Integration Interval Determination Algorithms for ... - Semantic Scholar

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 2010

Integration Interval Determination Algorithms for BER Minimization in UWB Transmitted Reference Pulse Cluster Systems Li Jin, Xiaodai Dong, Senior Member, IEEE, and Zhonghua Liang

Abstract—A recently proposed transmitted reference pulse cluster (TRPC) structure contains compactly spaced reference and data pulses, and enables a low complexity, robust and practical auto-correlation detector to be used at the receiver. Previous research indicated that the integration interval of the auto-correlation detector is critical to the performance of TRPC. Therefore, in this paper, three practical data-aided algorithms are introduced to determine the integration interval of the TRPC structure: the conventional threshold-crossing concept, the new bit error rate (BER) minimization based approach, and the new hybrid scheme that combines threshold-crossing and the BER minimization concepts. The performances of the three schemes are extensively evaluated by simulation. Results show that, the BER minimization based approach and the hybrid scheme demonstrate around 2 dB performance gain over the thresholdcrossing scheme in IEEE 802.15.4a channels. Moreover, the hybrid scheme yields close performance to the BER minimization based scheme with much reduced complexity. Index Terms—Ultra-wideband (UWB), transmitted reference pulse cluster (TRPC), integration interval determination, bit error rate (BER) minimization.

proposed in [14] to optimize the integration intervals (both starting and stop time) for conventional TR systems based on BER minimization. It should be pointed out that all these work were based on the conventional inter-frame interference and inter-pulse interference (IPI) free TR structures which require impractically long delay lines in the receivers. So far no literature has dealt with the integration interval (both beginning and end of the interval) determination of the TRPC structure which has its own distinctive characteristics not present in the TR structures. In this paper, practical algorithms are developed to determine the integration interval for the TRPC structure. Our study is based on the assumption that there is no intersymbol interference (ISI), which is a reasonable case for low data rate applications. Using the equivalent system model presented in [15], one can consider the ISI issue for TRPC systems. II. S YSTEM M ODEL OF TRPC

T

I. I NTRODUCTION

RANSMITTED reference (TR) technique is a topic that has been studied extensively for ultra-wideband (UWB) communications due to its robust performance and simple implementation. However, one major obstacle to the development of a conventional TR system is the need to implement long wideband delay lines, which is not practically feasible. In [1], a new TR pulse cluster (TRPC) system was proposed, where compactly and uniformly placed multiple pulses forming a pulse cluster are used and only very short delay lines are required. Moreover, this new TRPC structure achieves 2−3 dB and 1.3 − 2 dB power gain over conventional TR scheme and non-coherent pulse position modulation (PPM), respectively [1]. Previous study showed that in this TRPC system, the integration interval of the auto-correlation detector is critical to the bit error rate (BER) performance [1]. Integration length determination in different conventional TR type detectors has been studied in many publications [2]– [13], where the integration stop time but not the starting time was investigated. Recently, a practical algorithm was Manuscript received March 20, 2009; revised September 4, 2009, January 5, 2010, and April 15, 2010; accepted May 21, 2010. The associate editor coordinating the review of this letter and approving it for publication was M. Torlak. The material in this paper was presented in part at the IEEE Vehicular Technology Conference (VTC-Fall’08), Calgary, Canada, Sep. 2008. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada through grant 349722-07. The authors are with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC V8W 3P6, Canada (e-mail: {jinli, xdong, zhliang}@ece.uvic.ca). Digital Object Identifier 10.1109/TWC.2010.061810.090415

For simplicity, we first consider the single-user case. A TRPC signal can be written as [1] √ ∞ 𝐸𝑏 ∑ 𝑠𝑏 (𝑡 − 𝑚𝑇𝑠 ) (1) 𝑠˜(𝑡) = 2𝑁𝑓 𝑚=−∞ 𝑚

∑𝑁𝑓 −1 𝑔(𝑡 − 2𝑖𝑇𝑑) + 𝑏𝑚 𝑔(𝑡 − (2𝑖 + 1)𝑇𝑑 ), where 𝑠𝑏𝑚 (𝑡) = 𝑖=0 𝐸𝑏 is the average energy per bit, 𝑁𝑓 is the number of repeated dual pulse pairs in one cluster, 𝑇𝑠 is the symbol duration determined by the bit rate, 𝑔(𝑡) is the transmitter pulse with duration 𝑇𝑝 , and 𝑏𝑚 ∈ {+1, −1} is the 𝑚-th bipolar information bit. The UWB channel described by IEEE∑ 802.15.4a channel 𝐾−1 models can be generalized as [16] ℎ(𝑡) = 𝑘=0 𝛼𝑘 𝛿(𝑡 − 𝜏𝑘 ), where 𝛼𝑘 and 𝜏𝑘 are the complex amplitude and delay of the 𝑘-th multipath component (MPC), 𝐾 is the number of MPCs. After the transmitted signal passes through the UWB channel and the receiver lowpass filter, the received signal can be written as 𝑟(𝑡)

=

𝐾−1 ∑

𝛼𝑘 𝑠˜(𝑡 − 𝜏𝑘 ) + 𝑛(𝑡)

𝑘=0

√ =

𝐸𝑏 2𝑁𝑓

∞ ∑ 𝑚=−∞

𝑞𝑏𝑚 (𝑡 − 𝑚𝑇𝑠 ) + 𝑛(𝑡)

(2)

