Art%3A10 1007%2Fs11277-012-0848-3

Wireless Pers CommunDOI 10.1007/s11277-012-0848-3

A Radon-Multiwavelet Based OFDM System Designand Simulation Under Different Channel Conditions

Abbas Hasan Kattoush · Waleed Ameen Mahmoud Al-Jawher ·Osama Q. Al-Thahab

© Springer Science+Business Media, LLC. 2012

Abstract Finite Radon transform (FRAT) mapper has the ability to increase orthogonalityof sub-carriers, it is non sensitive to channel parameters variations, and has a small constel-lation energy compared with conventional fast Fourier transform (FFT) based orthogonalfrequency division multiplexing (OFDM). It is also able to work as a good interleaver whichsignificantly reduces the bit error rate (BER). Due to its good orthogonality, discrete Multi-wavelet transform (DMWT) is attractive for implementation in OFDM systems which reducesinter-symbol interference (ISI) and inter-carrier interference (ICI) and eliminates the needfor cyclic prefix and increases the spectral efficiency of the design. In this paper both FRATand DMWT are implemented in a new design for OFDM. The new structure was tested andcompared with conventional FFT-based OFDM, Radon-based OFDM, and DMWT-basedOFDM for additive white Gaussian noise channel, flat fading channel, and multi-path selec-tive fading channel. Simulation tests were generated for different channels parameters values.The obtained results showed that proposed system has increased spectral efficiency, reducedISI and ICI, and improved BER performance compared with other systems.

Keywords Discrete multiwavelet transform · Finite Radon transform · Radon basedOFDM · DMWT based OFDM · Critically sampling algorithm

A. H. Kattoush (B)EE Department, Tafila Technical University, Tafila, Jordane-mail: [email protected]

W. A. M. Al-JawherDean of Engineering College, Al-Isra University, Amman, Jordan

O. Q. Al-ThahabEE Department, University of Technology, Baghdad, Baghdad, Iraq

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A. H. Kattoush et al.

1 Introduction

The main problem in the design of a communications system over a wireless link is to dealwith multi-path fading, which causes a significant degradation in terms of both the reliabil-ity of the link and the data rate [1]. Multi-path fading channels have a severe effect on theperformance of wireless communication systems even those systems that exhibits efficientbandwidth, like orthogonal frequency division multiplexing (OFDM). There is always a needfor developments in the realization of these systems as well as efficient channel estimationand equalization methods to enable these systems to reach their maximum performance [2].The OFDM receiver structure allows relatively straightforward signal processing to combatchannel delay spreads, which was a prime motivation to use OFDM modulation methods inseveral standards.

In transmissions over a radio channel, the orthogonality of the signals is maintained only ifthe channel is flat and time-invariant, channels with a Doppler spread and the correspondingtime variations corrupt the orthogonality of the OFDM sub-carrier waveforms [3]. In a dis-persive channel, self-interference occurs among successive symbols at the same sub-carriercasing inter symbol interference (ISI), as well as among signals at different sub-carriers cas-ing inter carrier interference (ICI). For a time-invariant but frequency-selective channel, ICI,as well as ISI, can effectively be avoided by inserting a cyclic prefix before each block ofparallel data symbols at the cost of power loss and bandwidth expansion [4].

The Radon transform (RT) was first introduced by Johann Radon (1917) and the theory,basic aspects, and applications of this transform are studied in [5,6] while the finite Radontransform (FRAT) was first studied by [7]. RT is the underlying fundamental concept usedfor computerized tomography scanning, as well for a wide range of other disciplines, includ-ing radar imaging, geophysical imaging, nondestructive testing and medical imaging [6].Recently FRAT was proposed as a mapping technique in OFDM system [8–10].

