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Wireless Personal Communications (2006) 37: 387420
DOI: 10.1007/s11277-006-9077-y C Springer 2006
A Review of Wavelets for Digital Wireless Communication
M. K. LAKSHMANAN and H. NIKOOKAR
International Research Center for Telecommunications Transmission and Radar (IRCTR), Department of
Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4,
2628 CD Delft, The Netherlands E-mails: [email protected], [email protected]
Abstract. Wavelets have been favorably applied in almost all aspects of digital wireless communication sys-
tems including data compression, source and channel coding, signal denoising, channel modeling and design of
transceivers. The main property of wavelets in these applications is in their flexibility and ability to characterize
signals accurately. In this paper recent trends and developments in the use of wavelets in wireless communications
are reviewed. Major applications of wavelets in wireless channel modeling, interference mitigation, denoising,
OFDM modulation, multiple access, Ultra Wideband communications, cognitive radio and wireless networks are
surveyed. The confluence of information and communication technologies and the possibility of ubiquitous con-
nectivity have posed a challenge to developing technologies and architectures capable of handling large volumes of
data under severe resource constraints such as power and bandwidth. Wavelets are uniquely qualified to address this
challenge. The flexibility and adaptation provided by wavelets have made wavelet technology a strong candidate
for future wireless communication.
Keywords: wavelets, wireless communications, multi carrier modulation, OFDM, CDMA, cognitive radio, ultra
wideband communication, wireless networks
Abbreviations: ARQ, Automatic Retransmission Query; AWGN, Additive White Gaussian Noise;BER, Bit ErrorRate; BPSK, Binary Phase Shift Keying; CDMA, Code Division Multiple Access; CP, Cyclic Prefix; CR, Cogni-
tive Radio; CWT, Continuous Wavelet Transform; DFT, Discrete Fourier Transform; DS-CDMA, Direct Sequence
CDMA; DWT, Discrete Wavelet Transform; FCC, Federal Communications Commission; FDM, Frequency Divi-
sion Multiplexing; FDMA,Frequency Division Multiple Access; GI, Guard Interval; HiperLAN, High Performance
Radio Local Area Network; ICI, Inter-Carrier Interference; IOTA, Isotropic Orthogonal Transform Algorithm; IR,
Impulse Radio; ISI, Inter-Symbol Interference; LDPC, Low-Density Parity-Check; MANET, Mobile Ad hoc Net-
works; MB-OFDM, Multi-Band OFDM; MC-CDMA, Multicarrier CDMA; MC-DS-CDMA, Multicarrier Direct
Sequence CDMA;MCM, Multicarrier Modulation;OFDM, Orthogonal FrequencyDivisionMultiplexing; OWDM,
Orthogonal Wavelet Division Multiplexing; PAM, Pulse Amplitude Modulation; PAPR, Peak-to-Average Power
Ratio; PR, Pseudo Random; PR-QMF, Perfect Reconstructed Quadrature Mirror Filter; PSWF, Prolate Spheroidal
Wave Functions; PSD, Power Spectral Density; PSK, Phase Shift Keying; QAM, Quadrature Amplitude Modula-
tion; QMF, Quadrature Mirror Filters; QoS, Quality of Service; QPSK, Quadrature Phase Shift Keying; SCDMA,
Scale Code Division Multiple Access; S-CDMA, Synchronous Code Division Multiple Access; SNR, Signal-to-
Noise Ratio; SSA-UWB, Soft Spectrum Adaptation UWB; STCDMA, Scale Time Code Division Multiple Access;
TDM, Time Division Multiplexing; TDMA, Time Division Multiple Access; TDOA, Time Difference Of Arrival;
UWB, Ultra Wideband; V-BLAST, Vertical Bell Laboratories Layered Space Time; WP, Wavelet Packet; WPM,
Wavelet Packet Modulation; WDM, Wavelet Division Multiplexing; WDMA, Waveform Division Multiple access;
WPDM, Wavelet Packet Division Multiplexing; WPT, Wavelet Packet Transform
1. Introduction
The Wavelet transform is a way of decomposing a signal of interest into a set of basis wave-
forms, called wavelets, which thus provide a way to analyze the signal by examining the
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388 M. K. Lakshmanan and H. Nikookar
coefficients (or weights) of wavelets. This method is used in various applications and is be-
coming very popular among technologists, engineers and mathematicians alike. In most of
the applications, the power of the transform comes from the fact that the basis functions ofthe transform are localized in time (or space) and frequency, and have different resolutions in
these domains. Different resolutions often correspond to the natural behavior of the process
one wants to analyze, hence the power of the transform. These properties make wavelets and
wavelet transform natural choices in fields as diverse as image synthesis, data compression,
computer graphics and animation, human vision, radar, optics, astronomy, acoustics, seis-
mology, nuclear engineering, biomedical engineering, magnetic resonance imaging, music,
fractals, turbulence, and pure mathematics. A thorough review of the application of wavelets
in these fields can be found in [1, 2].
Recently wavelet transform has also been proposed as a possible analysis system when
designing sophisticated digital wireless communication systems, with advantages such as
transform flexibility, lower sensitivity to channel distortion and interference and better uti-lization of spectrum [35]. Wavelets have found beneficial applicability in various aspects of
wireless communication systems design including channel modeling, transceiver design, data
representation, data compression, source and channel coding, interference mitigation, signal
de-noising and energy efficient networking. Figure 1 gives a graphical representation of some
of the facets of wireless communication where wavelets have been used.
In this paper we attempt to collate the latest advancements and developments in the use
of wavelets and their tributaries, like wavelet packets, in the field of wireless communication.
Special emphasis is placed upon the applications related to transmission technology. It is worth
mentioning that this work is by no means a comprehensive review of all the existing literature,
Figure 1. The spectrum of wavelet applications for wireless communication.
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A Review of Wavelets for Digital Wireless Communication 389
but rather an attempt to appraise the possibilities and potentials that wavelets offer in the area
of wireless communications system design and development.
This paper is organized as follows. In Section 2, we give the theoryof wavelet representationof signals, and examine the properties and main aspects of wavelets. In Section 3 we highlight
a few advantages of using wavelets for digital wireless communication systems. A review of
recent applications of wavelets in wireless communication is provided in Section 4. Finally,
in Section 5 we conclude the paper by underlining the promising role of wavelet technology
for future generation wireless systems.
2. Wavelet Representation for Signal Analysis
In this section we give a very brief introductionto thetheory of waveletsand wavelet transforms,
and we mention a few properties that are important from the perspective of communication
system design. A thorough study of the subject can be found in [616].
2.1. WA V E L E T S A N D WA V E L E T TR A N S F O R M S
The word wavelet derives from the French researcher, Jean Morlet, who used the French word
ondelette meaning a small wave [6]. A little later it was transformed into English by trans-
lating onde into wave to thus arrive at the name wavelets. As the name suggests, wavelets
are small waveforms with a set oscillatory structure that is non-zero for a limited period of time
(or space) with additional mathematical properties. The wavelet transform is a multi-resolution
analysis mechanism where an input signal is decomposed into different frequency components,
and then each component is studied with resolutions matched to its scales. The Fourier trans-
form also decomposes signals into elementary waveforms, but these basis functions are sinesand cosines. Thus, when one wants to analyze the local properties of the input signal, such
as edges or transients, the Fourier transform is not an efficient analysis tool. By contrast the
wavelet transforms which use irregularly shaped wavelets offer better tools to represent sharp
changes and local features. The wavelet transform gives good time resolution and poor fre-
quency resolution at high frequencies and a good frequency resolution and poor time resolution
at low frequencies. This approach is logical when the signal on hand has high frequency com-
ponents for short durations and low frequency components for long durations. Fortunately, the
signals that are encountered in most engineering applications are often of this type.
Figure 2 illustrates the aforementioned concept to explain how time and frequency resolu-
tions should be interpreted. Every block in the figure corresponds to one value of the wavelet
transform in the time-frequency plane. All the points in the time-frequency plane that fall into
a block are represented by one coefficient of the wavelet transform. It is easy to infer from
the figure that at lower frequencies the width and height of the windows are long and short,
giving good frequency resolution and poor time resolution. At higher frequencies the width
and height of the windows are short and long respectively giving good time resolution and
poor frequency resolution.
