iota filter

8/10/2019 IOTA Filter

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OFDMA-IOTA OFDM/CDMA-IOTA

Performance Comparison between OFDMA-IOTA and

OFDM/CDMA-IOTA Systems

2005 8


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OFDMA-IOTA OFDM/CDMA-IOTA

Performance Comparison between OFDMA-IOTA and


2005 8


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: ________________________

: ________________________

: ________________________

2005 8


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Abstract

Although conventional OFDM modulation scheme using guard interval is robust

to channel induced ISI (Inter-Symbol Interference) in time-domain, it is very sensitive

to ICI (Inter-Carrier Interference) due to doppler effect in frequency domain.

OFDM/OQAM-IOTA modulation scheme utilizes IOTA (Isotropic Orthogonal

Transform Algorithm) filter that has orthogonality in time and frequency domain so

that it is robust to delay spread and doppler effect. OFDMA-IOTA system can increase

bandwidth efficiency since it does not use guard interval.

In this paper, we show the design of IOTA filter and the structure of

OFDM/OQAM-IOTA system and perfect channel estimation for it. The advantage and

performances of OFDMA-IOTA are investigated by the comparison with the

conventional OFDM modulation.

Simulation results show that OFDMA-IOTA system outperforms OFDM system.

And finally, OFDMA-IOTA and OFDM/CDMA-IOTA are compared on their BER

performance. Due to the inherent orthogonality in OFDMA-IOTA, OFDMA-IOTA

performs better than OFDM/CDMA-IOTA under fair comparison condition of the

same transmission data rate.


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Contents

Abstract ......................................................................................................................

Contents .........................................................................................................................ii

Lists of Figures ................................................................................................................iv

List of Tables ..................................................................................................................vi

Chapter 1 Introduction...................................................................................................1

Chapter 2 Wavelet Basics................................................................................................3

2.1 Wave & Wavelet .................................................................................................3

2.2 Scaling Function & Wavelet Function ..............................................................3

2.2.1 Scaling Function ........................................................................................ 3

2.2.2 Wavelet Function ....................................................................................... 5

2.3 QMF Filter Bank ................................................................................................7

2.4 Wavelet Transform & Wavelet Packets...........................................................10

2.4.1 Fast Wavelet Transform........................................................................... 10

2.4.2 Wavelet Packet ......................................................................................... 12

2.5 Polyphase for Filter Banks...............................................................................17

Chapter 3 IOTA Filter Design......................................................................................20

3.1 IOTA Filter ........................................................................................................20

3.2 Polyphase IOTA Filter Design..........................................................................24

Chapter 4 Design of OFDM/OQAM-IOTA System ....................................................28

4.1 OFDM/QAM System........................................................................................28

4.2 OFDM/OQAM-IOTA System...........................................................................29


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4.3 Complexity Analysis of OFDM/OQAM-IOTA System..................................32

4.3.1 OFDM System ......................................................................................... 32

4.3.1.1 Transmit Side.................................................................................. 32

4.3.1.2 Receive Side ................................................................................... 33

4.3.2 OFDM/IOTA System ............................................................................... 33

4.3.2.1 Transmit Side.................................................................................. 33

4.3.2.2 Receive Side ................................................................................... 36

4.3.3 Conclusions .............................................................................................. 38

Chapter 5 OFDMA-IOTA System vs. OFDM/CDMA-IOTA System.........................40

5.1 New Ideal Channel Estimation Method ..........................................................40

5.2 OFDM/CDMA-IOTA System Design ..............................................................43

5.3 Performance Comparison between OFDM and OFDM-IOTA System .........46

5.4 Performance Comparison between OFDMA-IOTA and OFDM/CDM

A-IOTA System................................................................................................48

Chapter 6 Conclusions.................................................................................................52

References ......................................................................................................................53

Acknowledgements ........................................................................................................55


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iv

Lists of Figures

Figure 2.1. The nested function spaces spanned by a scaling function............................5

Figure 2.2. The relationship between scaling and wavelet function spaces .....................5

Figure 2.3. (a) A two-band filter bank for one-dimentional subband coding and

decoding, and (b) its spectrum splitting properties.........................................8

Figure 2.4. An FWT analysis bank .................................................................................11

Figure 2.5. The inverse FWT synthesis filter bank ........................................................12

Figure 2.6. A two-stage or two-scale FWT analysis bank ..............................................13

Figure 2.7. A coefficient (a) and analysis (b) tree for the two-scale FWT analysis

bank of Fig. 2.6.............................................................................................14

Figure 2.8. A three-scale FWT filter bank: (a) block diagram; (b) decomposition

space tree; and (c) spectrum splitting characteristics....................................14

Figure 2.9. A three-scale wavelet packet analysis tree ...................................................15

Figure 2.10. The (a) filter bank and (b) spectrum splitting characteristics of a

three-scale full wavelet packet analysis tree.................................................17

Figure 2.11. (a) Direct form of the analysis bank and (b) polyphase form of the

analysis bank.................................................................................................18

Figure 3.1. IOTA waveform and its Fourier transform...................................................24

Figure 3.2. Fourier transforms of IOTA function ...........................................................24

Figure 4.1. OFDM/QAM system....................................................................................29

Figure 4.2. OFDM/OQAM time and frequency lattices (compared to OFDM w/o

guard interval) ...............................................................................................31


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Figure 4.3. OFDM/OQAM-IOTA system.....................................................................32

Figure 4.4. 512-point decimation-in-frequency IFFT..................................................35

Figure 5.1. 1 st order neighbour sub-carriers ...............................................................42

Figure 5.2. Symbol sequence for intrinsic ISI reduction............................................42

Figure 5.3. Symbol-sequence-based ideal channel estimation....................................43

Figure 5.4. OFDM/CDMA-IOTA system.....................................................................45

Figure 5.5. BER performance of OFDM-IOTA system vs. OFDM system..............47

Figure 5.6. Comparison scenario of OFDMA-IOTA vs. OFDM/CDMA-IOTA........48

Figure 5.7. BER performance of OFDM/CDMA-IOTA vs. OFDMA-IOTA.............50


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vi

List of Tables

Table 3.1. First bk,j coefficients .................................................................................23

Table 3.2. The IOTA filter polyphase buffer...............................................................25

Table 3.3. The buffer with L=2 and M=4...................................................................25

Table 3.4. The buffer after 1 symbol...........................................................................26

Table 3.5. The buffer after 2 symbol...........................................................................26

Table 4.1. Hardware space for real and imaginary value...........................................34

Table 4.2. Complexity comparison between OFDM and OFDM/IOTA(1) ...............38

Table 4.3. Complexity comparison between OFDM and OFDM/IOTA(2) ...............39

Table 5.1. General link level simulation parameters...................................................46

Table 5.2. Simulation parameters .................................................................................46

Table 5.3. Comparison of OFDM and OFDM-IOTA .................................................47

Table 5.4. General link level parameters .....................................................................49

Table 5.5. OFDMA-IOTA system parameters .............................................................49

Table 5.6. OFDM/CDMA-IOTA system parameters ...................................................49


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Chapter 1

Introduction

Wavelet has been a very hot topic in recent years. Its application ranges from the

function approximation, signal multiresolution representation, image compression to

signal processing and other fields. The popularity of wavelets is primarily due to the

interesting structure they provide based on dilation and location. A few investigators

have begun to exploit those features of wavelets that suggest their applications in

communications [1].