∑𝐾−1 ∑𝑁𝑓 −1 𝑔(𝑡− 2𝑖𝑇𝑑 − 𝜏𝑘 )+ 𝑏𝑚 𝑔(𝑡− where 𝑞𝑏𝑚 (𝑡) = 𝑘=0 𝛼𝑘 𝑖=0 (2𝑖 + 1)𝑇𝑑 − 𝜏𝑘 ), and 𝑛(𝑡) is the filtered additive complex Gaussian noise. The auto-correlation function of 𝑛(𝑡) is given by 𝑅𝑛 (𝜏 ) = 𝐸[𝑛∗ (𝑡)𝑛(𝑡 + 𝜏 )] = 𝑁0 𝑅𝑡𝑟 (𝜏 ), where 𝑅𝑡𝑟 (𝜏 ) = 𝐵sinc(𝐵𝜏 ), 𝐵/2 is the lowpass filter bandwidth and 𝑁0 is the

c 2010 IEEE 1536-1276/10$25.00 ⃝


power spectral density of the complex white Gaussian noise. The receiver performs auto-correlation on the received signal and its 𝑇𝑑 delayed version. The decision variable (DV) for the 𝑚-th bit is then given by ∫ 𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 ) =

𝑚𝑇𝑠 +𝑡𝑒

𝑚𝑇𝑠 +𝑡𝑏

𝑟(𝑡)𝑟∗ (𝑡 − 𝑇𝑑 )𝑑𝑡,

(3)

where 𝑡𝑏 and 𝑡𝑒 are the beginning and end of the integration interval that need to be determined in the receiver to achieve minimum BER. The receiver makes a decision on “+1” if Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )} > 0, and “-1” if Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )} < 0. To obtain the BER expression for TRPC, the decision variable can be approximated as a Gaussian RV and the mean of Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )} is given by [1] 𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )

= E [Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )}] ∫ 𝑡𝑒 𝐸𝑏 ≈ Re{𝑞𝑏𝑚 (𝑡)𝑞𝑏∗𝑚 (𝑡 − 𝑇𝑑 )}𝑑𝑡. (4) 2𝑁𝑓 𝑡𝑏

Following the analysis in [1], Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )} can be simplified to Var[Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )}] =

2 𝜎23 (𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )

≈2𝑁0 ⋅ ∣𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )∣ + (𝑡𝑒 −

the

variance

+ 𝜎42 (𝑡𝑏 , 𝑡𝑒 ) 𝑡𝑏 ) ⋅ 𝑌 ⋅ 𝑁02 .

of

(5)

The conditional BER on bit 𝑏𝑚 is then given by ( 𝑃 (𝑒∣h, 𝑏𝑚 , 𝑡𝑏 , 𝑡𝑒 )≈𝑄 ∣𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )∣ 1

)

. ⋅√ 2𝑁0 ⋅ ∣𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑏𝑚 )∣ + (𝑡𝑒 − 𝑡𝑏 ) ⋅ 𝑌 ⋅ 𝑁02

(6)

In the multi-access scenarios, both time division (TDMA type) and amplitude spreading (CDMA type) approaches have been studied in [17] and TDMA is found to be more suitable for TRPC. Due to the dispersive nature of the UWB channel, the received signal has a much longer duration for a single UWB pulse transmitted. When multiple pulses must be used for one bit to meet the FCC spectral mask, generally closely placing these ultra-narrow pulses into a compact cluster that occupies only a small duration can be viewed as a single composite pulse and will reduce the interference among users in comparison with scattering each user’s pulses into the whole bit duration. According to the anaysis in [17, Eqs. (3)–(6)], multi-access interference (MAI) will be removed completely for TDMA signaling, provided that 𝑇𝑢 ⩾ 2𝑁𝑓 𝑇𝑑 + 𝜏𝑚𝑎𝑥 where 𝑇𝑢 denotes the time slot occupied by each user and 𝜏𝑚𝑎𝑥 means the maximum delay spread of the channel 1 . Hence, in the MAI-free case, (6) can also be used as the BER expression for the desired user. Due to the limited space, hereafter, we only consider the single-user case. Our analysis in the single-user case can be easily extended to the the multiaccess scenarios. 1 Such

an assumption is equivalent to the ISI-free condition 𝑇𝑠 ⩾ 2𝑁𝑟 𝑇𝑑 + 𝜏𝑚𝑎𝑥 for the single-user case. Therefore, in the multi-access scenarios, the date rate for each user is reduced while the sum rate is equal to the single user case.