2.2. C O N T I N U O U S A N D D I S C R E T E WA V E L E T TR A N S F O R M
Wavelet transforms are broadly classified as continuous and discrete wavelet transforms. In
this section we will look into their operation.
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Figure 2. Time-frequency tiling of the wavelet transform [14]
2.2.1. Continuous Wavelet Transform (CWT)
The continuous wavelet transform (CWT) of a continuous signal x (t) is defined as the sum
of all time of the signal multiplied by scaled, shifted versions of the wavelet (t). This is
expressed as:
(a, b) = 1a
x(t)
t ba
dt (1)
where, is the result of the operation and consists of many wavelet coefficients, which are afunction of scale a and translation b. The original signal can be reconstructed using the inverse
transform:
x(t) = 1C
(a, b)
t b
a
da
db
|a|2 (2)
where, C =
|()|2|| d and () is the Fourier transform of(t).
2.2.2. Discrete Wavelet Transform (DWT)
The DWT analyzes the signal at different frequency bands with different resolutions by de-composing the signal into an approximation containing coarse and detailed information. DWT
employs two sets of functions, known as scaling and wavelet functions, which are associated
with low pass and high pass filters. The decomposition of the signal into different frequency
bands is simply obtained by successive high pass and low pass filtering of the time domain
signal. The original signal x [n] is first passed through a half-band high pass filter g [n] and
a half-band low pass filter h [n]. A half-band low pass filter removes all frequencies that are
above half of the highest frequency, while a half-band high pass filter removes all frequencies
that are below half of the highest frequency of the signal. The low pass filtering halves the
resolution, but leaves the scale unchanged. The signal is then sub-sampled by two since half
of the number of samples is redundant, according to the Nyquists rule. This decomposition
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A Review of Wavelets for Digital Wireless Communication 391
Figure 3. Decomposition of input signal [14].
can mathematically be expressed as follows:
yhigh[k] =
n x[n]g[2k n]ylow[k] =
n x [n]h[2k n]
(3)
whereyhigh[k]andylow[k] are the outputs of the high pass and low pass filters, after sub-sampling
by a factor of two.
This decomposition halves the time resolution since only half the number of samples
then come to characterize the entire signal. Conversely it doubles the frequency resolution,
since the frequency band of the signal spans only half the previous frequency band effec-
tively reducing the uncertainty by half. The above procedure, which is also known as sub-
band coding, can be repeated for further decomposition. At every level, the filtering and
sub-sampling will result in half the number of samples (and hence half the time resolution)
and half the frequency bands being spanned (and hence doubles the frequency resolution).
This procedure is illustrated in Figure 3, where x[n] is the original signal to be decom-
posed, and h[n] and g[n] are the respective impulse responses of the low pass and high passfilters.
2.3. WA V E L E T PA C K E T TR A N S F O R M
The wavelet packet transform is just like the wavelet transform except that it decomposes even
the high frequency bands which are kept intact in the wavelet transform. Figure 4 illustrates the
wavelet packet decomposition procedure. Here S denotes the signal while A and D denote the
respective approximations (high frequency terms) and decompositions (low frequency terms).
Figure 5 shows the corresponding time-frequency plane division.
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Figure 4. Wavelet packet decomposition tree of input signal [15].
Figure 5. Wavelet packets; (Left): Semi-arbitrary tree pruning; (Right): Time-frequency plane division [3]. Here
wi,j denotes the j th wavelet packet coefficient at the ith iteration level and Dt(wi,j ), Df(wi,j ) denote the length ofthe time and frequency division, of the wavelet packet wi,j .
2.4. WA V E L E T P R O P E R T I E S
The most important properties of wavelets are the admissibility and the regularity conditions
[9]. Given a wavelet (t), the admissibility condition may be stated as,
C =
0
| () |2|| d =
0
| () |2|| d < (4)
where, () is the Fourier transform of (t). The admissibility condition enables decompo-sition followed by the reconstruction of a signal without loss of information. This condition
implies that the Fourier transform of (t) vanishes at the zero frequency, i.e.
| () |2|=0 = 0 (5)
this means that the wavelets have a band-pass-like spectrum. Furthermore, a zero at zero
frequency means that the average value of the wavelet in the time domain is zero, i.e.
(t) dt= 0 (6)
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A Review of Wavelets for Digital Wireless Communication 393
The regularity conditions are imposed on the wavelet functions in order to make the wavelet
transform decrease quickly as the scale decreases. The regularity condition requires that the
wavelet be locally smooth and concentrated in both the time and frequency domains. Moreabout regularity and admissibility conditions can be found in [7, 16].
2.5. Q U A D R A T U R E M I R R O R F I L T E R S
An important feature of the discrete wavelet transform is the relationship between the impulse
responses of the high pass (analysis) and low pass (scale) filters. The relationship is stated
below:
g[L 1 n] = (1)nh[n] (7)
where g[n] and h[n] are the impulse responses of the high pass and low pass filters, and L isthe filter length. Filters satisfying this condition are commonly used in signal processing, and
they are known as the Quadrature Mirror Filters (QMF). The two filtering and sub-sampling
operations can be expressed by the expressions given in (3).
The reconstruction in this case is easy since the half-band filters form orthonormal bases.
The above procedure is followed in a reverse order for the reconstruction. The signals at every
level are upsampled by two, passed through the synthesis filters g[n], and h[n] (highpassand lowpass, respectively), and then added up. A nice feature to note here is that the impulse
responses of the analysis and synthesis filters are conjugate time reversed versions of one
another i.e.
h[n] = g[n] and g[n] = h[n] (8)Therefore, the reconstruction formula for each layer is given as:
x[n] =
k=(yhigh[k]g[2k n] + ylow[k]h[2k n]) (9)
2.6. S U B B A N D C O D I N G
Subband coding is a hierarchical coding scheme where the signal to be coded is successively
split into high and low frequency components. The wavelet transform can be regarded as a form
of subband coding where the signal to be analyzed is passed through a sieve of filter banks.The outputs of the different filter stages are then the wavelet and scaling function transform
coefficients.
The filter bank is built by using filters that iteratively split the spectrum into two equal parts,
high pass and low pass. The high-pass parts contain the smallest details and hence are not to
be processed any further. However, the low-pass part still contains some details and therefore
it is split again. This dyadic operation is repeated until the required degree of resolution is
obtained. Usually the number of subbands is limited by the amount of data or computation
power available. The process of splitting the spectrum is graphically displayed in Figure 6. The
advantage of this iterative dyadic implementation is that only two filters have to be designed
[14].
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Figure 6. Subband Coding; (Left): Frequency domain representation, (Right): Tree-structure [9].
3. Motivation for Using Wavelets for Wireless Communication
There are several advantages of using wavelets and wavelet packets, for wireless communica-
tion systems. Here we shall list a few desirable features of wavelets:
3.1. S E M I - A R B I T R A R Y D I V I S IO N O F T H E S I G N A L S P A C E
A N D M U L T I R A T E S Y S T E M S
Wavelet transform can create subcarriers of different bandwidth and symbol length. Since each
subcarrier has the same time-frequency plane area, an increase (or decrease) of bandwidth is
bound to a decrease (or increase) of subcarrier symbol length. Such characteristics of the
wavelets can be exploited to create a multirate system. From a communication perspective,
such a feature is favorable for systems that must support multiple data streams with different
transport delay requirements.
3.2. F L E X I B I L I T Y W I T H TI M E - F R E Q U E N C Y TI L I N G
Another advantage of wavelets lies in their ability to arrange the time-frequency tiling in
a manner that minimizes the channel disturbances. By flexibly aligning the time-frequency
tiling, the effect of noise and interference on the signal can be minimized. Wavelet based
systems are capable of overcoming known channel disturbances at the transmitter, rather than
waiting to deal with them at the receiver. Thus, they can enhance the quality of service (QoS)
of wireless systems.
