Computer Communications 32 (2009) 159–168 Contents lists available at ScienceDirect Computer Communications journal homepage: www.elsevier.com/locate/comcom An efficient initial ranging algorithm for WiMAX (802.16e) OFDMA q Hisham A. Mahmoud a,*, Huseyin Arslan a, Mehmet Kemal Ozdemir b a b Electrical Engineering Department, University of South Florida, 4202 E. Fowler Ave., ENB-118, Tampa, FL 33620, USA Logus Broadband Wireless Solutions, 3803 Corporex Park Pl. Suite 700, Tampa, FL 33619, USA a r t i c l e i n f o Article history: Received 17 March 2008 Received in revised form 23 September 2008 Accepted 25 September 2008 Available online 10 October 2008 Keywords: Initial ranging OFDM OFDMA WiMAX Synchronization a b s t r a c t Ranging is one of the most important processes in the mobile WiMAX standard. Power adjustment, timing offset estimation, and synchronization between a base station (BS) and all users within a cell are done through the ranging process referred to as initial ranging. In this paper, we discuss the details of initial ranging together with some of the proposed algorithms, as well as a novel algorithm to carry out a successful ranging process. Performance curves and computational complexity comparisons are presented. The system performance is evaluated for both additive white Gaussian noise (AWGN) channels and practical multipath fading channels with multiuser interference. It is shown that the proposed algorithm offers a reduced complexity ranging method that can be employed in practical WiMAX-based BSs. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction WiMAX, as a relatively new technology, has received the attention of researchers and wireless companies. This new wireless technology promises to deliver both high data rates and long-range coverage. With the approval of the mobile WiMAX standard (IEEE802.16e-2005) at the beginning of the year 2006, this technology became even more exciting. Unlike WiFi [1,2], which is designed for indoor applications and wireless local area network (WLAN), WiMAX is optimized for outdoor applications and wireless metropolitan area network (WMAN). One of the exciting aspects of WiMAX is that its medium access control (MAC) layer supports more than one physical layer (PHY) mode [3]. This feature not only enables companies to differentiate their products from each other, but also makes WiMAX an adaptive technology that can satisfy different needs depending on the application. One of the most promising PHY modes supported by WiMAX standard is orthogonal frequency division multiple access (OFDMA) PHY mode which enables a WiMAX base station (BS) to support multiple fixed or mobile users at the same time. In this mode, a BS system utilizes the channel by dividing available subcarriers into subchannels that can be assigned to multiple users in a sophisticated and adaptive way. As a matter of fact, users q This work has been partially presented in the IEEE Military Communications Conference, MILCOM’06, October 23–25, 2006, Washington, USA. * Corresponding author. Tel.: +1 813 974 5005. E-mail addresses: hmahmoud@mail.usf.edu (H.A. Mahmoud), arslan@eng.usf. edu (H. Arslan), kemal@loguswireless.com (M.K. Ozdemir). 0140-3664/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2008.09.032 can be assigned to different bandwidths, different time durations, and different modulation orders based on various parameters such as user carrier-to-interference-plus-noise ratio (CINR) and the available bandwidth. One of the problems that faces WiMAX systems, or OFDMA systems in general, is timing and synchronization between a BS and subscriber stations (SS) within the cell. While other orthogonal frequency division multiplexing (OFDM) receivers can easily synchronize to the received signal [4,5], this is not the case for OFDMA receivers. At an OFDMA receiver, where multiple users arrive at the same time, if users are not synchronized with the receiver, they will interfere with each other, and therefore the BS will not be able to recover individual signals of each user. Hence, for OFDMA PHY mode to work properly, all users should arrive at the BS at the same time with a considerably high timing accuracy. This can be achieved if all users are synchronized with the BS before the communication link is established. The standard states that for a user to join the channel, first, the round trip delay (RTD) between the SS and the BS must be known to the SS [3]. This delay estimation is used by the SS to synchronize its signal such that it arrives at the BS in its allocated time. The process in which this delay is estimated is called initial ranging, and this process is mandated for all SSs that desire to synchronize to the channel initially. Other types of ranging processes (e.g. periodic ranging and bandwidth request) exist, but only initial ranging is considered in this paper. As stated in the IEEE802.16e-2005 standard [3], the MAC layer at the BS defines a ranging channel as a group of six (or more) subchannels, where a subchannel is a group of subcarriers that are chosen according to a randomization formula. In addition, users 160 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 are allowed to collide in this ranging channel. Any SS that attempts to establish a communication link is required to carry out a successful initial ranging process with the BS over the ranging channel. Once a SS senses a BS, for network entry, it first scans for a downlink (DL) channel and synchronizes itself with the BS. Then, the SS shall acquire transmit parameters, which are included in the uplink channel descriptor (UCD), uplink (UL)-MAP, and DLMAP. Using acquired parameters, the SS initiates the initial ranging process by sending a ranging code over the UL frame. At the receiver side, the BS is required to detect different received ranging codes and estimate the timing offset and the power for each user that bears an initial ranging code. The BS then broadcasts the detected ranging codes with adjustment instructions for the timing and power level. The status notifications of either a successful ranging process or retransmission are also broadcasted. In initial ranging, the SS chooses one of the available ranging codes randomly and transmits it twice over two consecutive OFDM symbols with BPSK modulation. The SS should transmit