Available online at www.sciencedirect.com
ScienceDirect Procedia Technology 24 (2016) 775 – 781
International Conference on Emerging Trends in Engineering, Science and Technology (ICETEST - 2015)
Performance improvement of multicarrier systems using wavelet filter banks Job Chunkath*, Arjun S.S, V.S. Sheeba, Anso Raj S Government Engineering College, Thrissur, 680009, India University of Calicut
Abstract
The need for higher data rates with increased bandwidth efficiency has focussed the search for techniques which deliver better results than conventional Orthogonal Frequency Division Multiplexing (OFDM) system. A wavelet filter bank system is investigated as a multicarrier modulation system (MCM). Such a system is found to be flexible, efficient and has many advantages over the present OFDM systems. This paper deals with identifying the suitability of different wavelet families, which can be used to improve the performance parameters of existing systems. Different wavelets families Daubechies, Meyer and Battle-Lemarie, are used as filter coefficients for wavelet based OFDM system and it is found that Daubechies wavelet (Db4) based multicarrier system outperforms the other two. © Published byby Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2016 2016The TheAuthors. Authors.Published Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICETEST – 2015. Peer-review under responsibility of the organizing committee of ICETEST – 2015 Keywords: Multicarrier modulation; OFDM; FBMC; Wavelet filter bank; Wavelet packet modulation; BER; Stopband attenuation
1. Introduction During the pioneering years of digital communication single channel systems were used. The requirement of establishing high performance systems necessitated the introduction of multicarrier transmission systems. In these systems there exists a trade off between data rate and cost of performance, another between bandwidth and symbol
* Corresponding author. Job Chunkath Tel.: +919496234943. E-mail address:
[email protected]
2212-0173 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICETEST – 2015 doi:10.1016/j.protcy.2016.05.087
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interval [1]. Due to factors like inter symbol interference (ISI) , inter channel interference (ICI), channel delay spread which causes ISI, Doppler shift caused due to dynamics of transmitter/receiver and multipath propagation, much research attention is given for developing methods which would mitigate these problems. Methods which give good performance in noisy channels, better bandwidth efficiency and ability to overcome the above stated problems to some extend are explored. In this wake, OFDM[2], a multicarrier modulation system was introduced. In such Multicarrier systems, spectrum gets divided into many sub-channels which results in each sub-channel getting modulated at a lower data rate and each sub-channel getting less affected by noise and fading, thus each can be dealt separately. The delay spread can be reduced as its effect gets spread over all the sub-channels [1]. But the OFDM has large side lobes due to the rectangular pulse used. This results in poor performance. To tackle this drawback, cyclic prefix (CP) is used. But this comes at the price of reduced bandwidth efficiency as CP occupies some bandwidth without carrying useful information. Another drawback is large peak to average power ratio (PAPR). [3] Discusses methods like clipping, μlaw companding and A-law companding etc. for PAPR reduction in FBMC systems. Wavelet based systems provide enough reasons to be used for multicarrier modulation. Wavelets are localized in both time and frequency domain and have compact support [4, 22]. Wavelet based systems provide flexibility[5, 6], they exhibits lower PAPR [7] and have better spectral containment[6].It is mentioned that Daubechies wavelets based multicarrier system outperform OFDM when it comes to time dispersive channels as wavelets are better localized in time[8]. Since wavelet based system do not use CP, bandwidth efficiency is increased up to 35% [9, 10]. WPM[11, 12] is a wavelet based multicarrier system which using the power of multirate filter bank theory,[13] can be utilized