OFDM Based on Low Complexity Transform to Increase ... - IEEE Xplore

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 12, DECEMBER 2011

OFDM Based on Low Complexity Transform to Increase Multipath Resilience and Reduce PAPR Mohammed Sh. Ahmed, Student Member, IEEE, Said Boussakta, Senior Member, IEEE, Bayan S. Sharif, Senior Member, IEEE, and Charalampos C. Tsimenidis, Member, IEEE

Abstract—This paper introduces a new multicarrier system using a low computational complexity transform that combines the Walsh-Hadamard transform (WHT) and the discrete Fourier transform (DFT) into a single fast orthonormal unitary transform. The proposed transform is used in a new orthogonal frequency division multiplexing (T-OFDM) system, leading to a significant improvement in bit error rate (BER) and reasonable reduction in the peak-to-average power ratio (PAPR). Use of the proposed transform with OFDM has been found to attain high frequency diversity gain by combining all data samples resulting in the transmission over many subcarriers. Consequently, the detrimental effect arising from channel fading on the subcarrier power is minimized. Theoretical analysis of the uncoded T-OFDM performance over additive white Gaussian noise (AWGN), flat fading, quasi-static (fixed within the entire period of OFDM symbol transmission) frequency selective fading channel models with zero-forcing (ZF) and minimum mean square error (MMSE) equalizers is presented in this work. Moreover, the low superposition of the subcarriers passing through the T-transforms leads to a reduction in the high peak of the transmitted signal whilst preserving the average transmitted power and data rate. Analytical results confirmed by simulations demonstrated that the proposed T-OFDM system achieves lower PAPR and the same BER over AWGN and flat fading channels. Compared to OFDM, T-OFDM is found to have better BER when MMSE equalizer is used, but slightly worse when ZF equalizer is used. Index Terms—Fast Fourier transform (FFT), orthogonal frequency division multiplexing (OFDM), peak-to-average power ratio (PAPR), -OFDM, -transform, Walsh-Hadamard transform (WHT).

T

T

I. INTRODUCTION

M

ANY high data rate communications systems have adopted orthogonal frequency division multiplexing (OFDM) [1]. However, one of the major drawbacks of such a system is the high peak-to-average power ratio (PAPR) of the transmitted signal and the effect of multipath propagation [2]. Manuscript received April 06, 2011; revised July 07, 2011; accepted August 11, 2011. Date of publication August 30, 2011; date of current version November 16, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Maja Bystrom. This work was supported by the Royal Society of Engineering. The material in this paper was presented in part at the IEEE International Conference on Communications (ICC), Cape Town, South Africa, June 2010. The authors are with the School of Electrical, Electronic and Computer Engineering, Newcastle University, Newcastle upon Tyne, NE1 7RU, England, U.K. (e-mail: [email protected]; [email protected], [email protected]; [email protected]; charalampos.tsimenidis @ncl.ac.uk). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2011.2166551

Therefore, different schemes have been developed to minimize the effect of high PAPR in OFDM systems and increase its multipath resilience. In terms of PAPR reduction, in general there are two categories of PAPR reduction techniques: techniques with distortion and techniques without distortion [3]. Relating to our approach, [4] proposed a method for PAPR reduction in an OFDM system by combining selective mapping (SLM) and dummy sequence iteration (DSI) with the Walsh-Hadamard transform (WHT). The researchers achieved PAPR reduction, but at the expense of the high complexity of using many inverse fast Fourier transforms (IFFTs) and WHTs, and data rate losses through redundant SI. References [5] and [6] suggested a new method of PAPR reduction which added the WHT to the OFDM system. Although these techniques reduced the peak power of the transmitted signals, the complexity was high owing to the utilization of IFFT and the WHT separately. Regarding BER performance improvement, many techniques have been investigated to reduce the deleterious effect of multipath dispersion. Based on frequency diversity increasing, the spread OFDM (called WHT-OFDM) system has received a considerable amount of attention with the aim of improving the performance of such a system in these environments [7]. Moreover, subcarrier spreading by using a WHT-OFDM system is a more convenient approach to exploiting wideband channel diversity potential than using an adaptive system. Furthermore, WHTOFDM has lower complexity, better bandwidth efficiency, and a better data rate compared to adaptive systems [2]. In addition, the benefits of adding WHT to an OFDM system to reduce the influence of the selective fading channel on system performance have been demonstrated in [8]–[11]. However, this improvement is only achieved at the expense of an increase in the computational complexity of using WHT and the fast Fourier transform separately in a cascaded form. In a similar way to the method developed in [12] but with a different number of additions and multiplications processes, [13] has attempted to improve the implementation of combining WHT and the IFFT for multicarrier code division multiple access (MC-CDMA). However, the computational cost is still high. The number of additions is lower, but the number of multiplications is higher than the two transforms calculated separately. In this paper, a low computational complexity transform , which was introduced in [12], is considered in a new OFDM-based communications system to achieve significant BER performance improvement and considerable PAPR reduction. The high diversity gain of the -OFDM system increases its resilience over multipath propagation [14]. In addition,

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Fig. 1.