the ranging code during the UL frame as long as there is a ranging opportunity. UL-MAP shows if a ranging opportunity is available through the next UL frame. Another option is to send two consecutive ranging codes over four OFDM symbols to increase the probability of code detection [3]. In this paper, ranging over two symbols is considered. Synchronization for multiuser OFDM systems has been discussed in the literature. The use of filters matched to the intended user’s subcarriers (in our case, the ranging channel subcarriers) and then use of cyclic prefix (CP) redundancy to estimate the timing offset is proposed in [5]. However, in WiMAX standard, the ranging subcarriers are not necessarily adjacent, which makes the filtering process inapplicable. In [6], it is proposed to synchronize users to the BS one at a time, with the assumption that other users are already synchronized. This method cannot be used for OFDMA with multiple users colliding in the ranging channel. Finally, in [7,8] it is proposed to use a bank of correlators (corresponding to number of ranging codes) in time or in frequency domain to detect received ranging codes. The disadvantage of this approach is that the computational complexity increases as the number of codes increases. In this paper, a new algorithm for OFDMA initial ranging is proposed. The proposed algorithm is examined using theoretical analysis and computer simulations over additive white Gaussian noise (AWGN) and dispersive channels in the presence of multiuser interference. A performance and complexity comparison between the proposed algorithm and prior algorithms in practical system conditions are presented. It is demonstrated that the proposed algorithm offers a significant reduction in computational complexity while carrying out a successful initial ranging process. The reduction in complexity becomes especially important for certain practical WiMAX implementations. For example, the proposed algorithm can be attractive to low-cost OFDMA-based femtocell BS implementations where the number of users per BS is limited. The remainder of this paper is organized as follows. The system model is introduced in Section 2. Current and proposed ranging algorithms are discussed in detail in Sections 3 and 4, respectively. In Section 5, the computational complexity of the proposed algorithm is calculated and compared to current ranging algorithms. Simulation results and discussions demonstrating the performance of the proposed ranging algorithm compared to other algorithms are presented in Section 6. Finally, the conclusions are outlined in Section 7. After assigning DC and guard subcarriers, the remaining subcarriers, N d , are grouped into Q subchannels. Each subchannel has N Q ¼ N d =Q subcarriers, where Q is chosen such that N d is an integer multiple of Q. Each user in the UL is assigned one or more subchannels. The BS defines a group of six subchannels (or more) for ranging. Note that the subcarriers assigned to each subchannel are chosen randomly and thus they are not necessarily adjacent. The BS broadcasts all the ranging information (i.e. ranging opportunities, ranging channels, ranging codes and so on) in the UL-MAP. One ranging time slot spans two OFDMA symbol duration. The kth ranging user signal in frequency domain is denoted as ðkÞ ðkÞ ðkÞ ðkÞ cp ¼ ½cp ð1Þ; cp ð2Þ; . . . ; cp ðLÞT , where p is the index of the randomly chosen ranging code and L is the size of the ranging code. The signal is then extended to N t by inserting N t  L zeros, which T ðkÞ ðkÞ results in XpðkÞ ¼ ½X ðkÞ p ð1Þ; X p ð2Þ; . . . ; X p ðN t Þ . Note that X ðkÞ p ðmÞ ¼ ( ðkÞ cp ðnÞ if m ¼ ir ðnÞ; 0 otherwise; ð1Þ where ir ðnÞ is the index of the nth subcarrier within the ranging channel subcarriers set ir ¼ ½ir ð1Þ; ir ð2Þ; . . . ; ir ðLÞT . The vector XpðkÞ is then fed to an Nt -point inverse discrete Fourier transform (IDFT). The resulting signal in time domain is extended over two OFDMA ðkÞ symbols by repeating xp twice and adding the cyclic prefix with ðkÞ no phase discontinuity, where xp is the time representation of ðkÞ Xp . Note that the BS receiver uses an observation window of ðN t þ N CP Þ to acquire OFDMA symbols, where N CP is the size of the CP. Finally, the transmitted signal is denoted as ðkÞ ðkÞ ðkÞ ðkÞ sp ¼ ½sp ð1Þ; sp ð2Þ; . . . ; sp ð2Nt þ 2N CP ÞT , where ðkÞ ðkÞ sðkÞ p ¼ ½xp ðN t  N CP þ 1Þ; . . . ; xp ðN t Þ; ðkÞ xðkÞ p ð1Þ; . . . ; xp ðN t Þ; ð2Þ ðkÞ xðkÞ p ð1Þ; . . . ; xp ðN t Þ; T ðkÞ xðkÞ p ð1Þ; . . . ; xp ðN CP Þ : ðkÞ The transmitted signal sp is then received by the BS after being corrupted by the communication channel. In the following analysis, we assume that the channel is non-dispersive and that the received signals from multiple users are only affected by complex AWGN. However, we will evaluate the proposed ranging algorithm performance for both AWGN and dispersive channels. A similar approach has been used in [9] to analyze the proposed algorithm. All users other than the users performing initial ranging are assumed to be already synchronized to the BS. This is a valid assumption as synchronizing to the BS is mandatory before a SS can establish the communication link. Hence, it is guaranteed that there is no interference from synchronized user signals to the ranging channel. Note that this is not the case for ranging users as their unsynchronized signals can cause interference to synchronized users. As this situation is unavoidable, initial ranging SS is required by the standard to start the ranging process with minimum possible power level. Then, as long as the SS fails to get a response from the BS, the power is increased incrementally until a response is detected. If the maximum power level is reached and the SS still cannot get a response from the BS, the user starts from the minimum power level and the process is repeated. This shows how important it is for a BS to detect ranging users with the lowest signal levels possible. 