3.3. S I G N A L O R WA V E F O R M D I V E R S I T Y
Wavelets give a newdimension Waveform diversity, to the physical diversities currently ex-
ploited, namely, space, frequency and time-diversity. Signal diversity which is similar to spread
spectrum systems, could be exploited in a cellular communication system, where adjacent cells
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A Review of Wavelets for Digital Wireless Communication 395
can be designated different wavelets in order to minimize inter-cell interference. Another ex-
ample is the Ultra Wideband (UWB) communication system where a very large band with
reduced interference can be shared by users by clever use of transmitting pulse wave shapes.
3.4. S E N S I T I V I T Y T O C H A N N E L E F F E C T S
The performance of a modulation scheme depends on the set of waveforms that the carriers
use. The wavelet scheme therefore holds the promise of reducing the sensitivity of the system
to harmful channel effects like Inter-symbol interference (ISI) and Inter-carrier interference
(ICI).
3.5. F L E X I B I L I T Y W I T H S U B - C A R R I E R S
The derivation of wavelets is directly related to the iterative nature of thewavelet transform. Thewavelet transform allows for a configurable transform size and hence a configurable number
of carriers. This facility can be used, for instance, to reconfigure a transceiver according to a
given communication protocol; the transform size could be selected according to the channel
impulse response characteristics, computational complexity or link quality.
3.6. P O W E R C O N S E R V A T I O N
Wavelet-based algorithms have long been used for data compression. In the context of mobile
wireless devices, which are mostly energy starved, this has an added significance. By com-
pressing the data, a reduced volume of data is transmitted so that the communication power
needed for transmission is reduced.
4. Overview of Applications of Wavelets for Wireless Communication
In this section we give a review of the latest advancements and developments in the use of
wavelets for wireless communications. We have categorized the applications into seven broad
domains namely channel characterization, interference mitigation and de-nosing, modulation
and multiplexing, multiple access communication, Ultra Wideband (UWB) communication,
cognitive radio, and networking. Where necessary, the major domains are dissected into sub-
categories. Further, a brief theory on each technology is introduced at appropriate junctures.
4.1. C H A N N E L C H A R A C T E R I Z A T I O N
4.1.1. Modeling Wireless Channels with Wavelets as Bases
Existing wireless communication channel models are based on statistical impulse response
models derived from empirical results. While these models perform adequately for time in-
variant channels, they fail to accurately map time varying channels. Due to their inherent
joint time-frequency localization property and their ability to accurately characterize the time-
varying nature of the estimation problem, the wavelets offer various advantages for channel
modeling. Some of them are: accurate characterization of time-varying as well as frequency
selective multipath fading channels, fast convergence of estimate, representation of channel
with fewer number of coefficients, small output error, and clear interpretation of modeling
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error. In [17] an algorithm to model time variant systems using wavelets is proposed. Through
simulation and theoretical analysis, it is shown that time-variant channels can be represented
by two sets of discrete time wavelets and a constant coefficient vector. Extension of the work[17] to represent both time invariant and time variant systems with wavelet packets is reported
in [18]. The possibility of using wavelets for modeling channels for OFDM systems is studied
in [19], where the performance of wavelet based channel modeling is compared with that
of Karhunen-Loeve decomposition. In [20] a method for blind maximum-likelihood channel
estimation is proposed where the unknown channel time variations are decomposed using or-
thonormal wavelet bases. Such a characterization of the channel is shown to be effective in fast
fading environments as those found in macrocell wireless communication applications. Ker-
nels derived from orthonormal wavelets for multiresolution representation of direct-sequence
code-division multiple-access (DS-CDMA) channels under fading is proposed in [21]. This
modeling scheme is reported to outperform known approaches in rapidly fading multipath
channels. Lastly, guidelines for choosing the optimal wavelet basis for modeling a time-varyingand fading channels are suggested in [22].
We now discuss the mechanism to model wireless channels with wavelet packets as bases
[23]. For a time-invariant channel, the channel impulse response hTI(t) can be represented by
a set of wavelet packets Pj (t), j = 1, 2, . . . , N as
hTI(t) =N
j=1Aj Pj (t) (10)
Here A is a constants vector and Nthe number of wavelet packets. For time variant channels,
the impulse response hTV(t) can be represented by two sets of wavelets as:
hTV(t, ) =Di
j=1
Doi=1
Ai j Pj ()Qi (t) (11)
where, A is a constants vector of size DiDo, is the time delay, P and Q are sets of wavelet
packet bases of lengths Di and Do, respectively. These bases are used to provide the frequency
and time selection. Figure 7 is a block diagram of such a model.
By judiciously choosing the wavelet and the level of iteration for P and Q, a desired level
of accuracy in the frequency or time domain or both domains can be achieved. Because of
this adaptability, the model holds the promise of representing frequency-selective fading and
time-varying channels efficiently. With the wavelet packet based modeling, for a given input
x (t), the channel output y (t) can be written as
y(t) =Doi=1
Qi (t)
Dij=1
Ai j (Pjx)(t) (12)
4.1.2. Antenna Design and Electromagnetic Computations
One of the cornerstones of antenna theory and design is to describe antenna calculations
in terms of Maxwells electromagnetic equations. In many applications these equations are
difficult to solve explicitly and complex computations have to be employed. Wavelets have
spawned novel algorithms that help solve large electromagnetic field problems using modest
computational resources. These algorithms eventually help in improved design of antennas
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A Review of Wavelets for Digital Wireless Communication 397
Figure 7. Wavelet packet based modeling of time-varying channel [23].
and in the development of optimal waveforms for reliable signal transmission. A comprehen-
sive review of wavelet applications in engineering electromagnetics and in signal analysis is
provided in [24].
4.1.3. Speed Estimation in Wireless Systems
Several reasons motivate the need for estimating the speed in wireless systems [25]. Primary
amongst them is to determine the duration of the temporal window over which the received
signal is averaged to mitigate signal variation. Conventionally, an estimate of the speed is
made by using the statistics of the received signal such as average signal strength or maximumDoppler frequency [25]. In these approaches an appropriate temporal observation window
which depends on the unknown mobile speed is chosen to estimate the required quantities.
The disadvantage of these methods is that such statistics are not readily available. Moreover,
when the speed is varying and the received signal is non-stationary it becomes extremely
difficult to choose the right window. It is in this regard, that wavelets offer a solution. The
wavelet transform at different scales corresponds to different time window lengths and, hence,
obviates the need for adapting the duration of the temporal observation window.
Speed estimation using wavelets exploits three underlying principles [25]:
1. The small-scale spatial variation of the received envelope has a scale that is on the order of
a carrier wavelength.
2. The temporal variations of the received envelope are due to the spatial variations throughthe mobile speed.
3. An estimate of the speed as a function of time can be obtained by tracking the temporal
variations of the received envelop.
To illustrate this, consider Figure 8 where the plot of variation pattern of a typical received
signal with distance is shown. It is evident from the plot that the local minima of the signal
occur with a separation of a fraction of a carrier wavelength . The estimate of the speed may
be given as
v (t) = k/T (t) (13)
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Figure 8. Typical signal trace a function of distance [25].
Figure 9. Absolute value of CWT of signal given in Figure 8 [25].
where, k is the mean separation in distance between the local minima with 0 < k < 1, is the carrier wavelength, and T(t) is the mean time separation between two local minima
of the received signal in the neighborhood of the time t. An efficient estimate of the speed
depends on the accuracy with which T(t) can be measured.
An estimate ofT(t) is obtained by performing a continuous wavelet transform (CWT) of
the received signal. The CWT has the unique property of characterizing minima of the signal.
Thus by performing CWT at various scales, the mean time between two mean minima of the
received signal in the neighborhood of time tcan be efficiently identified. Figure 9 depicts the
absolute value of a CWT of the signal given in Figure 8. White represents large magnitude
and black represents small magnitude. From the CWT coefficients the minima and T(t) are
obtained. Finally, using (13) the speed is estimated.