Conventional OFDM/QAM systems are robust for multi-path channels due to the cycli-cally prefixed guard interval which is inserted between consequent symbols to cancel ISI.However, this guard interval decreases the spectral efficiency of the OFDM system as thecorresponding amount. Wavelet bases, which do not require the use of guard interval, havebeen introduced into the communication field as an alternative approach to Multi-carriermodulation (MCM) [11,12]. It is found that OFDM based on Haar-based orthonormal wave-lets (DWT-OFDM) are capable of reducing the ISI and ICI, which are caused by the loss inorthogonality between the carriers. Multiwavelet are designed to be simultaneously symmet-ric, orthogonal and having short supports with high approximation power, which cannot beachieved at the same time for wavelet using only one scaling function.

In this paper the idea of one dimensional serial Radon based OFDM proposed in [9]and the idea of Multiwavelet based OFDM proposed in [13] were developed farther to wordsincreasing spectral efficiency and reducing BER. Further performance gains and higher spec-tral efficiency were made by combining both FRAT and DMWT in the design of OFDMsystem. Simulation results show that proposed system has better performance than Fourier,Radon, and Multiwavelet based OFDM under different channel conditions.

2 The Radon-Based OFDM

Radon-based OFDM was recently proposed in [8–10], it was found that as a result of apply-ing FRAT, the bit error rate (BER) performance was improved significantly, especially inthe existence of multi-path fading channels. Also, it was found that Radon-based OFDM

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A Radon-Multiwavelet Based OFDM System Design

Serial toParallel

FRATMapper 1-D

IFFT

OFDM Receiver

OFDM Transmitter

InputData

Add GuardInterval

Parallel toSerial Channel +

AWGN

Serial toParallel

Removethe Guard

1-D FFTFRATDemapper

Parallel toSerial

1-D Serial OFDM SignalPilot

Symbols

ChannelCompensation

ChannelEstimator

Fig. 1 Serial Radon based OFDM transceiver

structure is less sensitive to channel parameters variation, like maximum delay, path gain,and maximum Doppler shift in selective fading channels as compared with standard OFDMstructure.

In Radon based OFDM system, FRAT mapping is used instead of QAM mapping [8] asshown in Fig. 1. The other processing parts of the system remain the same as in conventionalQAM OFDM system. It is known that fast Fourier transform (FFT) based OFDM obtainthe required orthogonality between sub-carriers from the suitability of inverse FFT (IFFT)algorithm [3,4]. Using FRAT mapping with the OFDM structure increases the orthogonalitybetween sub-carriers since FRAT computation uses one-dimensional (1-D) IFFT algorithm.Also FRAT is designed to increase the spectral efficiency of the OFDM system throughincreasing the bit per Hertz of the mapping. Sub carriers are generated using N points dis-crete Fourier transform (DFT) and guard interval (GI) inserted at start of each symbol is usedto reduce ISI. Guard interval may be zero-padded GI, cyclic prefix, or possibly other typesof guard intervals; one possible cyclic prefix type of GI is a half guard period, and the otherhalf is a zero padded. The channel estimator is used to estimate the channel effect so as toeliminate it by channel compensator.

The procedure steps of using the Radon based OFDM mapping is as follows:

Step 1. Suppose d(k) is the serial data stream to be transmitted using OFDM modulationscheme. Converting d(k) from serial form to parallel form will construct a onedimensional vector containing the data symbols to be transmitted,

d(k) = (d0d1d2 . . . dn)T . (1)

Where, k and n are the time index and the vector length respectively.Step 2. Convert the data packet represented by the vector d(k) from one-dimensional vec-

tor to a p × p two dimensional matrix D(k), where p should be a prime numberaccording to the matrix resize operation.

Step 3. Take the 2-D FFT of the matrix D(k) to obtain the matrix, F (r, s). For simplicityit will be labeled by F.