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is an inverse -transform (ITT) where and are the WHT matrix, and the normalized matrix, and inverse discrete Fourier transform (IDFT) matrix rearranged by column reverse order, respectively. The transmitted is padded with samples as vector a cyclic prefix (CP) and convolved circularly with -tap channel , and then corimpulse response (CIR), rupted by the additive white Gaussian noise (AWGN). Consequently, the received signal after removing the CP can be ex, where denotes circular convolupressed by tion and is the zero mean Gaussian noise with variance . At the receiver side, the vector is fed into the forward -transform block to acquire the distorted signal as

T-OFDM block diagram.

theoretical analysis of the uncoded -OFDM performance over AWGN, flat fading, quasi-static (fixed within the entire period of OFDM symbol transmission) frequency selective fading channel models with zero-forcing (ZF) and minimum mean square error (MMSE) equalizers is presented in this work. Furthermore, the statistical PAPR analysis of the -OFDM system is quantified. The proposed transform is found to be efficient in reducing the high peak power, hence, reducing the PAPR of the transmitted signal whilst preserving the average transmitted power and data rate. It should be noted that the proposed technique is different from all existing techniques in that it uses a single orthonormal transform as opposed to the cascading of the WHT and IFFT in other techniques. It is also, the only technique which reduced BER, PAPR and computational complexity together. The rest of this paper is organized as follows. Section II describes the uncoded OFDM system based on the proposed transform. The theoretical analysis of the -transform and its computational complexity when used with OFDM are introduced in Section III. In Section IV, BER performance and statistical PAPR analysis of the uncoded conventional OFDM and -OFDM are quantified. Section V presents and discusses the simulation results, and Section VI concludes the paper. Notation The notations used in this paper are as follows. The superscripts , and denote the transpose and expectation opand stand for the th element erators, respectively. of vector , and the th row and th column in matrix , respectively. II. PROPOSED SYSTEM DESCRIPTION It is important to note that in the remainder of this paper, -OFDM is referred to as the uncoded OFDM-based -transform. Fig. 1 shows a block diagram of a typical -OFDM system. This system is analyzed with the assumption of perfect channel response knowledge and perfect synchronization. complex modulated The transmitted bits are mapped to 1 , which are fed into signal vector, the inverse -transform to obtain the 1 vector of the discrete-time complex baseband signal as (1)

(2) where is the forward -transform (FTT) matrix, which will be described in detail in Section III, and is the nordiscrete Fourier transform (DFT) matrix rearmalized ranged with row reverse order. Subsequently, the distorted signals need to be equalized. In a conventional OFDM system the channel equalization is performed in frequency domain by multiplying the received signal with the equalizer sequence. Whereas in the proposed -OFDM system two channel equalization approaches can be utilized, as shown in Fig. 1. The first approach is similar to the frequency equalization in the OFDM system followed by the WHT of the equalized signal. The second method of channel equalization in the -OFDM system is accomplished by computing the WHT of the forward -transform block output and multiplying the result with frequency-domain equalizer elements. Then the WHT of the equalized signal is computed. The equalized signal can be expressed as (3) denotes point-by-point multiplication of the two vecvector is equal either to or for path 1 or 2, respectively, in the receiver of Fig. 1, is the frequency domain of and equalizer sequence. As will be verified in Section IV, -OFDM system achieves an outstanding BER performance when the equalizer based on the minimum mean square error (MMSE) criterion is utilized (i.e., its elements are , where is the element of CIR in the frequency domain, and stands for the SNR value of the received signal). The key point of using the MMSE criterion is its ability to suppress noise enhancement along with equalizing the channel [2]. Eventually, the resultant signal is given as where tors,

(4)

III. A.

-TRANSFORM

-Transform Structure

In this paper the size forward -transform (FTT), , which is used in the receiver of Fig. 1 is evaluated as (5)

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where , are the normalized DFT matrix rearranged by row reverse order, and the Walsh-Hadamard matrix, respec, where tively. Their elements are , are the bits representation of the integer values and , and , respectively. The rows and columns of the DFT matrix, can be recalculated based on , where stands for the modulo- operation. Similarly but with a reverse sign, the elements of the first half and the second half of the matrix enable us to express the for relationship among these elements as . Consequently, the matrix in row reverse order can be written as

times yields the general form given by (10) at the process bottom of the page. To show the block diagonal structure of in (note: this analysis (10), consider, for instance, the case can be performed for any size). Thus, see (11) at the bottom , is the Walsh-Hadamard of the page, where matrix, , , , , and are the submatrices of , which can be expressed as

(6) where and are the submatrices of . On the other hand, of size , where and is an integer value, can be computed as (7) where the matrix

is the 2

2 WalshHence, substituting , , , , and for their values as (12) at in (11) yields the -transform matrix for the bottom of the next page, where , , , , , , , , , , , , , and . This means that the matrix for can be written as in (13) at the bottom of the next page, . where Equations (12) and (13) have a block diagonal structure with more than two thirds of the elements being zero. The number of arithmetic operations to multiply by the -transform matrix are only 1/3 of those involved in multiplying by the DFT or IDFT matrices. Also, the -transform matrix can be transformed into a product of sparse matrices leading to a much faster algorithm for calculating the -transform and its inverse as follows.

denotes the tensor or Kronecker Hadamard matrix and matrix can be written as a function product. In general, the of lower order matrices as (8) Upon substituting (6) and (8) into (5) we get

(9) where the lower part of (9) permits the extension to a high order transform matrix and the upper part is the forward transform matrix of order and can be factorized further. Repeating the

..