This thesis, is the result of a research work that of a design of wavelet-based

OFDM systems, classical OFDM modulation uses a guard interval (or cyclic prefix) to

efficiently combat the multi-path effect, at the price of a loss of spectral efficiency. It is

robust to channel induced ISI (Inter-Symbol Interference) in time-domain, but sensitive

to ICI (Inter-Carrier Interference) due to doppler effect in frequency domain. So we

design a novel wavelet function and utilize it to the OFDM system called OFDMA-

IOTA system. Since this prototype functionIOTA(Isotropic Orthogonal Transform

Algorithm) filter is very well localized both in the time and frequency domain to limit

the inter-symbol and inter-carrier interferences, and can also guarantee orthogonality

between sub-carriers, so OFDMA-IOTA is very robust to delay spread and doppler

effect. It can increase bandwidth efficiency since it does not use guard interval.

OFDMA-IOTA is an interesting alternative to classical OFDM, it has very good

characteristic, but more complexity may be induced at the reception. Classical channel

estimation can be based on scattered pilots or on a preamble, when it is applied straight


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forwardly to OFDMA-IOTA modulation, an intrinsic Inter-Symbol-Interference is

observed. This severely degrades its performances, so we use a specific symbol

sequence based approach to reduce this interference. We verify through simulations that

this method provides reliable bounds of performances.

In this paper, we also suggest OFDM/CDMA-IOTA system for the purpose of

comparison with OFDMA-IOTA system.

This paper is organized as follows. First we give the introduction of wavelet basis,

then the method of IOTA filter design, and finally, the BER results of the 2 systems

under both AWGN and Rayleigh fading channels are shown by computer simulation.


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Chapter 2

Wavelet Basics

2.1 Wave & Wavelet

Wave is an oscillating function of time or space (eg. sinusoid). Fourier transform

uses the wave or sinusoid to represent a signal, the basis of fourier transform is

completely localized at the frequency domain, but not localized in time domain.

Wave analysis is useful for periodic, time-invariant, stationary signal.

Wavelet is a small wave (a pulse), which has its energy concentrated in particular

time. Wavelet transform uses the wavelets to represent a signal. Unlike the case of

Fourier transforms, there exists a large selection of wavelet families depending on the

choice of the mother wavelet. However, desirable properties such as orthogonality,

compactness of support, rapid decay, and smoothness impose a variety of restrictions.

Wavelet is a good tool to analyze transient, non-stationary, time-varying signal.

2.2 Scaling Function & Wavelet Function

2.2.1 Scaling Function

A signal or functioncan be expressed as a linear combination of expansion

functions

(2.1)( ) ( )k k k

f x x =


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where k is an integer index of the finite or infinite sum, the k are real-

valuedexpansion coefficients, and the ( ) xk are real-valued expansion functions.

Consider the set of expansion functions composed of integer translations and

binary scalings of the real, square-integrable function ( ) xk ; that is, the set ( ) xk

where [2]

(2.2)

Here k determines the position of ( ) xk j , along the x-axis, j determines ( ) xk j , s

widthhow broad or narrow it is along the x-axisand 22 j controls its height or

amplitude. Because the shape of ( ) xk j , changes with j , ( ) x is called a scaling

function. By choosing ( ) x wisely, ( ) xk j , can be made to span ( ) R L2 , the set of

all measurable, square-integrable functions.

The scaling function usually obeys the four fundamental requirements of multi-

resolution analysis [2]:

The scaling function is orthogonal to its integer translates.

The subspaces spanned by the scaling function at low scales are nested within

those spanned at higher scales.

The only function that is common to all jV is ( ) 0= x f .

Any function can be represented with arbitrary precision.

As can be seen in Fig. 2.1, subspaces containing high-resolution functions must

also contain all lower resolution functions. That is,

/ 2 2, ( ) 2 (2 ), , ( ) ( )

j j j k x k for all j k Z and x L R =


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V V V V V V 2101 (2.3)

Figure 2.1. The nested function spaces spanned by a scaling function.

2.2.2 Wavelet Function

Given a scaling function that meets the MRA requirements, we can define a

wavelet function ( ) x that, together with its integer translates and binary scalings,

spans the difference between any two adjacent scaling subspaces, jV and 1+ jV . The

situation is illustrated graphically in Fig. 2.2. We define the set ( ) xk j , of wavelets

(2.4)

Figure 2.2. The relationship between scaling and wavelet function spaces.

0V

2V

1V

0 1 2 ...V V V V ,{ ( )} j j k

k

V Span x =

0V

2V

1V

1 j j jV V W + = ,{ ( )} j j k k

W Span x =

0W 1W

/ 2, ( ) 2 (2 )

j j j k x k =


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for all Z k that spans the jW space in the figure. As with scaling function, we write

( ){ } xSpanW k jk j , = (2.5)

and note that if ( ) jW x f ,

( ) ( )=k

k jk x x f , (2.6)

The scaling and wavelet function subspaces in Fig. 2.2 are related by

j j j W V V =

+1 (2.7)

where denotes the union of spaces (like the union of sets). The orthogonal

complements of jV in 1+ jV is jW , and all members of jV are orthogonal to the

members of jW . Thus,

( ) ( ) 0, ,, = x x l jk j (2.8)

for all appropriate j , k , Z l .

Since wavelet spaces reside within the spaces spanned by the next higher

resolution scaling functions (see Fig. 2.2), any wavelet function can be expressed as a


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weighted sum of shifted, double-resolution scaling functions. That is, we can write

(2.9)

where the ( )nh are called the wavelet function coefficients and h is the wavelet

vector. Using the condition that wavelet spans the orthogonal complement spaces in Fig.

2.2, and that integer wavelet translates are orthogonal, it can be shown that ( )nh is

related to ( )nh by

( ) ( ) ( )nhnh n = 11 (2.10)

In the frequency domain, the space spanned by the scaling function is a lowpass

channel, while the space spanned by the wavelet is a bandpass channel.

2.3 QMF Filter Bank

Fig. 2.3 (a) shows the principal components of a two-band quadrature mirror filter

(QMF) bank. The input of the system is a one-dimensional, band-limited discrete-time

signal ( )n x for ,...;2,1,0=n the output sequence, ( )n x , is formed through the

decomposition of ( )n x into ( )n y0 and ( )n y1 via analysis filters ( )nh0 and ( )nh1 ,

and subsequent recombination via synthesis filters ( )n g 0 and ( )n g 1 . Note that filters

( )nh0 and ( )nh1 are half-band digital filters whose idealized transfer characteristics,

0 H and 1 H , are shown in Fig. 2.3(b). Filter 0 H is a low-pass filter whose output is an

( ) ( ) 2 (2 )n

h n x n =


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approximation of ( )n x ; filter 1 H is a highpass filter whose output is the high

frequency or detail part of ( )n x . All filtering is performed in the time domain by

convolving each filters input with its impulse responseits response to a unit

amplitude impulse function, ( )n . We wish to select ( )nh0 , ( )nh1 , ( )n g 0 , and ( )n g 1

(or, alternately 0 H , 0 H , 0G , and 1G ) so that the input can be reconstructed perfectly.

That is, so that ( ) ( )n xn x = [3].

(a)

(b)

Figure 2.3. (a) A two-band filter bank for one-dimensional subband coding and

decoding, and (b) its spectrum splitting properties.

The Z-transform, a generalization of the discrete Fourier transform, is the ideal tool

for studying discrete-time, sampled-data systems like the one in Fig. 2.3(a). The Z-

transform of sequence ( )n x for ,...2,1,0=n is

( ) ( )

= n z n x z X (2.11)

where z is a complex variable. Our interest in the Z-transform stems from the ease with

( )nh0

( )nh1

( )n g 0

( )n g 1

( )n x ( )n x( )n y1

( )n y0

( ) 0 H ( ) 1 H

2


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which it handles sampling rate changes. Downsampling by a factor of 2 in the time

domain corresponds to the simple Z-domain operation

( ) ( ) ( ) ( ) ( )[ ]212121

2 z X z X z X n xn x downdown +== (2.12)

where the double arrow indicates that the expressions on the left and right forma Z-

transform pair. In a similar manner, upsamplingagain by a factor of 2is defined by

the transform pair

(2.13)

If sequence ( )n x is downsampled and subsequently upsampled to yield ( )n x , Eqs.

