Implementation and optimization of Wavelet

in Additive Gaussian channels

R. Niazadeh, S. Nassirpour, M.B. Shamsollahi

Electrical Engineering Department, Sharif University of Technology, Tehran, Iran

Abstract- In this paper, we investigate the wavelet series ofa signal is the key tool for implementing implementation of wavelet modulation (WM) in a digital the Wavelet Modulation which is defined as follows [2]: communication system and propose novel methods to ju improve its performance. We will put particular focus on set) = I Ch,n ((Jj,n(t) +II Xj(n) Wj,n(t) (1) the structure of an optimal in AWGN channels n j=h n j j and address two main methods for inserting the samples Cj(n) =< CfJj,n(t),s(t) > ,CfJj,n(t) = 2 CfJ(2 t - n) (2) of the message signal in different frequency layers. xj(n) =< tPj,n(t),s(t) >,tPj,n(t) = 2jtP(2jt - n) (3) Finally, computer based algorithms are described in order to implement and optimize receivers and transmitters. Where cp(t) and t/J(t) are the scaling and wavelet functions respectively [3], and {xj(n)} are different Keywords Wavelet Modulation; versions of the message signal inserted in the modulated I. INTRODUCTION signal, set). It can be shown that if a signal in L2 (R) is Various analytical methods have been suggested for joint homogenous (which means that its scaled versions are time-frequency analysis. Among these analytical tools, the proportional to its original version), its detail coefficients wavelet transform is of particular interest due to its would be the same in all different scales [2]. Therefore, if distinctive properties that make it usable not only in signal \t} is supposed to be the set of MRA subspaces with ~ processing applications, but also in as the difference between these subspaces, and Cjl S are applications; for example for sending signals in a fractal arbitrarily selected to be zero, then set) can be modulation framework, in which we can send several considered as a signal in \t}u+1 subspace with zero versions of the message in different frequency layers and projection on \t}l' This modulated wave is transmitted therefore transmit the data with a lower bit error rate. This through the . The redundancy of modulation can guarantee a reliable communication through data in this signal allows the recovering of the message channels, in which we do not have any information about signal in receiver with less bit error rate. their exact band-width, bit interval and frequency properties. A fast algorithm for wavelet decomposition of the This powerful method has the property that if the message modulated signal at the receiver is the Mallat algorithm [4] signal gets distorted in the channel due to different which can be implemented using a filter bank structure, undesirable phenomena like lSI and fading, the information as described below: can still be retrieved in the receiver by demodulating the redundant data existing in other rates. In addition to all these, due to its fractal characteristics, a wavelet modulated cAn) =I hem - 2n) Cj+1(m) (4) m signal is noise like and hence can be used in secure data transmission. The Wavelet Modulation (WM) can Xj(n) =I g(m - 2n) Cj+1(m) (5) supersede error control coding methods in wireless m communication systems to eliminate undesirable effects Where {cj (n)} is the projection ofthe received signal on resulted from various phenomena including Doppler's {\t}(n)} and h(n) and g(n) are the scaling and quadrature effect, multipath effect, etc. [1]. filters, respectively. Figures 1 and 2 present this implementation (the coefficients C + (n) are the II. IMPLEMENTATION 1U 1 projections of the modulated signal on It]U+l which, due In Wavelet Modulation we construct a self-similar signal to the similarity of cp (2 Iu+1 t) and the Dirac delta by adding together a countable number of scaled and function, are approximately identical to the samples of modulated versions of our message signal. The discrete the received signal when sampled at a rate of R = 2]U+l):

ISBN 978-89-5519-139-4 -1940- Feb. 15-18,2009 ICACT 2009 Fig.l: Wavelet modulator

Fig.2: Wavelet demodulator

Since there are decimation blocks in this structure, it is r(n) consists of the related observation for a specific value of n in more efficient to put a version of the message signal different scales. It can be shown that by wavelet-based model for 1/f with fewer samples in lower scales to reduce the amount processes (such as z(t) in practice) Zj(n) S may be modeled as of memory required to implement the structure. In this independent Gaussian random variables with zero mean and a paper we will address two novel and efficient methods, variance of Oz 2 [5]. The observation vector would have a different which to our best knowledge are proposed for the first structure according to the algorithm used in the receiver. In the first time. In the first method, we will put decimated versions method, ifwe assume the block to be sufficiently large, it can be seen L of the 2 -point signal x(n) in different scales (i.e. that for a specific value ofn, the symbols of x(n) would repeat in all j Xj(n) = x(2ju- n), where x(n) is the message signal). scales (in fact, this approximation indicates the upper limit of the In the second method, we will put the L-point signal probability oferror). Therefore, the observation vector would be: x(n) with 2m time repetitions in xjz+m(n) (i.e. j Xj(n) = x(n mod l), n E {O,1,2, ... , 2 -jzL - I}). In r(n) = {Tj(n): j EJ = Uz,h + 1, ... juJ} (7) order to send a block of symbols with identical size, less memory is needed in the first method than the second L It can be proved that the optimum decision rule resulting from the ML one (L compared to 2 ), however, in the second method, would be as followed (assuming that H0 and H1 are equally every transmitted symbol will repeat at all rates. probable): In order to prove the efficiency of these methods, their performance is examined in an AWGN P(H Ir(n)) sZ~ P(H Ir(n)) channel which can be modeled as a limited bandwidth, o 1 ~ ~Tj(n) ~Hl limited duration channel. We have considered a binary => I - 0 (8) - ~jEJ 8/ '::SHo communication case in order to simulate the results. i.e. our message signal is a random sequence of binary In the second method, the observation vector can be expressed as: values and each bit has an energy value of Eo ,in other

