INTERNATIONAL CONFERENCE ON COMMUNICATION, COMPUTER AND POWER (ICCCP’09) MUSCAT, FEBRUARY 15-18, 2009

Studies on DWT-OFDM and FFT-OFDM SystemsKhaizuran Abdullah 1 and Zahir M. Hussain1, SMIEEE

Abstract— Comparative studies on DWT-OFDM and FFT-OFDM systems are presented. The model for DWT-OFDMincludes zero-padding and vector transpose for transmittingthe OFDM signal. MATLAB simulation commands are alsodescribed. The discrete wavelet transform-OFDM (DWT-OFDM)has to satisfy the orthonormal bases and the perfect recon-struction properties to be considered as wavelet transform.Different wavelet families have been used and compared withthe conventional FFT-OFDM system. It is found that the DWT-OFDM platform has less mean amplitude for transmitting thesignal as compared to conventional FFT-OFDM system. Resultsalso show that DWT-OFDM is superior as compared to FFT-OFDM with regards to the bit error rate (BER) performance,especially when it uses bior5.5 or rbior3.3 wavelet family.

-Keywords: Discrete Wavelet Transform, Fourier-basedOFDM, wavelet-based OFDM.

I. INTRODUCTION

An Orthogonal Frequency Division Multiplexing (OFDM)system is a multi-carrier system which utilises a parallelprocessing technique allowing the simultaneous transmissionof data on many closely spaced, orthogonal sub-carriers. In-verse fast Fourier transform (IFFT) and fast Fourier transform(FFT) in a conventional OFDM system are used to multiplexthe signals together and decode the signal at the receiverrespectively. The system adds cyclic prefixes (CP) beforetransmitting the signal. The purpose of this is to increasethe delay spread of the channel so that it becomes longerthan the channel impulse response. The purpose of this is tominimize inter-symbol interference (ISI). However, the CP hasthe disadvantage of reducing the spectral containment of thechannels.

Wavelet transforms have been considered as alternativeplatforms for replacing IFFT and FFT [1], [2], [5], [6], [7]. Byusing the transform, the spectral containment of the channelsis better since it does not use CP [1], [2], [5], [6]. One typeof wavelet transform is namely as Discrete Wavelet TransformOFDM (DWT-OFDM). It employs Low Pass Filter (LPF) andHigh Pass Filter(HPF) operating as Quadrature Mirror Filterssatisfying perfect reconstruction and orthonormal bases prop-erties. The transform uses filter coefficients as approximateand detail in LPF and HPF respectively. The approximatedcoefficients is sometimes referred to as scaling coefficients,whereas, the detailed is referred to wavelet coefficients [3].Sometimes these two filters can be called subband codingsince the signals are divided into sub-signals of low and highfrequencies respectively.

1 School of Electrical and Computer Engineering, RMITUniversity, Melbourne, Victoria 3000, Australia. E-mails: khaizu-ran.abdullah@student.rmit.edu.au, zmhussain@ieee.org.

The purpose of this paper is to perform simulation studyon the wavelet based OFDM particularly in DWT-OFDMas alternative substitution for Fourier based OFDM. Thestudy also includes zero padding and vector transposing fortransmitting DWT-OFDM signal. By zero padding the signals,the transmitted signal is up-sampled and yielded to have lessmean of amplitude. This paper is divided into three mainsections: section II will briefly explain conventional FFT-OFDM, section III will describe in detail the models for DWT-OFDM, and section IV will discuss the bit error rate (BER)performance of both transformed platforms.