F (r, s) =p−1∑

m=0

p−1∑

n=0

D(m, n)e− j (2π/p)rme− j (2π/p)ns (2)

Step 4. Redistribute the elements of the matrix F according to the following optimumordering algorithm first described in [14]. The optimum number of FRAT pro-jections is one projection for each column, and the best ordering of the 2D-FFTcoefficients in these projections can be achieved if the normal vectors are deter-mined as [14]:

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(ak, bk) = arg min∣∣(C p(ak),C p(bk))

∣∣

(ak, bk) ∈ {nuk : 1 ≤ n ≤ p − 1} (3)

st. C p(bk) ≥ 0

Here C p(x) denotes the centralized function of period p; C p(x) = x −p.round(x/p). Hence,

∥∥(C p(ak),C p(bk))∥∥ represents the distance from the origin

to the point (ak, bk) on the Fourier plane. The constraint C p(bk) ≥ 0 is imposedin order to remove the ambiguity in deciding between (a, b) and (−a,−b) asthe normal vector for the projection. As a result the optimal normal vectors arerestricted to have angles in [0, π). So, the dimensions of the resultant matrix willbe p × (p + 1) and will be denoted by the symbol Fopt .

Step 5. Take the 1D-IFFT for each column of the matrix Fopt to obtain the matrix of Radoncoefficients R :

R = 1

p

N−1∑

k=0

Fopt e( j2πkn/p) (4)

Step 6. Construct the complex matrix R̄ from the real matrix R such that its dimensionswill be p × (p + 1)/2 according to:

rl,m = ri, j + j ri, j+1, 1 ≤ i ≤ p, 1 ≤ j ≤ p (5)

where, rl,m refers to the elements of the matrix R̄, while ri, j refers to the ele-ments of the matrix R. The complex matrix construction is made for a purpose ofincreasing bit per Hertz of mapping before resizing mapped data.

Step 7. Resize the matrix R̄ to a one dimensional vector r(k) of length p × (p + 1)/2.

r(k) = (r0 r1 r2, . . . , rp(p+1)/2)T (6)

Step 8. Take the 1D-IFFT for the vector, r(k) to obtain the sub-channel modulation.

s(n) = 1

p(p + 1)/2

NC −1∑

k=0

r(k)e

j2πknp(p+1)/2

(7)

where NC number of carriers.Step 9. Finally, convert the vector s(n) to serial data symbols: s0, s1, s2, . . . , sn .

3 Fast Algorithm for Discrete Multiwavelet Transform Computation

As a natural extension of wavelet, Multiwavelet are designed to be simultaneously sym-metric, orthogonal and having short supports with high approximation power, which cannotbe achieved at the same time for wavelet using only one scaling function. The trick is toincrease the number of scaling functions to raise the approximation power rather than onescaling function. It enhances the performance of many wavelet applications such as imagecoding and de-noising [15,16].

In particular, whereas wavelets have an associated scaling function φ(t) and waveletfunction �(t), Multiwavelets have two or more scaling and wavelet functions. For nota-tional convenience, the set of scaling functions can be written using the vector notation�(t) ≡ [φ1(t), φ2(t) . . . φr (t)]T , where �(t) is called the multi-scaling function. Like-wise, the Multiwavelet function is defined from the set of wavelet functions as �(t) ≡

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[ψ1(t), ψ2(t) . . . ψr (t)]T . When r = 1, �(t) is called a scalar wavelet, or simply wavelet.While in principle r can be arbitrarily large. The Multiwavelets studied to date are primarilyfor r = 2 [17].

The Multiwavelet two-scale equations resemble those for scalar wavelets [18]:

�(t) = √2

∞∑

k=−∞Hk�(2t − k) (8)

�(t) = √2

∞∑

k=−∞Gk�(2t − k) (9)

Note, however, that {Hk} and {Gk} are matrix filters, i.e., Hk and Gk are (r × r)matricesfor each integer k. The matrix elements in these filters provide more degrees of freedomthan a traditional scalar wavelet. These extra degrees of freedom can be used to incorporateuseful properties into the Multiwavelet filters, such as orthogonality, symmetry, and highorder of approximation. However, the multi-channel nature of Multiwavelets also meansthat the sub-band structure resulting from passing a signal through a multi-filter bank isdifferent [17].

