Adaptive CA-CFAR Threshold for Non-Coherent IR ... - Semantic Scholar
IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 12, DECEMBER 2009
1
Adaptive CA-CFAR Threshold for Non-Coherent IR-UWB Energy Detector Receivers Abdelmadjid Maali, Ammar Mesloub, Mustapha Djeddou, Hassane Mimoun, Genevi`eve Baudoin, and Abdelaziz Ouldali
Abstract—In the present letter, a new adaptive threshold comparison approach for time-of-arrival (TOA) estimation in ultra wideband (UWB) signals is proposed. This approach can be used with non-coherent energy detector receivers in UWB systems. It exploits the idea of cell averaging constant false alarm rate (CA-CFAR) used in radar systems, where the threshold changes every energy block. The performance of several approaches are compared via Monte Carlo simulations using the CM1 channel model of the standard IEEE 802.15.4a. Both simulation results and comparisons are provided highlighting the effectiveness of the proposed approach. Index Terms—Time-of-arrival (TOA), ultra wideband (UWB), cell averaging constant false alarm rate (CA-CFAR).
Tb
Fig. 1.
Energy detector receiver scheme.
II. S IGNAL M ODEL In impulse radio UWB systems, the received signal can be expressed as 𝑟(𝑡) =
I. I NTRODUCTION
z(n)
( . )²
BPF
LNA
+∞ ∑
𝑏𝑖 𝑤𝑟𝑠 (𝑡 − 𝑖𝑇𝑠 ) + 𝑛(𝑡),
(1)
𝑖=−∞
U
LTRA-WIDEBAND (UWB) is a promising technology that offers many advantages. It is used in numerous applications and especially in wireless sensor networks. The most interesting advantage this method provides for localization problems is the high time resolution [1]. One of the problems for UWB receivers is to find the first arrived path. This is called time-of-arrival (TOA) estimation problem. The most commonly used UWB receivers are matched filter (MF) receivers and energy detection (ED) receivers. MF receivers operate at high sampling frequency and use complex algorithms, which make them inadequate for some applications. ED receivers work at sub Nyquist sampling frequency and use low complex algorithms that make them low cost and good candidates for many applications [2]. With ED receivers, the TOA estimation problem consists of detecting the first energy block containing the received signal energy. A simple way to deal with this problem is to choose the maximum energy block as a leading edge. This method is called maximum energy selection (MES) [2][3]. A second way compares each energy block to an appropriate threshold. This approach is called threshold comparison (TC) [2][3]. A third way called maximum energy selection with search back (MESSB) [2], which is, combines the previous two algorithms. In the present letter, a new adaptive threshold algorithm is derived for TOA estimation. It is an application of cell averaging constant false alarm rate (CA-CFAR) radar in noncoherent UWB energy detector receivers.