II. FOURIER-BASED OFDM (FFT-OFDM)An OFDM transceiver system is shown in Fig. 1. The

inverse and forward transform blocks are concerned in moreattentions since they can be FFT-based or DWT-based OFDM.The system model for FFT-based OFDM will not be discussedin detail as it is well known in the literature. Thus, wemerely present a brief description about it. The data generatorproduces dk in random binary form. It is first being processedby a constellation mapping. M-ary QAM modulator is usedfor this work to map the raw binary data to appropriateQAM symbols Xm. These symbols are then input into theIFFT block. This involves taking N parallel streams of QAMsymbols (N being the number of sub-carriers used in thetransmission of the data) and performing an IFFT operationon this parallel stream. The output in discrete time domain isas follows:

Xk(n) =1√N

N−1∑

i=0

Xm(i) exp(j2π

n

Ni)

(1)

where Xk(n)|0 ≤ n ≤ N − 1 is a sequence in the discretetime domain and Xm(i)|0 ≤ i ≤ N − 1 are complex numbersin the discrete frequency domain. The cyclic prefix (CP) islastly added before transmission to minimize the inter-symbolinterference (ISI). At the receiver, the process is reversed toobtain the decoded data. The CP is removed to obtain the datain the discrete time domain and then processed to FFT for datarecovery. The output of the FFT in the frequency domain isas follows:

Um(i) =N−1∑n=0

Uk(n) exp(− j2π

n

Ni)

(2)

An example of the signals that are processed by inverse andforward transforms for FFT-OFDM is shown in Fig. 2.

III. WAVELET-BASED OFDM (DWT-OFDM)We have discussed briefly about FFT-OFDM in the previous

section. This section discusses the alternative way to replace

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INTERNATIONAL CONFERENCE ON COMMUNICATION, COMPUTER AND POWER (ICCCP’09) MUSCAT, FEBRUARY 15-18, 2009

Fig. 1: An OFDM transceiver with inverse and forward transformblocks that can be substituted as FFT-, or DWT-OFDM.

the conventional OFDM using DWT. We will first explainthe DWT-OFDM implementation model and follow with thediscussion that this platform satisfies the orthonormal basesand perfect reconstruction properties.

The transceiver of DWT-OFDM is shown in Fig. 3. Inthe top part, the transmitter first uses a digital modulator(i.e 16 − QAM ) which maps the serial bits into symbolsconverting dk into Xm, within N parallel data stream Xm(i)

where Xm(i)|0 ≤ i ≤ N −1. The main task of the transmitteris to perform the discrete wavelet modulation by constructingorthonormal wavelets. Each Xmi is first converted to serialrepresentation having a vector XX which will next be trans-posed into CA. This mean that CA not only its imaginary parthave inverting signs but also its form is changed to a parallelmatrix. Then, the signal is 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. Inthe transmitter part, this signal is simulated using MATLABcommand [Xk] = idwt(CA,CD,wv) where wv is the typeof wavelet family. On the other hand, the reverse process issimulated using [ca, cd] = dwt(Uk, wv) in the receiver. Theca signal will be processed to the QAM demodulator fordata recovery. However, the cd signal is discarded becauseit does not contain any useful information. One example ofthis processing signals that pass through this block model isshown in Fig. 4. Note that the data CD has zeroes samplesdue to the zero-padding operation.

A. Orthonormal Base Property

The LPF and HPF coefficients are named as approximatecoefficients and detail coefficients respectively. These filtershave to satisfy orthonormal bases in order to operate as wavelettransform, which means that they must be orthogonal andnormal to each other. By assigning g as LPF filter coefficients

0 10 20 30 40 50 60 70−10

−5

0

5

Rea

l(dat

a)

0 10 20 30 40 50 60 70−4

−2

0

2

4

Subcarrier index

Rea

l(dat

a)

(a) Top: Data XK . Bottom: Data Xm

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−5

0

5

Rea

l(dat

a)

0 10 20 30 40 50 60 70−5

0

5

Subcarrier index

Rea

l(dat

a)

(b) Top: Data UK . Bottom: Data Um

Fig. 2: Samples of one OFDM symbol that is processed inFFT-OFDM system related to Fig. 1. Part (a): Signals in FFT-OFDM Transmitter. Part (b): Signals in FFT-OFDM Receiver.

and h as HPF filter coefficients, the orthonormal bases can besatisfied via four possible ways of the dot products as follows[3]:

< g.g∗ >= 1 (3)

< h.h∗ >= 1 (4)

< g.h∗ >= 0 (5)

< h.g∗ >= 0 (6)

where (3) or (4) is related to the normal property and (5) or (6)is for orthogonal property accordingly. The star superscriptsindicate that the coefficients are conjugated. Both filters arealso assumed to have perfect reconstruction property. Thismeans that the input and output of the two filters are expectedto be the same. A further discussion about this is in the nextsub-section.

