Performance Analysis of a Novel OFDM System Based on Dual – Tree
Complex Wavelet Transform (DT-CWT)
Mohamed H. M. Nerma1, Nidal S. Kamel2 and Varun Jeoti3
Electrical & Electronic Engineering Department,
University Technology PETRONAS, Malayisa
E-mail: firstname.lastname@example.org , email@example.com , and firstname.lastname@example.org
As demand for higher data rates is continuously rising, there is always a need
to develop more efficient wireless communication systems. The work
described in this paper is an effort in this direction. We have proposed a
novel OFDM system base on DT-CWT. In the proposed scheme, DT-CWT
is used in the place of FFT. The proposed scheme offers the best PAPR
performance than the conventional OFDM and wavelet packet modulation
(WPM) systems at the expense of acceptable computational complexity
without using any pruning techniques. The complementary cumulative
distribution function (CCDF) of PAPR for the proposed scheme signal
achieves about 3 dB improvement in PAPR over the traditional OFDM and
WPM signals at 0.1% of CCDF. Also the proposed scheme achieves
excellent improvements in BER over conventional OFDM and WPM
systems. The need for CP is eliminated in the system design due to the good
orthogonality and time frequency localization proprieties of the wavelet.
Keywords: OFDM, WPT, CWT, DT-CWT, FFT, MCM, BER, PAPR.
1.1 Wavelet Modulation
1 INTRODUCTION Wavelet Transform (WT) is a relatively new
transform compared to the discrete Fourier transform
Orthogonal frequency division multiplexing (DFT). WT provides the time-frequency
(OFDM) and wavelet packet modulation (WPM) representation of signals, whereas DFT gives only
have emerged as an efficient multicarrier modulation the frequency representation. The properties of
schemes for wireless, frequency selective, wavelet, such as localization in time and frequency,
communication channels. Ease of implementation, orthogonality across scale and translation presents to
high spectral efficiency, resilience to impulse noise a new perspective in digital communication. It is
and multipath are a few advantages of OFDM and used as a modulation technique in many
WPM systems. However, a major drawback in the communication fields including multicarrier
signals of these two systems are their large envelope modulation (MCM) and wireless communication .
function, which limits the efficiency of the non- The conventional multicarrier systems are DFT
linear power amplifiers specific to wireless based systems. In MCM, the broadband channel
communications by forcing them to operate at lower splits into larger number of sub-channels. The high
average power. This problem is quantified by the data rate bit-streams are divided into parallel sub
Peak – to – average power ratio (PAPR) and results streams with lower data rate .
from the superposition of a large number of usually However, DFT based MCM systems suffer from
statistically independent sub-channels that can the high side lobes due to rectangular shaped DFT
constructively sum up to high peaks. Also from the window and also these systems waste precious
central limit theorem (CLT) , this causes the bandwidth due to the redundant cyclic prefix (CP).
OFDM and WPM signals to have complex Gaussian Moreover, the pulse shaping function used to
process behavior and the instantaneous power is chi- modulate each subcarrier extends to infinity in the
square distributed. Various schemes have been frequency domain this leads to high interference and
developed to reduce high PAPR in OFDM  , lower performance levels. WT based MCM systems,
and WPM signals  . namely wavelet based OFDM systems (WOFDM)
can help to mitigate these problems. Therefore,
conventional DFT based orthogonal systems are
being replaced by wavelet based transceivers. The
UbiCC Journal, Volume 4, Number 3, August 2009 813
wavelet based transceiver uses quadrature mirror
filters (QMF) in the synthesis and analysis filter 2
banks (FBs). The wavelet filters posses the
advantages of having greater side-lobe attenuation To reduce the PAPR in OFDM and WPM
and requires no CP . WPT modulation in wireless systems, several techniques have been proposed,
communication has been proposed in . The which basically can be divided in three categories.
characteristics of multicarrier modulated signal are First, there are signal distortion techniques, which
dependent on the basis functions being used in a reduce the peak amplitudes simply by nonlinearly
modulation scheme. Therefore, using WPT as basis distorting the OFDM signal at or around the peaks.