The two-scale equations (8) and (9) can be realized as a matrix filter bank operating on rinput data stream and filtering them into 2r output data streams, each of which is down-sam-pled by a factor two.

One famous Multiwavelet filter is the GHM filter proposed by Geronimo, Hardian, andMassopust [19]. The GHM basis offers a combination of orthogonality, symmetry, and com-pact support, which can not be achieved by any scalar wavelet basis.

According to Eqs. (8) and (9) for GHM system Hk are four scaling matrices H0, H1, H2,and H3, and Gk are four wavelet matrices G0,G1,G2, and G3 [20].

H0 =[

35√

245

− 120 − 3

10√

2

], H1 =

[3

5√

20

920

1√2

], H2 =

[0 09

20 − 310

√2

], H3 =

[0 0

− 120 0

](10)

G0 =[ − 1

20 − 310

√2

110

√2

310

],G1 =

[9

20 − 1√2

− 910

√2

0

],G2 =

[9

20 − 310

√2

910

√2

− 310

],

G3 =[

− 120 0

− 110

√2

0

](11)

The Multiwavelet coefficients that the low-pass filter H and high-pass filter G consistsof are r × r matrices, and during the convolution step they must multiply vectors (insteadof scalars). This means that multi-filter banks need r input rows. In this case r = 2 andtwo data streams enter the multi-filter. To create the two data streams from an ordinary sin-gle-stream input of length N , there are several possibilities. An important one of them isto create consistent approximation based equations that yield two length N/2 streams anda “de-approximation” that returns a length N stream. This method maintains a criticallysampled representation (produces N/2 length-2 vectors). The multi-filter processes two N/2point data streams using an approximation method suggested by Geronimo [19].

The aim of preprocessing is to associate the given scalar input signal of length N (mustbe a power of 2) to a sequence of length-2 vectors in order to start the analysis algorithm.After the wavelet transform, inverse transform, and post-filtering should recover the inputsignal exactly if nothing else has been done. For critically sampled Multiwavelets there are

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two methods of approximation: First-order approximation method and second-order approx-imation method [13].

The first order approximation-based preprocessing [21] (where every two rows generatetwo new rows) can be summarized as follows:

• For any odd row

new odd-row = (0.373615)[same odd-row] + (0.11086198)[next even-row]+ (0.11086198)[previous even-row] (12)

• For any even-row

new even-row = (√

2 − 1)[same even-row] (13)

The second order approximation-based preprocessing [13] (where every two rows gener-ate two new rows) can be summarized as follows:

– For any odd row

new odd-row = (10/8√

2)[same odd-row] + (3/8√

8)[next even-row]+ (3/8√

2)[previous even-row] (14)

– For any even-row

new even-row = [same even-row] (15)

It should be noted that when computing the first odd row, the previous even-row in Eq. (12)is equal to zero. The same thing is valid for Eq. (14).

For computing discrete Multiwavelet transform, scalar wavelet transform matrices can bewritten as follows:

W =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

H0 H1 H2 H3 0 0 · · · 0 0 0 00 0 H0 H1 H2 H3 · · · 0 0 0 0...

......

......

... · · · ......

......

H2 H3 0 0 0 0 · · · 0 0 H0 H1

G0 G1 G2 G3...

... · · · 0 0 0 00 0 G0 G1 G2 G3 · · · 0 0 0 0...

......

......

... · · · ......

......

0 0 0 0 0 0 · · · G0 G1 G2 G3

G2 G3 0 0 0 0 · · · 0 0 G0 G1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

where Hi and Gi are the low and high-pass filter impulse responses. They are 2×2 matrices.By examining the transform matrices of the scalar wavelet equations and Multiwavelets

equation (16), one can see that in the Multiwavelet transform domain there are first and secondhigh pass filter coefficients rather than one low-pass coefficient followed by one high-passcoefficient. Therefore, if these four coefficients are separated, there are four sub-bands in thetransform domain.

