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Int. J. Communications, Network and System Sciences, 2010, 3, 755-766 doi:10.4236/ijcns.2010.39101 Published Online September 2010 (http://www.SciRP.org/journal/ijcns) Copyright © 2010 SciRes. IJCNS A New Effective and Efficient Measure of PAPR in OFDM Ibrahim M. Hussain1, Imran A. Tasadduq2, Abdul Rahim Ahmad3 1Department of Computer Engineering, Sir Syed University of Engineering and Technology, Karachi, Pakistan 2Department of Computer Engineering, Umm Al-Qura University, Makkah Al-Mukerrimah, Saudi Arabia 3Systems Design Engineering, University of Waterloo, Waterloo, Canada E-mail: ibrahimh@ssuet.edu.pk, iatasadduq@uqu.edu.sa, arahim@uwaterloo.ca Received June 28, 2010; revised August 2, 2010; accepted September 3, 2010 Abstract In multi-carrier wireless OFDM communication systems, a major issue is high peaks in transmitted signals, resulting in problems such as power inefficiency. In this regard, a common practice is to transmit the signal that has the lowest Peak to Average Power Ratio (PAPR). Consequently, some efficient and accurate method of estimating the PAPR of a signal is required. Previous literature in this area suggests a strong relationship between PAPR and Power Variance (PV). As such, PV has been advocated as a good measure of PAPR. However, contrary to what is suggested in the literature, our research shows that often low values of PV do not correspond to low values of PAPR. Hence, PV does not provide a sound basis for comparing and esti- mating PAPR in OFDM signals. In this paper a novel, effective, and efficient measure of high peaks in OFDM signals is proposed, which is less simpler PAPR. The proposed measure, termed as Partial Power Va- riance (PPV), exploits the relationship among PAPR, Aperiodic Autocorrelation Co-efficient (AAC), and PV of the transmitted signal. Our results demonstrate that, in comparison to PV, Partial Power Variance is a more efficient as well as a more effective measure of PAPR. In addition, we demonstrate that the computa- tional complexity of PPV is far less than that of PAPR. Keywords: Aperiodic Autocorrelation Co-Efficient, OFDM, PAPR, Power Variance, Partial Power Variance 1. Introduction Several communication systems and techniques have been used for transferring data and information reliably at high speed over wireless channel. One such technique is Orthogonal Frequency Division Multiplexing (OFDM) used for high data rate wireless transmission [1]. In OFDM, data bits are transmitted in parallel using various carriers. Although OFDM is a multi-carrier technology, it is very efficacious in mitigating the effects of multi- path delay spread over a wireless radio channel. However, a major drawback with OFDM is the high Peak-to-Aver- rage Power Ratio (PAPR) of the transmitted signal. The high PAPR mainly results from certain data sequences, such as those containing all zeros or all ones. Such OFDM signals with high peaks result in poor power effi- ciencies. Appropriate measures should be taken to tackle this problem. Otherwise, the high PAPR signals would substantially limit the usefulness of battery powered equipment such as portable wireless devices. In addition, these high peaks cause problems such as inter-symbol interference (ISI) and out-of-band radiation. Transmitting high PAPR signals by increasing the operating range of the power amplifier deteriorates the power efficiency of the transmission equipment. Hence, it is imperative to reduce these peaks in the transmission signals. This issue has been addressed by several researchers [2-19]. One widely accepted method for reducing the peaks in OFDM signals is based on using Power Variance (PV) as a measure of PAPR of the signal [2-4]. The computa- tional complexity of PAPR depends on the complexity of the IFFT block, which increases by increasing the num- ber of subcarriers. However, Power Variance is compu- tationally less complex than PAPR for specific range of subcarriers [2-4]. In this paper, we investigate the relationship between the PAPR and the PV of the transmitted OFDM signals. We show that a low value of PV in an OFDM sequence does not always correspond to low value of PAPR and vice versa. Therefore, it would not be generally correct to compare PAPR of different signals by comparing their corresponding PVs. We show that PV is not a good meas- 756 I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS ure for PAPR and hence, our results contradict the wide- ly accepted premise, as stated in [3,4,20-23]. In addition, the computational complexity of PV is a major drawback, which makes it a poor choice for PAPR performance measurement. Towards this end, we propose a new, ef- fective, and computationally efficient measure, called Partial Power Variance (PPV), for estimating the PAPR. Such approximate and partial computational techniques are quite popular, as well as shown effective in various domains [24,25]. We demonstrate the efficiency of PPV through experimental results. The rest of the paper is organized as follows. Section 2 introduces OFDM systems. Section 3 describes PAPR as a performance measure for OFDM signals. Section 4 presents various PAPR minimization techniques in OFDM. Section 5 discusses the relationship between AAC, PV and PAPR. Section 6 highlights the computa- tional complexity issues in calculating PV and PAPR. Section 7 proposes PPV as a more efficient and less complex estimator of PAPR. Section 8 concludes the paper with some interesting future research directions. 