Impulsive Noise Effects on DWT-OFDM versusFFT-OFDM

Khaizuran Abdullah, ∗Saidatul Izyanie Kamarudin, ∗Nadiatul Fatiha Hussin,Sigit PW Jarrot and Ahmad Fadzil Ismail

ECE Department, International Islamic University Malaysia, Kuala Lumpur, Malaysia{khaizuran, sigit, af ismail}@iium.edu.my, ∗{yanie izz, nadiatul f}@yahoo.com

Abstract—A performance study on wavelet-based OFDM, par-ticularly using DWT OFDM as substitutions for Fourier-basedOFDM with the focus on impulse noise effects is demonstrated.The models of the inverse and forward transforms are discussed.The details about each model and study the BER performance intwo scenarios when varying the Poisson recurrence parameter 𝑎from small to large are also included. The wavelet-based OFDM(DWT-OFDM) is assumed to have orthonormal bases and perfectreconstruction properties. Results show that a large value of 𝑎limits the impact of impulsive noise on the system. Comparisonsof impulsive noises in terms of BER for both OFDM platformsare also demonstrated.

-Keywords: Impulse noise, Discrete Wavelet Transform,Wavelet Packet Transform, Fourier-based OFDM, wavelet-based OFDM.

I. INTRODUCTION

Conventional FFT-OFDM system uses IFFT and FFT algo-rithms at the transmitter and receiver respectively to multiplexthe signals and transmit them simultaneously over a numberof subcarriers. It employs guard intervals or cyclic prefixes(CP) so that the delay spread of the channel becomes longerthan the channel impulse response to minimize inter-symbolinterference (ISI). However, the CP has the disadvantage ofreducing the spectral containment of the channels.

Using wavelet transform to replace FFT-OFDM is an al-ternative method as discussed in [1], [2], [5], [6], [7]. Byusing the wavelet transform, the spectral containment of thechannels is better since they are not using CP [1], [2], [5], [6].It can be considered as Discrete Wavelet Transform OFDM(DWT-OFDM) which employs Low Pass Filter (LPF) andHigh Pass Filter(HPF). These filters operate as QuadratureMirror Filters (QMF) satisfying perfect reconstruction andorthonormal bases properties. The transform uses filter coeffi-cients as approximate and detail in LPF and HPF respectively.The approximated coefficients are sometimes referred to asscaling coefficients, whereas, the detailed is referred to waveletcoefficients [3]. Sometimes these two filters can be calledsubband coding since the signals are divided into sub-signalsof low and high frequencies respectively.

The effects of impulse noise on the wavelet-based OFDMparticularly using DWT as substitutions for Fourier-basedOFDM are the main purpose of this paper. A recent work hasfocussed on the effect of impulse noise when wavelet packet

division multiplexing (WPDM) is used [8]. Some discussionsare related to the performance comparison between OFDM andtime division multiplexing. However, its focus is on WPDMand has no indication about DWT-OFDM. Although the stud-ies in [9] provides strong analysis of impulsive noise and itseffect on the performance of OFDM system, the discussion ismainly for the application in the power line communications(PLC). The work on the flexible transformed models of DWT-OFDM and WPT-OFDM as alternative replacements of FFT-OFDM under the effect of impulse noise are shown in [10].It is stated that the work includes DWT-, WPT- and FFT-OFDM. However, the WPT-OFDM system is a complexitysystem since it introduces the wavelet packet tree with manyblocks of application in the transmitter and receiver. As aresult, it yields to time consuming when performing to run thesimulation results. Therefore, this paper highlights on the nichearea between DWT-OFDM and FFT-OFDM systems. Thispaper is divided into five main sections: section II will brieflyexplain conventional FFT-OFDM, section III will describe indetail the models for DWT-OFDM, section IV will providethe impulse noise effects on the OFDM system, and section Vwill discuss the bit error rate (BER) performance consideringtwo different scenarios of impulse noise effects.