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III. I NTEGRATION I NTERVAL D ETERMINATION OF THE TRPC S YSTEM A. Determine the dominant data 𝑑 for the target channel As discussed in [1], due to the existence of IPI, the conditional BERs by transmitting +1 and -1 data are different in a TRPC system. If we name the bit inducing most errors as the dominant data 𝑑, it is apparently the one whose decision variable has a smaller absolute mean. By optimizing the integration region for the dominant data, it is expected the overall BER performance will be improved. The criterion in determining the dominant data 𝑑 can be simplified as } { (7) 𝑑 = −sgn 𝜇(𝑡𝑏 , 𝑡𝑒 ∣ + 1) + 𝜇(𝑡𝑏 , 𝑡𝑒 ∣ − 1) . By substituting (4) into (7) and integrating over the entire symbol duration [0, 𝑇𝑠 ], since the interval [𝑡𝑏 , 𝑡𝑒 ] is not known yet, we have { { ∫ 𝑇𝑠 ∗ 𝑞+1 (𝑡)𝑞+1 (𝑡 − 𝑇𝑑 ) 𝑑 = −sgn Re 0 }} ∗ + 𝑞−1 (𝑡)𝑞−1 (𝑡 − 𝑇𝑑 )𝑑𝑡 . (8) To get a simplified to (8), we first define a signal ∑𝑁alternative 𝑓 −1 structure 𝑥(𝑡) = 𝑖=0 𝑔(𝑡 − 2𝑖𝑇𝑑), which can be viewed as the TRPC with only ∑ reference pulses. Furthermore, we define 𝑁𝑓 −1 ∑𝐾−1 𝑦(𝑡) = 𝑥(𝑡)∗ℎ(𝑡) = 𝑖=0 𝑘=0 𝛼𝑘 𝑔(𝑡−2𝑖𝑇𝑑−𝜏𝑘 ). Hence, the integral in (8) is written as ∫ 𝑇𝑠 ∗ ∗ 𝑞+1 (𝑡)𝑞+1 (𝑡 − 𝑇𝑑 ) + 𝑞−1 (𝑡)𝑞−1 (𝑡 − 𝑇𝑑 )𝑑𝑡 0 ∫ 𝑇𝑠 =4 𝑦(𝑡)𝑦 ∗ (𝑡 − 𝑇𝑑 )𝑑𝑡. (9) 0

Therefore, the dominant data can be determined as { { ∫ 𝑇𝑠 }} 𝑑 = −sgn Re 𝑦(𝑡)𝑦 ∗ (𝑡 − 𝑇𝑑 )𝑑𝑡 .

(10)

0

Since 𝑦(𝑡) in (10) is noise free and not obtainable, we can only estimate 𝑦(𝑡) from the received signal through noise averaging. The training sequence is thus designed as √ 𝑁1 −1 𝐸𝑏 ∑ 𝑥(𝑡 − 𝑚𝑇𝑠 ) (11) 𝑓1 (𝑡) = 2𝑁𝑓 𝑚=0 where 𝑁1 is the training length. After the training sequence passes through the multipath channel, the received signal is written as √ 𝑁1 −1 𝐸𝑏 ∑ 𝑟1 (𝑡) = 𝑦(𝑡 − 𝑚𝑇𝑠 ) + 𝑛(𝑡). (12) 2𝑁𝑓 𝑚=0 The receiver performs the integration-and-dump (I&D) operation every 𝑇𝑠 seconds and obtains { ∫ (𝑚+1)𝑇𝑠 } 𝜒𝑚 = Re 𝑟1 (𝑡)𝑟1∗ (𝑡 − 𝑇𝑑 )𝑑𝑡 , 𝑚

𝑚𝑇𝑠

= 0, 1, . . . , 𝑁1 − 1.

(13)

Finally the dominant data can be determined as 1 −1 1 −1 } { 𝑁∑ } { 1 𝑁∑ 𝑑 = −sgn 𝜒𝑚 = −sgn 𝜒𝑚 . 𝑁1 𝑚=0 𝑚=0

(14)

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For the transmitter to know the dominant data, either the receiver feedbacks this information to the transmitter, or the receiver (instead of the transmitter) sends the training sequence to let the transmitter determine the dominant data directly. The latter approach relies on the channel reciprocity property. B. Determine the integration interval To determine the integration interval for the target channel, another training sequence 𝑓2 (𝑡) adopting the TRPC structure with dominant data 𝑑 is transmitted. That is, √ 𝑁2 −1 𝐸𝑏 ∑ 𝑓2 (𝑡) = 𝑠𝑑 (𝑡 − 𝑚𝑇𝑠 ), (15) 2𝑁𝑓 𝑚=0 where 𝑁2 is the training length. After it goes through the channel, similar to (2), the received signal can be written as √ 𝑁2 −1 𝐸𝑏 ∑ 𝑟𝑑 (𝑡) = 𝑓2 (𝑡)∗ℎ(𝑡)+𝑛(𝑡) = 𝑞𝑑 (𝑡−𝑚𝑇𝑠 )+𝑛(𝑡). 2𝑁𝑓 𝑚=0 (16) The received signal is first multiplied with its 𝑇𝑑 delayed version, and then the product is passed through an I&D device that integrates and dumps for every subinterval/bin of Δ seconds. The sampling rate of the I&D device output is 1/Δ and Δ can take values at one or several pulse width. The total number of samples is 𝑇𝑠 /Δ × 𝑁2 . The auto-correlation over the 𝑖-th bin in the 𝑚-th training symbol is calculated as ∫ 𝑚𝑇𝑠 +𝑖Δ Re{𝑟𝑑 (𝑡)𝑟𝑑∗ (𝑡 − 𝑇𝑑 )}𝑑𝑡, 𝑋𝑚 [𝑖] = 𝑚𝑇𝑠 +(𝑖−1)Δ

𝑖

= 1, 2, ..., 𝑇𝑠 /Δ;

𝑚 = 0, 1, ..., 𝑁2 − 1. (17)

Next we will present three different algorithms to locate the beginning and end points of the integration interval, all based on (17). 1) The Conventional Threshold-crossing Method Similar to the conventional energy detection with thresholdcrossing method, the adapted version for TRPC is given as follows. To mitigate the noise effect, the auto-correlation in each bin is first averaged over the training length to get 𝑍[𝑖] =

𝑁2 −1 1 ∑ 𝑋𝑚 [𝑖], 𝑁2 𝑚=0

𝑖 = 1, 2, ..., 𝑇𝑠 /Δ.