2. Orthogonal Frequency Division Multiplexing (OFDM) A typical OFDM system consists of a transmitter and a receiver, as shown in Figure 1(a) and Figure 1(b) respec- tively. Such a system works as follows: Serial stream of bits {b0,b1,b2,…} are encoded using an encoder such that bi = 0 or 1; for i = 0,1,2,… . If Tb represents the duration of a single bit then the data rate would be K= 1/Tb bits/second (bps). The serial bit stream at the output of channel encoder is fed into a serial to parallel converter block that forms a parallel stream. This is achieved by increasing the time period of each bit from Tb to NTb, where N represents the number of subcarriers used in the OFDM system. These bits are transmitted simultaneously to maintain the same data rate as the original rate of K bps. The number of bits entering a particular branch or subcarrier depends on the mapper (i.e. the digital mod- ulation block) which is used after the serial to parallel converter. The number of bits per subcarrier is given by L=log2 M where M is the constellation size used by the mapper and depends on the modulation scheme being used. Hence the duration of a symbol per subcarrier is given by Ts N where Ts = LTb represents the dura- tion of each subcarrier symbol at the output of the serial to parallel converter which also represents the duration of a single OFDM symbol. The output of the mapper consists of complex numbers representing the constellation points in a particular mod- ulation scheme. These complex numbers are given by 01 1 ,,, N Ddd d . For Quadrature Phase Shift Key- ing (QPSK) mapper, the values of dk can take one of the values from 1,1, ,jj . The Inverse Fast Fourier Transform (IFFT) block transforms the discrete complex signal into another discrete complex signal. A typical baseband signal at the output of the IFFT block is given by the following well known Inverse Discrete Fourier Transform (IDFT) Equation [2]: 12/ 0 1 ();,0,1, ,1 NjkqN k k sqde kqN N (1) (a) (b) Figure 1. (a) A typical OFDM transmitter; (b) A typical OFDM receiver. I. M. HUSSAIN ET AL. 757 Copyright © 2010 SciRes. IJCNS In Equation (1), k indicates the subcarrier index, q is the discrete time index, and d k represents the complex numbers at the output of the mapper. As indicated by (1), the signal at the output of the IFFT block is the result of summation of various complex sinusoids with varying amplitudes and phases. Hence, the baseband signal given by (1) can be represented as a row vector i.e. S = {s0,s1,…,sN−1}. The resulting signal is converted into a serial stream using parallel to serial block after which a cyclic prefix or guard interval of length G is appended to it. The discrete-time sequence S which is input to the guard interval block is cyclically extended to form the new symbol sequence which is indicated as 12101 1 S',,,, ,, NG NGNN ss ssss . The cyclically extended discrete-time sequence has new length of '1.NNG This guard interval helps in mitigating the effect of multipath fading in wireless channels. The use of guard interval results in a loss of data throughput as bandwidth is wasted on repeated data. However, in this tradeoff, the loss in data throughput is compensated by significant gains through mitigation in interference. The cyclically extended discrete-time sequence is passed through a digital-to-analog converter to form the baseband OFDM signal. Finally, the base- band OFDM signal is modulated using a carrier fre- quency for transmission through a wireless channel. Transmitting a signal through a wireless channel results in convolution of the signal with the impulse response h(q) of the channel. Consequently, the signal is distorted by the additive white Gaussian noise (AWGN) n(q) present in the channel. The convolution between the transmitted signal and channel’s impulse response is a circular convolution due to the guard interval. Thus, as seen by channel, the discrete-time sequence S' looks as if s is repeated periodically for all time. The process of recovering the transmitted data se- quence begins with the down conversion of the received signal performed by an IQ detector. The output of the IQ detector is the distorted version of the complex signal s(t), indicated as ˆ() s t. The signal ˆ() s t is passed through an analog-to-digital (A/D) converter to obtain a complex discrete signal ˆ() s q. Subsequently, the cyclic prefix is discarded and the signal becomes [2]: 1 0 ;0, ˆ1, ,1 N N qm qm m shqsN (2) where () N qm represents modulo N subtraction. In vector form, ˆq s is represented as 01 1 ˆˆˆ ˆ S,,, N ss s . After passing the received sequence through the FFT block, an estimate of transmitted complex symbols is obtained which is given by: 12/ 0 1;0,1, 1 ˆˆ, NjkqN kq q dsekN N (3) After substituting (2) into (3) k d becomes: ˆkkk H dd (4) here, k H represents the transfer function component of the channel and ˆk d is the received subcarrier informa- tion at the output of the FFT block. The complex se- quence at the output of the FFT block i.e. 01 1 ˆˆ ˆ,,, ˆN dDdd is then passed through the signal de-mapper and parallel-to-serial converter to obtain an estimate of the encoded information. The decoder is then used to arrive at an estimate of the information transmitted. 