(2.12) and (2.13) combine to yield

( ) ( ) ( )[ ] z X z X z X +=21 )

(2.14)

where ( ) ( ) z X Z n x )

1 = is the resulting downsampled-upsampled sequence. The

( ) z X term in this equation is the Z-transform of an aliased or modulated version of

sequence ( )n x . Its inverse Z-transform is

( )[ ] ( ) ( )n x z X Z n11 = (2.15)

With this brief introduction to the Z-transform, consider again the subband coding

and decoding system of Fig. 2.3(a). In accordance with Eq. (2.14), we can express the

,...4,2,0=n( ) ( )2 z X z X up =( ) ( )=

0

2n xn xup

otherwise


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systems output as

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] z X z H z X z H z G z X z H z X z H z G z X +++= 111000 21

21 )

(2.16)

For error-free reconstruction of the input, ( ) ( )n xn x = and ( ) ( ) z X z X = . Thus,

we impose the following conditions:

( ) ( ) ( ) ( )01100 =+ z G z H z G z H (2.17)

( ) ( ) ( ) ( )21100 =+ z G z H z G z H (2.18)

and, the following equations satisfy the requirement of quadrature mirror filters:

(2.19)

(2.20)

(2.21)

2.4 Wavelet Transform & Wavelet Packets

2.4.1 Fast Wavelet Transform

The fast wavelet transform (FWT) is a computationally efficient implementation of

the discrete wavelet transform (DWT) that exploits a surprising but fortunate

relationship between the coefficients of the DWT at adjacent scales [4].

( ) ( ) z H z H = 01

( ) ( ) z H z G 00 =

( ) ( ) z H z G = 01


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See Figure 2.3, with ( ) ( )nhnh = 0 and ( ) ( )nhnh = 1 , we can therefore write

( ) ( ) ( )0,2

,1, =+= k k nn jW nhk jW (2.22)

( ) ( ) ( )0,2

,1,=

+=k k n

n jW nhk jW (2.23)

where the convolutions are evaluated at instants k n 2= for 0k . Evaluating

convolutions at nonnegative, even indices is equivalent to filtering and downsampling

by 2.

( )nh

( )nh

2( )n jW ,

( )n jW ,

( )n jW ,1+

2

Figure 2.4. An FWT analysis bank.

An equally efficient inverse transform for the reconstruction of ( ) x f from

DWT/FWT approximation and detail coefficients, ( )k jW , and ( )k jW , , can also be

formulated. Called the inverse fast wavelet transform (FWT1

), it uses the scaling andwavelet vectors employed in the forward transform, together with the level j

approximation coefficients. Noting the similarity between the FWT analysis bank in Fig.

2.5 and the two-band subband analysis portion of Fig. 2.3(a), we can immediately

postulate the required FWT 1 synthesis filter bank. Figure 2.6 details its structure,

which is identical to the synthesis portion of the two-band subband coding and decoding


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system in Fig. 2.3(a). The synthesis and analysis filters must be time-reversed versions

of one another. Since the FWT analysis filters (see Fig. 2.5) are ( ) ( )nhnh =

0 and

( ) ( )nhnh = 1 , the required FWT 1 synthesis filters are ( ) ( )nhnh =0 and

( ) ( )nhnh =1 .

The FWT 1 filter bank in Figure 2.5 implements the computation

(2.24)

where upW signifies upsampling by 2 (i.e., inserting zeros between the elements of W

so that it is twice its original length). The upsampled coefficients are filtered, by

convolution with ( )nh and ( )nh , and added to generate a higher scale

approximation.

( )nh

2

2( )n jW ,

( )n jW ,

( )n jW ,1+ +

( )nh

Figure 2.5. The inverse FWT synthesis filter bank.

2.4.2 Wavelet Packet

The fast wavelet transform decomposes a function into a series of logarithmically

related frequency bands. That is, the low frequencies are grouped into narrow bands,

while the high frequencies are grouped into wider bands. If we want greater control over

0( 1, ) ( ) ( , ) ( ) ( , )UP UP

k W j k h k W j k h k W j k + = +


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the partitioning of the time-frequency plane (e.g., smaller bands at the higher

frequencies), the FWT must be generalized to yield a more flexible decomposition

called a wavelet packet. The cost of this generalization is an increase in computational

complexity from ( ) M O for the FWT to ( M O ) M log [5].

( ) ( )n J W n f ,

=

( )nh

2

2

( )nh

( )nh

( )nh

2

2

( )n J W ,1

( )n J W ,2

( )n J W ,2

( )n J W ,1

Figure 2.6. A two-stage or two-scale FWT analysis bank.

Consider the two-scale filter bank of Fig. 2.6, Figure 2.7(a) details the structure of

the tree and links the appropriate FWT scaling and wavelet coefficients [from Fig. 2.6]

to its nodes. The root node is assigned the highest-scale approximation coefficients,

which are samples of the function itself, while the leaves inherit the transforms

approximation and detail coefficients outputs. The lone intermediate node, ( )n J W ,1 ,

is a filter bank approximation that is ultimately filtered to become two leaf nodes. Note

that the coefficients of each node are the weights of a linear expansion that produces a

band-limited piece of root node ( )n f . Since any such piece is an element of a known

scaling or wavelet subspace, we can replace the generating coefficients in Fig. 2.7(a) by

the corresponding subspace. The result is the subspace analysis tree of Fig. 2.7(b).

Although the variable W is used to denote both coefficients and subspaces, the two

quantities are distinguishable by the format of their subscripts.


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subsampler is, to be accurate, ( )n J W ,1 , it has been labeled 1 J W the subspace of

the function that is generated by the ( )n J W ,1 transform coefficients. This subspace

corresponds to the upper-right leaf of the associated analysis tree, as well as the

rightmost or widest bandwidth segment of the corresponding frequency spectrum.

Analysis trees provide a compact and informative way of representing multiscale

wavelet transforms. They are simple to draw, take less space than their corresponding

filter and subsampler-based block diagrams, and make it relatively easy to spot valid

decompositions. The three-scale analysis tree of Fig. 2.8(b), for example, offers the

following three expansion options:

11 = J J J W V V (2.25)

122 = J J J J W W V V (2.26)

1233 = J J J J J W W W V V (2.27)

They correspond to the one-, two-, and three-scale FWT decompositions. In general, a

P-scale analysis tree supports P unique decompositions.

Figure 2.9. A three-scale wavelet packet analysis tree.

jV

1 jV 1 jW

D J W ,1 A J W ,12 jW 2 jV

DA jW ,1 DD jW ,1 AD jW ,1 AA jW ,1 D jW ,2 A jW ,23 jW 3 jV


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Analysis trees are also an efficient mechanism for representing wavelet packets,

which are nothing more than conventional wavelet transforms in which the details are

iteratively filtered. Thus, the three-scale FWT analysis tree of Fig. 2.8(b) becomes the

three-scale wavelet packet tree of Fig. 2.9. The additional subscripting is introduced.