(18)

Based on the averaged bin auto-correlation, a threshold 𝑍𝑡ℎ is defined as 𝑍𝑡ℎ = 𝜉 ⋅ 𝑍𝑚𝑎𝑥 = 𝜉 ⋅

max

1≤𝑖≤𝑇𝑠 /Δ

𝑍[𝑖]

(19)

where 𝜉(0 < 𝜉 < 1) is referred to as the normalized threshold. Therefore, the beginning (𝑡𝑏 ) and end (𝑡𝑒 ) are determined as the first and last bins exceeding the threshold 𝑍𝑡ℎ , that is, 𝜙ˆ𝑏 𝜙ˆ𝑒

= min{𝑖∣𝑍[𝑖] > 𝑍𝑡ℎ } = max{𝑖∣𝑍[𝑖] > 𝑍𝑡ℎ }.

(20)

Since 𝜙ˆ𝑏 and 𝜙ˆ𝑒 are the bin indexes, the actual time for the beginning and end points of the integration are 𝑡ˆ𝑏 = Δ ⋅ 𝜙ˆ𝑏 and 𝑡ˆ𝑒 = Δ ⋅ 𝜙ˆ𝑒 , respectively. In summary, the threshold-crossing method aims at including all the multipaths with significant energy (above the

absolute threshold) in the received signal to carry out the detection. However, the fixed threshold-exceeding criterion makes it vulnerable to unexpected noise spikes, especially at low SNR scenarios. 2) The BER Minimization Based Method Given (6) and the property of the Q function, minimizing the BER is equivalent to maximizing the received SNR, that is, the argument of Q in (6). Hence, the BER minimization method determines the beginning and end points of the integration interval as ∣𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑑)∣ . (21) [𝑡ˆ𝑏 , 𝑡ˆ𝑒 ] = arg max √ 𝑡𝑏 ,𝑡𝑒 2∣𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑑)∣/𝑁0 + (𝑡𝑒 − 𝑡𝑏 ) ⋅ 𝑌 To estimate 𝜇𝑑 (𝑡𝑏 , 𝑡𝑒 ∣𝑑), the received signal in (16) is multiplied with its 𝑇𝑑 delayed version, integrated over every symbol-long segments, and average the integration results to obtain 𝑁2 −1 ∫ 𝑚𝑇𝑠 +𝑡𝑒 1 ∑ 𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑) = 𝑟𝑑 (𝑡)𝑟𝑑∗ (𝑡 − 𝑇𝑑 )𝑑𝑡 𝑁2 𝑚=0 𝑚𝑇𝑠 +𝑡𝑏 √ ∫ 𝑡𝑒 [ 𝐸𝑏 𝐸𝑏 = 𝑞𝑑 (𝑡)𝑞𝑑∗ (𝑡 − 𝑇𝑑 ) + 𝑞𝑑 (𝑡)𝑛∗ (𝑡 − 𝑇𝑑 ) 2𝑁 2𝑁 𝑓 𝑓 𝑡𝑏 √ 𝐸𝑏 ∗ 𝑞 (𝑡 − 𝑇𝑑 )𝑛(𝑡) + 2𝑁𝑓 𝑑 +

𝑁2 −1 ] 1 ∑ 𝑛(𝑡 − 𝑚𝑇𝑠 )𝑛∗ (𝑡 − 𝑚𝑇𝑠 − 𝑇𝑑 ) 𝑑𝑡, (22) 𝑁2 𝑚=0

∑𝑁2 −1 where 𝑛(𝑡) = 𝑙=0 𝑛(𝑡+𝑚𝑇𝑠 )/𝑁2 ∼ 𝒩 (0, 𝐵𝑁0 /𝑁2 ). The purpose of averaging is to reduce the noise variance. Since the last three terms of (22) all have zero mean, we have ∫ 𝑡𝑒 𝐸𝑏 Re{𝑞𝑑 (𝑡)𝑞𝑑∗ (𝑡 − 𝑇𝑑 )}𝑑𝑡 E[Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑)}] ≈ 2𝑁𝑓 𝑡𝑏 = 𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑑). (23) Consequently, Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑)} is an unbiased estimate for 𝜇(𝑡𝑏 , 𝑡𝑒 ∣𝑑). Therefore, by substituting Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑)} into (21), the integration interval can be finally determined as { [𝑡ˆ𝑏 , 𝑡ˆ𝑒 ] = arg max ∣Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑)}∣ 𝑡𝑏 ,𝑡𝑒

1

}

⋅√ 2∣Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑)}∣/𝑁0 + (𝑡𝑒 − 𝑡𝑏 ) ⋅ 𝑌

.