3. Peak-to-Average Power Ratio (PAPR) As pointed out in the previous section, the baseband OFDM signal is the result of summation of sinusoidal waves at the output of the IFFT block. At some sample points of these sinusoidal signals, constructive summa- tions may occur, resulting in high peaks in the signal. When transmitting high peak signals through a non-linear power amplifier, distortion occurs within the transmitted signal at the output of the amplifier in the form of ISI and out-of-band radiation. Hence, the influence of high peaks is evident at the output of a non-linear power amplifier but the point of occurrence of these peaks is at the output of an IFFT block. For this reason, the non-linear amplifi- er is neither used for the analysis throughout the paper nor in simulations being carried out as our main concern revolves around the measurement of these high peaks at the point of occurrence. One of the widely used measures for the power of these peaks is Peak to Average Power Ratio (PAPR) which is mathematically expressed as: 2 01 12 0 max max( ) 1 qN N avg q s q Pq PAPR P s q N (5) here, P(q) represents the instantaneous power and Pavg represents the average power of the OFDM signal. For constant envelope signals, it can be shown that Pavg = N. In order to simulate such a system, samples of OFDM signals are needed. For better PAPR estimation, over- sampling is required to capture these peaks since in normal symbol spaced sampling; some of the peaks might be missed and may result in less accurate PAPR measure. Hence oversampling (1) by a factor of J where (J > 1) gives a better PAPR estimation. It has been shown that J = 4 is sufficient to capture the peaks [5]. The peak value of an OFDM signal and the corresponding time domain signal differs from one mapper to another (e.g. peak in 32−Quadrature Amplitude Modulation (QAM) is different from the peak value in 32−Phase Shift Keying (PSK)). 758 I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS For all phase shift keying, the maximum peak has the value of N2 and hence a maximum PAPR of N. Figure 2 shows all possible OFDM signals for N = 4 and BPSK mapper. It can be seen that the maximum normalized absolute peak in such signals is 4 as indicated in the se- quences 0000, 1001, 1010 and 1111. Since the transmitted bits are generated randomly, the transmitted OFDM signals are random in nature and therefore the envelope of an OFDM signal, as given by (1), can be considered a random variable. For large values of N, according to the central limit theorem, the expected amplitudes of OFDM signals follow a Gaussian distribution. In addition, P(q) has a Chi square probability density function with two degrees of freedom [5]. The PAPR performance of a system is usually measured by the Complementary Cumulative Distribution Function (CCD F) curves, a standard way of depicting and describing PAPR related statistics. The CCDF shows the probability of an OFDM sequence exceeding a given PAPR (PAP0). For PAPR, the lower bound on CCDF for a specific PAPR value (i.e., PAP0) is given by [7]: 0 Pr P APR PAP This results in the following relationship: 0 11 N PAP e (6) 4. Techniques for Minimizing PAPR The objective is to minimize PAPR as much as possible so as to obtain signals with smaller peaks. Various algo- rithms and methods have been proposed for reducing PAPR. One simple method for reducing PAPR is direct clipping of high peaks and subsequent filtering of the signal [7,8]. In addition, various modulation schemes are used for efficient transmission of signals such as Conti- nuous Phase Modulation (CPM) [9,10]. Recently, constellation and shaping methods have Figure 2. Peak values in OFDM signals for N = 4 by using BPSK mapper. been used to reduce PAPR. In these methods, a mapping between the original complex numbers and the finally transmitted complex numbers takes place based on an algorithm or a coding technique. One such method is the trellis shaping method using a metric which is based on the Viterbi algorithm [11]. One of the variants of this method uses a metric-based symbol predistortion algo- rithm resulting in some implementation complexity [12]. Another promising technique for reducing PAPR in- volves scrambling the incoming OFDM sequence using some rotation vectors resulting in multiple signals that represent the same original information. Among these