An Efficient Technique for OFDM System UsingDiscrete Wavelet Transform

W. Saad, N. El-Fishawy, S. EL-Rabaie, and M. Shokair

Dep. of Electronic and Communication Eng., Faculty of Electronic Engineering,El-Menufiya University, Egypt

waleedsaad100@yahoo.com, nelfishawy@hotmail.com, srabie1@yahoo.com,i_shokair@yahoo.com

Abstract. With the rapid expand of wireless digital communications,demand for wireless systems that are reliable and have a high spectralefficiency have increased too. Orthogonal Frequency Division Multiplex-ing (OFDM) has been recognized for its good performance to achievehigh data rates. Fast Fourier Transforms (FFT) has been used to pro-duce the orthogonal sub-carriers. Due to the drawbacks of OFDM-FFTbased system which are the high peak-to-average ratio (PAR) and thesynchronization, many works have replaced the Fourier transform partby wavelet transform. In this paper, an efficient technique for the OFDMsystem using wavelet transform is proposed. This system shows a supe-rior performance when compared with traditional OFDM-FFT systemsthrough an Additive White Gaussian Noise (AWGN) channel. The sys-tem performance is described in Bit Error Rate (BER) as a function ofSignal to Noise Ratio (SNR) and the peak-to-average ratio (PAR). Fur-thermore, the proposed system gives nearly a perfect reconstruction forthe input signal in the presence of Gaussian noise.

Keywords: Orthogonal Frequency Division Multiplexing, Fast FourierTransform, Discrete Wavelet Transform.

1 Introduction

OFDM is a multi-carrier transmission technique, which divides the availablespectrum into many carriers, each one being modulated by a low rate datastream. To implement the OFDM transmission scheme, the message signal mustfirst be digitally modulated. The carrier is then split into lower-frequency sub-carriers that are orthogonal to one another [1,2].

The message signal is first modulated using M-ary QAM. With the adventof cheap powerful processors, the sub-carriers can be generated using FFT. TheFFT moves a signal from the time domain to the frequency domain. While theinverse FFT (IFFT) performs the reciprocal operation. The cyclic prefix (CP)is a copy of the last part of the OFDM symbol, and is of equal or greater lengththan the maximum delay spread of the channel. inter-symbol interference (ISI)[2]. The system model for FFT-based OFDM will not be discussed in detail inthis paper as it is well known in the literature.

R.-S. Chang et al. (Eds.): GPC 2010, LNCS 6104, pp. 533–541, 2010.c© Springer-Verlag Berlin Heidelberg 2010

534 W. Saad et al.

OFDM has been chosen for several current and future communications ys-tems all over the world such as asynchronous digital subscriber line (ADSL),digital audio broadcast (DAB), terrestrial digital videbroadcast (DVB-T) sys-tems, WiMAX systems, etc....

OFDM has several advantages compared to other type of modulation tech-niques such as bandwidth efficiency, overcome the effect of ISI, and combatsthe effect of frequency selective fading and burst error. Nevertheless, the OFDMsystem has some weaknesses such as the high which means that the linear am-plifier has to have a large dynamic range to avoid distorting the peaks. Theother limitation of OFDM in many applications is that it is very sensitive tofrequency errors caused by frequency differences between the local oscillators inthe trasmitter and the receiver [2].

Due to the OFDM drawbacks, wavelet transforms have been considered asalternative platforms for replacing IFFT and FFT. In OFDM wavelet based,the spectral containment of the channels is better since it does not use CP and,hence throughput increases [3,4,5,6,7,8,9,10,11].

In this paper, an OFDM system based on wavelet transform is proposed.The proposed system is simply implemented by a one level Haar wavelet. Thesystem is comed with the traditional OFDM systems using MATLAB simulinkprograms.

This paper is organized as follows; Section 2, gives an overview of the con-tinuous and discrete wavelet transforms and its inverse definitions. Section 3,introduces a comprehensive discussion of the related previous works and theproposed OFDM based on wavelet transform architecture. The simulation re-sults and the comison tests between the proposed and traditional systems aremade in Section 4. Finally, conclusions are made in Section 5.

2 Discrete Wavelet Transform

In most Digital Signal Processing (DSP) applications, the frequency content ofthe signal is very important. The Fourier Transform is probably the most popu-lar transform used to obtain the frequency spectrum of a signal. But the FourierTransform is only suitable for signals whose frequency content does not changewith time. The Fourier Transform does not tell at which time these frequencycomponents occur. To solve this problem, the Wavelet transform, which was de-veloped in the last two decades, provides a better time-frequency representationof the signal than any other existing transforms [12,6,4,13].

The Continuous Wavelet Transform (CWT) is provided by eq. 1.

XWT (τ, s) =1√s

ˆx(t).ψ∗

(t− τ

s

)dt (1)

Where x(t) is the signal to be analyzed, ψ(t) is the mother wavelet or the basisfunction, τ is the translation ameter, and s is the scaling parameter. There are anumber of basis functions that can be used as the mother wavelet such as Haar,

An Efficient Technique for OFDM System Using Discrete Wavelet Transform 535

Daubechies, Symlets and Coiflets. Haar wavelet is one of the oldest and simplestwavelet. While Daubechies wavelets are the most popular wavelets.