Manuscript received July 29, 2009. The associate editor coordinating the review of this letter and approving it for publication was Y.-C. Wu. A. Maali, A. Mesloub, M. Djeddou, and A. Ouldali are with the Systems Communications Laboratory, Military School Polytechnic, BP 17, Bordj-ElBahri, Algiers, Algeria (e-mail: [email protected]). H. Mimoun and G. Baudoin are with the ESYCOM Laboratory, Universit´e Paris Est, ESIEE Paris (e-mail: {h.mimoun, g.baudoin}@esiee.fr). Digital Object Identifier 10.1109/LCOMM.2009.091579
where 𝑏𝑖 is the bit information, 𝑖 is the symbol index and 𝑇𝑠 is the symbol duration. 𝑛(𝑡) is an additive white Gaussian noise with variance 𝜎 2 , 𝑤𝑟𝑠 (𝑡) is the received symbol waveform given by 𝑁𝑓 −1
𝑤𝑟𝑠 (𝑡) =
∑
𝑤𝑟 (𝑡 − 𝑗𝑇𝑓 − 𝑐𝑗 𝑇𝑐 ),
(2)
𝑗=0
where 𝑁𝑓 is the number of frames per symbol, 𝑇𝑓 and 𝑇𝑐 are the frame and chip durations respectively, {𝑐𝑗 } are the time hopping codes used to avoid catastrophic collision between multiple users. 𝑐𝑗 ∈ {0, ..., 𝑁𝑐 − 1}, with 𝑁𝑐 being the number of chips per frame. Finally, 𝑤𝑟 (𝑡) is the received pulse waveform, which is expressed as 𝑤𝑟 (𝑡) =
𝐿 ∑
𝛼𝑙 𝑤(𝑡 − 𝜏𝑙 ),
(3)
𝑙=1
where {𝛼𝑙 , 𝜏𝑙 } are the attenuation and delay of 𝑙𝑡ℎ path respectively, 𝑤(𝑡) is the emitted pulse waveform, 𝜏𝑇 𝑂𝐴 = 𝜏1 is the delay of the first arrived path and 𝐿 is the number of multipath components. The signal at the receiver antenna is amplified by a low noise amplifier (LNA), passes through a band-pass filter (BPF), is processed with a square law device, and feeds an integrator before sampling as indicated in Fig. 1. The integrator output samples provide energy blocks denoted 𝑧(𝑛) and expressed as [2][4] : 𝑁𝑓 −1 ∫ 𝑗𝑇 +(𝑐𝑗 +𝑛)𝑇 𝑓 𝑏 ∑ ∣𝑟(𝑡)∣2 𝑑𝑡, (4) 𝑧(𝑛) = 𝑗=0
𝑗𝑇𝑓 +(𝑐𝑗 +𝑛−1)𝑇𝑏
where 𝑛 ∈ {1, 2, ..., 𝑁𝑏} stands for the block index, 𝑇𝑏 is the integration and sampling period, 𝑁𝑏 is the number of blocks contained in the time frame 𝑇𝑓 and half of the next frame,
c 2009 IEEE 1089-7798/09$25.00 ⃝
2
IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 12, DECEMBER 2009
Reference blocks E D Receiver
r(t)
Guard blocks
BUT
Guard blocks
...
...
Nb/2
Nb/2
Sum
Reference blocks
Sum
Sum
𝑧𝑠 (𝑛) ∼ 𝜒2 (𝑀, 𝜎 2 , 𝐸𝑛 ), Z(n)
U T
X
TU
comparator
Output
Fig. 2.
1) Fixed threshold: In [5], 𝑧(𝑛) assumed as a centralized chi-square distribution for noise blocks only and a noncentralized chi-square distribution for noise plus signal blocks, i.e., (7) 𝑧𝑛 (𝑛) ∼ 𝜒2 (𝑀, 𝜎 2 ),
Proposed algorithm scheme.
where 𝑧𝑛 (𝑛) and 𝑧𝑠 (𝑛) are the noise energy blocks and the noise plus signal energy blocks respectively. 𝑀 is the degree of freedom, which is approximated via 𝑀 ≈ 2𝐵𝑇𝑏 [5], where 𝐵 is the signal bandwidth. 𝐸𝑛 is the signal energy contained in 𝑧𝑠 (𝑛). The relationship between the false detection probability 𝑃𝑓 𝑎 and a fixed threshold 𝜉 can be expressed as 𝑃𝑓 𝑎
Noise block energy
=
CA-CFAR Threshold Fixed Threshold
T O A CA-CFAR
Tb
Tf Frame end
Frame start
CA-CFAR and fixed threshold.
3𝑇
𝑁𝑏 = 2𝑇𝑓𝑏 . In order to reduce the noise effect, an expectation over 𝑁𝑠 symbols can be done [5] 𝑧𝑒 (𝑛) =
𝑃 (𝑧𝑛 (𝑛) > 𝜉) 𝑚−1 𝜉 ∑ 1 𝜉 𝑘 ( 𝑒𝑥𝑝(− 2 ) ) , 2𝜎 𝑘! 2𝜎 2
(9)
𝑘=0
Real TOA
Fig. 3.