signals, the signal with the lowest PAPR is transmitted. Examples of such methods are Selected Mapping (SLM) [13-15] and Partial Transmit Sequence (PTS) [16,17]. In SLM, shown in Figure 3(a), from a single OFDM sequence of length N, U sequences are generated that represent the original information or OFDM sequence. The sequence having lowest PAPR value is transmitted. These sequences are generated by multiplying the original OFDM sequence with U different factors. These factors are given in vector form as ()() ()() 01 1 ,,, iiii N Bbbb where i = 1 to U represents the indices of these factors. After multiplying these factors by the original OFDM sequence D, we get 00111 1 ,,, iii i NN Xdbdbdb . These factors are phase rotations selected appropriately such that multiplying a complex number by these factors (a) (b) Figure 3. The SLM method (a) at the transmitter and (b) at the receiver. I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS 759 results in rotation of that complex number to another complex number representing a different point in the constellation. Hence, () () i n j i n be where () [0,2 ) i n . At the receiver end as shown in Figure 3(b), the orig- inal OFDM sequence is recovered by multiplying the received sequence by the reciprocal of the vector being used at the transmitter end. Hence, in SLM, it is essential to transmit the rotational vector B(i) as side information to recover the original sequence. On the other hand, in PTS method, as shown in Figure 4, the original OFDM sequence D is first partitioned into H disjoint sub-sequences. The length of each sub- sequence is still N but padded with zeros. Each sub- sequence is fed into a separate IFFT block of length N each. Hence, there are H number of IFFT blocks. The set of complex numbers at the output of each IFFT block is multiplied by a factor belonging to one of the rotation factors as indicated in SLM technique. These factors are optimized in such a manner that the PAPR of the com- bined sub-sequences are reduced. After multiplying by the factors, the complex numbers from all the IFFT blocks are added together carrier-wise, resulting in a final sequence. This final sequence has lower PAPR than the original one. Many other proposed algorithms tackle different parameters for reducing the PAPR indirectly. One such method is the Aperiodic Autocorrelation Coefficient (AAC) of the transmitted OFDM signals in which PAPR reduction is achieved using selective scrambling of the transmitted sequence, generating a number of statistically independent sequences [2]. A Selective Function (SF) is computed for every sequence and the sequence with the lowest SF, which also corresponds to the lowest PAPR, is transmitted. Another factor that plays an important role in reducing Figure 4. PTS method. PAPR is the Power Variance (PV) of an OFDM se- quence. It is indicated in the literature that there exists a strong relationship among AAC, PV and PAPR [2-4,20- 23,26]. In the next section, we study this relationship in OFDM signals. 5. PV and Aperiodic Autocorrelation Coefficients For a complex envelope signal given by (1), the instan- taneous power is given as: * PqqS q (7) where ‘*’ denotes conjugate of a complex signal. By combining (1) and (7) we get: 22 11 * 00 1 j kqj pq NN NN kp kp Pqdede N (8) Since the subcarriers in the OFDM signal are ortho- gonal, they satisfy the following condition [2]: 1 * 0 0, Φ,Φ,, N q ij iqjq Nij (9) such that, 2 2/2( 1)/ 2( 1)/2( 1)/ 11 1 1 Φ 1 jNjN N jN NjNN ee ee (10) By simple manipulation of (8) and using the orthogonality property of the subcarriers given by (9), the instantaneous power can be expressed as: 22 11 * 0 11 1jiq jiq NN NN ii ii PqR ReRe N (11) where Ri is called the ith Aperiodic Autocorrelation Co-efficient of the complex OFDM sequence D and given as [3]: 1 * 0 ;0 1 Ni ikki k RddiN (12) Note that for i = 0, (12) becomes 12 0 N kavg ko RdP . in case of a constant envelope mapper, Pavg = N. By com- bining (11) and (12) then dividing by Pavg, we get the normalized instantaneous power γ(q) given below: avg Pq qP 1 1 1* 1 122 cos sin 22 cos sin N i i avg N i i iq iq NR j NPN N iq iq Rj NN hence, γ(q) is given by: 760 I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS 11 * 11 122 2Re cos2Imsin NN ii ii avg q iq iq NRR NPN N (13) In compact form, (13) can be reduced into: * 1 1 0 1 12 2cos tan Re Ni i i avg i Img R iq qRR NPN R (14) Using Chebyshev polynomials, an upper bound on (14) is given by [19]: 1 0 1 1 Γ2 N i i avg qRR NP (15) By taking the difference between the instantaneous power and the average power, i.e. ΔP(q) = P(q) − Pavg, we get the Power Variance of OFDM signal using the following expression: 12 0 1N q PVP q N which can be further expanded using trigonometric properties into the following expression: 11 2 11 12 1 14 22 cos cos1 Nm im iN N imavg q PVR R NN iqmq NP NN (16) Hence, an upper limit on (16) becomes: 12 0 N norm i i PV PVR (17) here, PVnorm is the normalized power variance. It can be observed from (11-15) that P(q), γ(q) and Γ are all func- tions