3. Existing ranging algorithms 2. System model The system model is based on the IEEE802.16e-2005 standard [3]. The UL of an OFDMA system with Nt subcarriers is considered. To detect ranging codes at the BS, one approach would be to cross-correlate the received signal with all possible ranging codes in time domain [7]. To reduce the high computational complexity of this process, one can instead auto-correlate the received signal H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 with its delayed replica to exploit the repetition in the ranging code. However, for this approach to work properly, the system needs to extract other non-ranging user signals from the received signal as the ranging users are frequency-multiplexed with those synchronized users. Not only does such a process increase the complexity and delay of the algorithm, but it is also affected by the performance of non-ranging users signal estimator. In addition, the codes used for ranging are modulated in frequency domain and performing the correlation in time domain weakens the auto-correlation/cross-correlation properties of used codes. Another approach to detect ranging codes is to perform the cross-correlation on the frequency-domain signal at the output of the discrete Fourier transform, (DFT) [8]. In this case, a complete OFDMA ranging symbol in the observation window results in a correct ranging code in the frequency domain even if a timing offset exists. The effect of the timing offset is translated into a linear phase shift (also called a phase rotation) in the frequency domain. To estimate the timing offset of the ranging code, the system applies all possible linear phase shifts (corresponding to possible timing offsets) to the signal. Then, the received signal is correlated with all ranging codes. A threshold is set to detect the existence of a ranging code in the current observation window and its timing offset. One advantage of performing the cross-correlation process in frequency domain is that there is no interference from nonranging users to the ranging channel. Thus, the probability of a missed detection or a false alarm – due to multiuser interference – is reduced. Another advantage of this approach is that operating in frequency domain utilizes the auto-correlation/cross-correlation properties of used ranging codes. The main disadvantage of previously mentioned algorithms is the high computational complexity. If the total number of codes is K and the maximum RTD considered has an integer value of smax samples, then K smax cross-correlation operations are needed for every OFDMA symbol. In the wireless IEEE standard 802.16a [10] and 802.16a/b [11] considered in [7] and in [8], respectively, the number of long ranging codes used for initial ranging is only 16. However, for IEEE802.16e [3], the number of ranging codes is 256. These codes are divided into three categories: initial ranging codes, periodic ranging codes, and bandwidth-request codes. Initially, the BS is expected to assign more codes for initial ranging as users within the cell start entering the network. If we assume that 128 codes would be assigned for initial ranging, then for OFDMA systems based on IEEE802.16e-2005, the computational complexity for the ranging process would be eight times as much as the complexity of 802.16a/b. Other ranging algorithms were proposed in the literature (i.e. [12,13]). However, those algorithms are not applicable to the current standard in its present state. Hence, they are not considered in this work. 4. Proposed algorithm In the proposed algorithm, we concentrate more on the trade off between computational complexity and performance of the ranging process so that the algorithm can be realized in practical systems. We choose an observation window with one OFDMA symbol size and apply the algorithm in frequency domain for the advantages of this method that is mentioned earlier in Section 3. However, instead of directly cross-correlating with every possible code and every possible phase offset for every OFDMA symbol, we break the initial ranging process into three main steps. The first step is to find OFDMA symbols containing ranging codes. This step allows the system to find out which symbols it should process further and which ones the system should just drop so that no additional computations are performed on empty OFDMA symbols. Energy detectors, which are discussed in more detail in Section 161 4.1, are used to detect OFDMA symbols containing ranging codes in the ranging channel. In the second step, the algorithm finds how many codes there are within detected OFDMA symbols from step one and determine the timing offsets (or linear phase shifts) for each code. Again, in this step the system further reduces the computational complexity by first finding the timing offset of the codes before performing the cross-correlation with all possible codes for every possible linear phase shift. Section 4.2 discusses the details of this step. The last step in the algorithm is the identification of multiuser codes by cross-correlating detected codes with all possible ranging codes after removal of any timing offsets. This step is covered in Section 4.3. Using the proposed approach, the computational complexity is greatly reduced while the performance is still acceptable as it is shown in Sections 5 and 6. 4.1. Energy detector If there is a ranging opportunity in the next UL frame, the ranging channel will be available through the entire UL subframe duration. The BS samples the received signal and groups it into N t þ N CP samples. The CP is removed and the remaining N t samples are fed to the DFT unit. The ranging channel contains noise and energy from ranging users. The received unsynchronized signal of a ranging user consists of two ranging symbols with no phase discontinuity. The OFDMA symbols processed by the BS can be only one of three possible cases: (a) empty symbols containing only noise, (b) symbols containing incomplete parts of the ranging signal, which cause interference to subcarriers other than the ranging subcarriers and thus interfere with synchronized users, and (c) symbols that are entirely filled with the ranging signal, referred to as symbols with complete ranging signal. We are interested in detecting the third kind, as it contains the required information to detect the user ranging code. For every ranging user signal, at least one symbol with complete ranging signal should be received by the BS since the ranging signal spans two OFDMA symbols. Empty symbols should be ignored since they contain no information. The information included in OFDMA symbols with incomplete ranging signal (i.e. the ranging code, its timing offset, and