Fast DMWT computation for 1-D signal using critically sampling is achieved using thefollowing procedure:

• Checking input dimensions: Input vector should be of length N , where N must be powerof two.

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• Constructing a N/2 × N/2 transformation matrix W using GHM low and high passfilters matrices given in Eqs. (10) and (11) respectively. After substituting GHM matrixfilter coefficients values, an N × N transformation matrix results.

• Preprocessing rows: Approximation-based row preprocessing can be computed by apply-ing Eqs. (12) and (13) to the odd and even-rows respectively of the input N × N matrixfor the first order approximation preprocessing. For the second order approximation pre-processing, Eqs. (12) and (13) are used for preprocessing odd and even-rows of the inputN × 1 matrix P respectively.

• Transformation of input vector which can be done by applying matrix multiplication tothe N/2 × N/2 constructed transformation matrix by the N × 1 preprocessing inputvector. Transformation of input vector can be done as follows:

[Z ] = [W ] × [P]T

The inverse of the upper procedure for computing IFDMWT for 1-D signal using criticallysampling algorithm is as follows:

• Let X be the N × 1 Multiwavelet transformed vector, where N is power of 2.• Construct N/2 × N/2 reconstruction matrix W2 = W T .• Multiply a N/2×N/2 reconstruction matrix W2 with the N×1 Multiwavelet transformed

vector.• Apply post-processing by using the following:

• Compute post-processing first order approximation by:

odd-row = [[same odd-row] − (0.11086198)[next even-row] (17)

−(0.11086198)[previous even-row]]/(0.373615)

even-row = [same even-row]/(√

2 − 1) (18)

to the column reconstructed N × 1 vector matrix.• Compute post-processing second order approximation by applying equations:

odd-row = [[same odd-row] − (3/8√

8)[next even-row]

−(3/8√2)[previous even-row]](10/8

√2) (19)

even-row = [same even-row] (20)

to the column reconstructed N × 1 vector matrix P .

Reconstruction of input transformation vector can be done as follows: [z] = [W2]×[p]T .

4 Proposed System for Radon-Multiwavelet Based OFDM Transceiver

Due to good orthogonality of both DMWT and FRAT which reduce ISI and ICI, inproposed system there is no need of using cyclic prefix (CP). The block diagram of pro-posed Radon-Multiwavelet based OFDM system is depicted in Fig. 2 and the inverse dis-crete Multiwavelet transform (IDMWT) modulator and DMWT demodulator are shown inFig. 3.

The processes of serial to parallel (S/P) converter, signal demapper, and insertion of train-ing sequence are the same as in the system of FFT-OFDM. Also the zeros are added as in theFFT based case and for the same reasons. After that the IDMWT is applied to the signal. Themain and important difference between FFT based OFDM and DMWT based OFDM is that

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Serial to

parallel(S/P)

I/PData

FRATSignal

Mapper

ZeroPadding

(ZP)

Training (S/P)

IDMWTTraining

Insertion

RemoveZero

PadingChannelcompensation

ChannelEstimator

IFRAT(P/S)

(P/S)

Training

Separation (S/P)DataO/P DMWT

MultipathRayleighFading

AWGN+

*h(t)

Fig. 2 Block Diagram of FRAT-DMWT based OFDM system

DMWT-OFDM Modulator

IDMWTZeroPad

DMWTZero padremoval

DMWT-OFDM DeModulator

I/P toOFDM

Modulator

O/P fromOFDM

Demodulator

O/P fromOFDM

Modulator

I/P to OFDMDemodulator

Fig. 3 DMWT-OFDM modulation–demodulation

in DMWT based OFDM cyclic prefix is not added to OFDM symbols. Therefore the datarates in DMWT based OFDM is higher than those of the FFT based OFDM. At the receiver,the zeros padded at the transmitter are removed, and the other operations of channel estima-tion, channel compensation, signal demapping and parallel to serial (P/S) are performed inthe same manner as in FFT based OFDM.