B. Perfect Reconstruction Property

A simple construction of block diagram showing perfectreconstruction (PR) property is shown in Fig. 5. The first levelof analysis filter in the receiver part can be folded and thedecimator and the expander are cancelled out by each other.To satisfy a perfect reconstruction operation, the output Yk(i)

is expected to be the same as Xk(i). With the exception of a

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INTERNATIONAL CONFERENCE ON COMMUNICATION, COMPUTER AND POWER (ICCCP’09) MUSCAT, FEBRUARY 15-18, 2009

Fig. 3: The proposed model of DWT-OFDM for substituting into theInverse and Forward Transforms in Fig. 1. Top: The synthesis filters(transmitter part). Bottom: The analysis filters (receiver part) are atthe bottom.

0 10 20 30 40 50 60 70−5

0

5

0 10 20 30 40 50 60 70−1

0

1

0 20 40 60 80 100 120−5

0

5

Subcarrier index

Rea

l ( d

ata

)

(a) Top: data CA. Middle: data CD. Bottom: data Xk

0 10 20 30 40 50 60 70−5

0

5

0 10 20 30 40 50 60 70−1

0

1

2

0 20 40 60 80 100 120−5

0

5

Subcarrier index

Rea

l ( d

ata

)

(b) Top: data ca. Middle: data cd. Bottom: data Uk

Fig. 4: Example of the processed signals of one symbol inwavelet based OFDM system using bior5.5. Part (a): Signalsin DWT-OFDM transmitter. Part (b): Signals in DWT-OFDMreceiver.

Fig. 5: A two-channel filter bank illustrating a perfect reconstructionproperty with the superscript referring to the steps.

time delay, the input can be considered as Yk(i) = Xk(i−n)

where n can be substituted to an integer value. The steps toperform the mathematical operation of PR are given below [3].

1) Selecting the filter coefficients for ga, i.e., a and b. Thus,ga = {a, b}.

2) ha is a reversed version of ga with every other valuenegated. Thus, ha = {b,−a}. If the system has 4filter coefficients with ga = {a, b, c, d}, then ha ={d,−c, b,−a}.

3) hs is the reversed version of ga, thus hs = {b, a}.4) gs is also a reversed version of ha, therefore gs =

{−a, b}.

The above steps can be rewritten as follows:

ga = {a, b}, ha = {b,−a}, hs = {b, a}, gs = {−a, b} (7)

Considering that the input with delay is applied to ha and ga

in Fig. 5, then the output of these filters are

Zk(i) = b(Xk(i) − a(Xk(i−1)) (8)

Wk(i) = a(Xk(i) + b(Xk(i−1)) (9)

Considering also that Zk(i) and Wk(i) are delayed by 1, thenthe subscript i can be replaced by (i− 1) as follows

Zk(i−1) = a(Xk(i−1) + b(Xk(i−2)) (10)

Wk(i−1) = b(Xk(i−1) − a(Xk(i−2)) (11)

The output Yk(i) yields to

Yk(i) = gsZk(i) + hsWk(i) (12)

or,

Yk(i) = −aZk(i) + bZk(i−1) + bWk(i) + aWk(i−1) (13)

Substituting equations (8), (9), (10) and (11) into (13) yieldsto

Yk(i) = 2(a2 + b2)Xk(i−1) (14)

The output Yk(i) is the same as the input Xk(i) except that it isdelayed by 1 if we substitute the coefficient factor 2(a2 + b2)by 1. The PR condition is satisfied.

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INTERNATIONAL CONFERENCE ON COMMUNICATION, COMPUTER AND POWER (ICCCP’09) MUSCAT, FEBRUARY 15-18, 2009

TABLE I: FFT- and DWT-OFDM Parameter Values.