=
Signal plus noise block energy
TOA TC
(8)
𝑓 −1 ∫ 𝑖𝑇𝑠 +𝑗𝑇 +(𝑐𝑗 +𝑛)𝑇 𝑁𝑠 −1 𝑁∑ 𝑓 𝑏 1 ∑ ∣𝑟(𝑡)∣2 𝑑𝑡. (5) 𝑁𝑠 𝑖=0 𝑗=0 𝑖𝑇𝑠 +𝑗𝑇𝑓 +(𝑐𝑗 +𝑛−1)𝑇𝑏
where 𝑚 = 𝑀 2 ≈ 𝐵𝑇𝑏 . 2) CA-CFAR threshold: In our approach, the threshold changes ∑ every BUT 𝑧(𝑛). The used threshold is 𝜉 = 𝑇 𝑈 with 𝑈 = 𝑖 𝑧(𝑖). If the 𝑧(𝑖) are chi-square distributed so is their sum[7]. For each BUT 𝑧(𝑛), reference energy blocks are taken as noise blocks and they are chi-square distributed with 𝑀 degrees of freedom. The sum of the 𝑧(𝑖) denoted 𝑈 is chi-square distributed with 𝑁𝑏 𝑀 degrees of freedom. The false detection probability 𝑃𝑓 𝑎 can be expressed as 𝑃𝑓 𝑎
= 𝑃 (𝑧(𝑛) > 𝑇 𝑈 ) ∫ +∞ 𝑃 (𝑧(𝑛) > 𝑇 𝑢)𝑓𝑈 (𝑢)𝑑𝑢. =
(10)
0
After development, 𝑃𝑓 𝑎 can be expressed as follows [6] :
III. P ROPOSED A PPROACH : CA-CFAR T HRESHOLD The proposed approach is summarized in Fig. 2. The main idea is to apply the CA-CFAR technique used for threshold computation in radar systems [6]. Each energy block or block under test (BUT) 𝑧(𝑛) is compared to an adaptive threshold, which is formed by the sum of other blocks subsequently multiplied by a threshold multiplier 𝑇 . The first block exceeding this adaptive threshold is considered a leading edge block as illustrated in Fig. 3. Then the estimated TOA is given by
𝑃𝑓 𝑎 =
𝑚−1 ∑ Γ(𝑁𝑏 𝑚 + 𝑘) 𝑇 1 ( )𝑘 . 𝑁 𝑚 (1 + 𝑇 ) 𝑏 𝑘!Γ(𝑁𝑏 𝑚) 1 + 𝑇
(11)
𝑘=0
As for CA-CFAR radar, the false detection probability 𝑃𝑓 𝑎 only depends on 𝑇 , 𝑁𝑏 and 𝑀 . IV. S IMULATION , RESULTS AND DISCUSSION
∑ where 𝑈 = 𝑖 𝑧(𝑖) is the sum of energy blocks from both sides, excluding guard blocks. Generally, the guard blocks duration is chosen larger or equal to the channel mean delay [6].
In the subsequent simulations, the mean absolute error (MAE) for various approaches is analyzed. The CM1 (residential LOS) channel model of IEEE802.15.4a has been considered [8]. Channel realizations are sampled at 8GHz; one thousand distinct realizations are generated, each of which has a TOA uniformly distributed within [0, 𝑇𝑓 ]. The other parameters are 𝑇𝑓 = 200𝑛𝑠, 𝑁𝑓 = 𝑁𝑠 = 1, 𝑇𝑐 = 1𝑛𝑠, 𝐵 = 4𝐺𝐻𝑧 and the guard blocks duration is equal to 20𝑛𝑠.
A. Relation between thresholds and false detection probability
A. Properties of the threshold multiplier 𝑇 in CM1 channel model
This section gives the relationship between false detection probability 𝑃𝑓 𝑎 and threshold multiplier 𝑇 . In addition, it shows how 𝑃𝑓 𝑎 changes when 𝑇 either increases or decreases. The development is presented for two cases : fixed and CACFAR threshold.
Different values of the threshold multiplier 𝑇 are investigated in CM1 channel, and the results are depicted in Figs. 𝐸𝑏 , noise blocks are comparable to signal 4 and 5. For low 𝑁 0 plus noise blocks then 𝑈 takes large values. In order to detect the first signal plus noise block, 𝑇 must take smaller values
𝜏𝐶𝐴−𝐶𝐹 𝐴𝑅 = [min(𝑛∣{𝑧(𝑛) > 𝑇 𝑈 }) − 0.5]𝑇𝑏 , 𝑛
(6)
MAALI et al.: ADAPTIVE CA-CFAR THRESHOLD FOR NON-COHERENT IR-UWB ENERGY DETECTOR RECEIVERS 2
3
2
10
10
MES MESSB TC CA−CFAR
MAE(ns)
8 dB
1
MAE(ns)
10
1
10
26 dB Increasing Eb/N0 0
10 0.005
0.01
0.015
0.02
0.025
0.03
T 0
Fig. 4.
MAE versus 𝑇 for different values of
𝐸𝑏 𝑁0
10
(𝑇𝑏 = 1𝑛𝑠).
