A Wavelet based Filtered Multi-Tone

Roman M. Vitenberg

Wavetone Technologies Ltd roman@wavetonetech.com

Abstract This paper discusses the performance of the

Wavelet based Filtered Multi-Tone (WFMT) modulation. The

novel WFMT modulation was proposed in 2003 for improving

characteristics of Wireless and DSL multicarrier systems. In this

paper we describe a main idea of WFMT and discuss the

advantages of this novel modulation. The WFMT modulation in

comparison with OFDM and DMT has low level of out-of-band

side lobes, low sensitivity to narrowband RF interference and low

Peak-to-Average Power Ratio. The attractive feature of WFMT

is its low implementation complexity compared with known

multicarrier architectures, that are based on FIR synthesis and

analysis filter-banks. Several main characteristics of the WFMT

are illustrated by simulation results. The practically realized

WFMT system is described followed by the test results.

Keywords FMT, WFMT, filter-bank, wavelet, PAR, OFDM,

DMT.

I. Introduction

Filter-bank multicarrier modulation (FBMCM) has

received a significant interest in the area of broadband wired

(DSL) and wireless (LAN) access systems. In filter-bank

systems the data symbols are transmitted over a number of

independent sub-carriers, using a frequency division

multiplexing (FDM) [13]. This contrasts with OFDM or DMT

systems, where separations of sub-channel signals are

provided using the orthogonality of the sub-carriers [1].

Over the past three decades, many different filter-bank

based multicarrier systems have been introduced. All these

systems may be divided into three classes in accordance with

the type of sub-channel modulation.

The first class is a filtered multi-tone system (FMT) that

comprises a number of the spectrally non-overlapped

sub-channels. Each of them transmits information by double

side modulated QAM [2]. Because the spectrums of adjacent

sub-channels do not overlap, the inter-channel-interference

(ICI) in FMT system is very low. As a result, FMT may be

seen as a number of single-carrier QAM systems operating in

parallel [15]. Hence, characteristics of this system can be

simply defined. In [2] an efficient implementation of FMT

that uses polyphase techniques was described. One of the

FMT disadvantages is the complexity of the synthesis and

analysis filter-banks. It is clear that for achieving a high

performance, the FMT system must use the sub-channel

filters with roll off factor of 15.0≤β to avoid loosing more

than 15% of channel bandwidth. In case of digital

implementation, each sub-channel FIR filter can have a length

of about 512~1024. The second problem is a complexity of

FMT equalizer. The typical FMT receiver described in [2]

required one decision feedback equalizer per sub-channel.

Such an equalizer must use an adaptive FIR filter that has at

least 64 coefficients.

The second class of the filter-bank system comprises of

transceivers, which use sub-channel signals modulated by

single-side-band PAM modulation [15]. In [23] a system with

Sub-band Division Multiplexing (SDM) modulation

developed by Rainmaker Technology LTD was described.

The Discrete Wavelet Multi-Tone (DWMT) modulation was

proposed by AWARE Ltd [16]. Another version of SDM,

named Wavelet OFDM was proposed for wireless and power

lines [19]. The Cosine Modulated Multi-Tone (CMT)

modulation [22] and The Cosine Modulated Filter-Bank

(CMFB) modulation [17], were investigated in many

contributions. Recently a perfect-reconstructed Exponentially

Modulated Filter-Bank (EMFB) that uses the complex

numbers for filter coefficients was developed [14]. All these

filter-bank systems use an overlapped spectrum of an adjacent

sub-channel. The division of the adjacent sub-channels in the

receiver is provided by an orthogonality of transmitted signals

The third class of filter-bank systems proposed by

Salsberg [21] uses a special Offset Quadrature Amplitude

Modulation (OQAM) for the transmission of data over

sub-carrier bands spaced by symbol rate. In this system, the

adjacent bands have significant overlapping. Successful

separation of sub-channel signals is possible thanks to an

orthogonality of real and complex components of OQAM

[15].