The first subscript of a double-subscripted node identifies the scale of the FWT parent

node from which it descended. The seconda variable length string of As and Ds

encodes the path from the parent to the node. An A designates approximation filtering,

while a D indicates detail filtering. Subspaces DA J W ,1 , for example, is obtained by

filtering the scale 1 J FWT coefficients (i.e., parent 1 J W in Fig. 2.9) through an

additional detail filter (yielding D J W ,1 ), followed by an approximation filter (giving

DA J W ,1 ). Figures 2.10 (a) and (b) are the filter bank and spectrum splitting

characteristics of the analysis tree in Fig. 2.9. The evenly spaced frequency bands that

are characteristics of full packet decompositions.

The three-scale packet tree in Fig. 2.9 almost triples the number of decompositions

(and associated time-frequency tilings) that are available from the three-scale FWT tree.

The wavelet packet tree of Fig. 2.9 supports 26 different decompositions. For instance,

jV can be expanded as

DD J DA J AD J AA J D J A J J J J W W W W W W W V V ,1,1,1,1,2,233 = (2.28)

whose spectrum is shown in Fig. 2.10 (b).


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Figure 2.10. The (a) filter bank and (b) spectrum splitting characteristics of a three-

scale full wavelet packet analysis tree.

2.5 Polyphase for Filter Banks

The polyphase idea extends from one filter to a bank of filters. The polyphase form

does the downsampling first. In the block diagram of the filter bank, the decimators

move outside the filters. We can write C and D or 0 H and 1 H for the lowpass and

highpass filters:

( )nh

( )nh

2

2

2

2

2

2

2

2

2

2

2

2

2

2

( )nh

( )nh

( )nh

( )nh

( )nh

( )nh

( )nh

( )nh

( )nh

( )nh

( )nh

( )nh

DD J W ,1

3 J V

3 J W

A J W ,2

D J W ,2

AA J W ,1

DA J W ,1

AD J W ,1

( ) VJ x f

1 J W

1 J V

D J W ,1

A J W ,1

2 J W

2 J V

3 J V 3 J W A J W ,2 D J W ,2 AA J W ,1 AD J W ,1 DA J W ,1 DD J W ,1

J V

1 J V 1 J W

D J W ,1 A J W ,12 J W 2 J V

( ) H

8 4 2 0


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(a) Direct

(b) Polyphase

with

Delay

Figure 2.11. (a) Direct form of the analysis bank and (b) polyphase form of the

analysis bank.

The polyphase matrix multiplies ( ) z X 0 and ( ) z X z 11 to produce ( ) z V 0 and ( ) z V 1 :

( )

( )

( )

( )=

z D

z C

z V

z V

0

0

1

0 ( )

( ) z D

z C

1

1 ( )

( ) ( )

( )

( )=

z X z

z X z H

z X z

z X p

1

1

0

1

1

0 (2.29)

This defines and displays ( ) z H p . For FIR causal filters, the kind we expect to use, the

polyphase components are polynomials in 1 . When the input x is also causal, the

outputs are causal.

The indices in 0 X and 1 X refer to even and odd. The indices in 0V and 1V

refer to the two channels. This is normal for matrix multiplication, when ij H

multiplies j X and contributes to iV . Rows of ( ) z H p go with channels, and columns

of ( ) z H p go with phases.

In an M-channel bank, i is the channel index and j is the phase index in

2 ( )n0

( )n1

( )n x

( )n y0

( )n y1

C

D 2

( )n0

( )n1

( )n x ( )n x0

( )11 n x( ) z H p

2

2


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( ) z H ij . Then iV is the output from channel i , and j X is the th j phase of input. We

often reorganize a filter bank into its polyphase form.


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Chapter 3

IOTA Filter Design

3.1 IOTA Filter

IOTA (Isotropic Orthogonal Transform Algorithm) filter guarantees a quasi-optimal

localization in time/frequency domain; it is obtained by applying the IOTA to the

Gaussian function aiming at orthogonalizing this function. The idea of IOTA function is

to orthogonalize the Gaussian function, which is optimally localized in the time-

frequency space but is not orthogonal to its shifted (in time or frequency) versions. The

temporal orthogonalization of the Gaussian function ( ) f Gauss is defined as follows [6]:

( ) ( )( )

=

k auss

aussausst

k f G

f G f GO

200

(3.1)

Similarly, the frequency orthogonalization of a function ( )t x is defined as:

( ) ( )( )

=

k

f

k t x

t xt xO2

00 (3.2)

The IOTA filter is then defined as:

ausst f GOO=

(3.3)


30/64

Orthogonality of IOTA function is expressed as:

( ) ( ) '''' ,,,,Re nnmm R

nmnm dt t t =

(3.4)

with ( )02, nt ei ft inmnm = + .

Thus, the IOTA function has the following properties [7] [8]:

It is identical to its Fourier Transform, so the OFDM/IOTA signal is affected

similarly by the time and frequency spreading due to propagation conditions.

The time-frequency localization is quasi optimal as the IOTA function does

not differ a lot from the Gaussian function.

For the practical realization, we generate the IOTA function in the case:

(3.5)

We call this particular function21

, so we have:

(3.6)

with

(3.7)

2100 ==

41

21 2.)( f

t f eOOt =

( ) ( )

( ) =

k

t

k f X

f X f X O

2

221


31/64

and

(3.8)

here2

41

2)( f auss e f G = .

In practice, we cannot numerically compute perfectly orthogonal IOTA function as

given by (3.6), and we have to use an approximate expression given as a finite series

expansion. The approximate IOTA function, denoted21

, are given by [9] [10]:

( ) ==

+

+=

s s K

l l

K

k aussaussk

t l d

k t G

k t Gd t

021,

021,21 21

2cos21212

1 (3.9)

where ( ) ( )( ) 1,sup K K K s = , with ( ) K the required numbers of terms. Here

( ) 11= K .

( ),0

2,, 0

=

+=k j

j

k j jk k ebd

K k 0 (3.10)

where ( ) 2k K jk = , in Table 3.1, a list of jk b , coefficients corresponding to 14= K

is reported. This leads to an accuracy around 191079.0 [11].

( ) ( )

( ) =

k

f

k t x

t xt xO

2221


32/64

Table 3.1. First jk b , coefficients.

43

64105

256675

65536457107

104857612097169

419430470545315

1619

256213

83886084063017

167772165014575

2097152676039

13107246189

163846435

512231

1677721620050485

6553638753

10485761400487

41943049895893

20481395

131072202281

524288

1434705

1677721685037895

40967797

1638456163

209715213861065

838860887185895

6435

43

5121545

65536596277

2621443679941

20489765

16777216394159701

1

16384762331

12863

1024429

1310722067909

16384

16192564

2198

15

10246055

1048576

26060847

3276820691

262144374325

20971525297445

1024763

8192

13875262144790815

4194304

23600537

26214488179

41943042120495

3276812155

262144146289

128123

41943041300075

20971522641197

10242289

819234871

131075969375

85

4194304

51182445

j 0( to )7

k

0

to

14

524288277797

(

)

Figures 3.1 and 3.2 show the simulation results of various representations of the

IOTA function.


33/64

-4 -3 -2 -1 0 1 2 3 4-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Symbol duration

A m p

l i t u d e

IOTA

-4 -3 -2 -1 0 1 2 3 4-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Inter-carrier spacing

A m p

l i t u d e

IOTA Fourier Transform

Figure 3.1. IOTA waveform and its Fourier transform.

-4 -3 -2 -1 0 1 2 3 4-160

-140

-120

-100

-80

-60

-40

-20

0

20

A m p

l i t u d e

( d B )

Normalized intercarrier spacing

IOTA Function Fourier Transform (logarithmic)

Figure 3.2. Fourier transforms of IOTA function.