signal power) can be extracted from the symbols with complete ranging signal. Therefore, a missed detection of symbols with incomplete signal should not affect the performance of the algorithm. In order to detect symbols with a complete ranging signal we use an energy detector in frequency domain. The energy detector measures the energy within the ranging channel. This method has two advantages: (a) since the energy is measured in the frequency domain and since the ranging subcarriers are not adjacent, the likelihood of a pulse of noise triggering the energy detector by mistake is low, (b) the energy is already measured to obtain the noise variance of the channel and can also be used later to measure the signal power of ranging users. Therefore, no additional computational complexity is required for this step. The measured energy in the ranging channel is Eg ¼ L1 X jYðmÞj2 ; ð3Þ n¼0 where m ¼ ir ðnÞ and Y is the N t vector at the output of the DFT unit at the receiver side. After measuring the energy within the ranging channel, a threshold g1 is used to decide if the OFDMA symbol contains a ranging code or not. To find the best value for g1 , the probability of a false alarm P fa and the probability of missed detection Pmd are calculated. The probability of a false alarm is defined as the probability of noise energy in empty OFDMA symbols exceeding g1 . In the same manner, the probability of missed detection is defined as the probability of the energy of an OFDMA symbol not 162 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 n¼0 W 2R ðmÞ þ n¼0 L1 X W 2I ðmÞ; 1 where erfc is the complementary error function. For an OFDMA symbol containing a complete ranging signal, if a user k signal has a timing offset of sk samples,1 then the OFDMA symbol with complete ranging signal contains a copy of the ranging code which is cyclically shifted by sk samples. The OFDMA symbol in the frequency domain has a linear phase shift of 2pnsk =Nt , where n is the subcarrier index. In this case, ð6Þ n¼0 where /m ðsk Þ ¼ 2pmsk =N t . Since BPSK modulation is used, ðkÞ cp ðmÞ ¼ 1. Thus, Eg ¼ L1 n X 1 þ W 2R ðmÞ þ W 2I ðmÞ þ 2cpðkÞ ðmÞ cos½/m ðsk ÞW R ðmÞ n¼0 o þ 2cðkÞ p ðmÞ sin½/m ðsk ÞW I ðmÞ : ð7Þ Using the central limit theorem, the distribution of this energy can be approximated as a normally distributed random variable with mean l2 ¼ L þ LN 0 and variance r22 ¼ LN 20 þ 2LN 0 as shown in Appendix A. Hence, the probability of a missed detection becomes Pmd 0 1 Bg  l C ¼ 1  0:5 erfc@q1ffiffiffiffiffiffiffiffiffiffiffiffi2 A: 2pr22 0.5 0.3 ð5Þ 2 L1  X  ðkÞ  cp ðmÞej/m ðsk Þ þ WðmÞ ; 0.6 0.4 n¼0 Bg  l C Pfa ¼ 0:5 erfc@q1ffiffiffiffiffiffiffiffiffiffiffiffi1 A; 2pr21 Eg ¼ 0.7 ð4Þ where W R ðmÞ and W I ðmÞ are the real and imaginary parts of WðmÞ, respectively. The energy in this case is the sum of 2L samples of the square of normally distributed random variables with zero mean and N 0 =2 variance. Hence, the measured energy can be described as a random variable with Chi-square distribution having a mean l1 ¼ LN0 and variance r21 ¼ LN20 . Using the central limit theorem, if 2L is large enough, the energy distribution can be approximated as a normally distributed variable with the same mean and variance. Based on the standard [3], L or the ranging code length is equal to 144 which is large enough to validate the above approximation. The probability of a false alarm then becomes [14] 0 0.8 0.2 0.1 0 0 5 10 15 20 25 30 Normalized Threshold η1 (2/LN 0 ) 35 40 Fig. 1. P fa and P md for different noise levels. Normalized measured energy jWðmÞj2 ¼ L1 X 0.9 400 300 200 100 0 0 2 Symbols Eg ¼ L1 X 1 Probability exceeding g1 while containing a complete ranging signal. Note that the case of an OFDMA symbol containing incomplete ranging signal is ignored in the calculation of P md as missing this symbol does not affect the algorithm performance. In fact, detecting an incomplete ranging signal can additionally provide correct ranging information if the timing offset is relatively small. If the current OFDMA symbol contains no ranging code, then YðmÞ ¼ WðmÞ, where W is a vector of complex AWGN samples with zero mean and N0 =2 variance. From (3), 4 6 OFDMA symbol 8 S5 S4 S3 S3 S2 S1 0 2 10 12 S5 S4 S2 S1 4 6 Time 8 10 12 Fig. 2. Normalized measured energy on the ranging channel at SNR = 10 dB. users are shown in time domain. Also, the figure shows the normalized measured energy on the ranging channel for every OFDMA symbol and for an SNR of 10 dB. Note that in the above analysis, it is assumed that only one complete ranging signal exists in the received symbol. This assumption is based on worst case scenario, since P md gets even lower if more than one complete signal exists in the received symbol as shown in Fig. 2. 4.2. Timing offset estimation ð8Þ Fig. 1 shows Pfa and P md against normalized g1 (normalized by L and N 0 ) for different signal-to-noise ratio (SNR) levels. Note that P fa does not change as the SNR changes since g1 is normalized and since Pfa depends only on N 0 . In this work, N 0 is assumed to be available for choosing the best value of g1 . This is a valid assumption since the BS needs to estimate the noise level for calculations of different users SNR. An example of a received UL frame with ranging codes is shown in Fig. 2. The received symbols of ranging 1 We only consider timing offset that is integer multiple of the sampling time. The non-integer part of the delay is insignificant to the OFDM system performance and is usually incorporated as part of the communication channel. In the previous step, the system detects OFDMA symbols containing one or more complete ranging signals. The next step is to identify how many codes are in each symbol and estimate the timing offset for each of these codes. Since the proposed algorithm is applied in frequency domain, timing offset would be directly translated into a linear phase shift. In this case, what is estimated is actually the linear phase shift for each code in the current OFDMA symbol. This can be done by cross-correlating the ranging channel of the current OFDMA symbol with all possible codes after applying all possible linear phase shifts to the symbol. The correlator output is followed by a threshold detector to detect different ranging codes and their phase shifts within the current symbol. However, this