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4.2. I N T E R F E R E N C E M I T I G A T I O N A N D D E N O I S I N G
In a communication channel, an interference can be due to many reasons unintentional,intentional (Jamming), overlap of symbols due to temporal spreading (Intersymbol interference
or ISI), adjacent channel interference (Interchannel interference or ICI). Wavelets with their
inherent flexible nature, offer the keys to successfully discriminate the signal from the noise
(denoising), while mitigating the effects of interference and noise. In this section we give an
overview of the work carried out in this field.
4.2.1. Signal De-Noising
Wavelet thresholding is a powerful tool for denoising and recovery of true signals from noisy
data. Wavelet orthonormal bases are well-adapted to approximate piecewise-smooth functions,
and to effectively separate signal and noise. In the wavelet denoising method first a suitable
wavelet transform of the noisy data is performed. Then an adaptive threshold is applied.
Wavelet coefficients below the threshold comprise the noise and are eliminated. Finally, the
coefficients are padded with zeros to produce a legitimate wavelet transform and this is inverted
to obtain the signal estimate.
In [26] the profitable use of wavelet based denoising technique to improve power delay
profile estimates in indoor wideband environments is reported. In [27], a wavelet based de-
noising method to estimate the time difference of arrival (TDOA) for GSM signals in noisy
channels is demonstrated. In [127] denoising of signals with low SNR is performed using
composite wavelet shrinkage. Finally, in [28] wavelet based digital signal processing algo-
rithms to combat high power non-stationary noise in infrared wireless systems is proposed,
where through simulations, different wavelet methods are evaluated for denoising and their
efficiency verified.
4.2.2. Mitigation of Interference
In [3] a wavelet packet based wireless communication system that is less sensitive to multipath
channel distortion, synchronization error and non-linear amplification than its Fourier based
counterparts is described. A fuzzy scheme based on the Haar wavelet decomposition for reduc-
ing impulsive noise in direct sequence code division multiple access (DS-CDMA) systems is
demonstrated in [29]. By means of computer simulations, the authors verify the effectiveness
of the method in suppressing the interferences in both time and frequency domains. In [30] a
wavelet packet modulated direct sequence spread spectrum system is introduced. This system
implements an anti-jamming algorithm that helps suppress narrow band jamming signals. And
in [128], a wavelet packet transform-based approach for interference measurement in spread
spectrum wireless communication systems is suggested. This system is non-intrusive and is
reported to be capable of extracting the interference from the spread spectrum signal.
4.2.3. Mitigation of ISI and ICI
ISI and ICI are influenced by two effects [31]:
(i) time dispersion (due to multipath propagation) and frequency dispersion (due to the
Doppler and non-linear effects) of the mobile radio channel and
(ii) shape of the basic pulse.
While the channel effects cannot be controlled, the pulse shape can be carefully designed to
give minimum distortion for a given Doppler and delay spread. The wavelet transform allows
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more flexibility in the design of the waveforms thereby offering the potential to better handle
the channel effects. Investigations in [46] prove that the wavelet based multi-carrier schemes
are indeed superior in suppressing ICI and ISI when compared to the traditional Fourier basedsystems.
4.3. WA V E L E T S F O R M O D U L A T I O N A N D M U L T I P L E X I N G
Digital communication systems can be viewed as general transmultiplexer systems, which
consist of synthesis part and analysis part. The major element which plays an important role
in characterization of the system is the filter set used in both synthesis and analysis parts. The
time-frequency properties of these filters, i.e. time spread and frequency spread, will determine
the type of communication systems (TDMA, FDMA, CDMA, OFDM, MC-CDMA, MC-DS-
CDMA). Unlike contemporary implementations, the wavelet filters that characterize these
systems are derived from wavelet bases (i.e. non-Fourier basis).Recent developments in wireless communications are being driven primarily by the in-
creased demands for radio bandwidth. A key thrust of this drive has been to develop novel
signal transmission techniques that allow for significant increases in wireless capacity without
increases in bandwidth. With this increasing demand one is therefore entitled to wonder about
the possible improvements that more advanced transforms such wavelet transform, could bring
compared to the conventional configurations. Indeed there have been concerted efforts in this
direction and in this section we provide an overview of the various activities carried out towards
the use of wavelets for modulation and demodulation schemes.
4.3.1. Wavelets for Single Carrier Modulation
Figure 10 shows the blocks of a typical narrowband, single carrier communication system. Atthe transmitting end, a source generates an arbitrary stream of data derived from the source
alphabet. This stream of data is then linearly modulated by a pulse shaping filter S(f) and then
transmitted to the channel. At the receiver the received signal is demodulated and decoded by
a receiving filter U(f) and after further processing the data is estimated. Wavelets and scaling
functions can be used to derive the shaping pulse filter.
In [32, 33] give comprehensive notes on the theory and design of communication systems
based on wavelets and wavelet packets. In [34, 35] the use of wavelets and scaling func-
tions as envelop waveforms for modulation are proposed. By using Daubechies wavelets as
shaping pulse, the efficacy and potential of the scheme in improving bandwidth utilization is
demonstrated. This work is extended in [36] where wavelet packets are used for modulation.
While in [37] two types of wavelet based schemes to improve the spectral efficiency of a
digital communication system are presented. The first one is a wavelet based pulse shapingscheme. In this scheme, orthonormal wavelets and their translated versions are used as base
Figure 10. Baseband equivalent of a narrowband communication system [33].
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A Review of Wavelets for Digital Wireless Communication 401
band shaping pulses. This scheme is found to improve spectral efficiency and coding gain.
The other is a wavelet based digital modulation called Wavelet Shift Keying (WSK). In this
scheme, the user data stream is represented by a sequence of scaled and shifted wavelets. Thismodulation scheme is reported to offer a greater spectral efficiency and power efficiency. In
[38] a wavelet with very low sidelobes whose spectral occupancy in the frequency domain is
adaptive is presented. The associated scaling function of the wavelet is derived from the Gaus-
sian waveform and the function is found to be well suited as an elementary shaping pulse for
digital modulation. A survey of the applications of wavelets for modulation schemes is given
in [38]. The applications reviewed include wavelet based Pulse Amplitude Modulator (PAM),
and Wavelet packet modulator (WPM). The wavelet based schemes are reported to provide
improved spectral efficiency and superior performance while ensuring adequate quality of cov-
erage even under difficult conditions. In [40] the impact of clock errors and synchronization
errors on the bit-error rate performance of the optimum receiver for a wavelet-based multirate
signal is investigated. In this reference a maximum likelihood synchronizer is used for timeacquisition. And in [41] the properties of the scaling function are used in the derivation of the
acquisition function.
4.3.2. Wavelets for Multicarrier Modulation Wavelet Based OFDM (WOFDM)
Multi-Carrier Modulation (MCM) is a data transmission technique where the data-stream is
divided into several parallel bit streams, each at a lower bit rate, and by using these substreams
to modulate several carriers. Orthogonal Frequency Division Multiplexing or OFDM is a
MCM scheme where the sub carriers are orthogonal sine/cosine waves. The major drawback
of such an implementation is the rectangular window used, which creates high side lobes.
Moreover, the pulse shaping function used to modulate each sub-carrier extends to infinityin the frequency domain [44]. This leads to high interference and lower performance levels.
The effect of wave-shaping of OFDM signal on ISI and ICI is reported in [42]. In [43] the
optimal wavelet is designed for OFDM signal in order to minimize the maximum of total
interferences. The wavelet transform has longer basis functions and can offer a higher degree
of side lobe suppression [3]. With the promise of greater flexibility and improved performance
against channel effects, wavelet based basis functions have emerged as strong candidates for
MCM in wireless channels. For wired channels though the Fourier based systems are reported
to outperform their wavelet based counterparts [129].
A simplified block diagram of the wavelet packet based multicarrier communication system
is shown in Figure 11.
The transmitted signal in the discrete domain, x[k], is composed of successive modulated
symbols, each of which is constructed as the sum ofMwaveforms k[k] individually amplitude
Figure 11. Wavelet packet modulation functional block diagram [3].