3.2 Polyphase IOTA Filter Design

The IOTA function ( )t is defined on [ ] , but practically, it is truncated to the

interval [ ]00 , L L . Usually, to have both good performances and reduced complexity

L = 4. This choice leads to a prototype function of length 4 N where N is the FFT

size. The filtering is performed using polyphase filters. That is, with this length of

prototype filter, we would have N polyphase of length 8.

( )t


34/64

To perform the filtering, we compute coefficients of the

function , equally spaced. Here N is the FFT length and M=N /2. The ideal IOTA

filter would be infinite but at it vanishes rapidly; it can be truncated to a reduced number

of coefficients. They are denoted k with k = 0 12 L . We have for each k :

(3.11)

As the IOTA function is even we have k MLk = 12 .

Each of the M polyohase components of the filter has 2L coefficients. We use a

circulating buffer as defined in Table 3.2.

Table 3.2. The IOTA filter polyphase buffer.

0

M

M 2 M L )12(

1 1+ M 12 + M 1)12( + M L

M M M O M

1 M 12 M 13 M 12 LM

For example, with L=2 and M=4 (that is with an FFT size of N=2M=8), we would

have the buffer represented in Table 3.3.

Table 3.3. The buffer with L=2 and M=4.

0 4 8 12

1 5 9 13

2 6 10 14

3 7 11 15

ML L N

222

=

)(21 t

)22

(2

121

0

L

M

k k

=


35/64

If we denote nmC , the outputs of the IFFT where m is the sub-carrier index in

[02 M 1], an n the time index, we will have the following table after the first outputs

of the IFFT:

Table 3.4. The buffer after 1 symbol.

00,0 C C 4,04 C 0,08 C 4,012

10,1 C C 5,05 C 1,09 C 5,013

20,2 C C 6,06 C 2,010 C 6,014

30,3 C C 7,07 C 3,011 C 7,015

At each 0 the M outputs of the buffer corresponds to its first column, then the

columns of the buffer are shifted to the left.

Table 3.5. The buffer after 2 symbols.

C 4,04 + C 0,10 C 0,08 + C 4,14 C 4,012 + C 0,18 121,4 C

C 5,05 + C 1,11 C 1,09 + C 5,15 C 5,013 + C 1,19 131,5 C

C 6,0

6 + C 2,1

2 C 2,0

10 + C 6,1

6 C 6,0

14 + C 2,1

10141,6 C

C 7,04 + C 3,13 C 3,011 + C 7,17 C 7,015 + C 3,111 151,7 C

The general formula for the transmitted coefficients s i is:


36/64

i M qqk M iiqM L

qqk iikM C C s +++++

=+ += )12()12(,2

1

02, .. (3.12)

where C is the output of IFFT, and is IOTA filter coefficient.

At the receiver side, the same filter is applied to the signal before the FFT

operation, but the algorithm is slightly different, we give the general formula in the

below:

=+++ =

1

022,

L

qqM iqM jM i ji s R (3.13)

for i in [02 M 1] and j N .

With this simple example, we clearly see that with an FFT size of N , the number of

complex samples in one OFDM/IOTA symbols equals2

N M = , which illustrates that the

OFDM/IOTA symbol duration 0 is twice shorter than in classical OFDM.


37/64

Chapter 4

Design of OFDM/OQAM-IOTA System

4.1 OFDM/QAM System

The technique of Orthogonal Frequency Division Multiplexing (OFDM) is based

on the well-known technique of Frequency Division Multiplexing (FDM). The OFDM

technique differs from traditional FDM in the following interrelated ways:

Multiple carriers (called sub-carriers) carry the information stream

The sub-carriers are orthogonal to each other

A guard time may be added to each symbol to combat the channel delay spread

In OFDM/QAM modulation, an OFDM signal consists of a sum of subcarriers that

modulated by quadrature amplitude modulation (QAM), if nmd , are the complex QAM

symbols, the OFDM signal can be expressed as:

=

=

+=

=

=1

0

)2( )()( M m

mu

t f mimn

n

n

nT t g ed t s (4.1)

where nmd , is the transmitted OFDM symbol (Complex value) on thethm sub-carrier at

thn symbol, M is the number of sub-carriers.

A guard interval is added prior to each useful OFDM symbol. This guard time is

introduced to minimize the inter-OFDM-symbol-interference power caused by time-

dispersive channels. The guard interval duration T g must be hence sufficient to cover


38/64

the most of the delay-spread energy of a radio channel impulse response.

At the receiver, a computationally efficient Fast Fourier Transform(FFT) is used to

demodulate the multicarrier information and to recover the transmitted data.

Figure 4.1 shows the transmission/reception chain of OFDM/QAM system.

Figure 4.1. OFDM/QAM system.

4.2 OFDM/OQAM-IOTA System

Classical OFDM modulation using guard interval is well known for its robustness to

multi-path time varying propagation channels. OFDM/OQAM-IOTA modulation is an

alternative to it, which has the advantage not to require the use of a guard interval, this

leads to a gain in spectral efficiency .

For this purpose, the prototype function modulating each sub-carrier must be very

well localized in the time domain, to limit the inter-symbol interference. Moreover, it

can be chosen very well localized in the frequency domain, to limit the inter-carrier

interferences (doppler effects, phase noise). This function must also guarantee

orthogonality among sub-carriers and among multi-carrier symbols. Functions having

these characteristics exist, which guarantee the orthogonality only in real domain.


39/64

Consequently, the complex QAM data stream ( cmn) must be separated into its two real

components: real part ( amn) and imaginary part ( bmn.) (see Fig. 4.1), the imaginary part

being modulated with a half-symbol-duration ( T u /2 ) shifted version of the modulation

filter (thus the connotation Offset). It is important to notice that the density of the time-

frequency frame related to OFDM/OQAM equals 2, i.e. 21

00 = . In other words, this

modulation is as spectral efficient as OFDM/QAM without guard interval. Indeed, c.f.

Fig 4.1, for a given inter-carrier spacing 0 , on each sub-carrier, OFDM/OQAM carries

one real value each 0 while OFDM/QAM without guard interval carries one complex

value each 02 [12].

The classical OFDM signal (without cyclic prefix) can be expressed as (4.1), where

g(t) is a rectangular filter. By separating the two parts of ( cmn), the corresponding

OFDM/OQAM modulated signal can be written as:

( ) +=

=

=

=

++=

n

nu

u ft mimmn

M m

mu

ft mimmn nT

T t g eiibnT t g eiat s 2)()2(

1

0

)2( (4.2)

where ( )t g is the prototype function (noted )(t in the case of IOTA). In a more

concise writing, this gives:

nmnmnmn

M

m

ft minmnm bor ad nt eid t s ,,,

1

00

2, ,)()( ==

=

+ (4.3)

where nmd , : Real information value(Offset QAM) sent on thethm sub-carrier at


40/64

thn symbol, is the OFDM/IOTA symbol duration, is the sub-carrier spacing,

is the IOTA filter. Orthogonality is guaranteed if [13]

( ) ( ) '''' ,,,,,, Re nnmmnmnm Rnmnm dt t t =

=

(4.4)

Note that data is multiplied by nmi + prior to modulation in order to have

orthogonality in real domain [13] [14].

0

uT = 02

0

Figure 4.2. OFDM/OQAM time and frequency lattices

(compared to OFDM w/o guard interval).

Figure 4.2 illustrates the OFDM/IOTA system transmission /reception chain. The

demodulated data is obtained by taking the real part of the projection of the received

signal on the corresponding matched poly-phase IOTA filter. And at the reception,

0 0


41/64

specific channel equalization might be required in difficult propagation conditions,

which would be more complex than the simple operation of division required by

classical OFDM.

nmd , nmi +

nmi +

( )t s

nmd ,

Figure 4.3. OFDM/OQAM-IOTA system.