.

..

.

..

.

..

.

..

.

(10)

(11)

AHMED et al.: OFDM BASED ON LOW COMPLEXITY TRANSFORM

The resultant matrices of evaluated, respectively, as

and

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can be

(14) and (15) In the same way, the values of as

matrices are computed

(16)

Also, the matrix of

can be written as

(17)

(12)

(13)

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Thus, the matrix multiplication in (17) is computed as in (18) at the bottom of the page. Consequently, the decomposition of (18) in terms of sparse matrices is given by

Furthermore, the

matrix can be expressed as

(21)

(19)

Similarly, the final three submatrices of evaluated as

In the same way, the matrix as

can be factorized

are

(22)

Consequently, the method can be extended to any transform is given in length. A flowchart of the transform for Fig. 2.

(20)

B. Computational Complexity For any transform to be a good candidate for practical communication systems, it must be efficient in terms of computational time and speed. This section evaluates the computational

(18)

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Fig. 3. One butterfly structure of the T-transform. TABLE I REAL ARITHMETIC OPERATIONS IN THE DIRECT IMPLEMENTATION OF THE T-TRANSFORM AND WHT-IFFT BASED ON SINGLE BUTTERFLY IMPLEMENTATION

Fig. 2.

T-transform flowchart.

costs of the proposed -transform and compares it with similar techniques. 1) Evaluation of the -Transform: One advantage of the -transform is its simple structure and low computational comthe number of stages plexity. Usually for radix-2 and is four. However, as can be seen from Fig. 2, the -transform requires three stages only and in each stage many of the butterflies are zero and do not need to be calculated reducing the computational complexity. Also, scalability is much easier in the case of the -transform than other well known transforms, as increasing the transform length will involve adding the lower part only, as shown in Fig. 2. Equation (13) also shows that the matrix is a sparse matrix, with two thirds of its elements being zero; hence, its direct implementation will involve multiplications and additions, while the direct calculation of WHT-IDFT will involve complex multiplications and complex additions. This means that the direct implementation of -transand real multiplications and form requires real additions, respectively; whereas the direct implementation and of real multiof WHT-IDFT requires plications and real additions, respectively. Direct implementation of transforms is usually advantageous when the transform length is short or when a systolic array implementation is used. As shown in Table I, the -transform involves much less arithmetic operations. The length of the proposed -transform is power of two, so it is amenable to fast algorithms. It can be deduced from Fig. 2 that the -points -transform requires butterflies only; each individual butterfly can be expressed as shown in Fig. 3. The output and of each butterfly are calculated as data, and , respectively, where and are the input data, and is the twiddle factor which has in the inverse -transform and the value of range

in the forward -transform, with the index . The calculation of each butterfly

TABLE II REAL ARITHMETIC OPERATIONS IN THE FAST IMPLEMENTATION OF THE T-TRANSFORM AND WHT-IFFT BASED ON SINGLE BUTTERFLY IMPLEMENTATION

involves one complex multiplication and three complex additions. Therefore, the total number of complex multiplications and additions, including trivial operations for the calculation of the -transform, can be calculated as (23) (24) The real operations can be computed based on the fact that the complexity of one complex multiplication is equivalent to either the complexity of four real multiplications and two real additions (4/2) or the complexity of three real multiplications and three real additions (3/3). In this paper, the (4/2) is adopted. Moreover, one complex addition is equivalent to two real addi, and real tions. This means that the real multiplications, , can be computed as additions, (25) (26) while the use of WHT-IFFT will involve (27) (28) and denote the real multiplications and where additions, respectively. The number of multiplications and additions in both systems is given in Table II.