The DWT is just a sampled version of CWT. The signals are analyzed inCWT using a set of basis functions which relate to each other by simple scalingand translation. While in the case of DWT, a time-scale representation of thedigital signal is obtained using digital filtering techniques.

The DWT is computed by successive lowpass and highpass filtering of thediscrete time-domain signal as shown in figure 1.

Fig. 1. Three-levels wavelet decomposition tree (DWT)

Where n is an integer. The low pass filter is denoted by G0 while the high passfilter is denoted by H0. At each level, the high pass filter produces detail infor-mation, d[n], while the low pass filter associated with scaling function producescoarse approximations, a[n].

Fig. 2. Three-levels wavelet reconstruction tree (IDWT)

Figure 2 shows the reconstruction of the original signal from the wavelet coef-ficients or the inverse DWT (IDWT). Basically, the reconstruction is the reverseprocess of decomposition. In order to reconstruct the signal, a pair of reconstruc-tion filters G1 and H1 are designed such that output signal y[n] is identical to theinput x[n]. This is known as the condition for perfect reconstruction (PR) [13].

3 The Proposed Technique Description

Fourier based OFDM (OFDM-FFT) implementations have used conventionalFourier filters, and have accomplished data modulation and demodulation viathe IFFT and FFT operations respectively. Wavelet based OFDM provides bet-ter performance due to superior spectral containment properties. In this paper,

536 W. Saad et al.

the IFFT and FFT blocks are simply replaced by IDWT and DWT wavelet fil-ter blocks respectively. In OFDM-DWT data modulation and demodulation areaccomplished via IDWT and DWT operations respectively.

Several works have been done in this field. In [6] the design of the transmitterand receiver for wavelet modulation and its performance in an AWGN channelhave been outlined. It has showed that the BER performance of wavelet modu-lation as function of SNR in the AWGN channel is more accurate. In [8] differenttransmission scenarios concluded that DWT-OFDM performs much better thanDFT-OFDM. But they observed an error floor in DWT- OFDM systems. Theysuggested that it may be resulted from the Haar wavelet base. While in [4] theperformance of Multicarrier Code Division multiple Access communication (MC-CDMA) for wireless Environment investigated in three transmission scenarios.Whereas in [7] the performance of five different most widely used Wavelet basesOFDM schemes over wireless channels has been studied. BER versus SNR andan elapsed time for simulation of wavelet OFDM have been used as a measureof the system performance.

In this paper, the DWT process is simply implemented by a one level (2-band) reconstruction block as shown in figure 3. The input signal x[n] is splitby two filters G0 and H0 into a low pass component and a high pass componentrespectively , both of which are decimated (down-sampled) by 2. In order toreconstruct the signal, a pair of reconstruction filters G1 and H1 are designedaccording to the perfect reconstruction condition. Haar 2-tap wavelet has beenchosen for this implementation. That is because the Haar wavelet is the simplesttype of wavelets, fast, and memory efficient. The filters coefficients correspondingto this wavelet type using MATLAB are shown in Table 1.

Fig. 3. The 2-band reconstruction block

Table 1. Haar 2-tap Wavelet Coefficients

H0 G0 H1 G1

−1√2

1√2

1√2

1√2

1√2

1√2

−1√2

1√2

The basic block diagram for wavelet based OFDM transceiver is shown infigure 4. At the transmitter, the random data bits are mapped into symbols tobe modulated with Quadrature Amplitude Modulator (QAM). Afterwards, thecomplex symbols are modulated via IDWT instead of IFFT then it is transmittedthrough the channel.

An Efficient Technique for OFDM System Using Discrete Wavelet Transform 537

At the receiver, the opposite operation is done. Firstly, the received symbolsare demodulated using DWT then it is demapped by QAM demodulator. Finally,the symbols are converted to the bits.

As illustrated in figure 4, the OFDM-DWT modulator is implemented asfollows; the incoming data symbols are separated into even and odd samples thenthey are applied to Haar 2-tap IDWT. While the OFDM-DWT demodulator isexecuted by Haar 2-tap DWT followed by sample interpolation (up-sampled) by2. The delay unit by one sample time is inserted before the up-sampling unit tocompensate the timing between even and odd samples. Subsequently, the twobranches are added to reconstruct the original signal.

Instead of inserting high and low path filters before the IDWT to produce thedetail and the coarse information respectively [6], samples separation into evenand odd samples is employed. This simplifies the hardware implementation of thesystem. In addition, timing response is improved. Furthermore, the noise fromthe channel has less effect on the signal due to the signal higher instantaneousamplitude as in this case, the inputs to the IDWT are two coarse signals.

Fig. 4. OFDM-DWT transceiver

4 Simulation Results

In this section, the OFDM-DWT system behavior is studied through MATLABsimulink. In addition, comparisons between the OFDM-FFT systems and theproposed OFDM-DWT are done in terms of BER and.