{~, j words x(n) E -.fEJ. set) and ret) are the r(n) = {1j(n+ml): j E J = Ul,jl + 1, ..·ju},m E {O,1,2, ... 2 -jz - 1}} j modulated and the received signal respectively. In an n E NU) = {O,1,2, ... , 2 - h L - I} (9) AWGN channel, ret) = set) +z(t), where z(t) is a Gaussian random process with zero mean and a variance And the optimum decision rule would be modified as follows: 2 of Oz • In the receiver, the projection of r(t) on t'iU+l' sZ~ ~ would build the rj(n) coefficients. The observation P(Holr(n)) P(H1 Ir(n)) ~~u ~2j-h ~Tj(n+mL) ~Hl vector in the receiver would be as follows: I = • 0 (10) ~}=}l ~m=O 8/ '::SHo r(n) = {1j(n):j E},n E NU)} ,1j(n) = Xj(n) +Zj(n) (6)

ISBN 978-89-5519-139-4 -1941- Feb. 15-18, 20091CACT 2009 CanpIIi!im fi~ betleB1 BirlilyPMt BId Biray"" Ie'el) WI sirn.Ia1iJn With this decision rule, the more the repetitions of a 0.1r------T"""---r------.....---...... ------. symbol are, the less the probability of error is. So, in 0.6 sending blocks of the same length, the second method would have lower error probability but also With this 0.5 decision rule, the more the repetitions of a symbol are, g the less the probability of error is. So, in sending blocks :. 0.4 o ofthe same length, the second method would have lower ~ error probability but also lower spectral efficiency. ! 0.3 e When using the first method, the probability oferror can ~ be formulated as below: 0.2 J~~O 0.1 Pe =P{I > 0IH1)=Q( ) (11)

M =ju - h + 1, Q(x) =f; ke-x2/2dx

Comparison of P(E) between Binary PAM and Binary WM Type2(6 le\el) in simulation O. 7 .---~~-.----- And when using the second method, it would be: J~~o 0.6 Pe =P(/ > 0IH1) =Q( ) (12) 0.5 ju j g K = 2 -j,. Q(x) = ~e-~dx L r ~ 0.4 j=h x ,,(2rr) o ~ ~ 0.3 e The drawback of the wavelet modulation compared to a.. other digital is its relatively low spectral 0.2 efficiency (1]/ = 0.5) [2]. In figure 3, the probability of \ error vs. SNR is plotted for the first and the second 0.1 \f-PAM method together with the PAM modulation for o'------~---'------'-~--'" -- "\--- --~--- comparison. The relative data is resulted from the -150 -100 -SO 0 50 100 simulation of a 512-point signal transmission in 6 SNRlBit successive scales and the wavelet used in this simulation Comparison of P(E) between Binary PAM and Binary WM Type1 & Type2 in Simulation 0.7 ----,-----r------r-----r------, is Daubechies (N=4). As these figures imply, the first and the second method both have a significant SNR 0.6 improvement compared to PAM. But, the amount of

memory required to implement the first method is 0.5 remarkably less than the second one. These results are g summarized in Table 1. ~ 0.4 o :>. I Table I Comparison ofthe Two Methods I 031 t , i.~ Method Simulation SNR improvement in e I ~. 0.. 0.2. Time (1) comparison with PAM WM1r\ I at probability oferror = 0.1 WM2~1 WM1 2.3s 18.7dB 0.1 r I ·\+-PAM WM2 50.22s 13.4dB \ 1 O~ ----.J~~_~\.. I\.. (l)Indicates the amount ofmemory required to implement the -150 -100 -50 o 50 100 SNRlBit method Figure!

ISBN 978-89-5519-139-4 -1942- Feb. 15-18, 20091CACT 2009 References:

[1] Manglani, M. J. and Bell, A. E., "Wavelet modulation in [3] Burrus, C. S., "Introduction to wavelets and wavelet Gaussian and Rayleigh fading channels", Acoustics, Speech, transforms", New Jersey: Prentice Hall, 1998 and Signal Procesing Proceeding(ICASSP'02), IEEE [4] Mallet, S. G., "A theory for multiresolution signal International Conference, 2002 decomposition: The wavelet representation", IEEE [2] Oppenheim, A. V. and Womell, G. W., "Wavelet-based Transactions on Pattern Analysis and Machine Intelligence, representation for a class of self-similar signals with 2(7):674-692, 1989 application to fractal modulation", IEEE Transactions on [5] Womell, G. W., "A Karhunen-Loeve like expansions for IIf Information Theory, 38(2):785-800, 1992 processes via Wavelets", IEEE Transactions on Information Theory, 33: 859-861, 1990

ISBN 978-89-5519-139-4 -1943- Feb. 15-18,2009 ICACT 2009