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402 M. K. Lakshmanan and H. Nikookar
modulated. It can be expressed in the discrete domain as:
x[k] =
s
M
1m=0
(as,m m [k s M]) (14)
where, as,m is a constellation encoded sth data symbol modulating the mth waveform. To reduce
probability of error, the waveforms are made orthogonal. In OFDM, the discrete functions
m [k] are the complex basis functions ej 2 m
M . In the wavelet packet scheme, the subcarrier
waveforms are obtained through the wavelet packet transform (WPT). The transmitted symbol
is built by performing inverse transform while the forward transform is used to retrieve the
data symbol. The carrier waveforms are obtained by iteratively filtering the signal into high
and low frequency components. The relationship between the number of iterations J and the
number of carrier waveforms Mis given by M= 2J.In Negash and Nikookar [4] suggest replacing the conventional Fourier-based complexexponential carriers of a multicarrier system with orthonormal wavelets. The wavelets are
derived from a multistage tree-structured Haar and Daubechies orthonormal QMF bank. An
improved performance with respect to reduction of the power of ISI and ICI is reported. This
work is extended in [45] by realizing a high-speed digital communication system over low-
voltage powerline. With empirical investigations on a model obtained from the measurements
of a practical low-voltage powerline communication channel, the authors reaffirm the effec-
tiveness of wavelets for use in OFDM systems, especially with regard to ISI and ICI mitigation.
Another real time application of the system is reported in [46] where Wavelet based OFDM
for V-BLAST [47] (vertical Bell laboratories layered space time) is discussed. According to
[46] the bit error rate (BER) performance of the wavelet based V-BLAST system is superior
to their Fourier based counterparts. In the conventional systems, the ISI and ICI are reduced
by adding a guard interval (GI) using cyclic prefix (CP) to the head of the OFDM symbol.Adding CP can largely reduce the spectrum efficiency. Wavelet based OFDM schemes do not
require CP, thereby enhancing the spectrum efficiency. According to the IEEE broadband wire-
less standard 802.16.3 [48], avoiding the CP gives Wavelet OFDM an advantage of roughly
20% in bandwidth efficiency. Moreover, as pilot tones are not necessary for the wavelet based
OFDM system, they perform better in comparison to existing OFDM systems like 802.1la
or HiperLAN, where 4 out of 52 sub-bands are used for pilots. This gives WOFDM another
8% advantage over typical OFDM implementations. An advanced OFDM modulation scheme
called Isotropic Orthogonal Transform Algorithm (IOTA) for future broadband physical lay-
ers is proposed in [130]. This system uses isotropic Gaussian functions to generate the carrier
waves and gives good spectral efficiency by eliminating the use of a cyclic prefix. In [49] the
promise shown by a Haar WOFDM system with Hadamard spreading codes in reducing itspeak-to-average power ratio (PAPR) is reported. Jamin and Mahonen [3] expanded the realm
of wavelets for OFDM by using wavelet packets to create and detect the different sub-carriers.
The wavelet packets were found to be more flexible, and less sensitive to multipath channel
distortion, synchronization error and non-linear amplification. In [50] a wavelet packet based
OFDM transmission system, referred as orthogonal wavelet division multiplexing (OWDM),
for multirate integrated service is demonstrated. The performance of this system under impulse
noise and single tone interference is reported to be superior to existing Fourier based variants.
The requirements imposed in the design of usable wavelets and wavelet packets for multi-
carrier modulation are studied in [51]. According to this work, for perfect reconstruction of
data the wavelets have to satisfy bi-orthogonal property. In [52] the performances of wavelets
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A Review of Wavelets for Digital Wireless Communication 403
based OFDM and DFT based OFDM are compared and contrasted. Lastly, the performance
of wavelet based OFDM with popular channel coding schemes like Turbo and LDPC codes
for AWGN and Rayleigh channels is reported in [53, 54].
4.3.3. Fractal Modulation
Fractal modulation is a novel diversity strategy that is well-suited for transmission over un-
reliable channels and for broadcast applications. This works by embedding the information
stream into a self-similar waveform so that it is present on all time scales [55]. The major
attraction of this scheme is the user configurable quality of service. The spectral and frac-
tal characteristics of wavelets make them appealing candidates for use in fractal modulation.
Wavelet representations can be exploited to construct orthonormal self-similar bases for these
signals. In [55, 56] the theory of fractal modulation and the use of wavelet-based represen-
tations for this scheme are explained. In [57] two algorithms to choose wavelets for fractal
modulation are presented. In [5] a review of the role of multirate filterbanks and wavelets infractal modulation is provided. The performance analysis of various wavelet families for a
fractal modulation scheme has been examined in [58]. The results reported show the effec-
tiveness of the fractal modulation for utilization in data broadcasting. In [59] the development
of a Wireless Communication system using M-ary phase shift keying (PSK) technique by
fractal wavelet packet transform is discussed. The performance of this system in comparison
to conventional Fourier based systems is reported to be superior. And in [60] an algorithm for
time recovery in fractal modulation is proposed.
4.3.4. Multiplexing
The orthogonal properties of wavelets and wavelet packets make them excellent candidates
for use in multiplexing techniques. Recent developments to this effect are the wavelet division
multiplexing (WDM) and wavelet packet-division multiplexing (WPDM). These are high-capacity, flexible, and robust multiple-signal transmission technique in which the message
signals are waveform coded onto wavelet and wavelet-packet basis functions for transmission.
In [61, 62] multidimensional signaling techniques, and multirate wavelet based modulation
formats that can utilize existing Quadrature Amplitude Modulation. (QAM) channels are
presented. The advantages of these methods include dimensionality in both time and frequency
for flexible channel exploitation and an efficient all-digital filter bank implementation. In [63]
an expression for the probability of error for a WPDM scheme in the presence of both impulsive
and Gaussian noise sources is derived. Through simulations it is demonstrated that WPDM can
provide greater immunity to impulsive noise than both a time-division multiplexing (TDM)
scheme and an orthogonal frequency-division multiplexing scheme. In [64], the performances
of different families of wavelets for modulation over frequency division multiplexing (FDM)nonlinear satellite channels are discussed.
4.4. WA V E L E T S F O R M U L T I P L E A C C E S S C O M M U N I C A T I O N
Wavelets and wavelet packets possess unique properties that make them attractive for use in
multiple access communication. With their offer of greater flexibility in designing signature
waveforms, and their inherent orthogonality property, they can play a vital role in the design of
waveforms and receivers for multiple access systems. In [65] a detailed description of the de-
sign, implementation and analytical analysis of wavelet based multiple access communication
is system is provided. While [66] gives a preliminary analysis on the applicability of wavelet
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404 M. K. Lakshmanan and H. Nikookar
packets in multiple access systems. In [67] a wavelet based asynchronous multiple-access tech-
nique called waveform division multiple access (WDMA) scheme is introduced. This scheme
is reported to offer superior probability of error performance and bandwidth efficiency. In [68]the design and implementation of Wavelet Packet based filter banks for multiple access com-
munications with mathematical background is provided. An improved wavelet-packet-division
multiple access (WPDMA) system is presented in [69] where a wavelet design method based
on numerical optimization is employed to mitigate the effects of asynchronism in the system.
Wavelets and wavelet packets (WP) have also facilitated the design of user signature wave-
forms in code division multiple access (CDMA) communication systems. By randomly clip-
ping the wavelet construction tree, a complete and orthonormal basis is generated. This basis
eventually spawns spreading codes that are orthogonal to one another. Moreover, they display
greater capacity to suppress multiple access interferences [70]. The design and construction
of orthogonal signatures for use in a spread signature code division multiple access system
(CDMA) is discussed in [5]. In [71] a method to use wavelet packets as user signature wave-forms in code division multiple access (CDMA) communications is proposed. According to
[72] wavelet packets allow for simpler equalization and detection of CDMA signals at the
receiver. New spread spectrum codes, whereby spreading and dispreading is achieved by per-
fect reconstructed quadrature mirror filter (PR-QMF) filter banks is described in [73, 74]. In
[75] wavelet-packet-based signatures for asynchronous CDMA systems are proposed. The use
of wavelet transforms for delay detection of a new user entering a CDMA communications
systems is discussed in [76].