8
10
12
14
16 18 Eb/N0(dB)
20
22
Fig. 6. MAE for several algorithms under CM1 versus 1𝑛𝑠, 𝑤𝑠 = 30𝑛𝑠, 𝑇 = 0.015).
2
10
8dB
𝐸𝑏 𝑁0
24
26
(𝜉𝑛 = 0.4, 𝑇𝑏 =
2
MES MESSB TC CA−CFAR 1
10
MAE(ns)
MAE(ns)
10
26dB
1
10
Increasing Eb/N0 0
10 0.03
Fig. 5.
0.04
0.05
0.06 T
MAE versus 𝑇 for different values of
0.07
0.08
𝐸𝑏 𝑁0
(𝑇𝑏 = 4𝑛𝑠).
0.09
and cause several false detections which increase 𝑀 𝐴𝐸. For 𝐸𝑏 high 𝑁 values, noise blocks are lower than signal plus noise 0 blocks which makes 𝑈 smaller. 𝑇 should take larger values in order to avoid false detections which decrease 𝑃𝑓 𝑎 and 𝑀 𝐴𝐸. B. Performances of various TOA estimation algorithms The performances of various algorithms are assessed in IEEE802.15.4a CM1. Results are summarized in Figs. 6 and 7. 𝜉𝑛 = 0.4 and 𝑤𝑠 =30 ns are the normalized threshold and the searched back window respectively [2]. 𝑇 = 0.015 for 𝑇𝑏 = 1𝑛𝑠 and 𝑇 = 0.05 for 𝑇𝑏 = 4𝑛𝑠. We notice that there 𝐸𝑏 region, where the MES, MESSB are two regions a low 𝑁 0 and CA-CFAR algorithms give better results than the TC 𝐸𝑏 algorithm, and a high 𝑁 region, where the proposed approach 0 𝐸𝑏 gives the best results. In the low 𝑁 region, there is a problem 0 of false detections made by the noise energy blocks which 𝐸𝑏 are important and exceed different thresholds. In the high 𝑁 0 region, noise blocks are not important. Therefore CA-CFAR and TC give the best results. Our approach gives better results thanks to its adaptive threshold, which avoids false detections and gives a more accurate TOA estimation. V. C ONCLUSION In the presnet letter, a new algorithm for TOA estimation in UWB systems is proposed. This algorithm consists of using an adaptive CA-CFAR threshold, which is formed by the sum of energy blocks and a multiplication by a threshold multiplier
0
10
8
10
12
14
16 18 MAE(dB)
20
22
Fig. 7. MAE for several algorithms under CM1 versus 4𝑛𝑠, 𝑤𝑠 = 30𝑛𝑠, 𝑇 = 0.05).
𝐸𝑏 𝑁0
24
26
(𝜉𝑛 = 0.4, 𝑇𝑏 =
𝑇 . Simulation results show that our approach gives accurate results and better performances for high SNR. R EFERENCES [1] S. Gezici, T. Zhi, G. B. Giannakis, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios: a look at positioning aspects for future sensor networks,” IEEE Signal Process. Mag., vol. 22, no. 4, pp. 70-84, 2005. [2] I. Guvenc and Z. Sahinoglu, “Threshold based TOA estimation for impulse radio UWB systems,” in Proc. IEEE Int. Conf. UWB (ICU), Zurich, Switzerland, pp. 420-425, Sep. 2005. [3] R. Badorrey, A. Hernandez, J. Choliz, A. Valdovinos, and I. Alastruey, “Evaluation of TOA estimation algorithms in UWB receivers,” in Proc. 14th European Wireless Conference, June 2008. [4] I. Guvenc and Z. Sahinoglu., “Multiscale energy products for TOA estimation in IR-UWB systems,” in Proc. IEEE GLOBECOM, Nov. 2005. [5] I. Guvenc and H. Arslan, “Comparison of two searchback schemes for non-coherent TOA estimation in IR-UWB systems,” in Proc. Sarnoff Symposium, Mar. 2006. [6] B. R. Mahafza, Radar Systems Analysis and Design Using MATLAB. Chapman & Hall/CRC, 2000. [7] M. K. Simon, Probability Distributions Involving Gaussian Random Variables. Boston: Kluwer Academic Publishers, 2002. [8] A. F. Molish, K. Balakrishnan, C.-C. Chong, S. Emami, A. Fort, J. Karedal, H. Schantz, U. Schuster, and K. Siwiak, “IEEE 802.15.4a channel model-final report,” Tech. Rep., Sep. 2004, https://mentor.ieee.org/802.15/documents.