II. FOURIER-BASED OFDM

Fig. 1 shows a typical block diagram of an OFDM system.The inverse and forward blocks can be FFT-based or DWT-based OFDM. The system model for FFT-based OFDM is notdiscussed in detail in this section because it is well knownin the literatures. Thus, we merely present a brief descriptionabout it. The data 𝑑𝑘 is first being processed by a constellationmapping. 𝑀 -ary QAM modulator is used for this work tomap the raw binary data to appropriate QAM symbols. Thesesymbols are then input into the IFFT block by processing sub-carriers 𝑁 within parallel streams of QAM outputs. The IFFToutputs are in discrete time domain signal as presented in thefollowing equation

𝑋𝑘(𝑛) =1√𝑁

𝑁−1∑𝑖=0

𝑋𝑚(𝑖) exp(𝑗2𝜋

𝑛

𝑁𝑖)

(1)

where 𝑋𝑘(𝑛) is a sequence in the discrete-time domain and𝑋𝑚(𝑖) are complex numbers in the discrete frequency domain.

2011 17th Asia-Pacific Conference on Communications (APCC) 2nd – 5th October 2011 | Sutera Harbour Resort, Kota Kinabalu, Sabah, Malaysia

978-1-4577-0390-4/11/$26.00 ©2011 IEEE 488

Fig. 1: A Typical model of an OFDM transceiver with inverse andforward transforms that can be substituted as FFT-, or DWT-OFDM.

The cyclic prefix (CP) is lastly added before transmissionto minimize the inter-symbol interference (ISI). When thesystem receives the signal, the process is reversed to obtainthe decoded data. The CP is removed to obtain the data in thediscrete time domain. After the signal is free from the CP, itis then processed to FFT for data recovery. The output of theFFT is in frequency domain and can be expressed as

𝑈𝑚(𝑖) =𝑁−1∑𝑛=0

𝑈𝑘(𝑛) exp(− 𝑗2𝜋

𝑛

𝑁𝑖)

(2)

III. DISCRETE WAVELET TRANSFORM (DWT)

This section describes wavelet based OFDM or DWT-OFDM model. The DWT-OFDM transceiver is shown in Fig.2. The DWT transmitter uses a digital modulator such as16 − 𝑄𝐴𝑀 to map the serial bits into symbols. 𝑋𝑚 is theQAM parallel output after being converted from data 𝑑𝑘 inFig. 1. 𝑋𝑚 can be written as 𝑋𝑚(𝑖)∣0 ≤ 𝑖 ≤ 𝑁−1 where 𝑁 isnumber of OFDM sub carrier. The main task of the transmitteris to perform the discrete wavelet modulation by constructingorthonormal wavelets. 𝑋𝑚𝑖 is then converted into serial formwhich is represented by the vector 𝑋𝑋 before transposing into𝐶𝐴. The signal is then up-sampled and filtered by the LPFcoefficients or namely as approximated coefficients. Since ouraim is to have low frequency signals, the modulated signals𝑋𝑋 perform circular convolution with LPF filter whereas theHPF filter also perform the convolution with zeroes paddingsignals 𝐶𝐷 respectively. Note that the HPF filter containsdetailed coefficients or wavelet coefficients. Different waveletfamilies have different filter length and values of approximatedand detailed coefficients. Both of these filters have to satisfyorthonormal bases in order to operate as wavelet transform.This means that they must be orthogonal and normal to eachother. By assigning 𝑔 as LPF filter coefficients and ℎ as HPFfilter coefficients, the orthonormal bases can be satisfied viafour possible ways [3]:

< 𝑔, 𝑔∗ >= 1 (3)

< ℎ, ℎ∗ >= 1 (4)

< 𝑔, ℎ∗ >= 0 (5)

Fig. 2: An Inverse and Forward Discrete Wavelet Transform DWT-OFDM model. The synthesis filters (transmitter part) are at the topand the analysis filters (receiver part) are at the bottom.