(24)

The denominator of (24) consists of two terms, where the former, i.e., 2∣Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑)}∣/𝑁0 is proportional to the SNR, and the latter only depends on the integration length. When the SNR is low, we may neglect the first part and (24) is simplified to ∣Re{𝐷(𝑡𝑏 , 𝑡𝑒 ∣𝑑)}∣ √ [𝑡ˆ𝑏 , 𝑡ˆ𝑒 ] = arg max . 𝑡𝑏 ,𝑡𝑒 𝑡𝑒 − 𝑡𝑏

(25)

The BER minimization based scheme is suitable for practical implementation because it does not require analog averaging using long wideband delay lines nor digital averaging at Nyquist sampling rate. 3) The Hybrid Method


As summarized above, the threshold-crossing method is intuitive and simple, but vulnerable to the noise effect. The BER minimization based method determines the optimal integration interval from theoretical derivation and is thus more accurate. However, it requires a two-dimensional search with a computation complexity of 𝑂(𝑁 2 ) where 𝑁 = 𝑇𝑠 /Δ. To keep the low complexity characteristics in the thresholdcrossing method while at the same time utilizing the accurate integration interval obtained by the BER minimization based method, a hybrid method is proposed, which uses the threshold-crossing method to locate the integration starting point, and applies the one dimensional search method to determine the end point. Following (20) in the threshold-crossing method, the beginning point is determined as } { (26) 𝑡ˆ𝑏 = 𝜙ˆ𝑏 ⋅ Δ = min 𝑖∣𝑍[𝑖] > 𝑍𝑡ℎ ⋅ Δ. After substituting 𝑡ˆ𝑏 into (24), we find the ending point of the hybrid method by

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[14], the BER expression of the conventional TR system can be obtained as ( 𝑃𝑡𝑟 (𝑒∣h, 𝑡𝑏 , 𝑡𝑒 )≈𝑄 𝐸𝑏 ⋅ 𝐸𝑐 (𝑡𝑏 , 𝑡𝑒 ) ⋅√ (31) 2𝑁0 𝐸𝑏 ⋅ 𝐸𝑐 (𝑡𝑏 , 𝑡𝑒 ) + 2𝑁𝑓 ⋅ 𝐵 ⋅ (𝑡𝑒 − 𝑡𝑏 ) ⋅ 𝑁02 ∫𝑡 where 𝐸𝑐 (𝑡𝑏 , 𝑡𝑒 ) = 𝑡𝑏𝑒 ∣𝑞𝑡𝑟 (𝑡)∣2 𝑑𝑡 is the signal energy captured in the integration interval. Note that the three schemes given here for the conventional TR requires 𝑇𝑑,𝑡𝑟 long delay lines while the method in [14] needs frame long (2𝑇𝑑,𝑡𝑟 ) delay lines. 1) The Conventional Threshold-crossing Method Similar to the derivations given by (17)–(20), the following formulas for the conventional TR system are obtained: ∫ 𝑚𝑇𝑠 +𝑗𝑇𝑓 +𝑖Δ { } ∗ 𝑋𝑚,𝑡𝑟 [𝑖, 𝑗] = Re 𝑟𝑡𝑟 (𝑡)𝑟𝑡𝑟 (𝑡 − 𝑇𝑑,𝑡𝑟 ) 𝑑𝑡, 𝑚𝑇𝑠 +𝑗𝑇𝑓 +(𝑖−1)Δ

∣Re{𝐷(𝑡ˆ𝑏 , 𝑡𝑒 ∣𝑑)}∣ . (27) 𝑡ˆ𝑒 = arg max √ 𝑡𝑒 2∣Re{𝐷(𝑡ˆ𝑏 , 𝑡𝑒 ∣𝑑)}∣/𝑁0 + (𝑡𝑒 − 𝑡ˆ𝑏 ) ⋅ 𝑌

𝑍𝑡𝑟 [𝑖, 𝑗] =

Similarly, when the SNR is low, (27) can also be simplified as ∣Re{𝐷(𝑡ˆ𝑏 , 𝑡𝑒 ∣𝑑)}∣ √ 𝑡ˆ𝑒 = arg max . (28) 𝑡𝑒 𝑡𝑒 − 𝑡ˆ𝑏

𝑍𝑡ℎ,𝑡𝑟 [𝑗] =

Compared to the BER Minimization Based Method, the hybrid scheme only requires a one dimensional search. Thus the computational complexity is reduced from 𝑂(𝑁 2 ) to 𝑂(𝑁 ).

𝜙ˆ𝑏 [𝑗] = 𝜙ˆ𝑒 [𝑗]

+

𝑏𝑚 𝑔(𝑡 − (2𝑖 + 1)𝑇𝑑,𝑡𝑟 − 𝑚𝑇𝑠 )]

(29)

where 𝑇𝑠 = 2𝑁𝑓 𝑇𝑑,𝑡𝑟 and 𝑇𝑑,𝑡𝑟 ⩾ 𝜏𝑚𝑎𝑥 + 𝑇𝑝 .2 Consequently, the received signal is given by √ 𝑁𝑓 −1 𝐸𝑏 ∑ [𝑞𝑡𝑟 (𝑡 − 2𝑖𝑇𝑑,𝑡𝑟 − 𝑚𝑇𝑠 ) 𝑟𝑡𝑟 (𝑡) = 2𝑁𝑓 𝑖=0

+ 𝑏𝑚 𝑞𝑡𝑟 (𝑡 − (2𝑖 + 1)𝑇𝑑,𝑡𝑟 − 𝑚𝑇𝑠 )] + 𝑛(𝑡) (30) ∑ where 𝑞𝑡𝑟 (𝑡) = 𝑔(𝑡)∗ℎ(𝑡) = 𝐾−1 𝑘=0 𝛼𝑘 𝑔(𝑡−𝜏𝑘 ). Following the Gaussian approximation analysis for autocorrelation receivers 2 Note that this condition guarantees there is no IPI. Therefore, the conventional TR systems do not have asymmetry in the decision variable conditioned on data.

