10
-1
10
-1
10
-2
10
-3
:OFDM,fm=0.001
:OFDM,fm=0.005
:OFDM,fm=0.01
:OFDM,fm=0.05
:Haar WOFDM, fm=0.001,4 levels
:Haar WOFDM, fm=0.005, 4 levels
:Haar WOFDM, fm=0.01, 4 levels
:Haar WOFDM, fm=0.05, 4 levels
10
-4
0 2 4 6 8 10
SNR [dB]
12 14 16 18 20
BER performance in various Doppler shift scenarios.
10
-2
10
-3
0 2 4 6 8 10
SNR [dB]
12 14 16 18 20
Wavelets mother comparison in a WOFDM system.
 WOFDM has better results than OFDM
 For WOFDM, the time-localization of the carriers is the predominant factor which determines the BER performance
ISSCS Iasi 2009 ETC Timisoara

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
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
ISSCS Iasi 2009 ETC Timisoara

Marius Oltean & Alexandru Isar
C l i c k t o e d i t c o m p a n y s l o g a n .