functions, the sensitivity to multipath channel Examples of distortion techniques are clipping, peak
distortion, inter symbol interference (ISI), inter windowing and peak cancellation. The second
carrier interference (ICI), and synchronization can be category is coding techniques that use a special
reduced as compared to traditional OFDM and the forward-error correcting code set that excludes
system performance of WOFDM with reference to OFDM symbols with a large PAPR. The third
ISI, ICI and signal-to-noise ratio (SNR) is shown to technique is based on scrambling each OFDM
be far better than the conventional OFDM (DFT- symbol with different scrambling sequences and
OFDM) , . selecting that sequence that gives the smallest PAPR
1.2 Peak – to – Average Power Ratio (PAPR) This paper is organized as follows: In section II
The PAPR of the baseband transmitted signal we discuss the dual-tree complex wavelet transform
is defined as the ratio of the peak power (DT-CWT); in section III we discuss the half-sample
( max | |2 ); i.e., the maximum power of delay condition; in section IV we make a
comparisons between traditional OFDM and WPM
the transmitted signal over the average power
systems; in section V we discuss the OFDM based
( E | |2 ). In digital implementations of
on DT-CWT; in section VI we discuss the PAPR in
communications transceivers, rather than using the
OFDM based on DT-CWT; the BER results are
continuous time signal in PAPR computation,
presented in section VII; and we conclude this paper
we instead work with , the discrete time samples
in section VIII by discussion the simulation results.
of x(t), provided that an oversampling factor of at
least 4 is used. PAPR is then expressed as :
2 THE DUAL – TREE COMPLEX WAVELET
max | |
E | |
Since the early 1990s the WT and WPT have
received more and more attention in modern
where . denotes ensemble average calculated
communications and have been widely used in
over the duration of the OFDM or WPM symbols. wireless communication . A number of
In both OFDM and WPM systems, the signal modulation schemes based on wavelets have been
going into the channel is a sum of random proposed [15 - 23]. In fact, complex wavelet
symbols modulating orthogonal basis functions.
transform (CWT) is applied perfectly to digital
Based on the CLT, it is claimed that is complex image processing. Kingsbury [24 - 28] introduced
Gaussian and its envelope follows a Rayleigh and made a concrete description of DT-CWT. The
distribution. This implies a large PAPR. A high DT-CWT employs two real discrete WT (DWT); the
PAPR of the transmitted signals demands a very upper part of the FB gives the real part of the
linear transmission path and limits the practical transform while the lower one gives the imaginary
deployment of low-cost non-linear power amplifiers part. This transform uses the pair of the filters ( ,
forcing them to operate with reduced power the low-pass/high-pass filter pair for the upper
efficiency. Driving amplifiers operating close to FB respectively) and ( , the low-
saturation with signals of high PAPR results in the pass/high-pass filter pair for the lower FB
generation of unwanted spectral energy both in-band respectively) that are used to define the sequence of
and out-of-band, which in turn reduces the systems wavelet function and scaling function as
performance and gives rise to adjacent channel follows
In this work, the performance of the OFDM
√2 ∑ 2 3
based on DT-CWT in PAPR reduction is
demonstrated through the CCDF of PAPR, which is
√2 ∑ 2 4
a performance metric independent of the transmitter
amplifier. Given the reference level PAPR0 > 0, the
Where 1 1 0 , the wavelet
probability of a PAPR being higher than the
function , the scaling function and the
reference value is the CCDF and is expressed as
follows : high-pass filter for the imaginary part are
UbiCC Journal, Volume 4, Number 3, August 2009 814
defined similarly. The two real wavelets associated
with each of the two real transform are
and . To satisfying the perfect reconstruction
(PR) conditions, the filters are designed so that the
complex wavelet is
approximately analytic. Equivalently, they are
designed so that is approximately the Hilbert
transform of .
The analysis (decomposition or demodulation) and
Figure 2: The Inverse dual tree discrete CWT (IDT-
the synthesis (reconstruction or modulation) FBs
DCWT) Synthesis (modulation) FB.
used to implement the DT-CWT and their inverses
are illustrated in fig. 1 and fig. 2 respectively. The
inverse of DT-CWT is as simple as the forward
transform. To invert the transform, the real part and
the imaginary part are each inverted.
In practical implementation of the DT-CWT, the
3 THE HALF-SAMPLE DELAY CONDITION
delay condition (7) and (8) will be satisfied only
approximately; the wavelets and will
The two low pass filters should satisfy a very
simple property: one of them should be form only an approximate Hilbert pair; and the
approximately a half-sample shift of the other  complex wavelet will be only
approximately analytic. While the FT is based on
0.5 6 complex valued oscillating cosine and sine
components form a complete Hilbert transform pairs;
Since and are defined only on the i.e., they are 90 out of phase with each other.
integers, this statement is somewhat informal. Together they constitute an analytic signal that is
However, we can make the statement rigorous using supported on only one-half of the frequency axis
FT. In  it is shown that, if 0 .