Unfortunately the filter-bank systems described in

literature are very complex and costly so they practical

realization is difficult. Up to date only two filter-bank

techniques were implemented both based on wavelet

modulation proposed by Rainmaker Technologies. The first

is SDM that was used in prototype of the CATV

communication system [23] demonstrated by Rainmaker

Technologies (now Broadband Physics) on several technical

Fairs. The second practical realization of the filter-bank

technology is the Wavelet OFDM technique that was

proposed by Rainmaker [19] and improved by Matsushita

Electric [20].

Panasonic has demonstrated the first industrial product

based on Wavelet OFDM in 2005 (HD-PLC - a 200 Mb/sec

modem for Power Line Communications). One of problem of

the filter-bank technique is the high sensitivity to frequency

distortion in the communication line. This requires a using of

complex decision feedback equalizers in each sub-channel of

the system. Therefore, the filter-bank systems could be used

only in the case of low phase-frequency distortion, for

example in CATV or Power Line networks.

The Wavelet based Filtered Multi-Tone (WFMT)

modulation was proposed by R. M. Vitenberg ( 2003) [5]. The

WFMT characteristics were researched in during 2003-2005

in DATA JCE IC Design Center and in Wavetone

Technologies LTD 2005-2006.

This paper presents a novel WFMT technology and its

characteristics. The rest of the paper is organized as follows.

We present an overview of WFMT in Section II. In Section III,

we describe the practical system that was realized and tested

by our company in 2002 - 2003. This system was developed

for VDSL application and provided bit rates up to 24 Mbps in

the frequency band 138kHz~5200kHz; another system was

build for transmission of CATV signals over coaxial cable [9].

The picture of power spectrum density of the transmitted

WFMT signal illustrates the experimental results followed by

conclusions.

II. WFMT Overview

The Wavelet based Filtered Multi-Tone Modulation is a

version of the Filter-bank modulated communication system

described in detail in [2], [3], [4], in which the synthesis and

analysis of sub-channel wavelets were correspondingly

provided by IFFT and FFT cores. Unlike OFDM, the number

M of sub-channels in the WFMT system is significantly less

than N - the number of IFFT/FFT points. Each sub-channel

wavelet is generated in this case from K harmonics (IFFT/FFT

points) so there is a simple expression for the number of

sub-channels:

K

NM ≤ (1).

We start an explanation of the WFMT technique from a

comparison block schematics of OFDM and WFMT

transmitters shown correspondingly on Fig. 1 and Fig. 2.

IFFT

N pointP/S

0 N-1

0

X(t)

0

OFDM Signal

f

a0(iT)

a1(iT)

aN-3(iT)

aN-2(iT)

F

Figure 1. OFDM Transmitter

As we see, the OFDM Transmitter comprises a N point

IFFT core and parallel-to serial converter PS. The output

signal of the OFDM Transmitter comprises 2−N

orthogonal carriers with frequency shift: TF /1=∆ , where

T is the OFDM symbol period (without cyclic prefix). Each

carrier is modulated by a data stream: )(iTan in accordance

with a number of the sub-carrier n and the current time

iT ( ..)2,1,0=i .

IFFT

N pointP/S

0 N-1

0

X(t)

Synthesis

Filter-Bank

f

0

a0(iT

0)

K

K

K

D1

Non-overlapped Wavelets

K

G

G

G

G

G = [ G(0) , G(1), G(2),... G(K-1) ]

aM-1(iT0)

a1(iT

0)

aM-2

(iT0)

T0 = (N/L)T

Figure 2. WFMT Transmitter (without overlapping block)

The WFMT transmitter uses the same N point core IFFT

and transmits information over M data streams

110 ,..., −Maaa each at rate )/(/1 0 NTLT = , where L is an

overlapping coefficient - number of overlapped wavelets on

interval T [7]. As shown in Fig. 2, the data streams modulate

groups of K inputs of IFFT core to provide the sub-channel

wavelets with low level of spectrum side lobes. Some of the

outermost data streams can be set to zero for spectral

containment reasons.