of AAC. Low values of |Ri| correspond to low in- stantaneous power and hence low values of Γ. Based on this assessment, some authors use (15) as a parameter for measuring and comparing PAPR values of OFDM se- quences. In contrast, other authors show that PAPR analysis based on (14) and (15) gives misleading results [19]. Through an example, they show that sequences exist that have low PAPR values but high Γ values and vice versa. They conclude that Γ cannot serve as para- meters for PAPR comparison of sequences and hence Γ is not a good measure for PAPR. In addition, many authors concluded that PV is also a good measure of PAPR [2-4,20-23]. However, in subsequent paragraphs, we show that this is not always the case. Figures 5 and 6 show normalized PAPR and PV values respectively for different sets of 300 randomly generated OFDM sequences when eight subcarriers are used. Ap- parently, these figures suggest that PV is a good measure Figure 5. Normalized PAPR for randomly generated OFDM sequences with 8 subc ar riers. Figure 6. Normalized PV for randomly generated OFDM sequences with 8 subcarriers. of PAPR as the pattern of both the figures for the same set of symbols is almost the same. Hence, many authors concluded that PV is a good measure for PAPR [2-4,20-23]. However, we here demonstrate that this observation is misleading because not all low PAPR sequences have low PV and vice versa. To refute this assertion, we investigate the relation between PV and PAPR given as [4,5]: 11PAPR QQ PV PV (18) where Q(.)is the complementary error function and β denotes Pr (P(q) ≤ P(q)max ) which is given by (6). Figure 7 shows a plot of (18) for selective values of PAPR for 256 subcarriers with β = 0.7434. Two aspects in this Figure 7 are noteworthy. First, for a particular value of PAPR, there are two values of PV. For instance, at PAPR = 6.5 dB, the values of PV are 35 and 93.37. Second, PV does not vary in proportion to PAPR. For instance, at PAPR = 6dB, one of the values of PV is 79.89 whereas at PAPR = 7dB, one of the values of PV is 44.31, i.e. an in- I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS 761 Figure 7. Relationship between PAPR and PV for β = 0.7434 with 256 subcarriers. crease in PAPR does not always correspond to an in- crease in PV. It shows that PV cannot always constitute a basis for comparison of PAPR between two sequences. Hence, PV cannot be considered a reliable measure for PAPR. In addition, to emphasize our assessment of rela- tionship between PV and PAPR, CCDF plots are used for PAPR measure based on PV using SLM and PTS techniques. In the first simulation experiment, CCDF curves are plotted for 50,000 randomly generated OFDM sequences using SLM technique where U = 4, 16 and 64 as shown in Figure 8. In this approach, U different sequences are generated from a single OFDM sequence D using ran- domly generated phase rotation factors. The sequences having the lowest PV and the lowest PAPR are selected for transmission. In short, the selection decision for transmitting a sequence is based on both the lowest PV and the lowest PAPR. This simulation gives two sets of CCDF curves, as shown in Figure 8. It is clear that the reduction in PAPR based on PV is not the same as the one based on PAPR. For instance at CCDF = 0.001, when U = 64, the PAPR reduction using a PV-based de- cision is approximately 2dB while the PAPR reduction based on PAPR itself is around 3.8dB. A difference of almost 2dB is evident between the two transmission decisions. This difference in PAPR reduction is suffi- cient to show that PV is not a good and reliable measure for the purposes of PAPR reduction. For the second simulation experiment, CCDF curves are plotted for 50,000 randomly generated OFDM se- quences using PTS technique using 256 subcarriers and H = 16, shown in Figure 9. As it is the case in SLM, two curves are generated for both PV- and PAPR-based transmission decisions. A PAPR reduction of 2dB is achieved in case of PV-based decision whereas a reduc- tion of 3.5dB is achieved in case of PAPR-based decision. Once again, the difference in PAPR performance be- Figure 8. CCDF curves for SLM based on PAPR and PV for various values of U. Figure 9. CCDF curves for PTS technique based on PAPR and PV for H = 16. tween PV-based and PAPR-based transmission decisions is significant and obvious. All these results suggest that low values of PV do not always correspond to low PAPR values and vice versa. Consequently, PV is not a reliable measure of PAPR and it cannot be used as a parameter for OFDM sequence selection in both SLM and PTS techniques. Since OFDM sequences follow a random process, it is difficult to tell the range of PAPR values or sequences that correspond to high PV values and vice versa. It can also be noted that for a particular number of subcarriers N, the values of β vary with PAPR values. Although theoretically PAPR, PV and β can take any value, as indicated by (18), the actual values of these three parameters are finite and specific for a given num- ber of subcarriers. For