is computationally complex. We intended to reduce the complexity of this operation by exploiting the fact that timing off- 163 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 sets only affect the phase of the frequency domain signal. Since BPSK modulation is used, the signal ideally should not have an imaginary part. If there are K ranging users within the current OFDMA symbol, each user has a timing offset sk samples, where 0 < sk < smax and smax is the maximum RTD between the SS and the BS within the current cell. Then j/m ðsk Þ cðkÞ þ WðmÞ; p ðmÞe 0.8 Normalized energy level YðmÞ ¼ K1 X 1 ð9Þ k¼0 where m ¼ ir ð0Þ; ir ð1Þ; . . . ; ir ðL  1Þ. As shown in (9), if there were no timing offsets (i.e. sk ¼ 0 for k ¼ 0; 1; . . . ; K  1), the whole energy of the ranging user will be only in the real part of the signal and the imaginary part will contain only noise. By applying all possible linear phase shifts and taking the energy of the real part of the signal, we have Er ðuÞ ¼ L1 X n¼0 ¼ k¼0 i2 ^ cðkÞ p ðmÞ cos½/m ðsk  uÞ þ W R ðmÞ ; ~ r ðuÞ ¼ ½Er ðuÞ  r =r ; E ð10Þ ð11Þ where r ¼ 1 K Er ðuÞ: ð12Þ u¼0 As shown in Appendix B, EfEr ðuÞg   LðK þ 1 þ N0 Þ=2 for u ¼ sk ; LðK þ N0 Þ=2 ð13Þ for u–sk : From (12) and (13), we get r ¼ 0.2 0 −0.2   R YðmÞej/m ðuÞ ^ ^ R ðmÞ ¼ where u ¼ 0; 1; . . . ; smax , WðmÞ ¼ WðmÞej/m ðuÞ and W ^ RfWðmÞg. User timing offset sk , is modeled as a uniformly distributed random variable between 0 and smax . Since m is chosen randomly from the available subcarriers, the phase ½2pmðsk  uÞ=N t  can be approximated as a uniformly distributed random variable ^ and W have the same distribution, between p and p. Note that W average and variance. For sk ¼ u, cos½/m ðsk  uÞ ¼ 1 resulting in a peak in the measured energy Er ðuÞ. Therefore, a threshold g2 is set to detect those peaks and obtain an estimate of sk for all values of k. The measured real energy is normalized with respect to the average real energy of the symbol r such that K1 X 0.4 2 " L1 X K1 X n¼0 0.6 KEfEr ðuÞju ¼ sk g þ ðsmax  KÞEfEr ðuÞju–sk g smax ð14Þ The above equation is valid given that K < smax , which is a reasonable assumption since – depending on the cell radius – smax can go up to N t =2, while the number of ranging users within the same OFDMA symbol is usually much lower than this value. Fig. 3 shows e r ðuÞ j u–sk g for different values of K and e r ðuÞ j u ¼ sk g and Ef E Ef E for SNR = 10 dB. For simulated results, we use the system setup presented in Section 6.1. Assuming all ranging users are received with equal power, the signal-to-interference-plus noise ratio (SINR) levels for a given number of ranging users per OFDMA symbol and for an SNR level of 10 dB are presented in Table 1, where SINR is defined as the ratio of the power of one user signal to the power of the remaining users signals plus noise power. Fig. 3 shows that as the number of ranging users increases, the probability of a missed detection increases since the difference between the two means decreases. In addition, the variance of the measured energy also increases with the number of ranging users, since they act as interference noise for u–sk as shown in (10). Note that since K is 2 3 4 5 6 7 8 Ranging users per OFDMA symbol 9 10 e r ðuÞ j u ¼ sk g and Ef E e r ðuÞ j u–sk g for different numbers of ranging users. Fig. 3. Ef E Table 1 SINR vs number of users per OFDMA symbol. Number SINR of users (dB) Number SINR of users (dB) 1 2 3 4 5 6 7 8 9 10 10.0 0.4 3.2 4.9 6.1 7.1 7.9 8.5 9.1 9.6 usually very small compared to smax , then from (14), e r ðuÞ j u–sk g  0, as shown in Fig. 3. r  EfEr ðuÞ j u–sk g and Ef E 4.3. Code detector In the previous two steps of the proposed algorithm, OFDMA symbols containing ranging codes are detected and their timing offsets are estimated. The last stage is to identify which codes are transmitted out of the available ranging codes, P, where P is the number of codes assigned by the BS for initial ranging. In this step, the linear phase shift corresponding to each detected ranging code is removed and then a cross-correlation with all possible ranging codes is performed. The correlator output is EcðvÞ ðiÞ ¼ : 1 L1 X   R YðmÞej/m ðs^v Þ cðiÞ p ðnÞ; ð15Þ n¼0 ^v is the estimated timing offset of the where i ¼ 0; 1; . . . ; P  1 and s ðvÞ vth user. The vector c is calculated for every detected ranging ^  1, and K ^ is the total number of decode, v, where v ¼ 0; 1; . . . ; K ðvÞ tected ranging codes in the second step. c is then compared to a threshold, g3 , to identify which ranging code is within the current OFDMA symbol. g3 is normalized by the root mean square value ðvÞ of c . Since this step should not be reached unless at least one ranging code exists in the current OFDMA symbol, the code with maximum correlation could just be chosen for every value of v. However, a threshold is used so that the algorithm can detect multiple codes with the same timing offset which are interpreted as a ^ is upsingle code in the second step of the algorithm. In this case, K dated to reflect the increase in the number of detected ranging codes. In addition, if two users happen to use the same ranging code and both their signals are received in the same symbol but with different timing offsets, the system declares a collision and both codes are dropped. The corresponding users then have to retry in the next available ranging opportunity. 