, then .
The converse has been proven in  , making 3.1 Filter Design for the DT-CWT
the condition necessary and sufficient. The necessary There are various approaches to the design of
and sufficient conditions for biorthogonal case were filters for the DT-CWT, such as linear-phase
proven in . biorthogonal method, quarter shift method, and
common factor method. These filters are satisfied the
following desired properties:
1- Approximately half-sample delay property.
2- PR (orthogonal or biorthogonal).
3- Finite support (finite impulse response (FIR)
4- Vanishing moments/good stop-band.
5- Linear phase filters.
It turns out that the implementation of the DT-CWT
requires that the first stage of the dual-tree (DT) FB
requires a different set of filters from the succeeding
stages, and the succeeding stages can be used same
sets of filters. See fig. (1). If the same PR filters are
used for each stage, then the first several stages of
the FB will not be approximately analytic .
Figure 1: The dual tree discrete CWT (DT-DCWT)
Analysis (demodulation) FB. 4 OFDM AND WPM SYSTEMS
We can rewrite the half-sample delay condition in Traditionally, OFDM is implemented using FFT.
terms of magnitude and phase function separately as This transform however has the drawback that is
shown in (7) and (8) as follows. uses a rectangular window, which creates rather high
sidelobes. Moreover, the pulse shaping function used
to modulate each subcarrier extends to infinity in the
UbiCC Journal, Volume 4, Number 3, August 2009 815
frequency domain (FD) this leads to high receiver side a DT-CWT is used in place of FFT
interference and lower performance levels. ISI and block in conventional OFDM system or in place of
ICI can be avoided by adding a CP to the head of DWPT block in WPM system. Data to be transmitted
OFDM symbol. Adding CP can largely reduce the are typically in the form of a serial data stream. PSK
spectrum efficiency. The WPM system has a higher or QAM modulations can be implemented in the
spectral efficiency and providing robustness with proposed system the choice depends on various
regard to inter-channel interference than the factors like the bit rate and sensitivity to errors. The
conventional OFDM system, because of the out-of- transmitter accepts modulated data (in this paper we
band energy (low sidelobes). Moreover WPM is able use 16 QAM). This stream is passed through a serial
to decompose T-F plane flexibly by arranging FB to parallel (S/P) converter, giving N lower bit rate
constructions. In addition, using FFT in the data stream, and then this stream is modulated
traditional OFDM gives resolution only in the FD through an IDT-CWT matrix realized by an N-band
while using WPT in WPM system gives resolution in synthesis FB. Before the receiver can demodulate the
both FD and time domain (TD) . subcarriers, it has to perform the synchronization.
WPM system do not required CP, thereby For the proposed system, known data interleaved
enhancing the spectrum efficiency. According to the among unknown data are used for channel
IEEE broadband wireless standard 802.16.3, estimation. Then, the signal is down sampled by N
avoiding CP gives wavelet OFDM an advantage of and demodulated using elements of the DT-CWT
roughly 20% in bandwidth (BW) efficiency. matrix realized by an N-band analysis FB. The signal
Moreover as pilot tones are not necessary for wavelet is equalized after DT-CWT stage.
based OFDM system, they perform better in
comparison to existing OFDM systems like 802.11a
or HiperLAN, where 4 out of 52 sub-bands are used
for pilots. This gives wavelet based OFDM system
another 8% advantage over typical OFDM
implementations . We expected the OFDM based
on DT-CWT will take all the advantages of WPM
However, a major problem of the common
discrete WPT (DWPT) is its lack of shift invariance;
this means that on shifts of the input signal, the
wavelet coefficients vary substantially. The signal
Figure 3: DT-CWT modulation (DT-CWTM)
information may even not be stationary in the sub- functional block diagram.
bands so that the energy distribution across the sub-
bands may change . To overcome the problem of IDT-CWT works in a similar fashion to an IFFT
shift dependence, one possible approach is to simply or IDWPT. It takes as the input QAM symbols and
omit the sub-sampling causing the shift dependence. outputs them in parallel time-frequency
Techniques that omit or partially omit sub-sampling “subcarriers”. In fig. (2) as the synthesis process, it
are also known as cycle spinning, oversampled FBs can be shown that the transmitted signal, is
or undecimated WT. However, these transforms are constructed as the sum of M waveform individually
redundant , which is not desirable in multicarrier modulated with the QAM or PSK symbols , φ t
modulation. As an alternative, we used a non- is the scaling function, ψ t is the wavelet function,
redundant WT that achieves approximately shift
and a is kth symbol, i 1, 2, … , N as follows:
invariance , this transform yields to complex
wavelet coefficients that modulate the data stream in . 9
the same way that WPM do.