instance, through simulation we generate all possible OFDM symbols for 16 subcarriers using BPSK mapper and plot their corresponding PAPR values against PV values as shown in Figure 10. The total number of possible OFDM symbols is 65,536. The maximum and minimum PAPR values are 16 and 1.7071 respectively and the corresponding PV values for these PAPR values are 1240 and 24 respectively. Note that the plot is discontinuous through PAPR axis because the 762 I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS Figure 10. PV versus PAPR values for OFDM sequences with 16 subcarrier s and BP SK mapper . OFDM symbols have finite and specific values of PAPR, as mentioned earlier. It is interesting to note that our assessment regarding the relation between PV and PAPR is more pronounced for low values of PAPR than high values of PAPR (i.e. more concentration of points in the lower region of PAPR). It can be seen that for a single PAPR value in the lower range (approximately from 2dB to 5dB), there are more than one corresponding PV value. Similarly, for a single value of PV (approximately from 24 to 300), there are more than one corresponding PAPR value. Again, this plot is only possible for small number of subcarriers. For high number of subcarriers, it is difficult to find the distribution of PAPR and PV values for all possible OFDM sequences. 6. Computational Complexity of PV SLM algorithm based on PAPR comparison has moderate complexity. The main complexity is based on the com- putation of IFFT operation. This complexity increases as U increases. Hence, IFFT has to be computed for every sequence (i.e. U times) before transmitting the final selected sequence. Similarly, when evaluating PV, Equa- tion (17) needs to be evaluated for every sequence (i.e. U times) before transmitting the final selected sequence. So, it would be interesting to compare the complexity asso- ciated with computing PAPR with the complexity in evaluating PV. The complexity expressions in terms of complex addi- tions and multiplications for evaluating IFFT, and there- fore PAPR, are shown in Table 1 [28]. Similarly it could easily be shown that the complexity for evaluating Equa- tion (17) in terms of both complex additions and multip- lications is given as: 12 2 11 2 NN Complex Additions N ComplexMultiplicationsN (19) Table 1. Computational complexity of various parameters. ExpressionComplex Additions Complex Multiplica- tions IFFT JN(log2JN) JN(log2JN) ⁄ 2 IFFT and PAPR JN(log2JN) (JN(log2JN) ⁄ 2) + JN PV (N (N − 1) – 2 ) ⁄ 2 (N − 1) ( (N ⁄ 2) + 1) PPV (B (2N–B−1)−2) ⁄ 2 (B (2N–B−1) ⁄ 2) + B Table 2. Computational complexity of various parameters for different subcarriers. N Complex Additions PAPR (J = 1) PAPR (J = 4) PV PPV (B) PPV (B = 1) 4 8 64 5 4 (2) 2 8 24 160 27 21 (4) 6 1664 384 119 91 (8) 14 32160 896 495 219 (8) 30 64384 2048 2015 475 (8) 62 128896 4608 8127 1911 (16) 126 2562048 10240 32639 3959 (16) 254 N Complex Multiplications PAPR (J = 1) PAPR (J = 4) PV PPV (B) PPV (B = 1) 4 4 32 9 7 (2) 4 8 12 80 35 26 (4) 8 1632 192 135 100 (8) 16 3280 448 527 228 (8) 32 64192 1024 2079 448 (8) 64 128448 2304 8255 1928 (16) 128 2561024 5120 32895 3976 (16) 256 The computational complexity for both PAPR and PV for different values of N is presented in Table. 2. Note that, for comparison purposes, PAPR complexity is also evaluated when oversampling OFDM signals by a factor of 4. It is clear that the computational complexity of PV is lower than that of PAPR only for very small values of N (i.e. N = 2 and 4). For large values of N, the computa- tional complexity of PV is far greater than that of PAPR complexity even when J = 4. Hence, PV-based selection becomes impractical for large values of N. This is another disadvantage of using PV as a selection criterion in SLM or PTS techniques, pronouncing the need for an efficient measure of PAPR. In the next section, we propose one such measure of PAPR 7. AAC-SLM Algorithm and Partial Power Variance (PPV) In this section, we introduce a new parameter called Par- I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS 763 tial Power Variance (PPV) that can be effective in reducing PAPR. Indeed, such Soft Computing tools and approximate, but effective and efficient, computational techniques are rapidly gaining popularity in subjective and complex application domains [24,25]. We show that PPV is not only effective but also computationally effi- cient than both PV and PAPR. Towards this end, we also propose an algorithm for minimizing PAPR. The pro- posed algorithm is modeled on SLM technique and we call it AAC-SLM. As indicated by (17), the normalized PV expression consists of N − 1 aperiodic autocorrela- tion coefficients. The objective of AAC-SLM algorithm is to investigate these coefficients (i.e. R1,R2,…,RN−1) individually and find their contribution in reducing PAPR. For QPSK mapper, the proposed algorithm involves the following steps. From a single OFDM sequence, (N-1) different sequences are generated by reducing |Ri| of the sequence to its minimum