In conventional OFDM system, the length of input data frame is 60 symbols, andafter (S/P) conversion and QAM mapping the length becomes 30 symbols. Zero paddingoperation makes the length 64 symbols which are the input to IFFT (sub-carrier mod-ulation). After adding CP (usually 40 % of the length of the frame), the frame lengthbecomes 90 symbols. Since OFDM operations applied to training symbols are the sameas those applied to transmitted data (except the mapping operation), the length of trainingsymbols is also 90 symbols. The training and data frames are transmitted as one framestarting with training, so the length of transmitted frame is 180 symbols [22]. In pro-posed system, the length of the input data frame must be (p × p), where p is a primenumber. The closest number to 60 is (7 × 7) which makes the frame length 49 sym-bols. This is because the input of FRAT must be a two dimensional matrix with size(p × p).

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Table 1 System models

OFDM-system Parameters

FFT-OFDM FFT bins=64; guard interval=16 symbol (1/4);guard interval type=all zeros; number ofcarriers=28; and pilot-assisted channel estimatoris used; modulation: BPSK

Radon-OFDM FFT bins=64; FRAT window is 7 by 7; modulation:Radon mapping

DMWT-OFDM DMWT bins N = 64; modulation: BPSK

Radon-DMWT Radon mapping; DMWT bins N = 64; FFTbins=64; FRAT window is 7 by 7

Table 2 Channel models

Channel model Parameters

AWGN AWGN is added to the signal

Flat fading+AWGN Frequency components are affected by a constantattenuation and linear phase distortion. Thechannel has a Rayleigh’s distribution and Dopplerfrequencies are: 4, 300, 500 Hz

Selective fading+AWGN Multi-path Raleigh distributed selective-fading witha 2-ray’s Rayleigh-distributed multi-path fadingchannel, with the parameters corresponding tomultipaths are path gain of −8 dB, and a pathdelay of τmax = 0.1 µs. Doppler frequencies:4, 300, 500 Hz

5 Simulation Results of Proposed System

As shown in Table 1, four types of OFDM systems were simulated using MATLAB version7 and the BER performances of the four systems were found for different channel modelsshown in Table 2. The simulations are performed for a frame-based un-coded systems withthe following parameter: 64-point FFT, TS = 0.1 µs, FRAT window: 7 by 7, and DMWTbins N = 64. The FFT-OFDM frame duration worth’s 80 chips where 64 are for data while16 are cyclic prefix. This corresponds to an efficiency of 4/5. Out of the 64 narrow-bandsub-carriers, only 52 are carrying signal and other 12 are zeros (guard or null sub-carriers).Four of the 52 sub-carriers are used as pilots and the other 48 are used for data.

The multi-path Raleigh distributed selective-fading channel is assumed a 2-ray’s Ray-leigh-distributed multi-path fading channel, and the parameters of the second path are: pathgain equal −8 dB, and path delay τmax = 0.1 µs. The selective-fading channel is simulatedas 1-D FIR filter that adds multi-path effect and additive white Gaussian noise (AWGN) totransmitted symbols. The channel is assumed to be slowly varying, which doesn’t changewithin a packet frame. Thus, the estimation is done with the long preambles at the beginningof the frame. The channel frequency response is estimated by using training and receivedsequences as follows: H(k) = Received T raining Sample(k)

T ransmitted T raining Sample (k) , k = 0, 1, 2, The channel fre-quency response is used to compensate the channel effects on the data, and the estimateddata is found using the following equation:

Estimated-data(k) = H−1estimate(k) ∗ Received-data(k), k = 0, 1, 2, . . .