In this section we will review the application of wavelets to two new and promising variants
of CDMA, called SCDMA and MC-CDMA.
4.4.1. Scale Code Division Multiple Access (SCDMA)
Scale-code division multiple access is a new multiple access system which exploits the scaleorthogonality introduced by the wavelets in addition to the code and time orthogonality pro-
vided by traditional CDMA systems. In this scheme, each transmitter is assigned a specific
scale and a unique pseudo-random sequence that fits time slots in its scale. Each user encodes
its successive information symbols with time-shifted replicas of the same basic wavelet in
a specific scale and spreads its scaled and translated wavelets (information symbols) by its
pseudo noise sequence. For example, consider the dyadic case for a Haar wavelet as shown in
Figure 12. Here m denotes the scale and n the time slot and pi,l (t) the signature waveform of
the ith user. This waveform is of duration l time periods.
The first scale (the coarsest one, m= 0) uses only one time slot [0, T] using the basic wavelet.The following finer scale (m = 1) uses two time slots [0, T/2], [T/2, T]; the third one (m = 2)four time slots and so on. Therefore, in dyadic SCDMA, over the signaling interval [0, T] thefirst scale users transmit only one information symbol, the second scale users transmit two
information symbols, and the third scale users send four information symbols and so on. Thus,
a multirate system with different rate of information for different messages can be achieved
for the same channel.
Figure 13 shows the model fora SCDMA system. At the transmitter, the bit to be transmitted
bm,i (n), in the nth slot of mth scale by the ith user, is first spread using the orthonormal
wavelet wm,n (t) of scale m to obtain the data signal bm,i (t). A designated carrier cm,n,i (t) is then
modulated by the data signal to obtain the transmitted signal sm,i (t). The carrier is constructed
from a unique signature waveformpi,l (t). The transmitted signal therefore presents three levels
of orthogonality, namely code, time and scale, with respect to other users. At the receiver end,
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A Review of Wavelets for Digital Wireless Communication 405
Figure 12. SCDMA format [78].
Figure 13. SCDMA system model [78].
the transmitted signal is received after a time delay m,i with the addition of channel noise
n(t). The received signal r(t) is down converted, and decorrelated to obtain the baseband data.
Through use of a matched filter (integrator) and a threshold, an estimate of the transmitted
data bit bm,i (n) is obtained.
In [77] an efficient multirate multimedia system based on SCDMA is realized. In [78] the
performance of the SCDMA systems over asynchronous AWGN channels is described. In
[79, 80] a variant of the SCDMA called scale-time-code-division multiple access (STCDMA)is introduced. This system depends on the time, code, and scale orthogonality introduced
by wavelets and is reported to support a larger number of users than conventional DS-
CDMA (six or seven times that of DS-CDMA). In [81] the performance of the STCDMA
over the synchronous AWGN channel is analyzed. In all these studies the reliability of the
SCDMA/STCDMA system and its suitability for multirate multimedia communication is em-
phasized.
4.4.2. MC-CDMA
Multicarrier CDMA or MC-CDMA is a data transmissiontechnique that combines Multicarrier
modulation (MCM) and CDMA. It is a spread spectrum technology, where the spreading is
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406 M. K. Lakshmanan and H. Nikookar
Figure 14. Transmitter model of MC/BPSK-CDMA system [83].
performed in the frequency domain, unlike CDMA, where the spreading is done in the time
domain. By combining the best of MCM and CDMA, MC-CDMA promises high speed,
large bandwidth, better frequency diversity to combat frequency-selective fading and good
performance in severe multipath conditions [82]. MC-CDMA has thus emerged as a strongcandidate for future wireless systems. While conventional MC-CDMA systems use fast Fourier
transforms (FFT), we shall focus here on the role that wavelets can play in MC-CDMA systems.
In comparison to the conventional MC-CDMA systems, introducing wavelets to MC-CDMA
yields the following advantages [83]:
(i) They provide three levels of orthogonality:
between the subcarriers between the wavelets & scaling functions between the spreading sequences.
Therefore in comparison to conventional MC-CDMA systems, they provide new dimen-sions to combat multipath fading, ICI and narrowband interference or jamming signal,
(ii) Flexibility in choosing the spacing between the subcarrier frequencies,
(iii) Flexibility in choosing the wavelet family based on need.
The wavelet-based MC-CDMA transmitter works by first replicating the symbol to be
transmitted into Nc parallel branches. Each copy is then spread by multiplying it with a
pseudo-random chip Nc long and modulated by a wavelet and a subcarrier which is orthogonal
to its neighbors. The signal to be transmitted is finally obtained by adding the outputs of
all branches. At any instant, all the subcarriers convey the same information. For example,
Figure 14 shows the transmitter model of the wavelet-based MC/BPSK-CDMA system.
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A Review of Wavelets for Digital Wireless Communication 407
The wavelet-based MC/BPSK-CDMA is represented as [83]:
Sm(t) = 1Tb
N1i=0
cm [i ]am [k]
t kTb
Tb
+ cm [i ]bm [k]
t kTbTb
cos
2fct+ 2 i
F
Tbt
(15)
where, (t)and (t) are the wavelet and scaling functions, am [k]and bm [k] are two independent
data symbols at the kth bit interval of the mth user, cm [i] is the chip or spreading sequence for
the mth user, Nc is the length of the chip, Tb is the pulse duration, fc is the carrier frequency,
F/Tb is the spacing between neighboring subcarriers.
The implementation of the three wavelet-based MC-CDMA schemes (1) Wavelet-based
MC/BPSK CDMA, (2) Wavelet-based MC/QPSK CDMA and (3) Wavelet-based Fractal MC-
CDMA are given in [83]. In [84] compares and contrasts various MCM techniques including
the wavelet-based MC-CDMA systems. In [72] the development of a wavelet packet based
MC-CDMA system is described. In [85] the system performance is investigated for a multi-
path, slow Rayleigh fading channel. The performance of a wavelet packet based MC-CDMA
system in a correlated fading channel is evaluated in [86]. In [87] the performance of a
turbo-coded MC-CDMA system based on a complex wavelet packet in a Rayleigh fading
channel is analyzed where it is reported that the system copes well with multi-path fading and
multiple-access interference. The performance analysis of a wavelet-basedMC-CDMA system
for satellite communication channels affected by ionosphere scintillation phenomenon is ad-
dressed in [88]. In all these works, the wavelet-based schemes are reported to outperform their
Fourier-based counterparts in terms of bandwidth efficiency and interference immunity. In [89]
a wavelet orthogonal frequency division multiplexing (WOFDM) scheme is used in conjunc-tion with frequency hopping for synchronous code division multiple access (S-CDMA). In [90]
presents work done on an MC-CDMA scheme constructed using orthogonal and bi-orthogonal
filters for down-link cellular radio systems. In [91] the design of MC-CDMA systems based
on a bank of filters derived using the quadrature constrained least square algorithm and satis-
fying the PR-QMF theory, is discussed. In [92] synchronization of wavelet-Based MC-CDMA
systems is elaborated. While in [93] synchronization using wavelets in MC-CDMA systems is
detailed. A wavelet packet based MC-CDMA system is designed in [72]. In [94] a multistage
interference cancellation scheme for a multicarrier direct sequence code division multiple ac-
cess (MC-DS/CDMA) system using wavelet packet transform (WPT) as the basis function
is proposed. Through extensive computer simulations, it is shown that the new system is
suitable for high-rate wireless data transmission. A novel wavelet packet receiver design for
MC-CDMA communications is proposed in [95]. This system is reported to efficiently combat
multipath channel effects and to obviate the need for guard intervals. The performances of
equal gaining and maximal ratio combining equalization techniques for a wavelet packet based
MC-CDMA system are compared in [96]. Finally, the performance of MC-CDMA based on
wavelet packets in a Rayleigh multipath fading channel is studied in [97].