(32)

𝑁2 −1 1 ∑ 𝑋𝑚,𝑡𝑟 [𝑖, 𝑗], 𝑁2 𝑚=0

𝜉⋅

max

(33)

𝑍𝑡𝑟 [𝑖, 𝑗],

(34)

min{𝑖∣𝑍𝑡𝑟 [𝑖, 𝑗] > 𝑍𝑡ℎ [𝑗]},

(35)

1≤𝑖≤𝑇𝑠 /Δ

= max{𝑖∣𝑍𝑡𝑟 [𝑖, 𝑗] > 𝑍𝑡ℎ [𝑗]},

𝑁𝑓 −1 1 ∑ ˆ 𝜙ˆ𝑏 = 𝜙𝑏 [𝑗], 𝑁𝑓 𝑗=0

𝑡ˆ𝑏 = Δ ⋅ 𝜙ˆ𝑏 ,

IV. E XTENSION TO THE C ONVENTIONAL TR S YSTEM To further compare TRPC with the conventional TR in terms of performance when the three integration interval determination methods are employed, the algorithms in the previous section is extended to the conventional TR systems. A conventional TR signal for the 𝑚-th symbol can be written as √ 𝑁𝑓 −1 𝐸𝑏 ∑ [𝑔(𝑡 − 2𝑖𝑇𝑑,𝑡𝑟 − 𝑚𝑇𝑠 ) 𝑠˜𝑡𝑟 (𝑡) = 2𝑁𝑓 𝑖=0

)

1

𝜙ˆ𝑒 =

𝑁𝑓 −1

∑

𝜙ˆ𝑒 [𝑗],

(36) (37)

𝑗=0

𝑡ˆ𝑒 = Δ ⋅ 𝜙ˆ𝑒 ,

(38)

where 𝑇𝑓 = 2 ⋅ 𝑇𝑑,𝑡𝑟 , 𝑖 = 0, 1, ⋅ ⋅ ⋅ , 𝑇𝑓 /Δ, 𝑗 = 0, 1, ⋅ ⋅ ⋅ , 𝑁𝑓 − 1, 𝑚 = 0, 1, ⋅ ⋅ ⋅ , 𝑁2 − 1. 2) The BER Minimization Based Method According to (31), the integration interval for the conventional TR system can be determined via the BER minimization method as follows { [𝑡ˆ𝑏 , 𝑡ˆ𝑒 ] = arg max (𝐸𝑏 /𝑁0 )2 ⋅ 𝐸𝑐 2 (𝑡𝑏 , 𝑡𝑒 ) 𝑡𝑏 ,𝑡𝑒

} 1 ⋅ . (𝐸𝑏 /𝑁0 ) ⋅ 𝐸𝑐 (𝑡𝑏 , 𝑡𝑒 ) + 𝑁𝑓 ⋅ 𝐵 ⋅ (𝑡𝑒 − 𝑡𝑏 )

(39)

For the case that (𝐸𝑏 /𝑁0 ) ⋅ 𝐸𝑐 (𝑡𝑏 , 𝑡𝑒 ) is much smaller compared to 𝑁𝑓 ⋅ 𝐵 ⋅ (𝑡𝑒 − 𝑡𝑏 ), (39) can be simplified to [𝑡ˆ𝑏 , 𝑡ˆ𝑒 ] = arg max 𝑡𝑏 ,𝑡𝑒

𝐸𝑐2 (𝑡𝑏 , 𝑡𝑒 ) . (𝑡𝑒 − 𝑡𝑏 )

(40)

In practice, an estimate of 𝐸𝑐 (𝑡𝑏 , 𝑡𝑒 ) can be obtained by ˆ𝑐 (𝑡𝑏 , 𝑡𝑒 ) = 𝐸 ∫

𝑚𝑇𝑠 +𝑗𝑇𝑓 +𝑡𝑒

𝑚𝑇𝑠 +𝑗𝑇𝑓 +𝑡𝑏

𝑁∑ 𝑓 −1 2 −1 𝑁 ∑ 1 𝑁2 𝑁𝑓 𝑚=0 𝑗=0

{ } ∗ Re 𝑟𝑡𝑟 (𝑡)𝑟𝑡𝑟 (𝑡 − 𝑇𝑑,𝑡𝑟 ) 𝑑𝑡.

(41)

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 2010 CM1 channels

0

CM1 channels

0

10

10

Threshold Crossing

N =40 Bit Error Rate (BER)

Bit Error Rate (BER)

2

−2

10

−4

10

−6

10

12

ξ = 0.20 ξ = 0.25 ξ = 0.30 ξ = 0.35 ξ = 0.40

Δ = 2 ns, N = 10 2

−2

10

−4

10

−6

10

Hybrid −8

13

14

15

16 Eb/N0 (dB)

17

18

19

10

20

CM8 channels

0

12

BER minimization

Δ = 2 ns Δ = 4 ns Δ = 8 ns Δ = 16 ns 13

Hybrid 14

15

18

19

20

21

N2=40 Bit Error Rate (BER)


21

10

Threshold Crossing −2

10

−4

10

−6

10

20

CM8 channels

0

10

16 17 Eb/N0 (dB)

12

ξ = 0.20 ξ = 0.25 ξ = 0.30 ξ = 0.35 ξ = 0.40

Δ = 2 ns, N = 10

−2

10

−4

10

Hybrid

2

−6

13

14

15

16 E /N (dB) b

17

18

19

20

0

Fig. 1. Impact of 𝜉 on the BER performance for the hybrid and threshold crossing methods in IEEE 802.15.4a CM1 and CM8 channels, with 𝑁1 = 10, 𝑁2 = 10 and Δ = 2 ns.