164 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168  Algorithm 2: 5. Computational complexity For practical applications, the computational complexity of an algorithm is important. In this section, we evaluate the complexity of the proposed algorithm and compare it to other existing algorithms. The proposed algorithm is compared with the proposed algorithms in [7] and [8], which are referred to as Algorithms 1 and 2, respectively. For Algorithm 1, an observation window with the OFDMA symbol size is used. A bank of correlators, equal to the number of available ranging codes P, is used to separate ranging codes in the received signal. If the maximum possible delay is smax , then ðsmax þ 1ÞP cross-correlation operations are performed for every OFDMA symbol with ranging opportunity. Assuming the current UL consists of N UL OFDMA symbols, where N UL is required by the standard to be an integer multiple of 3, then the total number of correlation operations performed would be N UL ðsmax þ 1ÞP. The same number applies for threshold comparison operations. Algorithm 2, on the other hand, performs the cross-correlation in frequency domain. Thus, to perform the cross-correlation at every possible timing offset, a linear phase shift of /m ðuÞ where u ¼ 0; 1; . . . ; smax , is applied to the frequency domain signal prior to the cross-correlators bank. As a result, Algorithm 2 has an addition of smax þ 1 linear phase shifts added to its computational complexity. However, since the correlation is performed in the frequency domain, and since BPSK modulation is used, the correlations are done on real-signals unlike Algorithm 1 which has to perform complex-signal correlations. In the proposed algorithm, the energy of the ranging channel for every OFDMA symbol is calculated. As a result, N UL energy calculation operations and threshold comparisons are needed. The energy calculation is already performed for N 0 estimation. Assuming that every ranging symbol triggers the energy detector over two OFDMA symbols, then at most 2K OFDMA symbols will reach the next stage of the algorithm, where K is the total number of ranging users within the current UL frame. Of course, if one or more codes collide, then less OFDMA symbols will reach the next stage. In the second stage, all possible linear phase shifts are applied to the OFDMA symbol (i.e. from 0 to smax ). The energy of the real part of the signal is measured and compared to a threshold. Thus, ð1 þ smax Þ linear phase shifts, real energy calculation, and comparison operations are performed. A maximum of K ranging codes, within 2K ranging OFDMA symbols, will reach the third and last stage. A cross-correlation with all possible codes is performed. As a result, 2KP correlations and comparison operations (CMPs) are performed at this stage. Table 2 shows the number of additions (ADDs) and multiplications (MULs) needed for each operation. Using Table 2, the computational complexity of the algorithms under investigation is  Algorithm 1: NUL Pðsmax þ 1Þð4L  2ÞADD þ 4LNUL Pðsmax þ 1ÞMUL þ NUL Pðsmax þ 1ÞCMP: ðsmax þ 1Þð2L þ NUL PL  N UL PÞADD þ Lðsmax þ 1Þð4 þ N UL PÞMUL þ NUL Pðsmax þ 1ÞCMP:  Proposed Algorithm: ½NUL ð4L  2Þ þ 2Kðsmax þ 1Þð3L  1Þ þ 2KPðL  1ÞADD þ ½2LðKP þ 2Þ þ 10KLðsmax þ 1ÞMUL þ ½NUL þ 2Kðsmax þ P þ 1ÞCMP: Finally, the complexity of all three algorithms is calculated for a practical system.2 We assume NUL ¼ 12 OFDMA symbols, P ¼ 256 codes(maximum), L ¼ 144 bits(standard [3]), and smax ¼ 512 samples for Nt ¼ 1024. The number of cycles needed by each algorithm is as follows: Algorithm 1 : 3:629  109 cycles; Algorithm 2 : 9:088  108 cycles; Proposed Algorithm : 2:963  106 cycles for K ¼ 1; 5:909  107 cycles for K ¼ 20: The difference in computational complexity is evident. The complexity of the proposed algorithm is a few orders of magnitude lower than the complexity of Algorithms 1 and 2. While both of Algorithms 1 and 2 maintain fixed complexity regardless of the number of ranging users K, the proposed algorithm can update to K, which gives it the lowest limit in computational complexity. Note that the above results are optimistic as we assume there are no false alarms in the first and second stages of the proposed algorithm. A complexity comparison for a typical WiMAX system operating in a practical wireless environment is presented in the following section. 6. Simulation results 6.1. System setup An OFDMA system model based on [3] is used with the following parameters: N t ¼ 1024, N CP ¼ 128 samples, N UL ¼ 12 OFDMA symbols, L ¼ 144 bits. The system is assumed to be operating at 2 GHz with a bandwidth of 10 MHz and a sampling frequency of 11.2 MHz. The maximum RTD considered is 45.71 lS (smax ¼ 512 samples) allowing for a cell radius of 6.86 Km. The total number of ranging codes is 256 codes. In the considered system, all codes are assigned to initial ranging, i.e. P ¼ 256. The ranging channel is made up of six subchannels and spanning 144 subcarriers per OFDMA symbol. Ranging users choose two consecutive symbols randomly to send their ranging code during the UL frame with equal probability. In each simulation, 10,000 UL frames or 120,000 OFDMA symbols are used to evaluate the system performance. 6.2. Channel model Table 2 Proposed algorithm’s computational complexity. Operation Correlation, Correlation, Energy Cal., Energy Cal., Phase shift complex real complex real ADD MUL 2ðL  1Þ þ 2L L1 2ðL  1Þ þ 2L L1 2L 4L L 4L L 4L The performance of the proposed ranging algorithm is evaluated in both an AWGN channel and an outdoor dispersive channel. For the dispersive channel, we use one of the channel environments defined by ETSI for the evaluation of the UMTS radio inter- 2 Using CPU cycle counts based on XilinX DSP48 slice. 165 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 face proposals [15]. The time-varying channel impulse response for these models can be described by hðs; tÞ ¼ X ai ðtÞdðs  si Þ: 1 ð16Þ i Table 3 Characteristics of the ETSI ‘‘Vehicular A” channel environment. Tap Relative delay (nS) Average power (dB) 1 2 3 4 5 6 0 310 7104 1090 1730 2510 0.0 1.0 9.0 10.0 15.0 20.0 1 (b) AWGN channel, with timing error 2 Imaginary Imaginary (a) AWGN channel, no timing offset 2 0 −1 1 0 −1 −2 0 2 −2 0 2 Real Real (c) Dispersive channel, no timing offset (d) Dispersive channel, with timing error 2 2 1 1 Imaginary Imaginary −2 −2 0 −1 −2 −2 0 −1 0 Real 2 −2 −2 0 Real 2 Fig. 4. The effect of synchronization error on the constellation points of a typical BPSK-modulated OFDM symbol received over AWGN and dispersive channels. 