Now we explain the process of synthesis of the

sub-channel wavelets by Inverse Fourier Transform. First, we

discuss a process of generation of prototype wavelet. The

prototype wavelet is a base-band wavelet, whose spectrum is

centered on zero frequency and which has a property of shift

orthogonality. Each sub-channel wavelet can be constructed

from the prototype wavelet by shifting its spectrum to the

correspondent carrier frequency.

The ideal analytic wavelet for the digital communication

system is called “Modified Gaussian” was proposed in [24].

Following this idea, the ideal wavelet is constructed from

Gaussian waveform:

)4/( 222

)2/(1)( TteTts σπσ −= , where

T denotes the eventual shift orthogonality period of the

orthogonal pulse to be derived, and 224 Tσ is the pulse

variance in time.

The orthogonalization trick is performed on the Gaussian

waveform, which is not shift orthogonal [24]. The Fourier

transform of )(ts is ,)(222 )2( fTefS πσ−= applying

the orthogonalization trick to )( fS we obtain the function

∑∈

+−

−

=Φ

Zl

TlfT

fT

e

ef

2222

222

)/(8

)2(

)(πσ

πσ

whose inverse Fourier transform )(tφ can play role of a

scaling function.

A “Modified Gaussian“ wavelet )(tW can be processed as

)()()( tsttW ⋅=ψ ,

where )(tψ is the ortogonalization function that can be

calculated from )( fΦ and )( fS . Function )(tψ provides

necessary nulls in the time domain of )(tW .

Figure 3 illustrates a “Modified Gaussian” wavelet that

was developed for Cable TV applications [9]. This wavelet

has bandwidth roll-off coefficient about 18% and provides

overlapping coefficient 16=L ( −L a number of

overlapped wavelets). The Modified Gaussian Wavelet is an

Ideal waveform for the digital communication because

provides zero ISI and ICI distortion. However, Modified

Gaussian cannot practically realized because has non-limited

spectrum and length.

Figure 3. Modified Gaussian Wavelet for Cable

TV Applications

The WFMT Modulation uses a prototype wavelet that is

very close to the Modified Gaussian Wavelet. The prototype

wavelet is synthesized in the following way:

• FIRST, The selected Modified Gaussian is

represented by FOURIER Series:

)()( tiCOSAtW i ω∑∞

∞−

= ,

• SECOND, The prototype wavelet is presented by a

part of this FOURIER Series:

∑+

−

=m

m

i tiCOSAtPW )()( ω ,

where Km =+12 is a number of frequencies

used for prototype wavelet synthesis.

• THIRD, The interpolation errors defines:

∑−−

∞−

=1

22m

iAE

• IF an interpolation error is bigger than it is necessary

for the defined level of ISI, then the number of

frequency components K is increased by 2.

• The process is repeated while we get the necessary

level of ISI.

The prototype wavelet for the Cable TV System [9] that was

developed from Modified Gaussian Wavelet is shown in

figure 3. This wavelet was synthesized from 21- cosine

function and provides ISI distortion less then –56 dB. Such

a low ISI distortion makes it possible to transmit up to 13/14

bit of information per symbol.

As was shown in [7], the number of harmonics K

defines a quality of generated wavelets, in particular the

Inter-Symbol Interference (ISI) between wavelets and a

bandwidth of sub-channels.

We will characterize bandwidth losses of WFMT

sub-channel by the roll-off factor :β

SR

f∆= 2β , (2)

where f∆ - is an excess bandwidth and SR is a symbol

rate. As was shown in [7] the roll-off factor for the WFMT

system may be calculated by a simple formula: K

3=β .

This graph is shown in Figure 4.

Figure 4. The Roll-off factor of the WFMT sub-channel

The level of the Inter-Channel-Interference in WFMT

system is very low and must be theoretically equal to zero.

As we can see from Fig. 2, sub-channel wavelets in the

WFMT system comprises of a number of cosine functions

which all are orthogonal to cosine components of adjacent

sub-channel wavelets.