1
Adaptive CA-CFAR Threshold for Non-Coherent IR-UWB Energy Detector Receivers Abdelmadjid Maali, Ammar Mesloub, Mustapha Djeddou, Hassane Mimoun, Genevi`eve Baudoin, and Abdelaziz Ouldali
Abstract—In the present letter, a new adaptive threshold comparison approach for time-of-arrival (TOA) estimation in ultra wideband (UWB) signals is proposed. This approach can be used with non-coherent energy detector receivers in UWB systems. It exploits the idea of cell averaging constant false alarm rate (CA-CFAR) used in radar systems, where the threshold changes every energy block. The performance of several approaches are compared via Monte Carlo simulations using the CM1 channel model of the standard IEEE 802.15.4a. Both simulation results and comparisons are provided highlighting the effectiveness of the proposed approach. Index Terms—Time-of-arrival (TOA), ultra wideband (UWB), cell averaging constant false alarm rate (CA-CFAR).
Tb
Fig. 1.
Energy detector receiver scheme.
II. S IGNAL M ODEL In impulse radio UWB systems, the received signal can be expressed as 𝑟(𝑡) =
I. I NTRODUCTION
z(n)
( . )²
BPF
LNA
+∞ ∑
𝑏𝑖 𝑤𝑟𝑠 (𝑡 − 𝑖𝑇𝑠 ) + 𝑛(𝑡),
(1)
𝑖=−∞
U
LTRA-WIDEBAND (UWB) is a promising technology that offers many advantages. It is used in numerous applications and especially in wireless sensor networks. The most interesting advantage this method provides for localization problems is the high time resolution [1]. One of the problems for UWB receivers is to find the first arrived path. This is called time-of-arrival (TOA) estimation problem. The most commonly used UWB receivers are matched filter (MF) receivers and energy detection (ED) receivers. MF receivers operate at high sampling frequency and use complex algorithms, which make them inadequate for some applications. ED receivers work at sub Nyquist sampling frequency and use low complex algorithms that make them low cost and good candidates for many applications [2]. With ED receivers, the TOA estimation problem consists of detecting the first energy block containing the received signal energy. A simple way to deal with this problem is to choose the maximum energy block as a leading edge. This method is called maximum energy selection (MES) [2][3]. A second way compares each energy block to an appropriate threshold. This approach is called threshold comparison (TC) [2][3]. A third way called maximum energy selection with search back (MESSB) [2], which is, combines the previous two algorithms. In the present letter, a new adaptive threshold algorithm is derived for TOA estimation. It is an application of cell averaging constant false alarm rate (CA-CFAR) radar in noncoherent UWB energy detector receivers.