The comparisons module between the OFDM-FFT without cyclic prefix (CP),OFDM-FFT with CP, and the proposed OFDM-DWT are shown in figure 5.

The data source produces frames of data bits. The frame consists of 64 sam-ples each is converted into two bits per sample to be suited for QAM modulator.The IQ-Mapper converts the bits into samples and then performs QAM modu-lation. Afterwards, the complex data are generated which are the input to thethree systems. Thereafter, the output from the OFDM modulator is transmittedthrough AWGN channel which adds Gaussian noise to the transmitted signals.Subsequently, the signals are received and OFDM demodulated, IQ-demapped,and converted to bits respectively.

538 W. Saad et al.

Fig. 5. The comparisons module between the three systems

−35 −30 −25 −20 −15 −10 −5 0 510

−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

SNR vs BER for OFDM systems

OFDM−FFT without CPOFDM−FFT with CPOFDM−DWT

Fig. 6. SNR Versus BER for OFDM systems

Figure 6 displays BER performance as a function of SNR for all OFDM sys-tems. The more SNR, the less BER for the systsm due to the reduction of thenoise effect. The proposed system based on DWT shows better performancethan the other traditional systems. This is because of the use of IDWT insteadof IFFT. In addition, the samples separation implemented in the front of themodulator block gives more instantaneous amplitude to the transmitted signaland hence, it can immune the noise effect.

The two traditional OFDM systems base on FFT have nearly the same BERperformance. This is because the effect of the CP to remove the inter symbolinterference is expounded when multi-path propagation is tested. While the threesystems behaves the same performance with higher SNR above 5 dB.

The output waveforms from the three systems compared with the input signalbefore bit conversion are shown in figure 7. These values are obtained at SNR=1dB. The output from the proposed OFDM-DWT system gives nearly a perfectreconstruction for the input signal. This is because of the samples separations.Moreover, the filters are synthesized under the condition for perfect reconstruc-tion in DWT system. While the two other systems behaves nearly in the samemanner with worse performance compared with the proposed system.

An Efficient Technique for OFDM System Using Discrete Wavelet Transform 539

Fig. 7. The output waveforms compared with the input signal

The crest factor (CF) or peak-to-average ratio (PAR) or peak-to-averagepower ratio (PAPR) is a measurement of a waveform, calculated from the peakamplitude of the waveform divided by the Root Mean Square (RMS) value ofthe waveform

(CF = max|x(t)|2

x(t)2rms

). The definition of the PAR in dB should be

10log10 (CF ). The PAR values for the three OFDM systems are shown in Ta-ble 2. It is cleared that the proposed system gives the best PAR value among thesystems. This is because of the use of IDWT in place of IFFT. Then the averagevalue of the transmitted signal for the proposed system is reduced. The reasonfor this is, the proposed system can nearly stabilize the amplitude of the trans-mitted signal due to the use of the sample separation in the front of the IDWT.The proposed OFDM-DWT has 7.03 dB and 7.3 dB better PAR performancethan OFDM-IFFT without and with CP respectively.

The effect of the frame size on the three systems is revealed from figure 8.These values are obtained at SNR=1 dB for example. For OFDM-FFT basedsystems, the larger frame size, the more BER value and hence more systemdegradation. This is because that for larger frame size, the I/FFT size will be

540 W. Saad et al.

Table 2. PAR Value for the three OFDM systems

The system PAR value in dBOFDM-FFT without cyclic CP 8.53

OFDM-FFT with CP 8.8The proposed OFDM-DWT 1.5

0 100 200 300 400 500 60010

−4

10−3

10−2

10−1

100

Frame size (symbols)

BE

R

Frame size vs BER for OFDM systems

OFDM−FFT without CPOFDM−FFT with CPOFDM−DWT

Fig. 8. The effect of the frame size

increased, and the bit rate will be increased which degrades the system perfor-mance. While for the proposed OFDM-DWT system, it is not affected by theframe size. In this paper, Haar 2-tap I/DWT is used. Therefore, only two samplesare entered to the OFDM-modulator simultaneously unaffected with the framesize. In addition, the BER performance for the proposed ODFM-DWT systemis better than other systems for all frame sizes.

At any other value for the SNR, the same relation will be the same. This isbecause the proposed protocol is better than traditional system in all SNR as itis obviously appeared from figure 6.

5 Conclusions

In this paper, an efficient technique OFDM-DWT system was proposed. Extensivesimulation programs were performed to investigate the efficiency of the proposedsystem compared with traditional OFDM systems based on FFT. The proposedsystem showed a superior performance. It was simpler and less hardware than tra-ditional systems. In addition, it had less BER performance as a function of SNRthan traditional systems. Moreover, it donated a perfect reconstruction for theinput signal. Furthermore, it had less PAR value than traditional systems. In ad-dition, the frame size had no effect on the proposed system performance.

An Efficient Technique for OFDM System Using Discrete Wavelet Transform 541

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