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2) Evaluation of the -OFDM System: The complexity of the full -OFDM using the two approaches in Fig. 1 should be considered and compared to that of WHT-OFDM when calculated separately. The complexity of the first approach (FFT ( complex multiplications of and WHT) is of complex equalization process are considered) and multiplications and complex additions, respectively. The second method for channel equalization in the -OFDM system is accomplished by computing the WHT of the FTT block output elements and multiplying the results with the frequency-domain equalizer elements. Then the WHT of the equalized signal is computed. Thus, the computational complexity of this approach and of complex is multiplications and complex additions, respectively. Obviously, the first approach needs relatively more complex multiplications, whereas the second needs more complex additions. Consequently, the total computational complexity of the first approach at the receiver plus the inverse -transform (ITT) complexity at the transmitter, can be calculated as (29) (30) where and are the total complex multiplications and additions of the first equalization approach, respectively. , and real adThis means that the real multiplications, , of this approach are calculated as ditions, (31) (32) In addition, the complexity of the -OFDM system with the second equalization approach is computed as (33) (34) where and are total complex multiplications and additions, respectively. This means that the total real multiplica, and real additions, , of the -OFDM with tions, the second approach are calculated as (35) (36) On the other hand, the total real operations required to implement the WHT-OFDM with the equalizer involve the computation of the WHT and IFFT at the transmitter, WHT and FFT at the receiver, plus multiplications for the equalizer. Therefore, the total number of real multiplications and additions for the WHT-OFDM can be calculated as (37) (38) The number of real multiplications and additions for the proposed -OFDM for different numbers of subcarriers, with two equalization approaches and WHT-OFDM systems, is given in Table III. From this Table it is clear that the proposed -OFDM

TABLE III COMPARISON OF THE TOTAL REAL ARITHMETIC OPERATIONS (INCLUDING EQUALIZATION) IN THE PROPOSED T-OFDM AND WHT-OFDM SYSTEMS BASED ON SINGLE BUTTERFLY IMPLEMENTATION

involves less computational complexity than WHT-OFDM. In addition, it should be noted that in the case of WHT-OFDM we need to calculate two transforms successively, so the delay, time spent on indexing and data transfer is double of that in the case of the proposed -OFDM. Also, at the receiver of -OFDM system we only consider the methods which allow us to do equalization in the frequency domain. This allows us to use frequency equalization techniques similar to the OFDM system. This allows the -OFDM and OFDM systems to co-exist. In fact, we can use the -OFDM at the transmitter and OFDM at the receiver, and vice versa. However, other techniques which allow the equalization to be carried out in the -transform domain can be developed. IV. PERFORMANCE ANALYSIS To study the BER performance of the -OFDM system proposed in Section II, this Section will illustrate the performance analysis of the conventional OFDM and -OFDM systems across three different channels, such as AWGN, flat-fading, and quasi-static (fixed within the entire period of OFDM symbol transmission) frequency selective fading, with QPSK and 16-QAM data mapping. As will be verified mathematically and supported by simulation results, the BER performance of the -OFDM system is a function of the equalizer sequence. The performance of M-PSK and M-QAM systems over AWGN channel in terms of probability of error ( ) can be expressed as [15] (39) (40) where stands for the average number of nearest neighbours is the number of bits in each consignal points, is stellation sample, is the signal-to-noise ratio (SNR) , the power per symbol, stands for the Gaussian noise average power, and . By following the approach in [16], the performances analysis of the OFDM and the proposed -OFDM systems over various channel models will be achieved based on the use of (39) and (40), with the new computed values of . A. Conventional OFDM Performance Analysis Basically, the performance of M-PSK-OFDM and M-QAMOFDM systems over a white complex Gaussian noise channel is similar to the performance of M-PSK and M-QAM systems. In a conventional OFDM system across an AWGN channel, let

AHMED et al.: OFDM BASED ON LOW COMPLEXITY TRANSFORM

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be the transmitted data vector (the desired signal) with the assumption that each independent sample has a variance , so , where is the identity matrix, is the frethat quency domain of the uncorrelated Gaussian random variables with variance , and is the received data sequence. This means

Moreover, the same performance is achieved by using a minimum mean square error (MMSE) equalizer. To verify this through an example, the th subcarrier, after equalization based on the MMSE criterion over multipath transmission, can be expressed as (50)

(41) where SNR values

of (41) can be calculated as (42)

where denotes the expectation (statistical averaging) operation. With a flat fading channel, i.e., all subcarriers faded with the and zero-forcing (ZF) equalizer same channel coefficient sequence, the received equalized signal can be calculated as (43) , and is the th frequency domain channel where coefficient. The SNR of th subcarrier will be (44) Thus, the average SNR for the OFDM system over identical for all values of , i.e., flat fading can be expressed values of as

is the optimal approximation

of the MMSE criterion and . Therefore, the SNR of the th subcarrier in (50) can be computed as

(51) This verifies that the performance of conventional OFDM system over multipath channel with ZF and MMSE is identical. However, for the -OFDM, the two equalization criteria (ZF and MMSE) lead to different performances, as will be proven in the next subsection. B. Proposed -OFDM System Performance Analysis In this subsection, the effect of a -transform based OFDM system on the SNR value will be investigated. With the Gaussian noise channel, the received signal of the -OFDM system shown in Fig. 1 can be expressed as (52) Let

each

(45)

element of noise expresses as . Thus, the variance of

can be

written as By substituting (45) into (39) and (40), the theoretical BER of OFDM with flat fading channel can be computed as

(53a) (53b)

(46) (47)

Whereas, with quasi-static frequency selective (FS) fading channel, each individual subcarrier is faded by its own channel . Thus, the SNR for each subcarrier that faded coefficient individually can be written as in (44). Substituting (44) into (39) and (40), the average BER of OFDM with frequency selective fading channel can be expressed as (48)