5 OFDM BASED ON DT-CWT . 10
, , , ,
In this study, simulations are focusing on using
DT-CWT in the OFDM system as a non-redundant
WT that can achieves approximately shift invariance
[35 – 38]. Similar to the conventional OFDM and / , / , , , . 11
WPM systems, a functional block diagram of OFDM /
based on DT-CWT is shown in fig. (3). At the
transmitter an inverse DT-CWT (IDT-CWT) block is Where , is a constellation encoded data
used in place of inverse FFT (IFFT) block in symbol modulating the DT-CWT function.
conventional OFDM system or in place of inverse The IDT-CWT synthesis a discrete representation of
DWPT (IDWPT) block in WPM system. At the the transmitted signal as sum of M waveforms
UbiCC Journal, Volume 4, Number 3, August 2009 816
shifted in time that embed information about data sequence x n has high PAPR. Furthermore, to
symbols. Those waveforms are built by successive demonstrate the similarities between power spectrum
iterations of ^ , ^ and ^ , ^ . The DT-CWT at the density (PSD) characteristic of conventional OFDM,
receiver recovers the transmitted symbols , WPM and the proposed scheme, the simulated PSD
through the analysis formula exploiting orthogonality characteristics are presented in fig. 4. This fig. was
properties of DT-CWT and schematically shows that the proposed scheme performs better than
represented in fig. (1). the other two systems.
In the baseband equivalent OFDM transmitter with The simulation parameters are documented as
frame of N QAM symbols, , follows: Modulation type is 16-QAM; the number of
0,1, … , 1, the OFDM frame is given by: subcarriers is 64 subcarriers; a wavelet packet base is
Daubechies-1 (DAUB-1); maximum tree depth (D =
7); PAPR threshold is 2dB; shaping filter is Raised
12 Cosine (rolloff factor 0.001, upsampler = 4); DT-
CWT using different filters (LeGall 5,3 tap filters
(leg), Antonini 9,7 tap filters (anto), Near Symmetric
While for WPM system, the transmitted signal 5,7 tap filters (n-sym-a), and Near Symmetric 13,19
is constructed as the sum of M wavelet packet tap filters (n-sym-b)) for the first stage of the FB and
function individually modulated with the QAM (Quarter Sample Shift Orthogonal 10,10 tap filters
symbols as the case in this paper‡. (q-sh-06) only 6,6 non-zero taps, Quarter Sample
Shift Orthogonal 10,10 tap filters (q-sh-a) with 10,10
non-zero taps, unlike q-sh-06, Quarter Sample Shift
, 13 Orthogonal 14,14 tap filters (q-sh-b), Quarter Sample
Shift Orthogonal 16,16 tap filters (q-sh-c), and
Quarter Sample Shift Orthogonal 18,18 tap filters (q-
The construction of discrete versions of sh-d)) for the succeeding stages of the FB.
transmitted waveforms for the conventional OFDM
and WPM systems using (12) and (13) is quite 6 PAPR ANALYSIS RESULTS
similar. For any time index n, both waveforms are
sum of random symbols a or a , . Consider the simple case where all the sub-
symbols are independently and identically distributed
(i.i.d). then, by the CLT, the real and imaginary parts
of the N-point IFFT output have mutually
independent Gaussian probability distribution
function with zero mean and standard deviation to σ .
The instantaneous power of baseband signal , is
We can characterize the instantaneous power as a
chi-square distribution with two degrees of freedom
Figure 4: PSD of the Conventional OFDM, WPM
and OFDM based on DT-CWT. 1
, 0 15
In order to achieve fair comparisons, same
simulation parameters are used. The simulations If E | | is normalized to unity, then the CCDF
were carried out for conventional OFDM with a 64 of the PAPR is given by:
subcarriers using a 16 QAM modulation. The
Daubechies-1 (Daub-1) wavelet packet bases were Pr 1 1 16
used to construct the wavelet packet trees in WPM
system. For the proposed scheme, near symmetric
13,19 tap filters and quarter sample shift orthogonal Where N is the number of subcarriers.
14 tap filters were used to construct the real and the In order to analyze PAPR, we generate the
imaginary part of DT-CWT respectively. From the transmitted waveforms using 16 QAM modulation
CLT, x n is complex Gaussian distributed, and the with 64 subcarriers for all these systems. Fig. 5
shows, in the TD, the envelope of the proposed
system. For the comparison, we also plotted the
We use φ n for the wavelet packet function and ψ n for the envelope of the conventional OFDM and WPM
DT-CWT function to avoid any confusion.