Manuscript received July 29, 2009. The associate editor coordinating the review of this letter and approving it for publication was Y.-C. Wu. A. Maali, A. Mesloub, M. Djeddou, and A. Ouldali are with the Systems Communications Laboratory, Military School Polytechnic, BP 17, Bordj-ElBahri, Algiers, Algeria (e-mail: [email protected]). H. Mimoun and G. Baudoin are with the ESYCOM Laboratory, Universit´e Paris Est, ESIEE Paris (e-mail: {h.mimoun, g.baudoin}@esiee.fr). Digital Object Identifier 10.1109/LCOMM.2009.091579
where 𝑏𝑖 is the bit information, 𝑖 is the symbol index and 𝑇𝑠 is the symbol duration. 𝑛(𝑡) is an additive white Gaussian noise with variance 𝜎 2 , 𝑤𝑟𝑠 (𝑡) is the received symbol waveform given by 𝑁𝑓 −1
𝑤𝑟𝑠 (𝑡) =
∑
𝑤𝑟 (𝑡 − 𝑗𝑇𝑓 − 𝑐𝑗 𝑇𝑐 ),
(2)
𝑗=0
where 𝑁𝑓 is the number of frames per symbol, 𝑇𝑓 and 𝑇𝑐 are the frame and chip durations respectively, {𝑐𝑗 } are the time hopping codes used to avoid catastrophic collision between multiple users. 𝑐𝑗 ∈ {0, ..., 𝑁𝑐 − 1}, with 𝑁𝑐 being the number of chips per frame. Finally, 𝑤𝑟 (𝑡) is the received pulse waveform, which is expressed as 𝑤𝑟 (𝑡) =
𝐿 ∑
𝛼𝑙 𝑤(𝑡 − 𝜏𝑙 ),
(3)
𝑙=1
where {𝛼𝑙 , 𝜏𝑙 } are the attenuation and delay of 𝑙𝑡ℎ path respectively, 𝑤(𝑡) is the emitted pulse waveform, 𝜏𝑇 𝑂𝐴 = 𝜏1 is the delay of the first arrived path and 𝐿 is the number of multipath components. The signal at the receiver antenna is amplified by a low noise amplifier (LNA), passes through a band-pass filter (BPF), is processed with a square law device, and feeds an integrator before sampling as indicated in Fig. 1. The integrator output samples provide energy blocks denoted 𝑧(𝑛) and expressed as [2][4] : 𝑁𝑓 −1 ∫ 𝑗𝑇 +(𝑐𝑗 +𝑛)𝑇 𝑓 𝑏 ∑ ∣𝑟(𝑡)∣2 𝑑𝑡, (4) 𝑧(𝑛) = 𝑗=0
𝑗𝑇𝑓 +(𝑐𝑗 +𝑛−1)𝑇𝑏
where 𝑛 ∈ {1, 2, ..., 𝑁𝑏} stands for the block index, 𝑇𝑏 is the integration and sampling period, 𝑁𝑏 is the number of blocks contained in the time frame 𝑇𝑓 and half of the next frame,
c 2009 IEEE 1089-7798/09$25.00 ⃝
2
IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 12, DECEMBER 2009
Reference blocks E D Receiver
r(t)
Guard blocks
BUT
Guard blocks
...
...
Nb/2
Nb/2
Sum
Reference blocks
Sum
Sum
𝑧𝑠 (𝑛) ∼ 𝜒2 (𝑀, 𝜎 2 , 𝐸𝑛 ), Z(n)
U T
X
TU
comparator
Output
Fig. 2.
1) Fixed threshold: In [5], 𝑧(𝑛) assumed as a centralized chi-square distribution for noise blocks only and a noncentralized chi-square distribution for noise plus signal blocks, i.e., (7) 𝑧𝑛 (𝑛) ∼ 𝜒2 (𝑀, 𝜎 2 ),
Proposed algorithm scheme.
where 𝑧𝑛 (𝑛) and 𝑧𝑠 (𝑛) are the noise energy blocks and the noise plus signal energy blocks respectively. 𝑀 is the degree of freedom, which is approximated via 𝑀 ≈ 2𝐵𝑇𝑏 [5], where 𝐵 is the signal bandwidth. 𝐸𝑛 is the signal energy contained in 𝑧𝑠 (𝑛). The relationship between the false detection probability 𝑃𝑓 𝑎 and a fixed threshold 𝜉 can be expressed as 𝑃𝑓 𝑎
Noise block energy
=
CA-CFAR Threshold Fixed Threshold
T O A CA-CFAR
Tb
Tf Frame end
Frame start
CA-CFAR and fixed threshold.
3𝑇
𝑁𝑏 = 2𝑇𝑓𝑏 . In order to reduce the noise effect, an expectation over 𝑁𝑠 symbols can be done [5] 𝑧𝑒 (𝑛) =
𝑃 (𝑧𝑛 (𝑛) > 𝜉) 𝑚−1 𝜉 ∑ 1 𝜉 𝑘 ( 𝑒𝑥𝑝(− 2 ) ) , 2𝜎 𝑘! 2𝜎 2
(9)
𝑘=0
Real TOA
Fig. 3.