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Fig. 4 BER performance of FRAT–DMWT based OFDM in AWGN channel

5.1 Performance of Proposed OFDM System in AWGN Channel

Figure 4 shows the results of simulation of proposed system compared with other systems inAWGN channel. It is clearly seen that FRAT–DMWT based OFDM has better performancethan the other three systems: FFT-OFDM, DMWT-OFDM and FRAT-OFDM. This is dueto the high orthogonality of proposed system. To have BER = 10−4, FFT-OFDM requires28 dB, FRAT-OFDM requires 25 dB, DMWT-OFDM requires 18.6 dB, and FRAT–DMWTbased OFDM requires 15.2 dB. And to have BER = 10−5, FFT-OFDM requires 31.5 dB,FRAT-OFDM requires 28 dB, DMWT-OFDM requires 21.5 dB, and FRAT–DMWT basedOFDM requires 17 dB. From the results it can be noted that proposed system has 14.5 dBadvantage over FFT-OFDM, 9 dB over FRAT-OFDM, and 5 dB over DMWT-OFDM.

5.2 Performance of Proposed System in Flat Fading Channel with AWGN

In this channel, all signal frequency components are affected by a constant attenuation andlinear phase distortion, in addition to an AWGN. The channel was selected to be multi-pathand Rayleigh distributed. Doppler frequency used in simulation is calculated as follows:c = 300 × 106 m/s, in GSM system fc = 900 MHz so,

fd = fc × v

c= 900 × 106 × 1

s× v

300 × 108 m/s= 3/(1 m)× v (21)

The Doppler frequency used, is that corresponding to a walking speed (4.8 km/h), and it

has a value: fd = 31m × 4.8×1,000 m

3,600 s = 4 Hz .The results of simulations for 4 Hz Doppler frequency are shown in Fig. 5. From Fig. 5 it

can be seen that to have BER = 10−5, FFT-OFDM requires 33 dB, FRAT-OFDM requires30.5 dB, DMWT-OFDM requires 23 dB, and FRAT–DMWT based OFDM requires 18.5 dB.So proposed system offers 14.5 dB SNR-improvement compared with FFT-OFDM, 12 dBcompared with FRAT-OFDM, and 4.5 dB compared with DMWT-OFDM for this channelmodel. Other Doppler-Shift frequencies were used for proposed system simulation over the

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Fig. 5 BER performance of FRAT–DMWT based OFDM in flat fading channel at Doppler frequency 4 Hz

flat fading Rayleigh channel; the values used are 80 Hz corresponding to car speed (96 km/h),300 Hz corresponding to Helicopter speed (360 km/h), and 500 Hz corresponding to airplanespeed (600 km/h), and the same results were obtained for these frequencies. The reason forbest performance results of FRAT–DMWT based OFDM is the good orthogonality of Radontransform and the excellent orthogonality of DMWT.

5.3 BER Performance of Proposed OFDM System in Selective Fading Channelwith AWGN

In this section, the channel model is assumed to be selective fading channel. A second ray’sRaleigh-distributed multi-path fading channel is assumed, where the parameters of the mul-tipath channel are: path gain equal −8 dB and path delay τmax = 0.1 µs.

The BER performance of proposed system and the other OFDM systems over a selectivefading channel with Doppler frequency of 4 Hz is shown in Fig. 6. It can be seen that to haveBER = 10−5, FFT-OFDM requires 37.5 dB, FRAT-OFDM requires 35 dB, DMWT-OFDMrequires 25.2 dB, and FRAT–DMWT based OFDM requires 20 dB. So proposed system offersa large SNR-improvement compared with FFT-OFDM, FRAT-OFDM, and DMWT-OFDMfor this channel model.

The same performance characteristics of systems over selective fading channel with Dopp-ler frequencies 80, 300, and 500 Hz were simulated. Figure 7 shows BER performance ofFRAT–DMWT based OFDM in selective fading channel with Doppler frequency of 300 Hz.From Fig. 7, it is clearly seen that FFT based OFDM needs more than 40 dB of SNR to haveBER = 10−4, while FRAT based OFDM needs around 39 dB of SNR to reach BER = 10−5,DMWT based OFDM BER performance does not exceed 0.002425 with increasing SNR,whereas proposed FRAT–DMWT based OFDM has much better performance than the otherthree systems, it reaches BER = 10−5 at SNR=24.5 dB.