< ℎ, 𝑔∗ >= 0 (6)

where (3) or (4) is related to the normal property and (5) or(6) is for orthogonal property accordingly. Both filters are alsoassumed to have perfect reconstruction property. This meansthat the input and output of the two filters are expected to bethe same [3]. In the transmitter part, this signal is simulatedusing MATLAB command [𝑋𝑘] = 𝑖𝑑𝑤𝑡(𝐶𝐴,𝐶𝐷,𝑤𝑣) where𝑤𝑣 is the type of wavelet family. On the other hand, thereverse process is simulated using [𝑐𝑎, 𝑐𝑑] = 𝑑𝑤𝑡(𝑈𝑘, 𝑤𝑣) inthe receiver. The 𝑐𝑎 signal will be processed to the QAMdemodulator for data recovery. However, the 𝑐𝑑 signal isdiscarded because it does not contain any useful information.

IV. IMPULSIVE NOISE EFFECT

The general principles of the impulse noise when it affectsan OFDM system is described in this section. The recurrenceparameter of Poisson distribution which will affect the systemperformance is included. Assuming that the receiver havingperfect synchronization, 𝑟𝑘 indicated in Fig. 1 can be writtenas

𝑟𝑘 = 𝑦𝑘 + 𝑔𝑘 𝑘 = 0, 1, 2, ...𝐿− 1 (7)

where 𝑦𝑘 is the transmitted OFDM signal, 𝑔𝑘 is the noiseconsisting of AWGN and impulsive noise, and is given by

𝑔𝑘 = 𝑤𝑘 + 𝑖𝑘 (8)

where 𝑤𝑘 is the additive Gaussian process with mean zero andvariance 𝜎2

𝑤 and 𝑖𝑘 can be expressed as

𝑖𝑘 = 𝛽𝑘𝑧𝑘 (9)

where 𝛽𝑘 is the Poisson process indicating the arrival ofimpulsive noise and 𝑧(𝑛) is the white Gaussian process withmean 0 and variance 𝜎2

𝑧 . It is generally assumed that 𝜎2𝑧 is

much larger than 𝜎2𝑤. Note that the impulsive noise having

489

0 20 40 60 80 100 120 140−10

−5

0

5

10

15

time

Rea

l (r k)

GaussianGaussian + impulse noiseσ

z2 > σ

w2 when a=5

(a) Impulsive noise effect when a=5

0 20 40 60 80 100 120 140−10

−5

0

5

10

15

time

Rea

l (r k)

GaussianGaussian + impulse noise

σz2 > σ

w2 when a=50

(b) Impulsive noise effect when a=50

Fig. 3: Sequence Samples of two OFDM symbols with impul-sive noise effect, (a) 𝑎 = 5 and (b) 𝑎 = 50.

variance 𝜎2𝑧 amplitude occurs during the length of 𝐿 samples.

The occurrence of the impulsive noise generally is the Poissondistribution of a random variable 𝑋 can be expressed as [9]

𝑃 (𝑘) = 𝑃 (𝑋 = 𝑘) = exp[−𝑎(𝑎𝑘

𝑘!)] 𝑘 = 0, 1, 2, ..., 𝐿− 1

(10)where 𝑎 is the Poisson parameter which is the average valueof Poisson random variables.

V. EXPERIMENTAL RESULTS AND DISCUSSION

Table I shows the parameters that are used to obtain theresults. The number of samples for the subcarriers 𝑁 is 64,and the number of samples for the symbols 𝑛𝑠 is 1000. Othervariables are explained in sections II and III which are related

TABLE I: Simulation variables and their matrix values.

FFT-OFDM DWT-OFDMVariables Matrix Values Matrix Values

N 64 64ns 1000 1000

d (𝑁 × 𝑛𝑠) 64× 1000 64× 1000𝑋𝑚(𝑁 × 𝑛𝑠) 64× 1000 64× 1000

𝑥𝑥 1× 64000 1× 64000𝑋𝑘 64000× 1 128000× 1𝑈𝑘 64000× 1 128000× 1𝑢𝑢 1× 64000 1× 64000

𝑈𝑚 (𝑁 × 𝑛𝑠) 64× 1000 64× 1000

𝑑′(𝑁 × 𝑛𝑠) 64× 1000 64× 1000

to Figs. 1 and 2. The signal-to-noise ratio (SNR) for all thesimulations is determined as