4.5. UWB C O M M U N I C A T I O N
Over the last few years, Ultra-wideband (UWB) communication systems have received signif-
icant attention from industry, the media and academia. The reason for all this excitement is that
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this technology promises to deliver data rates that can scale from 110 Mbit/s at a distance of 10
m and up to 480 Mbit/s at a distance of two meters in realistic multi-path environments, while
consuming very little power and silicon area. It is expected that UWB devices will providelow cost solutions that can satisfy the consumers insatiable appetite for data rates while at the
same time satisfying new consumer market segments. Much of the increased attention paid to
UWB technology is due to the landmark ruling of the Federal Communications Commission
(FCC). In February 2002, the FCC opened up 7,500 MHz of spectrum (from 3.1 to 10.6 GHz)
for commercial UWB device. The FCC requires that UWB devices occupy more than 500
MHz of bandwidth in the 3.110.6-GHz band. Given the bandwidth from 3.1 to 10.6 GHz,
there are several ways to design a UWB communication system. Currently, two major tech-
nologies for UWB wireless communications are under discussion: the Impulse radio (IR) and
the Multi-Band OFDM. Wavelets find applicability in both schemes.
4.5.1. Impulse Radio (IR)IR technology is based on the transmission of very short pulses with relatively low energy. With
this method the entire bandwidth of 7500 MHz is utilized and the transmitted information is
distributed using spread spectrum or code-division multiple access (CDMA) techniques. The
main advantage of building UWB communication systems based on spread-spectrum tech-
niques is that these techniques are well understood and have been proven in other commercial
technologies (e.g. wideband CDMA). Wavelets, being small waves, are naturally suitable for
use in IR-UWB. Some of the properties of wavelets that make them desirable for IR-UWB
communication include [98101]:
(i) Localization in time,
(ii) Localization in frequency,
(iii) Finite energy of the wavelet pulses,(iv) The orthogonal property of wavelets that can be used to separate multipath signals from
two or more sources and to combat interference,
(v) Extensibility of bandwidth.
In [98] a review of various time and band limited waveforms that can be used to create
ultra wideband signals is given. Through simulations the suitability of both scaling and wavelet
functions for use in UWB communication is demonstrated. In [99] is an extension of the work
[98] where the expressions and methods for generating the wavelet functions of the UWB
signals are elaborated. It is also proved that the convolution and addition of signal spectra can
be used to expand UWB bandwidth with wavelet functions by orders of more than 4.5 times. In
[100] wavelet packets for use in ultra wideband communications are proposed, and in [101] asimple circuit for generating wavelets required in UWB impulse radio receiver is presented. In
[132] a multi-user communication system based on UWB technology is studied. In this study,
orthogonal waveforms based on Gegenbauer and Hermite functions have been proposed as
basis functions for the pulse shape. In [134] time of arrival estimation of IR-UWB signals
based on energy detection is discussed. This system uses signal conditioning techniques based
on a bank of cascaded multi-scale energy collection filters and wavelets. In [102] discusses
the reduction of interference from coexisting narrowband signals to UWB impulse radio. The
effect of the interferers is reduced by using a modified template waveform that is constructed
with the multicarrier type template thus removing sub-carrier pulses that are close to the
interfering spectrum.
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A Review of Wavelets for Digital Wireless Communication 409
4.5.2. Multi-Band OFDM (MB-OFDM)
Multi-band OFDM UWB systems have a different philosophy of operation. Instead of using the
entire band to transmit information, the spectrum is divided into several sub-bands with a 10-dBbandwidth of at least 500 MHz. The information is then interleaved across sub-bands and then
transmitted through multi-carrier (OFDM) techniques. The advantage is that the information
is processed over a much smaller bandwidth, thus reducing the complexity of the design and
the power consumption. Other advantages of this approach include its lower cost, improved
spectral flexibility, simplified digital complexity, ability to handle narrowband interference at
the receiver end and compliance to existing standards [103]. While current approaches to MB-
OFDM use the Fourier-based OFDM in each band, introducing wavelet-OFDM can bring new
benefits. For example, one of the disadvantages of the Fourier based OFDM scheme is the use
of the cyclic prefix to negotiate ISI and ICI. In addition to reducing the capability of the system,
the cyclic prefix also causes ripples in the power spectral density (PSD) of the UWB signal
thus resulting in a transmit power back-off of about 1.5 dB [103]. The wavelet-based schemedoes not use a cyclic prefix and hence does not produce the ripples, thereby obviating the need
for power back-offs. Further the UWB systems operate in a very large bandwidth and share the
spectrum with other users as well as with the existing communication systems. Consequently,
interference may occur. Wavelets make it possible to sculpt the UWB signal characteristics to
operate in minimum interference zones. By replacing the traditional sinusoid carriers of the
OFDM system with suitable wavelets, the interferencepower canto a large extent, be mitigated.
In [104] discusses the effects of interference on MB-OFDM UWB radio systems. To reduce
the effects of interference, multicarrier type transmission pulses and template waveforms are
proposed. Simulation studies have proven that the proposed interference mitigation technique
is effective, allowing for coexistence with different wideband systems.
4.6. C O G N I T I V E R A D I O I N T E L L I G E N T WI R E L E S S C O M M U N I C A T I O N S Y S T E M
Oneof the challenges contending communication engineersand designersalikeis that of how to
handle spectrum congestion. Furthermore, an FCC study revealed that spectrum congestions
are more due to the sub-optimal use of spectrum than to the lack of free spectrum [105].
The design of an intelligent communication system that estimates the channel and adaptively
operates in regions with minimum interference is highly desirable. Cognitive radio (CR) is an
attempt in that direction. Cognitive Radio is an advanced technology for the efficient use of
under-utilized spectrums. CR senses the spectrum and detects the presence of primary users.
It adaptively changes the parameters of radio transmission and learns from the environment
to adapt transmission. The two primary objectives of CR are [105]:
to produce highly reliable communications whenever and wherever needed, efficient utilization of the radio spectrum.
The key to the successful operation of cognitive radio of course is to efficiently estimate
the spectrum and scavenge for no-interference zones. Much research effort has been into
the designing of such systems. The transfer domain communication system (TDCS) [106,
107] is one such effort that has displayed the potential for interference avoidance capability.
The system works by gauging the local environment and sculpting the signal to avoid areas
in the basis where interference is present. An offshoot of this system is the wavelet domain
communication system (WDCS) [108110]. Wavelets are used in this scheme to identify and
establish an interference-free spectrum. Wavelet based spectrum estimation has been used for
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410 M. K. Lakshmanan and H. Nikookar
Figure 15. WDCS Transmitter block [106].
the following reasons [108]:
1. Increased adaptability over a larger class of interfering signals
2. Finer high-frequency resolution
3. Allow implementation of M-ary orthogonal signaling
The WDCS uses a packet-based transform to estimate the electromagnetic spectrum [108].
Through the use of adaptive thresholds and notches, sub-bands containing the interference
are effectively canceled. From this estimate, a unique communication basis function A() in
the transform domain is generated so that no (or very little) energy-bearing information is
contained in the areas occupied by primary users. These functions are then multiplied with a
pseudorandom (PR) phase vector ej () to generate Bb(). The PR code is used to randomizethe phase of the spectral components. The resulting complex spectrum is then scaled C to
provide the desired energy in the signal spectrum. A time domain version b(t) of the basis
function is then obtained by performing an inverse transform (wavelet/Fourier). Finally, the
basis function is modulated with data and transmitted. The block diagram of a WDCS general
process is shown in Figure 15.
Another novelty spawned by wavelets is in the design of intelligent UWB systems called
Soft-Spectrum Adaptation UWB (SSA-UWB) [131], which attempts to wed the best of Cog-
nitive radio and IR/multi carrier UWB. The basic unit of this scheme is to dynamically adjust
the transmitted signals spectrum by a proper choice of pulse waveforms and codes, so as to
minimize interference from other systems. In [131] it is suggested that prolate spheroidal wave
functions (PSWF) be used in the design of time-limited and band-limited pulse waveforms for
SSA-UWB systems.