4.3 Complexity Analysis of OFDM/OQAM-IOTA System

4.3.1 OFDM System

The complexity of classical OFDM system mainly determined by 2 parts: IFFT in

the transmitter and FFT in the receiver.

4.3.1.1 Transmitter Side

In the transmitter, complexity of IFFT and FFT is composed of complex

multiplication and complex addition.

IFFT (Complex multiplication) :


42/64

Complex Multiplication/symbol ( N -point IFFT & FFT, Radix-2, 512= N ):

( ) ( ) N N 2log2 = ( ) 2304512log256 2 = ;

Complex addition/symbol:

( ) N N 2log = ( ) 4608512log512 2 = ;

4.3.1.2 Receiver Side

Complexity of the receive side is almost the same to the transmit side.

FFT (Complex multiplication) :

Complex Multiplication/symbol (Radix-2):

( ) ( ) N N 2log2 = ( ) 2304512log256 2 = ;


( ) N N 2log = ( ) 4608512log512 2 = ;

4.3.2 OFDM/IOTA System

The use of polyphase IOTA filter brings more complexity to both of the

transmitter and receiver of this novel scheme, the main parts including symbol

dephase part, IFFT, polyphase IOTA transmitter filter in the transmitter, and the

corresponding parts in the receive side.

4.3.2.1 Transmitter Side

In the transmitter, there are three parts, complexity increase mainly due to

the IFFT and polyphase IOTA filter.


43/64

Symbol Dephase Part :

Symbol dephase part is to multiply the transmitted signal with nmi +

after separating the complex, where m is the thm sub-carrier, n is the

time index.

The value of nmi + has 4 cases: i , i ,1,-1, hardware has already saved

these 4 values, so there is no need multiplication in symbol dephase part.

Hardware space retain 2 spaces for saving real & imaginary value, for

example, 2 complex value: bja + , djc + , we will save in the hardware space in

the follows:

Table 4.1. Hardware space for real and imaginary value.

Real Imag

Multiplicand a b

Multiplier c d

Product ac-bd ad+bc

IFFT :

The OFDM/IOTA symbol is obtained by separating the complex

OFDM symbol ( )bja + into 2 real components before IFFT (see Figure

4.4).


44/64

[ ]260 x[ ]259 x

0 N W

0 N W

0 N W

0 N W

2 N W

0 N W

0 N W

0 N W

0 N W

[ ]256 x

2 N W

2 N W

2 N W

0 N W

0 N W

0

N W

0 N W

248 N W

[ ]0 x

[ ]0 x

[ ]4 x[ ]3 x[ ]2 x[ ]1 x

[ ]6 x[ ]5 x

[ ]7 x

[ ]4 x

[ ]3 x

[ ]2 x

[ ]1 x

[ ]6 x

[ ]5 x

[ ]7 x

[ ]258 x

[ ]263 x[ ]262 x[ ]261 x

[ ]257 x

[ ]256 x

[ ]258 x

[ ]260 x

[ ]262 x

[ ]257 x

[ ]259 x

[ ]261 x

[ ]263 x255

N W

250 N W

253 N W

254 N W

252 N W

251 N W

249 N W

Figure 4.4. 512-point decimation-in-frequency IFFT.

Complex Multiplication/symbol (Radix-2):

( ) ( )( )1log2 2 N N = 20488256 = ;

Real Multiplication/symbol:

5122256 = ;


( )( )1log 2 N N = ( ) ( )( ) 40961log512 2 = N ;

Real addition/symbol:

512 ;

Poly-phase Transmit Filter :

The general formula for transmitted coefficients i s is:


45/64

(4.5)

wherek iC , : the outputs of the IFFT (Complex value);

: IOTA filter coefficients (Real value);

i : the sub-carrier index in [ ]12...0 M ;

k : the time index;

is equal to N .

The IOTA function ( )t is truncated to the interval [ ]00, L L , to have

both good performances and reduced complexity, we choose 4= L .

IOTA filter taps = 2048425622 == L N ;


40968256282 == N ;

Complex Addition/symbol:

179272567 == N ;

Memory for poly-phase coefficients :

4.3.2.2 Receiver Side

In receiver side, the algorithm of IOTA receive filter is a little different, but

the complexity of FFT and polyphase receive filter is not much different

compared to that of the transmit side.

i M qqk M iiqM

L

qqk iikM C C s +++++

=+ += )12()12(,2

1

02, ..

ts;coefficienfor bitsof #:Cwhere: bits2048 C


46/64

FFT :

Complex multiplication/symbol (Radix-2):

( ) ( ) N N 2log2 = ( ) 2304512log256 2 = ;


( ) N N 2log = ( ) 4608512log512 2 = ;

Poly-phase Receive Filter Part :

At the receiver side, the same filter is applied to the signal before the

FFT operation, but the algorithm is slightly different.

=+++ =

1

022,

L

qqM iqM jM i ji s R (4.6)

where s : Received signal after multi-path channel

: IOTA filter coefficients

for i in [ ]12...0 M and N j .

Filter taps = 2048425622 == L N ;


;40968256282 == N

Complex Addition/symbol:

;204882568 == N

Memory for poly-phase coefficients :

ts;coefficienfor bitsof #:Cwhere: bits2048 C


47/64

4.3.3 Summary

In the following, we give the total real multiplication/symbol and real

addition/symbol of OFDM system and OFDM/IOTA system separately.

Total real multiplication/symbol of OFDM system:

( ) ( ) 24log2 2 N N 18432= ;

Total real addition/symbol of OFDM system:

( ) 22log 2 N N 18432= ;

Total real multiplication/symbol of OFDM/IOTA system:

( ) ( )( ) ( ) ( )2 22 log 1 4 2 8 2 log 4 2 8 N N N N N N N + + + + 26112= ;

Total real addition/symbol of OFDM/IOTA system:

( )( ) ( )2 2log 1 2 7 2 log 2 8 2 N N N N N N N + + + + 25600= ;

Complexity comparisons between OFDM and OFDM/IOTA are given in Table 4.2

and Table 4.3.

Table 4.2. Complexity comparison between OFDM and OFDM/IOTA (1).

OFDM OFDM/IOTA

Tx Rx Tx Rx

IFFT FFT IFFT

Polyp

hase fi

lter

Polyphasefilter

FFT

R e a l

Multiplication

N=51

2

8 N

= 4 0 9

6

8 N

= 4 0 9

6

Compl

ex

multiplication

( ) N 2log2

=2304

( ) N log2

=2304

( ) 22 (log N

=2048

( ) ( N 2log2

=2304


48/64

Real add

ition

N=512

Complex

additio

n

( N 2log

=4608

( N N 2log

=4608

2(log N N

=4096

2 N

=1792

82 N

=2048

( ) N N 2log

=4608

Table 4.3. Complexity comparison between OFDM and OFDM/IOTA (2).

OFDM OFDM/IOTA Ratio

Total # of equivalent

real multiplication 18,432 26,112 1.42

Total # of equivalen

t

real addition

18,432 25,600 1.39

OFDM/IOTA system increases complexity compared to classical OFDM system,

this is mainly due to the IOTA filter part.


49/64

Chapter 5

OFDMA-IOTA System vs. OFDM/CDMA-

IOTA System

5.1 Ideal Channel Estimation Method

OFDMA-IOTA is less robust to Rayleigh fading channel. When classical channel

estimation used for OFDM modulation is applied straight forwardly to OFDM / OQAM

modulation, an intrinsic Inter-Symbol-Interference is observed. This severely degrades

its performances, so we use a specific symbol sequence based approach to reduce this

interference.

