marius oltean, alexandru isar, faculty of electronics and telecommunications, timisoara, romania

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


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


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)


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



Effective bandwidth and duration

Two measures are introduced:

There isn’t “perfect” localization in time and frequency simultaneously:


2 22 2

2 2

2 2

( ) ( ), and (3)

( ) ( )t

t x t dt X d

x t dt X d

2t

(4)


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


OFDM

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


22

t

2


WOFDM

…….When time meets frequency.When time meets frequency

Cardinal sine

Daub20

Daub4

Haar


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


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


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


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


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]


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


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


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

Click to edit company slogan .Marius Oltean & Alexandru Isar

marius oltean, alexandru isar, faculty of electronics and telecommunications, timisoara, romania

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