10

12

BER minimization

Δ = 2 ns Δ = 8 ns Δ = 16 ns 13

Hybrid 14

15

16 17 Eb/N0 (dB)

18

19

Fig. 3. The effect of bin width on the BER performance of different integration interval determination methods for the TRPC system in CM1 and CM8 channels, with 𝑁1 = 10 and 𝑁2 = 40.

CM1 channels

0

10

Δ=2 ns, N =10 Bit Error Rate (BER)

2

−2

10

−4

10

−6

10

12

Threshold−crossing BER minimization BER minimization simplified, using Eq. (25) Hybrid Hybrid simplified, using Eq. (28) 13

14

15

16 17 E /N (dB) b

18

19

20

21

0

CM8 channels

0

10


Δ=2 ns, N2=10

−2

10

−4

10

−6

10

12

Threshold−crossing BER minimization BER minimization simplified, using Eq. (25) Hybrid Hybrid simplified, using Eq. (28) 13

14

15

16 17 Eb/N0 (dB)

18

19

20

21

Fig. 2. BER performance of the TRPC system in CM1 and CM8 channels, with 𝑁1 = 10, 𝑁2 = 10 and Δ = 2 ns.

In the next simulation section, we assume that the noiseless quantity 𝐸𝑐 (𝑡𝑏 , 𝑡𝑒 ) is available at the receiver to get the ideal solution for (39) or (40), and then the ideal solution is used as the representative of the performance of the BER minimization or the hybrid method for the conventional TR system. 3) The Hybrid Method For the hybrid mehod, 𝑡ˆ𝑏 is determined by (38) first. Then 𝑡ˆ𝑒 can be obtained by solving the optimization problem formulated by (39) or (40) with 𝑡ˆ𝑏 substituted into it. V. S IMULATION R ESULTS In this section, we present simulation results for the performance of the proposed integration interval determination schemes in IEEE 802.15.4a line of sight (LOS) CM1 channels

and non-LOS CM8 channels [16]. All parameters for the TRPC and the conventional TR systems take the same values as those in [1]. Moreover, the selection of the value of 𝜉 is crucial to the the hybrid and threshold crossing methods. Usually we select the value of 𝜉 from 0.2 to 0.4. Too small 𝜉s can include more noise energy in the decision variable, while too lage 𝜉s may result in insufficient signal energy captured. Although we know that the optimal 𝜉 is channeldependent, analytical derivation of 𝜉 is not tractable due to the dense multipath channel and IPI. Therefore, 𝜉s are only studied heuristically by simulation, in which we find that the performance of the hybrid method is always more robust to different values of 𝜉 than that of the thershold crossing method (see Fig. 1 as an example). Finally, we use a modest value of 𝜉 = 0.3 according to our experimental results. First, Figs. 2–7 present the results for the TRPC system. Fig. 2 plots the BER performances of the three investigated methods in CM1 and CM8 channels respectively. We see that method outperform the threshold-crossing method by 2.3 dB and 2.2 dB, respectively, at BER = 10−3 . After omitting the SNR related part in the optimization function, the simplified schemes still save around 1-1.5 dB compared to the threshold-crossing method. In CM8 channels, the BER minimization based method and the hybrid method outperform the threshold-crossing method by 2.0 dB, at BER = 10−3 . Fig. 3 depicts the impact of the bin width Δ on the BER performances of the BER minimization based and the hybrid methods. The bin width Δ determines the sampling rate required in the receiver. From the implementation perspective, larger Δ means lower sampling rate and all the benefits brought forth in the receiver development. In CM1 channels, increasing Δ has very little impact on the optimal BER minimization based method. However, for the hybrid method, a performance loss of 2.0 dB in CM1 channels at BER= 2 × 10−4 is observed when Δ increases from 2 ns to 16 ns. In CM8 channels, both schemes have very

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 2010 0

CM1 channels

0

10

10

Threshold−crossing

Δ=2 ns

−2

10

CM8

−1

10

−4

10

−2

10

2

10

N2 = 20 N2 = 30

−8

10

12

Δ=2 ns, N2=10

BER minimization

N = 10 −6

N2 = 40 13

14

15

16 17 E /N (dB) b

18

19

20

21

0

CM8 channels

0

10



2413

−3

10

CM1 −4

10

Δ=2 ns Bit Error Rate (BER)

Threshold−crossing −5

10

−2

10

Using exact N0 Using overestimated N ×(1+φ), φ = 10% −4

10

−6

N2 = 10

10

BER minimization

N2 = 20

Using underestimated N ×(1−φ), φ = 10% 0

N = 30 2

−6

10

12

N2 = 40 13

0

Using overestimated N0×(1+φ), φ = 20%

−7

10

14

15

16 17 Eb/N0 (dB)

18

19

20

12

Using underestimated N0×(1−φ), φ = 20% 13

14

15

21

Fig. 4. Impact of the training length on the threshold-crossing and BER minimization based method for the TRPC system in CM1 and CM8 channels, with Δ = 2 ns.