UbiCC Journal, Volume 4, Number 3, August 2009 817
waveforms corresponding to the same information stage of the FB and quarter sample shift orthogonal
symbol pattern. The transmitted envelopes for the (q-sh) 14 tap filters in the succeeding stages, OFDM
conventional OFDM and WPM systems illustrate DT-CWT_2 using (n-sym 13,19 with q-sha 10 (10
approximately similar behavior, where the peak is non zero taps) filters), OFDM DT-CWT_3 using
about 2.25, while the transmitted envelope for the (antonini (anto) 9,7 tap filters with q-sh0 10 (only 6
proposed system demonstrates better behavior than non zero taps) filters), OFDM DT-CWT_4 using
the other two systems, where the peak is only about (anto 9,7 with q-sh 14 filters), OFDM DT-CWT_5
1.25 and this is the reason for that the proposed using (n-sym 5,7 with q-sh 14 filters), OFDM DT-
system gives better result for PAPR than the other CWT_6 using (LeGall (leg) 5,3 tap filters with q-sh
two systems. 14 filters), OFDM DT-CWT_7 using (n-sym 5,7
with q-sh 16 filters) and OFDM DT-CWT_8 using
(leg 5,3 with q-sh 18 filters).
The results in fig. 7 show that there is no
observed degradation as a result of using different set
of mismatching filters in the design of the proposed
Figure 5. The Envelope of the Conventional OFDM,
WPM and OFDM based on DT-CWT.
Figure 7: The effect of using different set of filters
in design of the OFDM based on DT-CWT.
Figure 6: CCDF the Conventional OFDM, WPM
and OFDM based on DT-CWT.
The results for PAPR are best quantified using
CCDF. In fig. 6, for 64 subcarriers with 16 QAM
modulation, the CCDF plots for the proposed system,
conventional OFDM and WPM system are shown. Figure 8: CCDF of PAPR for 16-QAM modulated
conventional OFDM symbol with various values of
This figure shows that the OFDM based on DT-
CWT offers the best PAPR performance without
using any reduction techniques. The proposed
Again, the results for PAPR were repeated in
scheme signal achieves about 3 dB improvement in
both fig. 8 (for the conventional OFDM system) and
PAPR over the traditional OFDM and WPM signals
fig. 9 (for the proposed system) using different
at 0.1% of CCDF while the other two systems, are
numbers of subcarriers (64, 128, 256, 512, and 1024)
approximately given same results of PAPR.
with 16QAM modulation. We observe from these
The simulation results for PAPR were also
figures that the PAPR increases as the number of
repeated with 16 QAM modulation and 64
subcarriers numbers (N) increases. As shown in fig.
subcarriers (in fig. 7) using different set of filters.
6, 7 and in fig. 8, 9 the PAPR performance of DT-
OFDM DT-CWT_1 illustrate the PAPR when using
CWT based OFDM systems is better than the
near-symmetric (n-sym) 13,19 tap filters in the first
UbiCC Journal, Volume 4, Number 3, August 2009 818
conventional OFDM system or even WPM systems.
As we saw in figure 5 showing the time-domain
signal envelop of all the systems, this improvement
in PAPR performance is explained from lower peaks
of DTCWT systems observed in the fig. 5.
Figure 11: The effect of using different type of
filters in BER for OFDM based on DT-CWT.
Figure 9: CCDF of PAPR for 16-QAM modulated
OFDM based on DT-CWT symbol with various In this paper a new OFDM scheme that is based
values of subcarriers (N). on DT-CWT is proposed. Comparing the proposed
scheme in terms of PAPR and BER with the
7 BER ANALYSIS RESULTS traditional OFDM and WPM systems we see that the
proposed scheme offers 3dB better PAPR
The results given in this section compare the performance over the conventional OFDM and
BER in the OFDM based on DT-CWT, with that for WPM systems at 0.1% of CCDF. While the
traditional OFDM, and WPM. Also, same simulation conventional OFDM and WPM systems shows
parameters are used to achieve a fair comparison. similar behavior. Simulation results shows that there
The results of BER in OFDM based on DT-CWT is no observed PAPR and BER degradation as a
using different set of filters are also shown in this result of using different set of mismatching filters in
section. DT-CWT based system. The proposed scheme
The results for BER are shown that the proposed outperforms the traditional OFDM and WPM
scheme gives excellent improvements in BER over systems in term of BER. Also we found that the
conventional OFDM and WPM systems. At the same conventional OFDM system gives better results of
time the conventional OFDM outperform the WPM BER than WPM system.
system in term of BER. As shown in fig. 10.