𝑆𝑁𝑅 =𝑃𝑥

𝑃𝑛=

𝑃𝑥

𝜎2𝑤 + 𝑃𝑖

(11)

where 𝑃𝑥 is the mean power of the transmitted OFDM signal,𝜎2𝑤 is the mean power of the Gaussian noise and 𝑃𝑖 is the

mean impulsive noise power. The ratio 𝑟 between 𝑃𝑖 and 𝜎2𝑤

is defined as 𝑟 = 𝑃𝑖/𝜎2𝑤. Since the impulsive noise follows

the Poisson distribution in timing, the equation (11) can berewritten as

𝑆𝑁𝑅 =𝑃𝑥

𝑃𝑛=

𝑃𝑥

𝜎2𝑤 + 1

𝑎𝜎2𝑧

(12)

where 𝑎 is the Poisson parameter as indicated in equation(10) or it is the average value for Poisson random variablesoccurring during 𝐿 = 𝑁 × 𝑛𝑠 length of samples. In oursimulation, we are interested to vary the values of 𝑎 with thevalue of 𝑟 = 10. When 𝑎 is small, the received OFDM signal𝑟𝑘 has many impulse noise samples as compared to when 𝑎is large. Examples of typical samples of the received OFDMsignals having different 𝑎 = 5 (small value) and 𝑎 = 50 (largevalue) are shown in Figs. 3a and 3b. We have divided thissection in two parts; scenario I: 𝑎 = 5 and 𝑟 = 10, andscenario II: 𝑎 = 50 and 𝑟 = 10. Note that the value of 𝑟,which is the ratio value of 𝑃𝑖 over 𝑃𝑛, is not varied becauseit will not take much effect of BER results even though thetwo platforms are different. On the other hand, 𝑎 will havesignificant change of performance since it is related to theexponential function as indicated in equation (10).

A. Results: Scenario I

Fig. 4 shows the results of Scenario I when 𝑎 = 5 and𝑟 = 10 for FFT- and DWT-OFDM platforms. The conditionof Poisson recurrence parameter 𝑎 = 5 can be referred toas a heavily disturbed since the system has more frequentimpulse noise peaks as indicated in Fig. 3a. At SNR of 20dB, the BER of DWT-OFDM is about 0.15 as comparedto 0.4 of FFT-OFDM respectively when there is impulsenoise (𝜎2

𝑧 >>> 𝜎2𝑤). This also means that DWT-OFDM

produces about 25 less errors in 100 of the received samples ascompared to FFT-OFDM respectively. It is possible to obtainthat result because the wavelet OFDM produces wavelet basis

490

0 5 10 15 20 25 30 35 40 45 5010

−4

10−3

10−2

10−1

100

SNR per bit, or EbN0 in dB

Bit

Err

or

Ra

te FFT−OFDM: NO Impulse noise FFT−OFDM: Impulse noise presence DWT−OFDM: NO Impulse noise DWT−OFDM: Impulse noise presence

Fig. 4: BER performance of Scenario I when 𝑎 = 5 and 𝑟 = 10.

functions at the receiver having about similar basis functionsas in the transmitter satisfying the perfect reconstruction andorthonormal bases properties. In addition, the DWT-OFDMalso has the characteristics of forming the wavelet basisfunction which is the result of the splitting process of scalingand wavelet coefficients corresponding to the LPF and HPFcoefficients as discussed in section III.

B. Results: Scenario II

The results of Scenario II when 𝑎 = 50 and 𝑟 = 10 areshown in Fig. 5. In this scenario, the condition of impulsenoise effect is less impacted since the value of 𝑎 is large asillustrated in Fig. 3b. To be specific, the BER of DWT-OFDMis about 0.03 as compared to 0.3 OF FFT-OFDM respectivelyat SNR of 20 dB in the presence of impulse noise. This alsoshows that the DWT obtain about 3% of errors compared to30% of FFT platforms accordingly. Comparing the BER ofScenarios I and II, we can point out that the performance forthis section is better since it obtained the result of 3% erroragainst 15% error in Scenario I within the same SNR valuewhen impulse noise is presence. This shows that our resultsmatch with the values of 𝑎 as discussed in section IV becausethe effect of the impulsive noise is less when the value of 𝑎is large.

