Transcript Time-frequency localization
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 k
a k
: data symbols, carriers
x k
(
t
): orthogonal 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 2
dt dt
2
X X
2 2
d 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 The carriers
w m
0 ,
nt
0
m n a
w m
0 ,
nt
0
nt
0
e jm
0
t
p: rectangular window, m: subcarrier index
WOFDM
The signal The carriers
j
t
k
j
0
J k
k a
d
j
(
t
J
(
t
k
)
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
WOFDM
Daub4 Haar
H
t
1 3
H
N
Dau
t N M
3
N m
3 max
N
2
Dau
3
sc
sc
t
2 14 3 / 3 Daub20
N
lim
N
Dau
N
lim
N
ISSCS Iasi 2009
2 .
sc
Cardinal sine ….When time meets frequency ETC Timisoara
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 .