The simulation results for BER were repeated
using different set of filters (OFDM DT-CWT-1, ACKNOWLEDGMENTS
OFDM DT-CWT-2, …, OFDM DT-CWT-8) and is Authors wish to thanks Prof. Nick G. Kingsbury
shown in fig. 11. We see there is degradation of BER (University of Cambridge, United Kingdom) and Dr.
as a result of using different set of mismatching Mohan Baro (Dalhousie University, Canada) whose
filters in the design of the proposed scheme. comments and suggestions have largely contributed
to improve this paper.
A. CCDF Approximations
We can characterize the instantaneous power as a
chi-square distribution with two degrees of freedom
, 0 17
As a result, the cumulative distribution function
(CDF) is defined as:
Figure 10: BER performance of conventional
OFDM, WPM and OFDM based on DT-CWT.
UbiCC Journal, Volume 4, Number 3, August 2009 819
1 18 √2 √2
If E | | is normalized to unity, then the CCDF Note that the identity matrix on the right-hand side of
of the PAPR is given by: (22) is twice the size of those on the left-hand side.
Therefore if the two real DWTs are orthonormal
transforms, then the DT-CWT satisfies . ,
1 1 19 where * denotes the conjugate transpose.
Where N is the number of subcarriers, however, this
approximation is not close to the experimental results 9 REFERENCES
as the assumption made in deriving CCDF that the
 Papoulis A., Pillai S. U., “Probability
samples should be mutually uncorrelated is no longer Random Variables and Stovhastic
valid when oversampling is employed . There Processes,” New York: McGraw-Hill Inc.,
has been several attempts to determine the closed for 2002.
approximations for the distribution of PAPR. Some  M. Breiling, S. H. muller, and J. B. Huber,
of the approximations are shown in Table 1. “SLM peak power reduction without
explicit side information”, IEEE Commun.
Lett., vol. 5, no. 6, pp. 239-241, 2001.
Table 1: Approximation to CCDF of PAPR.  J. Tellado, Ed., Multicarrier Modulation
with low PAPR Application to DSL and
CCDF Remarks wireless. New York: Kluwer Academic
= 2.8 and  M. Baro and J. Ilow, “PAPR reduction in
N 64 wavelet packet modulation using tree
 pruning”, in 2007 IEEE 65th Vehicular
Technology Conference VTC 2007 –
spring, Dubin, Ireland, Apr. 2007.
 M. Baro and J. Ilow, “PAPR reduction in
 OFDM using wavelet packet pre-
processing”, Consumer Communications
and Networking Conference, 2008. CCNC
2008. 5th IEEE.
 M K Lakshmanan, H Nikookar, "A review
 of Wavelets for Digital Communication,"
Wireless Personal Communication (2006)
/ N and are 37: 387- OFDM* 420, Springer 2006.
√ large   J. A. C. Bingham, "Multicarrier modulation
for data transmission: An Idea Whose Time
Has Come," IEEE Communications
/ Magazine, vol. 28, no.5, pp. 5-14, 1990.
 H. M. Newlin, "Developments in the use of
wavelets in communication systems," TRW
Systems & Information Technology Group,
 A. Jamin, P. Mahonen, "Wavelet packet
B. DT-CWT and IDT-CWT modulation for wireless communications,"
If the two real DWTs are represented by the Wireless Communications and Mobile
Computing 5 (2): 123-137 (2005).
square matrices for the upper part and for the  C. V. Bouwel, J. Potemans, S. Schepers, B.
lower part, then the DT-CWT can be represented by Nauwelaers and A.V. de Capelle, "Wavelet
the following form. Packet Based Multicarrier Modulation,"
Proc. IEEE Benelux Symposium on
1 Communications and Vehicular
. 20 Technology, Leuven, Belgium, 19 October
 C. Schurgers and M. B. Srivastava, “A
And the IDT-CWT is given by systematic approach to peak – to – average
power ratio in OFDM,” in SPIE’s 47th
Annual Meeting, San Diego, CA, 2001, pp.
. 21 454-464.
√2  S. H. Han and J. H. Lee, “An Overview of
to peak – to – average power ratio reduction
Note that the complex sum/difference matrix in (20) techniques for multimedia transmission,”
is unitary (its conjugate transpose is its inverse). IEEE Wireless Communications, vol. 12,
no. 2, pp. 56-65, 2005.