VI. CONCLUSIONS

A performance study on DWT-OFDM as substitution forFFT-OFDM with the focus on the effects of impulse noise ispresented. MATLAB commands regarding DWT transmissionand received signals are also included. Performance in termof BER is also obtained for all techniques while varying thePoisson distribution parameters. Our results show that impulsenoise has less impact on the system when its recurrenceparameter 𝑎 is large. Although there is no mitigation techniquedirectly demonstrated to minimize the impulsive noise, theresults show that DWT-OFDM platform yields to have abetter performance compare to its counterpart. Future work

0 5 10 15 20 25 30 35 40 45 5010

−4

10−3

10−2

10−1

100

SNR per bit, or EbN0 in dB

Bit

Err

or

Ra

te

FFT−OFDM: NO Impulse noise FFT−OFDM: Impulse noise presence DWT−OFDM: NO Impulse noise DWT−OFDM: Impulse noise presence

Fig. 5: BER performance of Scenario II when 𝑎 = 50 and 𝑟 = 10.

may include combination of a novel mitigation technique withDWT-OFDM to minimize the effect of impulsive noise.

REFERENCES

[1] R. Mirghani and M. Ghavami, ”Comparison between Wavelet-based andFourier-based Multicarrier UWB Systems”, IET Communications, Vol. 2,Issue 2, pp. 353-358, 2008.

[2] R. Dilmirghani and M. Ghavami, ”Wavelet Vs Fourier Based UWBSystems”, 18th IEEE International Symposium on Personal, Indoor andMobile Radio Communications, pp.1-5, Sep. 2007.

[3] M. Weeks, Digital Signal Processing Using Matlab and Wavelets, InfinityScience Press LLC, 2007.

[4] S. R. Baig, F. U. Rehman and M. J. Mughal, ”Performance Comparisonof DFT, Discrete Wavelet Packet and Wavelet Transforms in an OFDMTransceiver for Multipath Fading Channel”, 9th IEEE InternationalMultitopic Conference, pp. 1-6, Dec. 2005.

[5] N. Ahmed, Joint Detection Strategies for Orthogonal Frequency Divi-sion Multiplexing, Dissertation for Master of Science, Rice University,Houston, Texas. pp. 1-51, Apr. 2000.

[6] S. D. Sandberg and M. A. Tzannes, ”Overlapped Discrete MultitoneModulation for High Speed Copper Wire Communications”, IEEE Jour-nal on Selected Areas in Communications, vol. 13, no. 9, pp. 1571-1585,1995.

[7] A. N. Akansu and L. Xueming, ”A Comparative Performance Evaluationof DMT (OFDM) and DWMT (DSBMT) Based DSL CommunicationsSystems for Single and Multitone Interference”, Proceedings of the IEEEInternational Conference on Acoustics, Speech and Signal Processing,vol. 6, pp. 3269 - 3272, May 1998.

[8] K. M. Wong, W. Jiangfeng, T. N. Davidson, J. Qu and P. C. Ching,”Performance of wavelet packet-division multiplexing in impulsive andGaussian noise”, IEEE Transactions on Communications, vol. 48, no.7,pp. 1083-1086, 2000.

[9] Y. H. Ma, P. L. So and E. Gunawan, ”Performance analysis of OFDMsystems for broadband power line communications under impulsive noiseand multipath effects”, IEEE Transactions on Power Delivery, vol. 20,no. 2, pp. 674-682, 2005.

[10] K. Abdullah, and Z.M. Hussain, ”Impulsive Noise Effects on DWT- andWPT-OFDM versus FFT-OFDM”, International Conference on Commu-nication, Computer and Power, pp. 373-377, Feb 15-18 2009.

[11] M. Ghosh, ”Analysis of the effect of impulse noise on multicarrier andsingle carrier QAM systems”, IEEE Transactions on Communications,vol. 44, no. 2, pp. 145-147, 1996.

491