(49)

(53c) Due to orthogonality of the basis functions of WHT, the second term of the right hand side of (53c) is equal to zero. Consequently, (54a) (54b) (54c)

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Therefore, the performances of -OFDM and conventional OFDM systems over an AWGN channel are identical. On the other hand, for simplicity, the performance analysis of a -OFDM system with a fading channel will be performed only on path-1 at the receiver of Fig. 1. The received equalized signal can be computed as (55a) (55b)

denote the desired and noisy signals, respecwhere and tively. Consequently, the general form of the average SNR at the receiver side of a -OFDM system can be expressed as (56) In a flat fading channel, the values of are identical for every value of . Therefore, can be computed as

By applying (56), the SNR values in (60b) can be evaluated as

(61)

Equation (61) verifies that the -transform will average the on all subcarriers, which at first may be helpful, but value ), the value of will with a null spectral channel (i.e., be infinity, which causes destruction of the received samples. As a result, the serious degradation in the BER performance owing to the use of the ZF with the -OFDM system should be taken into account. Alternatively, the MMSE criterion can be employed in order to resolve the null channels challenge. Thus, parts of (56) can be recalculated for the signal defined in (55b) with MMSE criterion over frequency selective channel coefficients as

(57a) (62)

(57b) and the noise variance can be computed as

Also, the noise variance can be computed as (58a)

(58b) (58c)

(63a)

(63b)

Substituting (57b) and (58b) into (56), we obtain

(59a)

(63c)

(59b) It is clear from (59b), the SNR value of -OFDM system over flat fading channel with MMSE criterion is similar to that with ZF. Also, it is similar to the SNR value of the conventional OFDM system. Consequently, the performances of both systems over flat fading channel are identical with MMSE and ZF equalization criteria. On the other hand, as will be verified, with a frequency selective fading channel the performance of a -OFDM system is altered dramatically when using the ZF instead of the MMSE equalizer, or vice-versa. Based on the ZF criterion, the received equalized signal can be expressed as (60a) (60b)

(63d)

(63e)

(63f)

(63g)

(63h)

AHMED et al.: OFDM BASED ON LOW COMPLEXITY TRANSFORM

For simplicity, let as

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. Thus, (63h) can be rewritten

(64) Upon substituting (62) and (64) into (56), we obtain

is the peak power of the

where

, is the average discrete-time signal, transmitted power, is the number of the subcarrier, and is the oversampling factor. The purpose of oversampling is to deliver a better approximation of the PAPR. The optimal value for oversampling factor was proved in [18] to be four (i.e., ). In the case of OFDM, oversampling can be achieved by . Simpadding the frequency-domain signal with ilarly, the same approach can be used with -OFDM by times as duplicating the signals

(65a)

(65b)

(72) where are copies of the signals . The -transform allows the preservation of the average transmitted power, thus, the average power of (1) can be calculated as

Basically,

(73a) (73b) (66) The

element of

can be written as

Thus, the denominator of (65b) can rewritten as (67a) (74)

(67b) By substituting (67b) into (65b), the average SNR of -OFDM with MMSE criterion of equalization and across frequency selective channel can be expressed as

Due to orthogonality of be simply written as

and

, the expectation of (74) can

(75) (68)

Eventually, the BER performance of -OFDM over a quasistatic (fixed within the entire period of OFDM symbol transmission) frequency selective fading channel with QPSK and 16-QAM, using the MMSE criterion, can be evaluated as (69) (70)

C. PAPR Distribution The discrete-time PAPR of the transmitted signal can generally be expressed as [3] (71)

The average power is preserved by the -transform, hence the PAPR reduction will depend on peak power reduction. In OFDM-based systems, owing to the superposition of the stages of the input signals when processed through IFFT, the peak power of the output signals can be high when compared to their average power. According to the central limit is considered signifitheorem (as a rule of thumb, cant enough for a good approximation of Gaussian distribution); is large, we can approximate the distribution of each when peak of the transmitted signal using a Gaussian distribution. Therefore, the overall time domain sample distribution can also be approximated as Gaussian distribution. Moreover, as shown stages only and in Fig. 2, the -transform involves has a block diagonal structure with sections and two direct paths. The maximum number of non-zero elements . As a result, the superposition of the input in each section is signals in the case of the -transform is less than in the case of the IFFT leading to a lower PAPR, as shown in Fig. 6. However,

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Fig. 4. Histogram for peak power of the conventional OFDM and T-OFDM systems (N = f64; 1024g). (X-axis represents the peaks range in dB and Y-axis represents the number of peaks that equal to each range in X-axis.)

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Fig. 6. CCDF for the OFDM and T-OFDM systems, with various data size.

the original power spectrum of the OFDM signals illustrated in [19]. V. SIMULATION RESULTS AND DISCUSSION A.