 Richard Van Nee and Ramjee Prasal,
“OFDM for Wireless Multimedia
UbiCC Journal, Volume 4, Number 3, August 2009 820
Communications”, P. cm Artech House Vancouver, BC, Canada, Sept. 10–13, 2000,
Universal Personal Communications Series, vol. 2, pp. 375–378.
Inc. Boston. London, 2000.  N.G. Kingsbury, “Complex wavelets for
 Panchamkumar D Shukla, “Complex shift invariant analysis and filtering of
wavelet Transforms and Their signals,” Appl. Comput. Harmon. Anal.,
Applications” Master Thesis 2003. Signal vol. 10, no. 3, pp. 234–253, May 2001.
Processing Division. University of  I W. Selenick, “Hilbert transform pairs of
Strathclyde Department of Electronic and wavelet bases,” IEEE Signal Processing
Electrical Engineering. Lett., vol. 8, no. 6, pp. 170-173, June 2001.
 M. Guatier, J. Lienard, and M. Arndt,  H. Ozkaramanli and R. Yu, “on the phase
“Efficient Wavelet Packet Modulation for condition and its solution for Hilbert
Wireless Communication”, AICT’07 IEEE transform pairs of wavelet bases,” IEEE
Computer Society, 2007. Trans. Signal Processing, vol. 51, no. 12,
 A. Jamin, and P. Mahonen, “Wavelet Packet pp. 2393–3294, Dec. 2003.
Modulation for Wireless Communications”,  R. Yu and H. Ozkaramanli, “Hilbert
Wiley Wireless Communications and transform pairs of orthogonal wavelet bases:
networking, Journal, vol. 5, no. 2, pp. 123- Necessary and sufficient conditions”, IEEE
137, Mar. 2005. Trans. Signal Processing IEEE Transactions
 Xiaodong Zhang and Guangguo Bi, on Signal Processing, vol. 53, no. 12, Dec.
“OFDM Scheme Based on Complex 2005.
Orthogonal Wavelet  R. Yu and H. Ozkaramanli, “Hilbert
Packet”,http://ieeexplore.ieee.org/iel5/7636/ transform pairs of biorthogonal wavelet
20844/00965270.pdf. bases”, IEEE Transactions on Signal
 M. Guatier, and J. Lienard, “Performance of Processing, VOL. 54, NO. 6, JUNE 2006.
Complex Wavelet packet Based  I. W. Selesnick, “the Double Density Dual-
Multicarrier Transmission through Double Tree DWT”, IEEE Transactions on Signal
Dispersive Channel”, NORSIG 06, IEEE Processing, 52(5): 1304 – 1315, May 2004.
Nordic Signal Processing Symposium  J. M. Lina,”Complex Daubechies Wavelets:
(Iceland), June 2006.
Filter Design and Applications”, ISAAC
 C. J. Mtika and R. Nunna, “A wavelet- Conference, June 1997.
based multicarrier modulation scheme,” in  Mohamed H. M. Nerma, Nidal S. Kamel,
Proceedings of the 40th Midwest and Varun jeoti, “PAPR Analysis for
Symposium on Circuits and Systems, vol. 2, OFDM Based on DT-CWT” Proceeding of
August 1997, pp. 869–872.
2008 Student Conference on Research and
 N. Erdol, F. Bao, and Z. Chen, “Wavelet Development (SCOReD 2008), 26-27 Nov.
modulation: a prototype for digital 2008, Johor, Malaysia.
communication systems,” in IEEE Southcon
Conference, 1995, pp. 168–171.  Mohamed H. M. Nerma, Nidal S. Kamel,
and Varun jeoti, “On DT-CWT Based
 A. R. Lindsey and J. C. Dill, “Wavelet OFDM: PAPR Analysis” Multi-Carrier
packet modulation: a generalized method Systems & Solutions 2009, V. 41, pp. 207-
for orthogonally multiplexed 217, SpringerLink April 26, 2009-
communications,” in IEEE 27th Dordrecht, Netherlands.
