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
Mar 15, 2016
On the Time-Frequency Localization of the Wavelet Signals, with Application to Orthogonal
ModulationsMarius Oltean, Alexandru Isar,
Faculty of Electronics and Telecommunications, Timisoara, Romania
ISSCS Iasi 2009ISSCS Iasi 2009 ETC TimisoaraETC Timisoara
Contents
Conclusions
Results
OFDM and WOFDM
Time-frequency localization
Orthogonal modulations concept
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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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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 ISIThe orthogonality allows demodulation:
k kk
s t a x t
, (2)k ka s t x t
(1)
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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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Effective bandwidth and duration
Two measures are introduced:
There isn’t “perfect” localization in time and frequency simultaneously:
Time-frequency localization
2 22 2
2 2
2 2
( ) ( ), and (3)
( ) ( )t
t x t dt X d
x t dt X d
2t
(4)
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OFDM and WOFDM
Properties & Representations
OFDM WOFDM
0 0, ,m n m nt
m ns t a w tThe signal
The carriers
00 0, 0
jm tm ntw t p t nt e
p: rectangular window, m: subcarrier index
The signal
The carriers
/ 22 2j jj
k Zt k t k
0
,
,
( ) ( )
( )
j k jj J k
J k Jk
s t d t k
a t k
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OFDM
Balyan-Low theorem: for all the time windows p(t) that gate complex exponential to generate orthonormal basis of L2(R), we have:
Time-frequency localization
22
t
2
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WOFDM
…….When time meets frequency.When time meets frequency
Cardinal sine
Daub20
Daub4
Haar
Time-frequency localization
2 1 3Ht 2H
Dau
3 32 2max
3N N N
t N DauM m
t
Daulim N scNt t
2 2lim .N sc
N
2
2 314 / 3
sct
sc
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Results
4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
Daubechies mother effective duration
No vanishing moments
4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
Daubechies mother effective bandwidth
No vanishing moments
4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7Daubechies mother time-frequency tradeoff
No vanishing moments
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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Orthogonal modulation chain
The channel is flat, and time-variantThe variability in time is related to the
maximum Doppler shiftIFFT implements the OFDM modulator and
IDWT implements the WOFDM modulator
Orthogonal modulation in flat, time-variant channels
[west]
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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BER results
WOFDM has better results than OFDMFor 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.
0 2 4 6 8 10 12 14 16 18 2010
-4
10-3
10-2
10-1
100
SNR [dB]
BE
R
: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
Wavelets mother comparison in a WOFDM system.
0 2 4 6 8 10 12 14 16 18 2010
-3
10-2
10-1
100
SNR [dB]
BE
R
: 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
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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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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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