possible values of 0 (for even i) and 1 (for odd i) where i = 1, 2,…, N−1. These sequences represented by Y1, Y2,…, YN−1 have the following property: the generated sequence Yi is the result of minimizing |Ri| of the original OFDM sequence to its minimum value (i.e. either 0 or 1 depending on i). Hence, for each new sequence, only its respective |Ri| is minimized while other |Ri|’s for that sequence may not have minimum value. For example, a sequence generated by minimizing |R3| i.e. Y3 may or may not have minimum values of other |Ri|, i.e. |R1|,|R2|,|R4|,…etc. From these sequences, the one with the lowest PAPR is transmitted. The method for reducing the coefficients to their mini- mum values is given in the Appendix. The CCDF curves for AAC-SLM algorithm using 256 subcarriers and 100,000 randomly generated OFDM sequences are shown in Figure 11. It is clear that a maximum reduction of 3.8 dB is achieved when all sequences from Y1 through Y 255 are included. The se- quence with the lowest PAPR is selected. Further, when reducing the number of sequences to 64 (i.e. Y1,Y2,…,Y64) and selecting the one with lowest PAPR, a negligible degradation in PAPR occurs. This suggests that the higher order sequences Y65 through Y 255 only have a marginal and negligible contribution in reducing PAPR. Similarly, when considering only Y1 and Y 2, a PAPR reduction of 3.2dB is achieved. In fact, only by reducing the first AAC (i.e. Y1 only), almost 3dB reduc- tion is achieved in PAPR performance. In other words, all the generated sequences in the AAC-SLM algorithm other than Y1 have a minimal contribution in PAPR reduction. It follows that all AAC in the PV expression have a minimal contribution in PAPR reduction except for |R1|. Based on the previous observations, Figure 8 is re- Figure11. AAC-SLM with 256 subcarriers. Figure 12. PPV in PAPR reduction for different values of B with N = 256. plotted for U = 4, shown in Figure 12. Around 100,000 OFDM symbols are randomly generated to carry out this experiment. Similar to Figure 8, CCDF curves for both PV- and PAPR-based decision are shown in Figure 12. The two additional CCDF curves are a result of making the transmission decision based on PPV, where PPV is the truncated version of the PV expression given by (17). The first curve is generated when the decision is based only on the first 64 AAC (R1,R2,…,R64) in (17) instead of using the whole expression for PV. Hence in this case,. From Figure 12 it is clear that a negligible degradation in PAPR performance results using this truncated ex- pression. In fact, when a single AAC is used i.e. PPV = |R1|2, a degradation of only 0.2dB occurs from the PV-based CCDF curve. These results also support our earlier observations for the efficacy of the AAC-SLM algorithm. In other words, we can say that AAC terms in PV expression do not contribute in reducing PAPR signif- icantly except for the first term i.e. R1 that has the maxi- mum contribution in reducing PAPR. In short, we can use PPV instead of PV while achieving almost the same 764 I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS PAPR reduction. In addition, the computational complexity of PPV is less than both PV and PAPR. To compare the computa- tional complexities of the PV- and PPV-based decisions, we write a general expression for PPV as follows: 2 1 B B i i PPV R (20) where B indicates the number of AAC terms included in the PPV expression. Note that when B = N − 1, PPV expression becomes PV. In terms of B, the complexity of PPV is given below (and also shown in Table 1): 212 2 21 .2 BNB Complex Additions BNB ComplexMultipB (21) In case of B = 1, the complexity reduces to N − 2 and N complex additions and multiplications, respectively. This shows a considerable reduction in computational com- plexity when PPV is used for transmission decision in- stead of PV or even PAPR itself. The reduction in com- putational complexity for different values of B is shown in Table 2. Once again, it can be seen that the maximum reduction in complexity is achieved when B = 1. For instance, in case of 256 subcarriers, the number of com- plex additions in PAPR (with J = 1) is 2048, whereas in case of PPV1 it reduces to 254, a reduction by a factor of nearly 8. The computational complexity is decreased considerably compared with IFFT complexity. Hence PPV is faster to implement both in hardware using digital signal processing techniques and software than IFFT. 8. Conclusions In this paper, we have established that PV is not a good measure of power efficiency as has been claimed in the literature. Our results clearly show that using PV as the power efficiency measure gives misleading results. Further, we show that PV is computationally more complex than PAPR and hence cannot be used as a power efficiency measure for OFDM. In addition, we have developed a new, effective and efficient measure for power efficiency called PPV which is computationally less complex than PAPR. The amount of reduction achieved in terms of complex additions and multiplications for a 