Fig. 5. PSD for baseband signals of the conventional OFDM [19], and T-OFDM systems with N = 256, and 16-QAM mapping. (a) OFDM. (b) T-OFDM.

when the number of subcarriers increases, the number of sections with Gaussian distributions increases, so the overall probability density function (PDF) of the -transform can also be approximated as Gaussian and the peak of the -OFDM system is closer to that of the OFDM system for larger . Moreover, a good estimation of the power density of the conventional OFDM and -OFDM systems can be acquired from the histograms plot of the peak power of such systems, as shown in Fig. 4. As is clear in this figure, variance change of the -OFDM peaks is smaller than that of the conventional OFDM; therefore, this figure also clearly shows how the -OFDM system outperforms conventional OFDM by having the fewest signals with high peaks, i.e., lower PAPR. Many PAPR reduction techniques cause in-band and outband distortions for the spectrum of the OFDM signal as a consequence of slower spectrum roll-off, more spectrum sidelobes, and higher adjacent channel interference. As shown in Fig. 5, the -OFDM system has no detrimental effect on

-OFDM Performance in Terms of PAPR Reduction

Employment of -transform in OFDM systems reduces the superposition of the subcarriers due to the sparsity and block diagonal structure of the -transform and its lower summation processes, as can be seen from (12), (13), and Fig. 2. Thus, the peak of the transmitted signals will be reduced and the transmitted average power preserved. Furthermore, this PAPR reduction is achieved without any redundant side information (i.e., no data rate losses). Fig. 6 depicts the complementary cumulative distribution function (CCDF) (a statistical description most commonly used to measure the PAPR of the transmitted signal) of the PAPR for OFDM and -OFDM signals with various subcarrier numbers and oversampling factor equal four (i.e., ). As clearly shown in Fig. 6, the simulation results show that the PAPR values of the transmitted signal in the -OFDM system are less than that of the OFDM system by a range of . B.

-OFDM BER Performance

It should be noted that BER performance comparisons are accomplished with the assumptions of perfect knowledge of channel response and perfect frequency and time synchronization. Also, the semi-analytical curves in Figs. 7–10 represent the numerical evaluation of (39), (40), (46), (47), (48), (49), (69) and (70). In Fig. 8, the -OFDM BER performance represents the numerical evaluation of (39) and (40) after computing the SNR using (61). It is perhaps worth emphasizing that in the evaluation we average over the statistics of different channel realizations. In our simulations results, the performance of the -OFDM and conventional OFDM systems are investigated over AWGN, flat fading, and quasi-static frequency selective fading channel models. As was proven in Section IV and shown in Fig. 7, due to the orthogonality of the unitary -transform, the -OFDM and

AHMED et al.: OFDM BASED ON LOW COMPLEXITY TRANSFORM

Fig. 7. BER performance of the T-OFDM and OFDM systems across AWGN and flat fading channel models, with QPSK and 16-QAM mapping.

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Fig. 9. BER performance of the

precoder depth

T-OFDM,

precoded-OFDM with the

2 f16; 64; 1024g, and the conventional OFDM systems

across ITU-B (Pedestrian) channel model, with QPSK and 16-QAM mapping using MMSE equalizer.

Fig. 8. BER performance of the T-OFDM and OFDM systems across ITU-B (Pedestrian) channel model, with QPSK and 16-QAM mapping using ZF equalizer.

OFDM systems have an identical performance across AWGN and flat fading channel models with QPSK and 16-QAM mapper, when using either ZF or MMSE equalizers. Moreover, in order to investigate the BER performance of the -OFDM and OFDM systems across the multipath fading channel, the quasi-static (within OFDM frame period) ITU-B (Pedestrian) and ITU-A (Vehicular) channel models [20] are adopted. In addition, the performance is evaluated with ZF and MMSE criteria of equalization. It is important to mention that the realization of these channel models are achieved with nor, where denotes malized path power (i.e., the expectation (statistical averaging) operator, and is the amplitude of tap at instance ). The parameters required to investigate the performance of such systems over these environments are as follows: the subcarriers number is 1024, the CP length is 256, the sample time is 88 ns (which represents the paths delay as a sample-spaced), the system bandwidth is 10 MHz, are transmitted for each SNR value. and

Fig. 10. BER performance of the T-OFDM, precoded-OFDM with the precoder depth 2 f16; 64; 1024g, and the conventional OFDM systems across ITU-A (Vehicular) channel model, with QPSK and 16-QAM mapping using MMSE equalizer.