Figure 8 shows BER performance of FRAT–DMWT based OFDM in selective FadingChannel with Doppler frequency of 500 Hz. From Fig. 8, the following conclusion can bestated: When Doppler frequency exceeds 500 Hz, proposed system suffers from the same

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Fig. 6 BER performance of FRAT–DMWT based OFDM in selective fading channel at Doppler frequency4 Hz


problem that DMWT based OFDM system suffer from, the performance of proposed systemdoes not increase with increasing SNR when the Doppler frequency exceed 500 Hz. It isseen that OFDM systems are very sensitive systems to the variation of Doppler frequency inselective fading channel.

The effect of Doppler frequency value on BER performance for proposed system is pro-vided in Fig. 9. It can be seen from Fig. 9 that the critical value of Doppler frequency forproposed system is around 420 Hz.

6 Conclusions

In this paper a novel OFDM generation method is proposed, simulated, and tested. Pro-posed system uses Radon-DMWT mapping instead of QAM mapping which increases the

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Fig. 9 BER performance of FRAT–DMWT based OFDM in selective fading channel at different Dopplerfrequencies

orthogonality. The optimal ordering (best direction) in the Radon mapper can be consid-ered as a good interleaver which serves in error spreading. In proposed system there is noneed for using CP because of excellent orthogonality offered by FRAT and DMWT, andsubsequently, reduces the system complexity, increase the transmission rate, and increasespectral efficiency. Simulation results of proposed Radon-DMWT based OFDM show a verygood SNR gain improvement and a BER performance as compared with DMWT-OFDM,FRAT-OFDM, and FFT-OFDM in an AWGN, a flat fading, and a selective fading channels.It offers more than 15 dB SNR improvement compared with FFT-OFDM for selective fadingchannel at Doppler frequency 4 Hz. From the simulation results, it can be seen that pro-posed Radon-DMWT based OFDM has the smallest sensitivity to variations of the channelparameters.

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Author Biographies

Abbas Hasan Kattoush received his M.S., and Ph.D. degrees in com-munication Eng. from USSR in 1979 and 1984 respectively. For 10years Dr. Kattoush was a technical manager of a leading SAKHERcomputers company. He was a pioneer in computer networking andsoftware engineering in Jordan. From 1993 to 2000 he worked atApplied Science University Amman Jordan where he was a found-ing member and a head of the department of electrical and com-puter engineering. In 2000 Dr. Kattoush moved to Al-Isra University,Amman, Jordan where he worked as an associate professor and a headof department of Electrical and Computer Engineering until October2008. Currently he is an associate professor and a head of departmentof Electrical and Computer Engineering at Tafila Technical Univer-sity, Tafila, Jordan. His areas of research interest include DSP, dig-ital communication systems, phase unwrapping, interferometric SARimage analysis, filtering, and interpretation. He has authored severaltenths of research articles, textbooks, and computer software systems.

Waleed Ameen Mahmoud Al-Jawher Dean of Engineering Col-lege, University of Isra, Jordan. He received a School of Researchin Digital Signal Processing (2005). He received his Ph.D. in Dig-ital Signal Processing from University of Wales University Collegeof Swansea, United Kingdom (1986). He has a teaching experiencein engineering for 32 years. A total of (15) National Awards. HePublished over (224) papers, Supervised (202) M.Sc and PhD Stu-dents. He was the First professor Award of University of Baghdad,the First professor Award of the Ministry of Higher Education &Scientific Research of Iraq (National first Professor Award). His pres-ent areas of research interest are the field of DSP and Communica-tion.

Osama Q. Al-Thahab was born in Baghdad in 1975. He received his B.Sc. in EE Department Universityof Technology Baghdad in 1997, his M.Sc. in electronic and communication from University of Baghdad2000, and his Ph.D. in electronic and communication from University of Technology Baghdad in 2006. Heis working as a lecturer in the electrical engineering department in the University of Technology Baghdad.His areas of research interest are the field of DSP and Communication.

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