256-sub- carrier system is more than 8 times as compared to PV and 3.5 times as compared to PAPR in order to achieve the best power efficiency. In fact, based upon the flavor used for PPV, the reduction in complexity may go down as low as 40 times as compared to PAPR at a nominal degradation in power efficiency. Hence, PPV is a more useful, realistic and cost-effective measure for power efficiency of OFDM signals. In this paper, we demon- strated the efficacy of the new measure by applying it on PTS and SLM techniques. The proposed measure can also be applied on other established algorithms to de- crease the computational complexity. Further, the per- formance of the new measure can also be tested for more practical systems where the number of subcarriers may go beyond 2048. In this paper the bit error rate (BER) performance of PPV is not discussed as the main objective was to inves- tigate the PAPR performance of PPV. As a future work, it would be interesting to find out the BER performance of PPV both in AWGN and multipath fading channels. 9. References [1] Special Issue on 4G Mobile Communications: “Toward Open Wireless Architecture, IEEE Wireless Communica- tions, Vol. 11, No. 2, April, 2004. [2] P. Van Eetvelt, G. Wade and M. Tomlinson, “Peak to Average Power Reduction for OFDM Schemes by Selec- tive Scrambling,” Electronics Letters, Vol. 32, No. 21, October 1996, pp. 1963-1964. [3] H. Nikookar and R. 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A Tasadduq, “PAPR Analysis in OFDM Signals Based on Power Variance,” Proceedings of Wireless Communications, Networking and Mobile Computing, Dalian, October 2008, pp. 1-4. [27] M. Negnevitsky, “Artificial Intelligence: A Guide to In- telligent Systems,” Pearson, Sydney, 2008. [28] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs: Prentice Hall, 1989. 766 I. M. HUSSAIN ET AL. Copyright © 2010 SciRes. IJCNS Appendix As indicated in (17), the normalized PV expression con- tains N – 1 AAC terms. AAC term can be represented by (12). In case of constant envelope mapper (e.g. QPSK which is used throughout our simulations), (12) can be modified into: 1 0 ;0 1 Ni k i kki d RiN d 031 12 123 1 Ni i iii iN ddd dd Rddd dd (22) Hence each expression of Ri contains N − i terms of kind (dk / dk+i) . Using (22) and (17) it can easily be shown that the total number of such terms in the power variance expression is 1 1 N iNi . Similarly, the total number of pairs of adjacent terms in R i as indicated by parenthesis in (22) is /2Ni. Now one way to reduce AAC to their minimum values (i.e. 0 or 1 depending on i) is to make the summation of each and every pair of terms in (22) equal to zero. Let 1 1 pp pipi dd dd be any pair in (22) where p = 0,2,4,…,N – 2 – i (for even i) and p = 0,2,4,…,N – 3 – i (for odd i). In case of i = 1,2,3,…,N – 1, to minimize the values of all such pairs to zero, we have to multiply every second complex number (i.e. subcarrier) starting from dp+i by a factor gi,v. This factor is obtained as shown below: For the first /2i number of pairs: 1 ,1 0 pp iv pipi dd gd d 1 , 1 ; p pi iv p pi dd gdd (23) For rest of the pairs when i is even: 1 , 1 ; p p i iv p pi dd gdd (24) For rest of the pairs when i is odd 1 , 1 ; ppi iv ppi dd d gd (25) where 1 ,/21 ,/2 , pp p iv ipivi dgd gdd and 1 2 p v. For example, in case of N = 16 subcarriers and i = 4, the fourth AAC becomes: 03567 12 4 4 4 56 7891011 ddddd dd d Rdd dddd dd 89 10 11 12 131415 dd dd dd dd This expression has 6 pairs i.e.164 / 2 and as many factors and each factor is used to minimize the corres- ponding pair to zero. Since i is even, then p = 0,2,4,…,10. For instance, the third pair in R4 is 5 4 89 d d dd where v = 3 and p = 4. In order to find the six factors used to mi- nimize each pair, (23) is used for the first 2 pairs i.e. 05 4,1 14 dd gdd and 27 4,2 36 dd gdd . Rest of the factors are calculated using (24) as i is even. Hence, 49 4,3 58 d gd dd , 611 4,4 710 d gd d d , 813 4,5 911 d gd d d and 10 15 4,6 914 d gd d d ,where 444,1 dd g , 664,2 dd g , 884,3 dd g and 10 104,4 dd g . Further, each alter- nate complex number starting from d4 (i.e. d4, d6, d8, d12, d14, d16) will be multiplied by g4,1, g4,2, g4,3, g4,4, g4,5 and g4,6 respectively. In this manner |R4| will be minimized to zero. It is clear that p d or 1p d means that before mi- nimizing the corresponding pair to zero, updation of p d into p d has to take place first. In the above example, for the third pair, updation of d4 (i.e.444,1 dd g ) has to take place first after which minimization of that pair takes place. Thus, the third pair has the form 45 89 dd dd rather than 5 4 89 d d dd . Now in case of i = 1 i.e. R1 which is an exception case, factor expression is given by (23) for the first pair and for the rest of the pairs, it is given by (24) where ,1 p p iv dgd . Further, each com- plex number starting from d1 is going to be multiplied by the corresponding factor. In this manner, we can minim- ize the coefficients of any sequence to their respective minimum values using the procedure above. |