17

18

19

20

21

Eb/N0 (dB)

Fig. 6. The effect of error in 𝑁0 estimation on the BER performance of the BER minimization based method for the TRPC system, with Δ = 2 ns, 𝑁1 = 10 and 𝑁2 = 10.

0

0

10

10

CM8 −1

CM8

−1

10

10

−2

Δ=2 ns, N =10 2

−2

10

10



16

−3

10

CM1

−4

10

Δ=2 ns

−5

10

−3

10

CM1

−4

10

−5

10

Using exact N0 −6

10

N2 = 10

−6

10

N2 = 20

10

12

N2 = 40 −7

13

Using overestimated N0×(1+φ), φ=20% Using underestimated N0×(1−φ), φ=10%

N2 = 30 −7

Using overestimated N0×(1+φ), φ=10%

14

15

16

17

18

19

20

21

Eb/N0 (dB)

Fig. 5. Impact of the training length on the hybrid method for the TRPC system in CM1 and CM8 channels, with Δ = 2 ns.

limited performance loss when Δ increases to 16 ns. Although not shown here, the threshold-crossing method is the most sensitive to the change in Δ. Figs. 4 and 5 show how the training length 𝑁2 affects the BER performance of the three investigated methods. In CM1 channels significant performance degradation is observed in the threshold-crossing method by decreasing 𝑁2 from 40 to 10, while on the other hand, the BER minimization based method is not much affected even adopting 𝑁2 = 10. In CM8 channels, the training length has little impact on both the threshold-crossing and the BER minimization based methods. The performance of the hybrid method is not much affected by varying the training length from 𝑁2 = 10 to 𝑁2 = 40 in both CM1 and CM8 channels as shown in Fig. 5. Because 𝑁0 needs to be estimated in practical systems, the

10

12

Using underestimated N0×(1−φ), φ=20% 13

14

15

16

17

18

19

20

21

Eb/N0 (dB)

Fig. 7. The effect of error in 𝑁0 estimation on the BER performance of the hybrid method for the TRPC system, with Δ = 2 ns, 𝑁1 = 10 and 𝑁2 = 10.

effects of 𝑁0 estimation error on the BER performance of the BER minimization based method and the hybrid method are presented in Figs. 6 and 7, respectively. Overall the schemes are quite robust to 𝑁0 estimation error, and the performance degradations are slightly higher in CM1 than in CM8 channels and higher for the hybrid method than for the BER minimization based method. Fig. 8 presents the results for the conventional TR system in CM1 and CM8, respectively. Due to the limited space, we only show the BER performance of the three investigated methods with Δ = 2 and 𝑁2 = 30 for the threshold-crossing method. Although not shown here, the performances of the BER minimization based method and the hybrid scheme are very close to those of their simplified versions given by (40),

2414


CM1 channels

0


10

−2

10

−4

10

Threshold−crossing, N = 30 2

−6

10

12

BER minimization, ideal solution using Eq. (36) Hybrid, ideal solution using Eq. (36) 13

14

15

16 17 Eb/N0 (dB)

18

19

20

21

CM8 channels

0


10

terval in the auto-correlation receiver for the TRPC system. Among them, the threshold-crossing method has the lowest complexity but inferior performance, and requires a long training length and relatively high sampling rate. The BER minimization based method has the best BER performance and the lowest requirement on sampling rate and training length but requires a two-dimensional search for the optimal interval. The hybrid method has been shown to yield similar performance to the BER minimization based method in most cases, with much reduced complexity. It is therefore a suitable candidate for a wide range of applications.

−2

10

R EFERENCES −4

10

Threshold−crossing, N2 = 30 −6

10

12

BER minimization, ideal solution using Eq. (36) Hybrid, ideal solution using Eq. (36) 13

14

15

16

17 Eb/N0 (dB)

18

19

20

21

22

Fig. 8. BER performance of the conventional TR system in CM1 and CM8 channels, with 𝜉 = 0.3, 𝑁2 = 30 and Δ = 2 ns.

when the 𝐸𝑏 /𝑁0 ranges from 12 to 22 dB. Fig. 8 shows that in CM1 channels, the BER minimization based method and the hybrid method outperform the threshold-crossing method by about 0.8 dB and 0.3 dB, respectively, at BER = 10−4 . However, the three investigated schemes achieve nearly the same BER performance in CM8 channels. This phenomenon can be explained by the severe dispersion of the non-LOS channels. In CM8 channels, to collect enough signal energy, the integration interval needs to be as long as possible due to the relatively low energy in each multipath components. The determined integration intervals tend to be the same for the three investigated algorithms. However, for TRPC systems, the compact cluster structure can provide much more signal energy compared to conventional TR systems, and therefore, the integration interval obtained by the BER minimization method or the hybrid scheme is usually different from that derived by the threshold-crossing approach. Therefore, by comparing the results presented in Figs. 4, 5 and 8, we see that the BER minimization based method and the hybrid scheme are more effective in the TRPC system than in the conventional TR system, especially in CM8 channels. We also see that there is only a slightly different bechavior of the three investigated algorithms in the TRPC system for CM1 and CM8 channels. That means for different channels, the performances of the three investigated algorithms in the TRPC system are more robust than those in the conventional TR system. Hence, the TRPC system still obtains a better performance compared to the conventional TR system, even considering the effect of the integration interval determination algorithms. VI. C ONCLUSION In this paper, three new practically implementable algorithms have been proposed to determine the integration in-

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