=
Signal plus noise block energy
TOA TC
(8)
𝑓 −1 ∫ 𝑖𝑇𝑠 +𝑗𝑇 +(𝑐𝑗 +𝑛)𝑇 𝑁𝑠 −1 𝑁∑ 𝑓 𝑏 1 ∑ ∣𝑟(𝑡)∣2 𝑑𝑡. (5) 𝑁𝑠 𝑖=0 𝑗=0 𝑖𝑇𝑠 +𝑗𝑇𝑓 +(𝑐𝑗 +𝑛−1)𝑇𝑏
where 𝑚 = 𝑀 2 ≈ 𝐵𝑇𝑏 . 2) CA-CFAR threshold: In our approach, the threshold changes ∑ every BUT 𝑧(𝑛). The used threshold is 𝜉 = 𝑇 𝑈 with 𝑈 = 𝑖 𝑧(𝑖). If the 𝑧(𝑖) are chi-square distributed so is their sum[7]. For each BUT 𝑧(𝑛), reference energy blocks are taken as noise blocks and they are chi-square distributed with 𝑀 degrees of freedom. The sum of the 𝑧(𝑖) denoted 𝑈 is chi-square distributed with 𝑁𝑏 𝑀 degrees of freedom. The false detection probability 𝑃𝑓 𝑎 can be expressed as 𝑃𝑓 𝑎
= 𝑃 (𝑧(𝑛) > 𝑇 𝑈 ) ∫ +∞ 𝑃 (𝑧(𝑛) > 𝑇 𝑢)𝑓𝑈 (𝑢)𝑑𝑢. =
(10)
0
After development, 𝑃𝑓 𝑎 can be expressed as follows [6] :
III. P ROPOSED A PPROACH : CA-CFAR T HRESHOLD The proposed approach is summarized in Fig. 2. The main idea is to apply the CA-CFAR technique used for threshold computation in radar systems [6]. Each energy block or block under test (BUT) 𝑧(𝑛) is compared to an adaptive threshold, which is formed by the sum of other blocks subsequently multiplied by a threshold multiplier 𝑇 . The first block exceeding this adaptive threshold is considered a leading edge block as illustrated in Fig. 3. Then the estimated TOA is given by
𝑃𝑓 𝑎 =
𝑚−1 ∑ Γ(𝑁𝑏 𝑚 + 𝑘) 𝑇 1 ( )𝑘 . 𝑁 𝑚 (1 + 𝑇 ) 𝑏 𝑘!Γ(𝑁𝑏 𝑚) 1 + 𝑇
(11)
𝑘=0
As for CA-CFAR radar, the false detection probability 𝑃𝑓 𝑎 only depends on 𝑇 , 𝑁𝑏 and 𝑀 . IV. S IMULATION , RESULTS AND DISCUSSION
∑ where 𝑈 = 𝑖 𝑧(𝑖) is the sum of energy blocks from both sides, excluding guard blocks. Generally, the guard blocks duration is chosen larger or equal to the channel mean delay [6].
In the subsequent simulations, the mean absolute error (MAE) for various approaches is analyzed. The CM1 (residential LOS) channel model of IEEE802.15.4a has been considered [8]. Channel realizations are sampled at 8GHz; one thousand distinct realizations are generated, each of which has a TOA uniformly distributed within [0, 𝑇𝑓 ]. The other parameters are 𝑇𝑓 = 200𝑛𝑠, 𝑁𝑓 = 𝑁𝑠 = 1, 𝑇𝑐 = 1𝑛𝑠, 𝐵 = 4𝐺𝐻𝑧 and the guard blocks duration is equal to 20𝑛𝑠.
A. Relation between thresholds and false detection probability
A. Properties of the threshold multiplier 𝑇 in CM1 channel model
This section gives the relationship between false detection probability 𝑃𝑓 𝑎 and threshold multiplier 𝑇 . In addition, it shows how 𝑃𝑓 𝑎 changes when 𝑇 either increases or decreases. The development is presented for two cases : fixed and CACFAR threshold.
Different values of the threshold multiplier 𝑇 are investigated in CM1 channel, and the results are depicted in Figs. 𝐸𝑏 , noise blocks are comparable to signal 4 and 5. For low 𝑁 0 plus noise blocks then 𝑈 takes large values. In order to detect the first signal plus noise block, 𝑇 must take smaller values
𝜏𝐶𝐴−𝐶𝐹 𝐴𝑅 = [min(𝑛∣{𝑧(𝑛) > 𝑇 𝑈 }) − 0.5]𝑇𝑏 , 𝑛
(6)
MAALI et al.: ADAPTIVE CA-CFAR THRESHOLD FOR NON-COHERENT IR-UWB ENERGY DETECTOR RECEIVERS 2
3
2
10
10
MES MESSB TC CA−CFAR
MAE(ns)
8 dB
1
MAE(ns)
10
1
10
26 dB Increasing Eb/N0 0
10 0.005
0.01
0.015
0.02
0.025
0.03
T 0
Fig. 4.
MAE versus 𝑇 for different values of
𝐸𝑏 𝑁0
10
(𝑇𝑏 = 1𝑛𝑠).
8
10
12
14
16 18 Eb/N0(dB)
20
22
Fig. 6. MAE for several algorithms under CM1 versus 1𝑛𝑠, 𝑤𝑠 = 30𝑛𝑠, 𝑇 = 0.015).