Southeastern Symposium on System
Theory, 1995, pp. 392–396.  Mohamed H. M. Nerma, Nidal S. Kamel,
and Varun jeoti, “BER Performance
 A. R. Lindsey, “Wavelet packet modulation Analysis of OFDM System Based on Dual –
for orthogonally multiplexed Tree Complex Wavelet Transform in
communication,” IEEE Transaction on AWGN Channel” Proceeding of 8th
Signal Processing, vol. 45, no. 5, pp. 1336– WSEAS International Conference on
1339, May 1997. SIGNAL PROCESSING (SIP '09), Istanbul,
 C. V. Bouwel, J. Potemans, S. schepers, B. Turkey, May 30 - June 1, 2009.
Nauwelaers, and A. Van Caelle, “wavelet  Mohamed H. M. Nerma, Nidal S. Kamel,
packet Based Multicarrier Modulation”, and Varun jeoti, “An OFDM System Based
IEEE Communication and Vehicular on Dual Tree Complex Wavelet Transform
Technology, SCVT 2000, pp. 131-138, (DT-CWT)” Journal: Signal Processing:
2000. An International Journal, Volume: 3, Issue:
 Ivan W. Selesnick, Richard G. Baraniuk, 2, Pages: 14-21, May 2009.
and Nick G. Kingsbury, “The Dual-Tree  M. K. Lakshmanan and H. Nikookar, “A
Complex Wavelet Transform,” IEEE Signal Review of Wavelets for Digital Wireless
Processing Mag, pp. 1053-5888, Nov 2005. Communication”, Wireless Personal
 N.G. Kingsbury, “The dual-tree complex Communications Springer, 37: 387-420,
wavelet transform: A new technique for Jan. 2006.
shift invariance and directional filters,” in  R. Van Nee and A. De Wild, “reduction the
Proc. 8th IEEE DSP Workshop, Utah, Aug. peak-to average power ratio of OFDM” 48th
9–12, 1998, paper no. 86. IEEE Vehicular Technology Conference,
 N.G. Kingsbury, “Image processing with Ottawa, Canada, May 18-21, 1998; 2072-
complex wavelets,” Philos. Trans. R. Soc. 2076, IEEE New York, USA.
London A, Math. Phys. Sci., vol. 357, no.  H. Ochiai and H. Imai, “On the disitribution
1760, pp. 2543–2560, Sept. 1999. of the peak-to-average power ratio in
 N.G. Kingsbury, “A dual-tree complex OFDM signals”, IEEE Trans. Commun.
wavelet transform with improved 2001; 49, 282-289.
orthogonality and symmetry properties,” in  S. wei, D. L. Goeckel, and P. E. Kelly “A
Proc. IEEE Int. Conf. Image Processing, modern extreme value theory approach to
UbiCC Journal, Volume 4, Number 3, August 2009 821
calculate the distribution of the peak-to- Nidal S. Kamel, received his
average power ratio in OFDM systems”, Ph.D degree (Hons) in
IEEE International Conference on
Communications, New York, Apr. 28- May telecommunication and
2, 2002; vol. 3, 1686-1690; IEEE New statistical signal processing
York, USA. from the Technical
 H. Yu, M. Chen and G. wei “Distribution of University of Gdansk,
PAPR in DMT systems”, Electron. Lett.
2003; 39, 799-801. Poland, in 1993. His research
 R. Rajbanshi, A. M. Wyglinski and G. J. is focused on linear
Minden, “Multicarrier Transceivers: Peak- estimation, noise reduction,
to-average power ratio reduction”, IEEE pattern recognition, optimal filtering, and
Commun. Society. WCNC 2008 telecommunications. Currently he is working for
Universiti Teknologi PETRONAS, Malaysia. He is
senior member of IEEE.
Varun Jeoti Jagadish,
Mohamed H. M. Nerma, was received his Ph.D. degree
born in Khartoum, Sudan. He from Indian Institute of
received the B.Sc. degree in Technology Delhi India in
Electrical Engineering 1992. He worked on several
(Control) from Sudan sponsored R&D projects in
University of Science and IIT Delhi and IIT Madras
Technology (SUST), Sudan in during 1980 to 1989
1999. He received the M.Sc. developing Surface Acoustic
degree in Communication Wave Pulse Compression filters, underwater optical
Engineering from Karary Academy of Technology, receivers etc.. He was a Visiting Faculty in
Sudan in 2002. From 2002 to 2006 he was lecturer in Electronics department in Madras Institute of
the Sudan University of Science and Technology. He Technology for about 1 year during 1989 to 1990
is currently PhD. Student in University Technology and joined Delhi Institute of Technology for next 5
PETRONAS, Malaysia. years till 1995. He moved to Electrical & Electronic
Engineering (E&E Engg) department of Universiti
Sains Malaysia in 1995 and joined E&E Engg of
Universiti Teknologi PETRONAS in 2001. His
research interests are in Wireless LAN and MAN
technologies, DSL technology and related signal
UbiCC Journal, Volume 4, Number 3, August 2009 822