0.8 Probability This equation defines the impulse response of a tapped-delay channel with every tap i having a delay of si and gain of ai ðtÞ. In this paper, we consider the ‘‘Vehicular A” channel environment [15]. The taps’ relative delays and average powers are shown in Table 3. The channel taps ai ðtÞ are complex independent Rayleigh-fading variables. Different user channels are assumed independent and the channels are also independent between UL frames. The multipath fading channel is expected to degrade the performance of the ranging algorithms under investigation. For the proposed algorithm, only the second and third steps are significantly affected by the multipath channel. The impact of channel models on the third step, where the signal is correlated with all possible codes, has already been discussed in [8] where Algorithm 2 is presented. Therefore, we are mainly interested in examining the effect of a dispersive channel on the second step where we detect ranging codes within the OFDMA symbol and estimate their timing offsets. As discussed earlier, the subcarriers of an OFDM symbol with a timing error exhibit a linear phase shift. Therefore, by applying the appropriate linear phase shift to the received symbols, this effect is compensated. Fig. 4 shows that effect on the constellation points of a typical BPSK-modulated OFDM symbol received over AWGN channel with SNR ¼ 10 dB and over the dispersive channel referred to earlier. As seen in the figure, the linear phase shift causes the symbol energy to be distributed equally between the real and imaginary parts of the signal. On the other hand, at the correct tim- 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 e r ðuÞ j> g j u ¼ sk g and Prfj E e r ðuÞ j> g j u–sk g for different values of Fig. 5. Prfj E 2 2 g2 over non-dispersive and dispersive channels. ing offset, the energy is more concentrated into a single axis. The effect of the dispersive channel is that at the correct timing offset, the signal is more noisy and there is also a random gain and phase shift. However, the correct timing offset is still detectable compared to the same signal with wrong timing offset. The real part of the signal contains 50% of the total symbol energy when a timing offset is present regardless of the channel model . In the above example and with no timing offset, the real part of the signal holds 99% and 69% of the total symbol energy over AWGN and multipath fading channels, respectively. To further illustrate this effect, computer simulations were used to calculate the probabilities e r ðuÞ j> g j u–sk g for different e r ðuÞ j> g j u ¼ sk g and Prfj E Prfj E 2 2 values of g2 over non-dispersive and dispersive channels. The ree r ðuÞ j as to avoid sults are shown in Fig. 5. Note that we consider j E negative values that can result from phase shifts introduced by the e r ðuÞ j> g j u ¼ sk g is significantly channel. As expected, the Prfj E 2 e r ðuÞ j> g j u–sk g for a given threshold level higher than the Prfj E 2 e r ðuÞ j> g2 . The figure shows that for both channels, Prfj E g2 j u–sk g exhibits the same behavior where it decays at a fast rate e r ðuÞ j> g j u ¼ sk g on the other hand, deas g2 increases. Prfj E 2 creases as g2 increases only for dispersive channels. 6.3. Proposed algorithm performance The system performance is evaluated at an SNR = 10 dB over both AWGN and ‘‘Vehicular A” dispersive channels. The proposed algorithm performance is compared to that of Algorithm 2. The performance is investigated for different numbers of ranging users per UL frame. To be able to fairly evaluate the system performance for different numbers of ranging users, it is assumed that all ranging signals are received with equal powers. However, both the proposed algorithm and Algorithm 2 can operate when different users have different power levels. For the results presented in this paper, the SINR corresponding to a given number of users per OFDMA symbol is shown in Table 1. Based on our analysis and the simulation results shown in Figs. 1 and 5, the thresholds (g1 ; g2 ; g3 ) for the proposed algorithm are chosen to be (3, 0.2, 6) for AWGN channels and (3, 0.1, 6) for dispersive channels. Figs. 6 and 7 show the probability of missed detection and the probability of false alarm, respectively, for both the proposed algorithm and Algorithm 2, and for different numbers of ranging users per UL frame. Over AWGN channels, both algorithm performances are almost identical with zero probability of false alarm. Over dispersive channels, the proposed algorithm suffers from a higher probability of missed 166 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 0.35 10 Proposed alg., dispersive Alg. 2, dispersive Proposed alg., AWGN Alg. 2, AWGN 8 7 0.25 Standard deviation Probability of missed detection 0.3 Proposed alg., dispersive Alg. 2, dispersive Proposed alg., AWGN Alg. 2, AWGN 9 0.2 0.15 0.1 6 5 4 3 2 1 0.05 0 0 1 2 3 4 5 6 7 8 Number of ranging users per UL frame 9 −1 10 1 2 Fig. 6. Probability of missed detection. Probability of false alarm 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 1 2 3 4 5 6 7 8 Number of ranging users per UL frame 10 9 10 Fig. 7. Probability of false alarm. detection compared to Algorithm 2. The difference between the two algorithms increases as the number of users increases. On the other hand, the proposed algorithm shows a much lower probability of false alarm than Algorithm 2 especially for low number of ranging users. Even with up to eight ranging users per UL frame, the proposed algorithm is able to detect more than 75% of received codes with a false alarm probability less than 0.2%. An important measure of the ranging algorithm quality for WiMAX systems is the timing estimation accuracy. For dispersive channels, the timing error definition is ambiguous [5]. In this paper, we consider the timing error relative to the channel tap with maximum average power, or first tap in the considered dispersive channel. The standard deviation for the timing estimator errors for both the proposed algorithm and Algorithm 2 is shown in Fig. 8. Both algorithms show zero timing errors for AWGN channels. Over dispersive channels, the proposed algorithm timing estimator shows a higher error standard deviation than that of Algorithm 2. Overall, both algorithms show an accurate timing estimation capability with an average standard deviation of 5 and 3 for the proposed algorithm and Algorithm 2, respectively. A more important measure of ranging accuracy is the probability of the timing error falling outside a given interval. For example, if we assume the tim- ing errors to be a normally distributed random variable with zero mean [5], the probability of the timing error to exceed 32 samples (25% of the CP) is less than 1% for both algorithms. Next, we consider the computational complexity of the proposed algorithm. For both algorithms under consideration, correlating the received signal with all possible ranging codes constitutes the majority of the algorithm computations. As such, the number of times this function is called by the ranging algorithm is used as a measure of how complex the algorithms are. In this regard, Algorithm 2 has a fixed computation complexity regardless of the number of ranging opportunities or the number of ranging users within the received UL frame as shown in Section 5. The reduction of computational complexity gained by using the proposed algorithm over Algorithm 2 is evaluated and the results are shown in Fig. 9. The computational complexity are normalized by the number of detected codes ð1  P md Þ to take into consideration the different probabilities of missed detection between both methods. The proposed algorithm reduces the computational complexity by over 98% over AWGN channels. Over dispersive channels, the complexity reduction ranges from 96% to 80% depending on the number of ranging users per UL frame. 