FFT-OFDM DWT-OFDMVariables Matrix Values Matrix Values

N 64 64ns 1000 1000CP 8 0wv - bior3.3, bior5.5, db2, db4

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

IV. SIMULATION RESULTS

Table I shows the simulation variables and their matrixvalues that are used for all simulations. The number of samplesfor the subcarriers N is 64, and the number of samples forthe symbols ns is 1000. If the number of samples for thesubcarriers and symbols are larger, the time for running thesimulations will be longer. By performing a large number ofsamples for the symbols will not make significantly different.Some other variables are listed according to their use as shownin Figs. 1 and 3.

Fig. 6 shows the OFDM symbols in time domain for bothOFDM platforms. Some of the simulation parameters relatedto this figure are: the OFDM symbol period To = 9 ms, thetotal simulation time t = 10 × To = 90 ms, the samplingfrequency fs = 71.11 kHz, the carriers spacing 4N = 1.11kHz and the bandwidth B = 4N × 64 = 71.11 kHz. It isinteresting to see that the average of amplitude vectors in timedomain of DWT-OFDM is smaller than that of FFT-OFDM.It has mean = −9.6673e−4 as compared to FFT-OFDM withmean = 1.4270. This is due to the fact that zero - padding isperformed in the DWT (transmitter) system model. As a result,most samples in the middle of its symbol is almost zeroes.

The BER performances can be observed from Figs. 7 and8. The wavelet families Biorthogonal, Reverse-biorthogonaland Daubechies are compared with FFT-OFDM. These figuresshow that the Biorthogonal and Reverse-biorthogonal produceresults opposite to each other. It is shown that bior5.5 is supe-rior among all in Fig. 7. It outperforms FFT and Daubechiesby about 2 dB, and bior3.3 by 8 dB at 0.001 BER. On the otherhand, Fig. 8 shows the result of using Reverse-biorthogonal.The wavelet family rbior3.3 shows the least error as comparedto others. At BER target of 0.001, it outperforms FFT andDaubechies approximately 4 dB, and rbior5.5 about 6 dB. Itis also interesting to observe that the Daubechies families (db2and db4) perform overlapping curves with FFT-OFDM. This ispossible since their orthogonality follows FFT’s orthogonalityat small filter length such as 2 and 4 in the presence of additivewhite Gaussian noise. The result might have been different ifwe could apply that to a higher filter length with multipathchannel or an impulsive noise environment. However, this willmake the system more complex and might produce more errorsat the receiver. In this case, the system may require mitigation

technique to reduce the errors.

V. CONCLUSIONS

We presented the simulation approaches for DWT-OFDM asalternative substitutions for FFT-OFDM. At some points, wediscussed in details the MATLAB commands regarding theDWT-OFDM platform models and also provided detail aboutorthonormal base and perfect reconstruction properties. Theresults in terms of BER performance were also obtained forall of them. The DWT-OFDM system is superior to others,especially when the system uses bior5.5 or rbior3.3 waveletfamily.

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, M. Ghavami, “Wavelet Vs Fourier Based UWB Systems”,18th IEEE International Symposium on Personal, Indoor and MobileRadio Communications, pp.1-5, September 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, April 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.

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INTERNATIONAL CONFERENCE ON COMMUNICATION, COMPUTER AND POWER (ICCCP’09) MUSCAT, FEBRUARY 15-18, 2009

Fig. 6: OFDM symbols in Time domain for FFT-OFDM (Top) and DWT-OFDM (Bottom).

0 5 10 15 20 25 3010

−3

10−2

10−1

100

SNR per bit, or EbN0 in dB

Bit E

rror R

ate

FFT−OFDMDWT−OFDM: bior3.3DWT−OFDM: bior5.5DWT−OFDM: db2DWT−OFDM: db4

Fig. 7: BER performance for FFT-OFDM and DWT-OFDM (Biorthogonal and Daubechies families).

0 5 10 15 20 25 3010

−3

10−2

10−1

100

SNR per bit, or EbN0 in dB

Bit E

rror R

ate

FFT−OFDMDWT−OFDM: reverse−bio3.3DWT−OFDM: reverse−bio5.5DWT−OFDM:db2DWT−OFDM:db4

Fig. 8: BER performance for FFT-OFDM and DWT-OFDM (Reverse-biorthogonal and Daubechies families).

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