As was shown in (61), -transform will average the value of noise enhancement on all subcarriers, which with a deep ) will be very sever and causes fading subchannel (i.e., destruction of the received samples. As a result, the serious degradation in the BER performance owing to the use of the ZF equalizer with the -OFDM system, as shown in Fig. 8, should be taken into account. Therefore, the MMSE criterion is employed in order to resolve the null subchannels challenge. Thus, as shown in Figs. 9 and 10, the -OFDM system outperforms the conventional OFDM system by achieving lower BER at much lower SNR over ITU-B (Pedestrian) and ITU-A (Vehicular) channel models, respectively. The outstanding BER improvement is a consequence of frequency diversity attained by -transform which combines the effect of WHT and IDFT. This diversity helps to mitigate the deep fade effect on individual subcarriers. Also, as stated in [21], the conventional OFDM has

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the drawback that the symbol transmitted on the th subcarrier can not be recovered when it is hit by a channel zero (i.e., ). This is not the case in the -OFDM system, as each symbol is transmitted over many subcarriers. The precodedOFDM system [22] performance plots shown in Figs. 9 and 10 are for comparison purpose only. The Walsh-Hadamard matrix is utilized as a linear precoder with various column sizes, i.e., various precoder depths, such as 16, 64, and 1024. Evidently, Walsh-Hadamard mathe precoded-OFDM that utilizes trix as a linear precoder and the -OFDM achieve identical performances. VI. CONCLUSION In this paper, we have proposed an uncoded -OFDM system that uses a low complexity -transform. -transform has the ability to mitigate the serious challenge of high peak power in the OFDM-based system whilst ensuring multipath resilience without any data rate losses or an increase in the average transmitted power. The new -OFDM system has the same performance as the WHT-OFDM but involves much less computational complexity. The computational analysis demonstrated the low computational complexity requirements of the proposed -OFDM system compared with the conventional WHT-OFDM system. Furthermore, the length of the -transform was shown to be the same as the number of subcarriers, thus there is no bandwidth expansion (i.e., no data rate losses) when utilizing the -transform with the transmission techniques. Theoretical performance analysis of the considered uncoded systems over AWGN, flat fading and quasi-static frequency selective fading channel models is presented. Analytical results confirmed by simulations demonstrated that the proposed -OFDM system outperforms the OFDM system in the presence of multipath transmission when MMSE criteria is used, and also achieves considerable PAPR reduction. Consequently, a multicarrier system utilizing the -transform will benefit from the reduced PAPR, SNR and computational complexity reduction. The -transform is easy to scale up or down for different transform lengths or subcarriers. Also, the -OFDM system can coexist with the existing OFDM systems. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments which helped improving the quality of this paper.

[5] M. Park, H. Jun, J. Cho, N. Cho, D. Hong, and C. Kang, “PAPR reduction in OFDM transmission using Hadamard transform,” in Proc. IEEE ICC2000, New Orleans, LA, Jun. 2000, vol. 1, pp. 430–433. [6] Y. Wu, C. K. Ho, and S. Sun, “On some properties of Walsh-Hadamard transformed OFDM,” in Proc. 56th IEEE VTC02-Fall, 2002, vol. 4, pp. 2096–2100. [7] Z. Lei, Y. Wu, C. K. Ho, S. Sun, P. He, and Y. Li, “Iterative detection for Walsh-Hadamard transformed OFDM,” in Proc. IEEE Veh. Technol. Conf. (VTC), Apr. 2003, pp. 637–640. [8] S. Wang, S. Zhu, and G. Zhang, “Walsh-Hadamard coded spectral efficient full frequency diversity OFDM system,” IEEE Trans. Commun., vol. 58, no. 1, pp. 28–34, Jan. 2010. [9] Z. Dlugaszewski and K. Wesolowski, “WHT/OFDM- An improved OFDM transmission method for selective fading channels,” in Proc. IEEE Symp. Commun. Veh. Technol., Oct. 2000, pp. 144–149. [10] X. Huang, “Diversity performance of precoded OFDM with MMSE equalization,” in Proc. IEEE Int. Symp. Commun. Inf. Technol., Oct. 2007, pp. 802–807. [11] B. Gaffney and A. D. Fagan, “Walsh Hadamard transform precoded MB-OFDM: An improved high data rate ultra wideband system,” in Proc. IEEE 17th Int. Symp. Pers., Indoor Mobile Radio Commun., Sep. 2006, pp. 1–5. [12] S. Boussakta and A. G. J. Holt, “Fast Algorithm for calculation of both Walsh-Hadamard and Fourier transforms (FWFTs),” IEE Electron. Lett., vol. 25, no. 20, pp. 1352–1354, Sep. 1989. [13] H. Bogucka, “Application of the joint discrete Hadamard-Inverse Fourier transform in a MC-CDMA wireless communication systemPerformance and complexity studies,” IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 2013–2018, Nov. 2004. [14] M. S. Ahmed, S. Boussakta, B. Sharif, and C. C. Tsimenidis, “OFDM based new transform with BER performance improvement across multipath transmission,” in Proc. IEEE Int. Conf. Commun. (ICC), Jun. 2010, pp. 1–5. [15] J. G. Proakis and M. Salehi, Digital Communications. New York, 2008: McGraw-Hill. [16] Y.-P. Lin and S.-M. Phoong, “BER minimized OFDM systems with channel independent precoders,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2369–2380, Sep. 2003. [17] G. Robinson, “Logical convolution and discrete Walsh and Fourier power spectra,” IEEE Trans. Audio Electroacoust., vol. AU-20, no. 4, Oct. 1972. [18] K. D. Wong, M.-O. Pun, and H. V. Poor, “The continuous-time peak-to-average power ratio of OFDM signals using modulation schemes,” IEEE Trans. Commun., vol. 56, no. 9, pp. 1390–1393, Sep. 2008. [19] P. Tan and N. C. Beaulieu, “A comparison of DCT-based OFDM and DFT-based OFDM in frequency offset and fading channels,” IEEE Trans. Commun., vol. 54, no. 11, pp. 2113–2125, Nov. 2006. [20] Guidelines for the Evaluation of Radio Transmission Technologies for IMT-2000 1997, Recommendation ITU-R M.1225. [21] B. Muquet, Z. Wang, G. B. Giannakis, M. de Courville, and P. Duhamel, “Cyclic prefixing or zero pading for wireless multicarrier transmissions?,” IEEE Trans. Commun., vol. 50, no. 12, pp. 2136–2148, Dec. 2002. [22] X.-G. Xia, “Precoded and vector OFDM robust to channel spectral nulls and with reduced cyclic prefix length in single transmit antenna systems,” IEEE Trans. Commun., vol. 49, no. 8, pp. 1363–1374, Aug. 2001.