100% Dispersive AWGN 98% Computational complexity reduction Proposed alg., dispersive Alg. 2, dispersive Proposed alg., AWGN Alg. 2, AWGN 0.04 9 Fig. 8. Standard deviation of timing errors. 0.05 0.045 3 4 5 6 7 8 Number of ranging users per UL frame 96% 94% 92% 90% 88% 86% 84% 82% 80% 1 2 3 4 5 6 7 8 9 Number of ranging users per UL frame 10 Fig. 9. Computational complexity reduction using the proposed algorithm. 167 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 7. Conclusions 0.35 Probability of missed detection, Nt=512 Probability of missed detection, Nt=1024 0.3 Probability of false alarm, N =512 t Probability of false alarm, Nt=1024 Probability 0.25 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 Number of ranging users per UL frame 9 10 Fig. 10. Probabilities of missed detection and false alarm for N t ¼ 1024 and N t ¼ 512. A novel algorithm for OFDMA initial ranging process based on IEEE802.16e-2005 standard is proposed. The proposed algorithm performs multiuser code detection and timing offset estimation for ranging users. The algorithm is divided into three stages. In the first stage, the system detects OFDMA symbols carrying ranging users. The second stage is responsible for detection of codes and estimation of timing offsets for each user within current OFDMA symbol. Finally, the last stage identifies user ranging codes. The proposed algorithm performance is evaluated for both AWGN channels and dispersive channels. A complexity comparison between the proposed algorithm and other existing algorithms is carried out as well. The results show that proposed algorithm reduces the computational complexity by 80–96% depending on the number of ranging users while maintaining the timing error standard deviation under 5% of the guard interval. Simulation results showed that the proposed algorithm can perform well with as high as 10 users per ranging channel in a given UL frame. Hence, it is believed that the proposed algorithm can be realized in practical mobile WiMAX BSs. Appendix A 10 9 Proposed alg., Nt=1024 8 Proposed alg., N =512 From (7), the expected value of Eg is Alg. 2, Nt=1024 " t Alg. 2, Nt=512 7 Standard deviation E½Eg  ¼ E L þ 6 þ2 5 W 2R ðmÞ þ n¼0 L1 X L1 X W 2I ðmÞ n¼0 cðkÞ p ðmÞ cos½/m ðsk ÞW R ðmÞ n¼0 4 þ2 3 L1 X cðkÞ p ðmÞ sin½/m ð k ÞW I ðmÞ s n¼0 2 # : ðA:1Þ Note that 1 E½cðkÞ p ðmÞ ¼ 0; 0 —1 L1 X ðA:2Þ E½W R ðmÞ ¼ E½W I ðmÞ ¼ 0; 1 2 3 4 5 6 7 8 Number of ranging users per UL frame 9 10 Fig. 11. Standard deviation of timing errors for N t ¼ 1024 and N t ¼ 512. and since m values are randomly chosen between 0 and Nt  1, /m ðsk Þ can be approximated as a uniformly distributed random variable between p and p. Thus E½cos½/m ðsk Þ ¼ E½sin½/m ðsk Þ ¼ 0: Finally, another set of system parameters from the WiMAX system profiles is considered to verify the performance of the proposed algorithm. The new set of system parameters are: N t ¼ 512, N CP ¼ 32 samples. In accordance to the standard, the signal bandwidth, in this case, is 5 MHz and the sampling frequency is 5.6 MHz. For fairness of comparison, the maximum RTD, the number of OFDMA symbols per UL frame, and the number of frames used to evaluate the simulation results remain constant. The probabilities of missed detection and false alarm for the first and second sets of system parameters are compared in Fig. 10. In both system profiles, the performances are almost identical. This is due to the fact that WiMAX standard maintains fixed subcarrier spacing regardless of the signal bandwidth or FFT size. As a result, the ranging channel, which consists of a fixed number of subcarriers, has a fixed bandwidth for different signal bandwidths. In Fig. 11, the standard deviations of timing errors for both cases as well as for both considered ranging algorithms are compared. Note that as the bandwidth is reduced, the sampling time is increased which leads to a reduction in timing errors (in samples). In this case, the timing estimation performances of the proposed algorithm and Algorithm 2 are almost equal. ðA:3Þ Therefore, the expected value of the last two terms in (A.1) goes to zero for large values of L. Then " # " # L1 L1 X X 2 2 E½Eg  ¼ L þ E W R ðmÞ þ E W I ðmÞ ¼ L þ LN0 : n¼0 ðA:4Þ n¼0 Next, we calculate E½E2g  from (7). Note that 2 2 E4 2 E L1 X E4 L1 X n¼0 cðkÞ p ðmÞ cos ½/m ðsk ÞW R ðmÞ n¼0 " !2 3 5 ¼ 1 LN2 þ 1 L2 N 2 ; 0 2 0 4 3 !2 W 2R ðmÞ L1 X n¼0 W 2R ðmÞ ! L1 X n¼0 !# W 2I ðmÞ ðA:5Þ 5 ¼ LN 0 ; ðA:6Þ ¼ ðA:7Þ 1 2 2 L N0 : 2 Then E½E2g  ¼ L2 þ LN 20 þ L2 N20 þ 2LN0 þ 2L2 N0 : ðA:8Þ 168 H.A. Mahmoud et al. / Computer Communications 32 (2009) 159–168 Appendix B References From (10) Er ðuÞ ¼ ( L1 K1 X X n¼0 þ cðkÞ p ðmÞ cos½/m ðsk  uÞ k¼0 L1 X n¼0 ¼ )2 " ^ R ðmÞ  2W K 1 X þ L1 X cðkÞ p ðmÞ cos½/m ð k s  uÞ k¼0 L1 X K1 X cos2 ½/m ðsk  uÞ þ n¼0 k¼0 ^ 2 ðmÞ W R n¼0 # L1 X K1 X n¼0 k¼0 K1 n o X ðvÞ  cðkÞ p ðmÞc p ðmÞ  cos½/m ðsk  uÞ cos½/m ðsv  uÞ v¼0 v–k þ L1 X n¼0 þ L1 X n¼0 ^ 2 ðmÞ W R ( ^ R ðmÞ cos½/ ðuÞ  2W m K1 X cðkÞ p ðmÞ cos½/m ð k ) s  uÞ : k¼0 ðB:1Þ The expected value of the second and last terms in (B.1) is zero. Therefore E½Er ðuÞ ¼ E E½Er ðuÞ ¼ " L1 X K1 X ( n¼0 k¼0 1 LK 2 1 LK 2 # cos2 ½/m ðsk  uÞ þ E " L1 X n¼0 # ^ 2 ðmÞ : W R ðB:2Þ þ 12 L þ 12 LN0 for u ¼ sk for only one value of k; þ 12 LN 0 for u–sk for all values of k: ðB:3Þ The assumption that u ¼ sk for only one value of k implies that in the same symbol and for a give timing offset u, only one code exist. 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