on the time-frequency localization of the wavelet signals, with application to orthogonal...
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![Page 1: On the Time-Frequency Localization of the Wavelet Signals, with Application to Orthogonal Modulations Marius Oltean, Alexandru Isar, Faculty of Electronics](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649eab5503460f94bb11f6/html5/thumbnails/1.jpg)
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
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 ISI
The 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 n
s t a w tThe signal
The carriers
0
0 0, 0jm t
m 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 Dau
M mt
Daulim N scN
t 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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Marius Oltean & Alexandru Isar