effectively to improve the present competitive scenario for higher data rates. However most of the previous works are based on single wavelet family. In some of the works, the BER performance is not analyzed with regard to communication systems, and in some cases lower BER is achieved with higher SNR. In this paper, a 6 level wavelet based OFDM system is simulated for 64 users and different wavelet families are tried as filter coefficients, their performances are compared so as to analyze the BER performance. A wavelet which gives a good stopband attenuation is Battle-Lemarie[14] is also tested. it is found that Daubechies filter bank multicarrier system gives better performance than the other two. Rest of the paper is organized as follows. Section II presents a brief discussion on FT, STFT and wavelets. Section III discusses multicarrier modulation which is followed by simulation results in section IV. Concluding remarks are given in section V. 2. Fourier Transform, STFT and Wavelets It is well known that complex exponentials are used as basis functions to analyze periodic and non-periodic signals in Fourier representation. A signal can be written as a sum of complex exponentials as given by , λ
ሺɘሻ ൌ න ሺሻെɘ
(1)
െλ
where ܠሺܜሻ is a continuous time signal and ܆ሺሻ is the Fourier transform of ܠሺܜሻ. Fourier transform lacks time localization which is very important in certain application like speech processing. This shortcoming is solved by Windowed Fourier transform (WFT) or Short Time Fourier Transform(STFT) in which time localization is introduced. STFT of ܠሺܜሻ is given by λ
ܺሺ߱ǡ ߬ሻ ൌ න ݔሺݐሻ݁ െ݆߱ݓ ݐሺ ݐെ ߬ሻ݀ݐ
(2)
െλ
where ܟሺǤ ሻ is the window, ૌ is time-shift and ܆ሺǡ ૌሻ is the STFT of ܠሺܜሻ. The time axis is thus divided into many equally spaced windows implying fixed resolution throughout the time-frequency plane. Thus both time and
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frequency resolution cannot be achieved at the same time as stated by uncertainty principle. However, a trade off between time and frequency resolution is possible which is achieved with Wavelet Transform (WT) [15, 16, 17]. Consider ૐሺܜሻ to be the mother wavelet, then, ߰ܽǡܾ ሺݐሻ ൌ
ͳ
ݐെܾ ߰൬ ൰ ǡ ܽ ܴ אǡ ܾ ܽ ඥȁܽȁ
(3)
where ܊is shifting factor, ܉is scaling factor and ૐ܉ǡ ܊ሺܜሻ are shifted and scaled versions of mother wavelets known as daughter wavelets. Mother wavelet and daughter wavelets are orthogonal to each other, forming basis function. So variable length basis functions are obtained using wavelets. Wavelet transform of a signal ܠሺܜሻ is given by, ܹሺܽǡ ܾሻ ൌ
λ ݐെܾ න ݔሺݐሻ߰ ൬ ൰ ݀ݐ ܽ ඥȁܽȁ െλ
ͳ
(4)
where ܉ൌ െ ܒǡ ܊ൌ ܓǡ ܒǡ ܈ א ܓfor discrete wavelet transform(DWT). 3. Multicarrier Modulation 3.1. FFT based OFDM OFDM is a multi-carrier transmission technique, which divides the available spectrum into many sub-channels for sub-carriers, each sub-channel getting modulated by a low rate data stream. Data streams before modulating the subcarrier get mapped using digital modulation schemes like PSK, QAM etc. For M symbols to be transmitted, M subcarriers which are orthogonal to each other are used. Orthogonal sub-carriers are produced using the IFFT matrix. The output of IFFT processing is the OFDM symbol as given below:
ሺሻ ൌ
ͳ
െͳ
ሺሻቀ
ʹɎ ቁ
ξ ൌͲ
(5)
where ܠሺܖሻ is OFDM signal, ܖൌ ǡ Ǥ Ǥ ǡ ۻെ . Cyclic prefix is added with OFDM symbol before transmission so that ISI is removed at the receiver end.
Fig.1. Block schematic of OFDM based communication system.
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3.2. FBMC Drawbacks in OFDM system leads to development of FBMC system. It is a MCM technique which requires the designing of a prototype filter to tackle channel interference. These prototype filters are of larger order having good stop band attenuation. Figure 2 shows a FBMC system. Where ݃݉ ሺ݊ሻ is synthesis filter and ݄݉ ሺ݊ሻ is analysis filter. The transmitted signal ݔሺ݊ሻ is given as ܯെͳ
ݔሺ݊ሻ ൌ ݉ݏǡ݈ ݃݉ ሺ݊ െ ݈ܮሻ ݈
(6)
݉ ൌͲ
Here ܯis the total number of subcarriers, ݉ݏǡ݈ is ݈ ݄ݐsymbol in ݉ ݄ݐsubcarrier and ܮis the number of symbols transmitted per symbol spacing. Filters ݃ሺ݊ሻ and ݄ሺ݊ሻ are designed in such a way as to satisfy perfect reconstruction (PR) condition.ݕሺ݊ሻ is the received signal.