The level of the Inter-Symbol Interference between

wavelets depends on a form of the prototype wavelet and

decreases very rapidly with an increase in the number of the

wavelet components, as illustrated in Fig. 5. The minimal

number of cosine functions that may be used for wavelet

synthesis is 9. In this case, the Signal/ISI ratio is about 36 dB

and wavelet may carry up to 8 bits of information. In the

practical VDSL system, we used the wavelets that were

constructed of 11 cosine components.

Signal / ISI

K5 10 15 20 25 300

dB60

55

50

40

45

Figure 5. Inter-Symbol Interference between

the wavelets in a WFMT Sub-channel

One of the significant advantages of the WFMT

modulation is a very precise equalization algorithm that

allows compensating the distortion of the communication

channel. This algorithm includes an independent correction

for each frequency component of a transmitted wavelet [7].

FFT

N pointsS/P

y(nT)

H0*G

0*

aM-1(lT

0)

HK-1

*GK-1

*

+ DetectHMK-K

*G0*

a0(lT

0)

HMK-1

*GK-1

*

+ Detect

0

0

Figure 6. WFMT Receiver

The WFMT receiver (Figure 6) consists of a matched

filter bank. Filters are matched to the equivalent sub-channel

response [7]. Instead of well-known polyphase filter-bank, the

WFMT receiver includes 1024 point FFT core that provides

analysis of frequency components of received wavelets. Each

received wavelet component on output of FFT is multiplied

on an equalizer coefficient jH for compensation of

distortion in communication channel. The information data

ia is calculated for each sub-channel by summing the

weighted )( iG wavelet components

III. Experimental WFMT System

For the first practical implementation of WFMT

modulation, the VDSL Application was chosen [8]. Figure

3 shows the developed WFMT VDSL Transceiver. The

Transceiver comprises of three PCB boards: AFE (analog

front-end) board, FPGA board, and Microprocessor board.

The AFE board is based on the Analog Devices chip AD9876

that comprises of 12bit ADC and DAC. The low noise VCO

oscillator, placed on AFE board is controlled by 16bit DAC

MAX5204 and provides clock and symbol synchronization.

The Microprocessor board comprises of the ARM7 processor

that provides control of transceiver and its interface with

external equipment. The FPGA Board provides all digital

signals processing necessary for synthesis and analysis of

wavelets. Two XILINX FPGA are placed on the FPGA Board.

One of them comprises all the components of WFMT

transmitter, second- all the components of WFMT receiver.

Figure 7. WFMT VDSL Transceiver

The transmitter prototype was realized in FPGA VERTEX2

(3000), and uses about 1 million gates. In this design standard

Xilinx cores for 1024-point IFFT, RAMs, and multipliers were

used. A 1024-point core IFFT was used to obtain 44

sub-channels in the frequency band 200 kHz – 3 MHz. The

bandwidth of each WFMT channel is about 59 kHz.

Constellations from 2 to 12 were used. The system clock of

transmitter is 88 MHz. The transmitter test has shown a close

coincidence with the simulation results. The power spectrum

density of a realized WFMT transmitter is shown on Figure 8.

Figure 8. PSD of WFMT VDSL downstream.

As can be seen from Figure 4 the WFMT Spectrum does not

comprises side lobe components practically. One of 33

sub-channels was disabled for demonstration of possible

rejection of RF narrowband noise. The rest of 32 sub-channels

were modulated by QAM-64 random symbols. The WFMT

receiver is realized in FPGA XILINX VERTEX2 (3000) and

uses about 1.5 million gates. A constellation diagram shown on

figure 9 illustrates the WFMT receiver performance.

Fig.9. WFMT Receiver Constellation Diagram

IV. Conclusions

We have presented results of a study of a novel WFMT

modulation. During 2003~2006 main characteristics of the

Wavelet based Filtered Multi-tone modulation have been

defined. It was shown that WFMT modulation has significant

advantages in comparison with others filter-bank systems such

as low complexity and simple equalizing. The FPGA based

hardware implementation of a WFMT transceiver was

developed and tested. The experimental results are very close

to the simulations and show good spectral characteristic of the

new technology. The practical realization of a novel WFMT

modulation shows that WFMT modulation may be a real

candidate for a new line code for future DSL and Wireless

System.

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