2
10
8dB
𝐸𝑏 𝑁0
24
26
(𝜉𝑛 = 0.4, 𝑇𝑏 =
2
MES MESSB TC CA−CFAR 1
10
MAE(ns)
MAE(ns)
10
26dB
1
10
Increasing Eb/N0 0
10 0.03
Fig. 5.
0.04
0.05
0.06 T
MAE versus 𝑇 for different values of
0.07
0.08
𝐸𝑏 𝑁0
(𝑇𝑏 = 4𝑛𝑠).
0.09
and cause several false detections which increase 𝑀 𝐴𝐸. For 𝐸𝑏 high 𝑁 values, noise blocks are lower than signal plus noise 0 blocks which makes 𝑈 smaller. 𝑇 should take larger values in order to avoid false detections which decrease 𝑃𝑓 𝑎 and 𝑀 𝐴𝐸. B. Performances of various TOA estimation algorithms The performances of various algorithms are assessed in IEEE802.15.4a CM1. Results are summarized in Figs. 6 and 7. 𝜉𝑛 = 0.4 and 𝑤𝑠 =30 ns are the normalized threshold and the searched back window respectively [2]. 𝑇 = 0.015 for 𝑇𝑏 = 1𝑛𝑠 and 𝑇 = 0.05 for 𝑇𝑏 = 4𝑛𝑠. We notice that there 𝐸𝑏 region, where the MES, MESSB are two regions a low 𝑁 0 and CA-CFAR algorithms give better results than the TC 𝐸𝑏 algorithm, and a high 𝑁 region, where the proposed approach 0 𝐸𝑏 gives the best results. In the low 𝑁 region, there is a problem 0 of false detections made by the noise energy blocks which 𝐸𝑏 are important and exceed different thresholds. In the high 𝑁 0 region, noise blocks are not important. Therefore CA-CFAR and TC give the best results. Our approach gives better results thanks to its adaptive threshold, which avoids false detections and gives a more accurate TOA estimation. V. C ONCLUSION In the presnet letter, a new algorithm for TOA estimation in UWB systems is proposed. This algorithm consists of using an adaptive CA-CFAR threshold, which is formed by the sum of energy blocks and a multiplication by a threshold multiplier
0
10
8
10
12
14
16 18 MAE(dB)
20
22
Fig. 7. MAE for several algorithms under CM1 versus 4𝑛𝑠, 𝑤𝑠 = 30𝑛𝑠, 𝑇 = 0.05).
𝐸𝑏 𝑁0
24
26
(𝜉𝑛 = 0.4, 𝑇𝑏 =
𝑇 . Simulation results show that our approach gives accurate results and better performances for high SNR. R EFERENCES [1] S. Gezici, T. Zhi, G. B. Giannakis, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios: a look at positioning aspects for future sensor networks,” IEEE Signal Process. Mag., vol. 22, no. 4, pp. 70-84, 2005. [2] I. Guvenc and Z. Sahinoglu, “Threshold based TOA estimation for impulse radio UWB systems,” in Proc. IEEE Int. Conf. UWB (ICU), Zurich, Switzerland, pp. 420-425, Sep. 2005. [3] R. Badorrey, A. Hernandez, J. Choliz, A. Valdovinos, and I. Alastruey, “Evaluation of TOA estimation algorithms in UWB receivers,” in Proc. 14th European Wireless Conference, June 2008. [4] I. Guvenc and Z. Sahinoglu., “Multiscale energy products for TOA estimation in IR-UWB systems,” in Proc. IEEE GLOBECOM, Nov. 2005. [5] I. Guvenc and H. Arslan, “Comparison of two searchback schemes for non-coherent TOA estimation in IR-UWB systems,” in Proc. Sarnoff Symposium, Mar. 2006. [6] B. R. Mahafza, Radar Systems Analysis and Design Using MATLAB. Chapman & Hall/CRC, 2000. [7] M. K. Simon, Probability Distributions Involving Gaussian Random Variables. Boston: Kluwer Academic Publishers, 2002. [8] A. F. Molish, K. Balakrishnan, C.-C. Chong, S. Emami, A. Fort, J. Karedal, H. Schantz, U. Schuster, and K. Siwiak, “IEEE 802.15.4a channel model-final report,” Tech. Rep., Sep. 2004, https://mentor.ieee.org/802.15/documents.