We suppose that the OFDM/OQAM modulation parameters are chosen to ensure

that the channel is frequency selective on the signal bandwidth and can be modeled as a

complex Gaussian random coefficient on each sub-carrier:

( ) ( ) ( )

=

+=n

M

mnmnmnm t bt a H t r

1

0,,, (5.1)

where nma , denotes the real information value (Offset QAM) sent on thethm sub-

carrier at the thn symbol, is the IOTA function. If ( )00 , nm denotes a reference

sub-carrier, we must look at the complex projection of the received signal ( )t r on

00 ,nm

to estimate 00 ,nm H :


50/64

( ) ( )00

00

00

,

,,

nm

nmnm a

dt t g t r H

= (5.2)

In classical OFDM, this extimation is optimal. In OFDMA-IOTA, the complex

projection induces an intrinsic ISI due to the real orthogonality of the subcarriers. To

illustrate this phenomenon, lets suppose that the channel is ideal: ( ) ( )t st r = .

( ) ( )( ) ( )

+=

00

000000,,

,,,,,nmnm

nmnmnmnmnm aadt t t s (5.3)

The second item00 ,nm

I represents the intrinsic interference, whose real part is null.

As the prototype function is chosen well localized, the main interfering term is due to

the sub-carriers adjacent to the ( )00 , nm sub-carrier.

( )

=

0,01

0000

,

,,,,1

nmnm

nmnmnmnm aC (5.4)

where 00 ,1 nm represents the indexes of these 8 adjacent sub-carriers (1st order neighbor

sub-carriers). So we propose to reduce the intrinsic interference by annulling 00 ,1 nmC on

each sub-carrier used for channel estimation.

If n 0 is the OFDM/OQAM symbol that is looked at for channel estimation, we

propose a symbol sequence that guarantees:


51/64

{ },2,...,1 M m 00,1 =nmC (5.5)

This structure depends on the parity of n 0. Therefore, symbol sequence for the case of

even centre symbol is illustrated on Fig. 5.2(a), and for the case of odd symbol, symbol

sequence to reduce intrinsic ISI can be similarly formulated. Symbol sequence maybe

composed of BPSK symbols { }ee + ; .

It can be demonstrated that these structures set every N th order intrinsic ISI

N nmC 00 , to zero ( )2 N [15].

f

0

10 n

0n

10 +nt ( ) carrier subnm 00 ,

0m10 m 10 +m

0

Figure 5.1. st 1 order neighbour sub-carriers.

0n

20

n

30 n

30 +n20 +n10 +n

10 n

e+ e

0n

20 n

30 n

30 +n20 +n10 +n

10 n

e+ e

a) even centre symbol b) odd centre symbol

Figure 5.2. Symbol sequence for intrinsic ISI reduction.


52/64

Based on the specific symbol sequence, the synoptic of this new ideal channel

estimation is illustrated on Figure 5.3.

k n H ,2

k n H ,12 +

k n H ,

Figure 5.3. Symbol-sequence-based ideal channel estimation.

In the following, we verify through simulations that this method provides reliable

bounds of performances. We test this symbol-sequence based channel estimation

method under AWGN and 1-path Rayleigh fading channel environment. It can obtain

good results, but under multipath channels, error appears. In this case, we can try to

apply similar method to realistic channel estimation with scattered pilot symbols [16].

We will go on this research work in the future.

5.2 OFDM/CDMA-IOTA System Design

The OFDM-CDMA transmitter spreads the original signal using a given spreading

code in the frequency domain. The basic transmitter structure of OFDM-CDMA scheme

is similar to that of a normal OFDM scheme. The main difference is that the OFDM-

CDMA scheme transmits the same symbol in parallel through many sub-carriers,

whereas the OFDM scheme transmits different symbols.

Similar to OFDMA-IOTA system, OFDM/CDMA-IOTA uses IOTA filter as the

pulse waveform, and IOTA filter is implemented by the poly-phase technique as used in


53/64

OFDMA-IOTA system. Figure 5.4 illustrates the scheme of OFDM/CDMA-IOTA

system, where we use Hadamard Walsh code as an optimum orthogonal set, with the

processing gain 4= MC G .

Similar to OFDMA-IOTA, the transmitted signal of the j th user of OFDM/CDMA-

IOTA is written as:

( ) ( )=+

=n

G

m

ft m j jm

nm jnm

j MC

MC

nt ecid t s1

02

,

(5.6)

where d m,n j denotes real information value (Offset QAM) sent on the mth sub-carrier at

the nth symbol of user j, cm j is Walsh code for the jth user, f is the sub-carrier

separation, and is the IOTA filter.

In receiver side of OFDM/CDMA, after the serial-to-parallel conversion, the mth

sub-carrier is multiplied by the gain qm to combine the received signal energy scattered

in the frequency domain. We believe this is the main advantage of OFDM/CDMA

scheme over other schemes. This is also used in our OFDM/CDMA-IOTA scheme. The

decision variable is given by [17]

=

= MC G

mm

jm

j yq D1

'' (5.7)

The gain for the maximum ratio combining (MRC) is given by

'*'' jm jm jm z cq = (5.8)


54/64

where*' j

m z is conjugate of channel coefficient.

In the case of one user, the maximum ratio combining method can minimize the

BER [18].

Figure 5.4. OFDM/CDMA-IOTA system.


55/64

5.3 Performance Comparison between OFDM and OFDM-IOTA

Systems

Table 5.1. General link level Simulation parameters.

Parameters Values

Carrier frequency 2GHz

Signal bandwidth 4.8MHz

Sampling frequency 7.68MHz

Vehicle speed 50 Km/hFast fading model Jakes spectrum

Modulation QPSK/OQPSK

Channel coding

Convolutional Encoder

/ Viterbi Decorder

(K=7,[133 171])

Code rate

(QPSK/OQPSK)1/2

Interleaver Block Interleaver(504 OFDM Symbols)

Channel estimation Perfect Compensation

Table 5.2. Simulation parameters.

arameters Values

Useful OFDM symbol duration 66.66 us

Cyclic prefix duration 16.67 us

Overall OFDM symbol duration 83.33 us

Inter-carrier spacing 15 kHz

FFT size 512

Modulated sub-carrier 320


56/64

Table 5.3. Comparison of OFDM and OFDM-IOTA.

Modulation Code Rate

OFDM/QPSK Information

Bit Rate[Mbps]

OFDM/ OQPSK -IOTA

Information Bit Rate[Mbps]

Increase of Spectral

Efficiency

QPSK 1/2 3.84 4.8 25%

0 2 4 6 8 10 12 14 16 18 2010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No

OFDM/QPSK vs OFDM-OQPSK/IOTA

OFDM-OQPSK/IOTAOFDM/QPSK

BER

(a) AWGN

0 5 10 15 20 2510 -6

10-5

10-4

10 -3

10-2

10 -1

100OFDM/QPSK vs OFDM/OQPSK -IOTA in 1 path Rayleigh fading

OFDM/OQPSK-IOTAOFDM/QPSK

Eb/N0

BER

(b) 1-path Rayleigh fading channel (50 km/h)

Figure 5.5. BER performance of OFDM-IOTA system vs. OFDM system.


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Simulation results show that OFDM-IOTA system outperforms OFDM system. In

AWGN channel, OFDM-IOTA system outperforms OFDM system 1dB, and in 1-path

Rayleigh fading channel, at target BER 10 4 , OFDMA-IOTA outperforms OFDM

almost 3 dB. That is because although OFDM modulation scheme is robust to channel

induced ISI in time domain, it is very sensitive to ICI (Inter-Carrier Interference) due to

doppler effect in frequency domain. But OFDM-IOTA modulation scheme utilizes

IOTA filter that has orthogonality in time and frequency domain, it guarantees a quasi-

optimal localization so OFDM-IOTA is robust to delay spread and doppler effect.