Fig.2. A general Filter Bank Multicarrier System 3.3. Wavelet based Systems 3.3.1. Wavelet Packet Transform(WPT) Wavelets are widely used in multiresolution analysis of signals [18]. Any signal can be split into two spaces namely V and W i.e, its approximation and details space respectively such that ܄ ؿ ڮെ ܄ ؿ ܄ ؿ ڮ ؿ and ܑ܄ൌ ܑ܄െ ْ ܑ܅െ ’܄s are made up of scaling function ሺܜሻ and its translates and ’܅s are made up of wavelet function ૐሺܜሻ and its translates. The basis functions of spaces ܒ܄and ܒ܅are respectively given as; Ԅ ሺሻ ൌ ʹȀʹ ሺሻԄሺʹ െ ሻ
(7)
ɗ ሺሻ ൌ ʹȀʹ ሺሻɗሺʹ െ ሻ
(8)
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where ܐሺܖሻ and ሺܖሻ are coefficients such that ሺܖሻ ൌ ሺെሻܐ ܖሺሺ ۼെ ሻ െ ܖሻ, ۼbeing number of coefficients in ܐሺܖሻ. They are QMF pair as well, ܐሺܖሻ being lowpass filter coefficients and ሺܖሻ being highpass filter coefficients. Signals can be represented using the scaling and wavelet function as: ݔሺݐሻ ൌ ݆ܽ ሺ݇ሻʹ݆ Ȁʹ ߶ሺʹ݆ ݐെ ݇ሻ ݆݀ ሺ݇ሻʹ݆ Ȁʹ ߰ሺʹ݆ ݐെ ݇ሻ ݇
݆
(9)
݇
Where ݆ܽ ሺ݇ሻ and ݆݀ ሺ݇ሻ are approximation and detail coefficients respectively. 3.3.2. Wavelet Packet Modulation(WPM) Multicarrier modulation based on wavelet packets is known as wavelet packet modulation [19]. A WPM transmitter is shown below with level ܔൌ i.e, total sub-channels ۻൌ ܔൌ . Filters used are ܘሺܖሻ andܙሺܖሻ, such that ܘሺܖሻ ൌ ܐሺܖሻ andܙሺܖሻ ൌ ሺܖሻ.
Fig.3.a. A two level WPM transmitter
Fig.3.b. A two level WPM receiver
The impulse response of each sub-channel of WPM say ܴ݉ ሺݖሻ where ݉ ൌ Ͳǡͳǡ Ǥ Ǥ ǡ ܯെ ͳ after doing some rearrangement using noble identities[20, 21], is given as: (10) ܴͲ ሺݖሻ ൌ ܲሺ ʹ ݖሻܲሺݖሻ ܴͳ ሺݖሻ ൌ ܳሺ ʹ ݖሻܲሺݖሻ
(11)
ܴʹ ሺݖሻ ൌ ܲሺ ʹ ݖሻܳሺݖሻ
(12)
ܴ͵ ሺݖሻ ൌ ܳሺ ʹ ݖሻܳሺݖሻ
(13)
The signal from transmitter section is passed through AWGN channel. The receiver is given below and is just the reversed version of transmitter section. Filters used are h(n) and g(n), which are LPF and HPF respectively 3.3.3. Wavelet packet based multicarrier system The IDWT and DWT blocks shown in Figure 4 is realized using a 6 level structure similar to 2 level WPM transmitter and receiver shown Figure 3.a and Figure 3.b respectively. The input signal is modulated using certain digital modulation schemes like BPSK, QPSK, M-QAM etc and data is passed through WPM transmitter followed by a channel. Received data is passed through WPM receiver and demodulated.
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Fig.4. Block schematic of wavelet packet based multicarrier system. 4. Simulation and results A wavelet packet based multicarrier system is developed in MATLAB. Different wavelets ݖ݅ݒǤ Db4, Meyer and Battle-Lemarie, are used as filter coefficients. The input data is NRZ-BPSK modulated and given to the WPM multicarrier system. The channel is approximated as an AWGN channel. Comparison is made between OFDMOQAM system [3] and W-OFDM system with 64 subcarriers or users. ͳͲ samples are transmitted and BER performance is evaluated. Figure 5 shows the BER performance of the system.
Fig.5. BER comparison between W-OFDM system with Db4, Meyer and Battle-Lemarie and FBMC-OQAM system It is found that wavelet based OFDM system which uses Db4 as filter coefficients gives better performance compared to FBMC-OQAM system. Comparison among various wavelets families are also made, namely Daubechies, Meyer and Battle-Lemarie, and it can be shown from the BER plot that system based on Db4 is better than the other two. 5. Conclusion A wavelet packet based multicarrier system is simulated in this paper where wavelets like Db4, Meyer and Battle-Lemarie were used as filter coefficients with ͳͲ bits transmitted through AWGN channel. The performance of the system is evaluated using BER plot. The simulation results show that wavelet packet based multicarrier system performs better than OFDM-OQAM system when Db4 is used as filter coefficient. Among the three wavelets compared namely Db4, Meyer and Battle-Lemarie, Db4 performs better. Future works can be focused on finding more wavelets with better performance, and with suitability in reducing PAPR.
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