4.7. WA V E L E T S F O R N E T W O R K I N G
Wireless interoperable communication networks are now a reality. They have spawned many
new and exciting applications like mobile entertainment, mobile internet access, healthcare and
medical monitoring services, data sensing in sensor networks, smart homes, combat radios,
disaster management, automated highways and factories. In each of these applications the
system requirements, network capabilities, and device capabilities have enormous variations
giving rise to significant wireless network design challenges. Further, in mobile applications
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A Review of Wavelets for Digital Wireless Communication 411
there are severe constraints on resources like battery power, radio spectrum and buffer space.
Wavelets can help address many of these challenges effectively. In this section we will look
into some of the research activities where wavelets have been used to resolve networkingissues.
4.7.1. Wavelet-Based Adaptive and Energy Efficient Data Processing for Mobile Services
Limited battery power availability is one of the major constraints of mobile communication.
To offset this problem, data compression is often considered. By compressing data, the volume
of data to be transmitted is reduced and hence the communication energy consumed by the
RF circuitry of the device is also reduced. However, the advantage gained by data compres-
sion is offset by the energy utilized by the data compression process. There is a need for a
mechanism that can intelligently trade between the size of the computations and the quality
of the compression desired, so as to deliver good compression even while consuming limited
energy. Wavelet transform based algorithms can be used to select a dynamic parameter that canhandle this trade-off by minimizing energy consumption based on constraints like bandwidth,
quality of compression, and latency. In [111] the details of an adaptive and energy efficient
wavelet-based algorithm for image compression, and for mobile multimedia data services, are
presented. Through experiments, it is shown that statically configured wavelet transform-based
codec can be used to significantly reduce both computation burdens, by minimizing the com-
putations needed to compress an image, and the communication energy, consumed by the RF
component of the mobile appliance. In [112] the applicability of wavelet-based compression
techniques to overcome the bandwidth limitations imposed by low power wireless radios is
considered. And in [133] a wavelet based data compression scheme for source broadcast in
sensor networks is discussed.
4.7.2. Wavelets for Traffic Analysis, Prediction and Load BalancingThe resource constraints in wireless (especially mobile) networks have meant that the dis-
tribution of network traffic must be shared equally between the constituent nodes and that
no host is overloaded. Heavily loaded hosts may cause congestion, long latencies and even
depletion of energy. Traffic predictions and the balancing of the load (traffic) of the network
therefore gain great importance. The wavelet has many advantages for use in traffic analysis,
and prediction and load balancing. Some of them are scale invariance, zero (or minimum)
correlation between wavelet coefficients and short range dependence [113]. Wavelet-based
traffic predictions and estimations have been suggested in quite a few research works. In [114]
a timescale decomposition approach to real time traffic prediction is proposed. The advantage
of using time decomposition with wavelets is that it can capture the correlation structure of
the traffic better than when examining the raw data directly. Using Haar wavelet transform,the raw traffic data is first decomposed into multiple timescales. The wavelet coefficients and
the scaling coefficients on each scale are predicted independently using an autoregressive
integrated moving average model. The predicted wavelet coefficients and scaling coefficient
are then combined to give the predicted traffic value. In [115] a traffic prediction method that
combines recursive least square adaptive filtering with wavelet transform is proposed. In [115,
116] wavelet-based fast, adaptive and convergent algorithms to predict Video traffic are sug-
gested. A wavelet based traffic prediction and adaptive load balancing mechanism for mobile
ad hoc networks (MANET) is proposed in [117]. Finally, in [118] a spatial traffic analysis
based on the wavelet transform is reported. This approach has been found to be effective when
obtaining succinct views of an entire networks traffic load, to gain insight into a networks
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412 M. K. Lakshmanan and H. Nikookar
global traffic response to a link failure, and to localize the extent of a failure event within the
network.
4.7.3. Wavelet Based Data Reconstruction Scheme
The variable nature of wireless environments results in data corruption and loss of informa-
tion. Existing schemes handle this problem by using retransmission protocols like automatic
retransmission query (ARQ) protocols. However, retransmits in noisy environments is unde-
sirable because repeated retransmits leads to an increase in traffic and power consumption.
Reconstructing lost data by using the available surrounding information is therefore an impor-
tant consideration. In [119] a fast scheme for the wavelet-domain interpolation of lost image
blocks in wireless image transmission is presented. The lost data block is reconstructed in
the wavelet domain using the correlation between the lost block and its neighbors. An im-
age transmission method for fading communication channels that provides an image of good
perceptual quality at the receiver in spite of the channel impairments is presented in [120].
4.7.4. Wavelets for Modeling Network Traffic
Modeling of the traffic in wireless networks is a complex task. This is because unlike wired
networks, where the distribution patterns are usually Gaussian, in a wireless environment
the network traffic possesses diverse statistical properties which are non-Gaussian in nature
and possess complex temporal correlation. The main motivation for introducing wavelets
for developing network traffic model comes from the property that although the network
traffic has a complex short and long range temporal difference, the corresponding wavelet
domain representation are all short range dependent. Moreover, the wavelet based scheme
comes with low computational complexity. In [121] wavelet models are proposed to represent
heterogeneous network traffic. Low order Markov wavelet models for Gaussian and fractional
Gaussian noise traffics are developed and investigated. Further, the short-range dependenceamong wavelet coefficients is reported.
4.7.5. Wavelets for Adaptive Distributed Data Processing in Wireless Sensor Networks
Wireless sensor networks are constructed using a large number of small, low-powered wireless
nodes. They are used for information gathering, monitoring and physical control from remote
locations and are usually employed in inaccessible areas and harsh climates. Naturally, they
are limited by their computation, communication, and sensing abilities. In order to maximize
their abilities, wireless sensor networks use two major optimizations:
Wireless sensor networks work on the principle of adaptive distributed load sharing amongst
the constituent nodes,
They exploit spatial correlation between the data collected between nodes that arephysicallyclose together.
Wavelets are used to develop distributed multiresolution algorithms to reconstruct the in-
formation gathered by the nodes with the sensors spending as little energy as possible. In
[122] the design of wavelet based adaptive distributed processing algorithms in large sensor
networks for power efficient data gathering through use of spatially correlated data is provided.
In [123] a distributed wavelet algorithm, based on the lifting scheme, as a means to decorre-
late data at the nodes by exchanging information between neighboring sensors is proposed.
A network architecture that efficiently supports multi-scale communication and collaboration
among sensors for energy and bandwidth efficient communication is reported in [124] where
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A Review of Wavelets for Digital Wireless Communication 413
an initial evaluation of the design under simulation is shown to lead to reduced communication
overhead, and save energy. In [125] an application of wavelet based neural-network has been
proposed for use in wireless sensor networks. The abilities of neural networks have been foundtranslatable to wireless sensor network platforms to yield: simple parallel distributed compu-
tation, distributed storage, data robustness and auto-classification of sensor readings. In [125,
126] report usage of wavelet- based neural networks for data processing of the sensory inputs
at different resolutions. This mechanism is shown to result in reduction of dimensionality, thus
resulting in high energy saving and lower cost.
5. Conclusion
Wavelets provide promising potential applications in wireless communications, ranging from
source and channel coding to transceiver design and from wireless physical channel to net-work and higher layers. The major property of wavelets for these applications is their ability
and flexibility to characterize signals with adaptive time-frequency resolution. The aim of this
paper has been to provide an overview of applications of wavelets in wireless communica-
tions portraying the myriad potentials and possibilities that the wavelets can offer to wireless
communications. Looking ahead, the convergence of information, multimedia, entertainment
and wireless communications has raised hopes of realizing the vision of ubiquitous commu-
nication. To actualize this there is a challenge of developing technologies and architectures
capable of handling large volumes of data under severe constraints of resources such as power
and bandwidth. Wavelets are uniquely qualified to address this challenge. They have strong
advantage of being generic schemes whose actual characteristics can be widely customized to
fulfill the various requirements and constraints of advanced mobile communications systems.The wavelet technology is the choice for smart and resource aware wireless systems. The flex-
ibility gained by wavelet technology cannot be fully exploited with the current systems and
technologies. Therefore, it is foreseen that the wavelet technology would be a strong candidate
for future generation wireless systems.
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