REFERENCES [1] J. G. Andrews, A. Ghosh, and R. Muhamed, Fundamentals of WiMAX, Understanding Broadband Wireless Networking. Englewood Cliffs, NJ: Prentice-Hall, 2007. [2] L. Hanzo, M. Munster, B. J. Choi, and T. Keller, OFDM and MC-CDMA for Broadband MultiUser Communications WLANs and Broadcasting. London, England: Wiley, 2003. [3] T. Jiang and Y. Wu, “An overview: Peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Trans. Broadcast., vol. 54, no. 2, pp. 257–268, Jun. 2008. [4] S. Kim, J. Chung, and H. Ryu, “PAPR reduction of OFDM signal by the SLM-based WHT and DSI method,” in Proc. IEEE TENCON06, Region 10 Conf., Nov. 14, 2006, pp. 1–4.

Mohammed Sh. Ahmed (S’10) received the B.Sc. and M.Sc. degrees in computer engineering from the University of Technology, Baghdad, Iraq, in 1995 and 2002, respectively. Currently, he is working as a Ph.D. student under the supervision of Prof. Boussakta at the school of Electrical, Electronic and Computer Engineering, University of Newcastle upon Tyne, U.K. His research interests are in the areas of computer engineering, networks communications, and digital signal processing (DSP).

AHMED et al.: OFDM BASED ON LOW COMPLEXITY TRANSFORM

Said Boussakta (S’89–M’90–SM’04) received the “Ingenieur d’Etat” degree in electronic engineering from the National Polytechnic Institute of Algiers (ENPA), Algeria, in 1985 and the Ph.D. degree in electrical engineering (signal and image processing) from the University of Newcastle upon Tyne, U.K., in 1990. From 1990 to 1996, he was with the University of Newcastle upon Tyne as a Senior Research Associate in digital signal and image processing. From 1996 to 2000, he was with the University of Teesside, Teesside, U.K., as Senior Lecturer in communication Engineering. From 2000 to 2006, he was with the University of Leeds as Reader in digital communications and signal processing. He is currently a Professor of communications and signal processing at the school of Electrical, Electronic and Computer Engineering, University of Newcastle upon Tyne, where he is lecturing in advanced communication networks and signal processing subjects. His research interests are in the areas of fast DSP algorithms, digital communications, communications networks systems, cryptography and security, digital signal/image processing. He has authored and coauthored more than 250 publications. Prof. Boussakta is a Fellow of the IET.

Bayan S. Sharif (M’93–SM’02) received the B.S. degree in electrical and electronic engineering from Queen’s University of Belfast, Belfast, Ireland, in 1984 and the Ph.D. degree in electrical and electronic engineering from Ulster University, Belfast, Ireland, in 1988. He was appointed as Lecturer at Newcastle University, Newcastle Upon Tyne, U.K., in 1990, and then as a Professor in Digital Communications in 2000. His research interests are mainly in radio and acoustic wireless transceiver structures and networks, where he has published more than 300 journal and conference papers.

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Charalampos C. Tsimenidis (M’05) received the M.Sc. (with distinction) and Ph.D. degrees both in communications and signal processing from the University of Newcastle Upon Tyne, U.K., in 1999 and 2002, respectively. His doctoral research was in the area of arrayed multiuser communications over multipath fading channels with applications to underwater acoustic data transmission. He is a Senior Lecturer in Signal Processing for Communications, School of Electrical, Electronic and Computer Engineering (EECE), Newcastle University. Currently, he conducts research in digital communications, signal processing, detection and estimation theory, and communication networks. He has made contributions in the area of receiver design to several underwater communications-related European funded research projects including long range telemetry in ultrashallow channels (LOTUS), shallow water acoustic network (SWAN), and acoustic communication network for monitoring the underwater environment (ACME). He has published more than 110 journal and conference papers.