5.4 Performance Comparison between OFDMA-IOTA and


In Fig. 5.6, comparison scenario of OFDMA-IOTA system and OFDM/CDMA-

IOTA system are given.

(a) OFDMA-IOTA system (b) OFDM/CDMA-IOTA system

Figure 5.6. Comparison scenario of OFDMA-IOTA vs. OFDM/CDMA-IOTA.

Table 5.4, 5.5 and 5.6 give the simulation parameters of link level, OFDMA-IOTA


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system and OFDM/CDMA-IOTA system respectively.

Table 5.4. General link level parameters.

Parameters Values

Carrier Frequency 2 GHz

Signal Bandwidth 4.8 MHz

Sampling Frequency 7.68 MHz

Vehicle Speed 60 Km/h

Modulation QPSK

Fast Fading Model Jakes Method

Channel EstimationIdeal Channel

Estimation

Table 5.5. OFDMA-IOTA system parameters.

Parameters Values

FFT Size 512

Number of Modulated Sub-carriers 320

Inter-carrier Spacing 15 KHz

OFDM/IOTA Symbol Duration 66.66 us

Table 5.6. OFDM/CDMA-IOTA system parameters.

Parameters Values

FFT Size 512

Inter-carrier Spacing 15 KHz

MC/CDMA-IOTA Symbol Duration 66.66 us

Spreading Factor 4

Number of Users 4

BER results of OFDMA-IOTA versus OFDM/CDMA-IOTA are provided in Fig.

5.7.


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0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100OFDMA-IOTAOFDM/CDMA-IOTA (1 user)OFDM/CDMA-IOTA (4 users)

Eb/N0

BER

(a) AWGN channel

0 5 10 15 20 25 30

10-4

10-3

10-2

10-1

100

OFDMA-IOTAOFDM/CDMA-IOTA (1 user)OFDM/CDMA-IOTA (4 users)

Eb/N0

BER

(b) 1-path Rayleigh fading channel (60km/h)

Figure 5.7. BER performance of OFDM/CDMA-IOTA vs.OFDMA-IOTA.

The performance of OFDM/CDMA- IOTA (1 user) is similar to that of OFDMA-

IOTA under both AWGN and 1-path Rayleigh fading channel. But in this case, due to


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the spreading factor of 4, total transmission data rate of 1 user OFDM/CDMA-IOTA is

equal to the 1/4 of total transmission data rate of 1 user OFDMA-IOTA.

However, OFDM/CDMA-IOTA (4 users) performs worse compared to OFDMA-

IOTA. In AWGN, at target BER 10 3 , OFDMA-IOTA outperforms OFDM/CDMA-

IOTA (4 users) almost 6 dB, in 1-path Rayleigh fading channel, at target BER 10 3 ,

OFDMA-IOTA outperforms OFDM/CDMA-IOTA 5 dB. This comparison is under the

condition of same total transmission data rate. Because spreading factor of 4 is

adaptable for OFDM/CDMA-IOTA, so total data rate of 4 users case of

OFDM/CDMA-IOTA is same to total date rate of OFDMA-IOTA, if whole number of

subcarriers is divided equally to the 4 users in OFDMA-IOTA system.

The combining technology used in OFDM/CDMA-IOTA is not optimal, and the

interference between different users also make OFDM/CDMA-IOTA performs a little

worse compared to OFDMA-IOTA.


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Chapter 6

Conclusions

This paper provides two novel systems: OFDMA-IOTA and OFDM/CDMA-IOTA

that use IOTA function as their prototype filter. IOTA filter is well known for its good

characteristic that can guarantee a quasi-optimal localization in Time/Frequency domain.

But IOTA filter only guarantee orthogonality on real values, that is why we use offset

QAM modulation(In this paper, we do the simulation by QPSK modulation).

OFDMA-IOTA and OFDM/CDMA-IOTA are very spectral efficient, but require

specific treatment at the reception, to limit what we called the intrinsic ISI. We adapt a

new channel estimation method that based on a specific symbol sequence. Finally

OFDMA-IOTA and OFDM/CDMA- IOTA are compared on their BER performance.

Through comparison, OFDMA-IOTA system outperforms OFDM system in both

AWGN and 1-path Rayleigh flat fading channel.

Due to the inherent orthogonality in OFDMA-IOTA, OFDMA-IOTA performs

better than OFDM/CDMA-IOTA under fair comparison condition of the same

transmission data rate.

It is an interesting challenge to utilize OFDM-IOTA as the novel wavelet platform

of 4G environment.


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References

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for personal communications systems, in Proc. IEEE Int. Conf. on Acoustics,

Speech, and Signal Processing, May. 1996, vol. 3, pp. 1443-1446.

[2] Rafael C.Gonzalez and Richard E. Woods, Digital image processing, Prentice Hall,

Apr. 1995.

[3] P.P Vaidyanathan, Multirate systems and filter banks, Prentice Hall, Sept. 1992.

[4] Gilbert Strang, Truong Nuyen and T. Ngugen, Wavelets and filter banks,

Wellesley-Cambridge Press, Oct. 1996.

[5] Alan V.Oppenheim and Ronald W.Schafer, Discrete-Time signal processing,

Prentice Hall, Feb. 1995.

[6] 3GPP TSG-RAN-1 Meeting #31, Technical description of the OFDM/IOTA

modulation, Feb. 2003.

[7] 3GPP TR 25.892 v 0.4.1, Oct. 2003.

[8] 3GPP TSG-RAN-1 Meeting #30, IOTA, an advanced OFDM modulation for

UTRAN enhancement: presentation and potentials, Jan. 2003.

[9] Pierre Siohan and Christian Roche, Derivation of extended Gaussian functions

based on the Zak transform, IEEE signal processing letters , vol. 11, pp. 401-403,

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[10] Christian Roche and Pierre Siohan, A family of extended gaussian functions with

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[14] Joo Heo, Xiao Zhou and KyungHi Chang, OFDM/OQAM-IOTA System, in Proc.

KONI Conf, Oct. 2004, vol. 8, No. 2, pp. 179-183.

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Acknowledgements

This thesis is completed under my professor KyungHi Changs guidance and

encouragement. First I give my deepest regards and thanks to him, my dear professor,

who teach me to study in the field of mobile telecommunication for Masters degree in

Korea. During these two years, he encourages me to study hard, directs good life road

for me, gives me great help on guiding my studying and research, and supports my life.

During the process of writing this thesis, he is concerned all along and gives me many

valuable comments and suggestions. And I also want to say thanks to professors of

Graduate School of Information and Telecommunications, Inha University for their

directions and help.

My lab mates also give me a lot of good suggestions on my study and research.

When I meet difficulties in study and research, they always help me patiently. Here I

want to express my sincere thanks to them. They are Mr. Joo Heo, Mr. InSuk Cha, Mr.

SungHo Park, Mr. JungJu Gim, Mr. JungHo Noh, Mr. YuPeng Wang and Mr. YiXian

Chen. Thanks to my Chinese friends Mr. GuangYun Jin, Mr. WeiHua Zhang, Mr.

QingHai Yang, Mr. HuaMin Zhu and Mrs. ShaoYi Xu for their kind help on my study

and my 2 years study abroad life.

I am extremely grateful to my dear parents, whose love, endurance and sacrifice

has made my research possible. Finally, I dedicate the thesis to them for their hard

fostering, great direction and eager hope.

iota filter

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