NEW TIME DELAY ESTIMATORS FOR CODED ANTI-COLLISION RADAR J.Zaidouni, A.Rivenq, S.Niar, Y. Elhillali, L. Sakkila ∗
Institut d’Électronique et de Microélectronique du Nord, DOAE. Université de Valenciennes et de Hainaut Cambrésis, France
ABSTRACT This paper deals with coded anti-collision radar based on numerical correlation receiver (CORR) associated to binary pseudo-random sequences (BP RS). To give the distance of the target, the radar must estimate the time delay between the transmitted and the received signal using the correlation function. In order to increase the performances of this time delay estimation, algorithms based on Higher Order Statitistics (HOS) are presented and compared. Simulations show that the proposed algorithm T OCE1 (Third Order Cumulant Estimator 1) gives good results, compared to the CORR algorithm. An adapted HOS signal processing unit has been implemented in FPGA device. KEYWORDS :
1
Radar, Pseudo random sequences, Correlation Receiver, Higher Order Statitistics.
Introduction
Time delay estimation between received signals at two sensor locations remains an important problem in several domains such in sonar, radar and biomedicine. Various methods have been proposed over the years for time delay estimation. The traditional approaches is based on computing the cross correlation between the two sensor measurements at various instants and searching the peaks [Tug91, Tug93, HC90]. However, these methods give the ambiguous results when the noise terms at the two sensors are correlated. To alleviate this problem when the noise process is jointly gaussian and the useful signal in non gaussian, several approaches have been proposed [Tug93]. One of them is appropriate when the non gaussian signal is known to be a linear process and the other suitable for an arbitrary non gaussian process. There are two crucial assumptions that must be satisfied for the proposed methods to succeed. One is that the signal is non gaussian with non zero bispectrum and/or 1
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Figure 1: The principle of the anti-collision radar. trispectrum. This last propriety is verified because any digital communications signal is non gaussian. The other crucial assumption is that the noise at the receivers is gaussian. In the anti collision radar system proposed by [A.M96], the problem of the time delay estimation is resolved by using the traditional cross correlation. In this technique, the pseudo random sequences are used because there are very adapted to detection over noisy communication channels and could be easily generated. In opposite to the system studied in [Tug93], our system contains only one sensor. The received signal is modelled by the sum of two signals: the first one is the reference code delayed by an unknown time period and the second one is an added noise. The estimator applied on this signal and the reference code will give the peak at the instant corresponding to the delay. The studied radar uses Binary pseudo random sequences [A.M96]. In order to perform the detection, we propose in this paper new approaches for time delay estimation based on Higher Order Statistics (HOS). The reference code and input noise, respect the principal assumptions as mentioned above.
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Principle of coded anti-collision radar
The bloc diagram of coded anti-collision radar is given in figure1. The main idea is to generate a binary pseudo random sequence (code) c(i) that modulates a microwave carrier at 76-77 GHz and to transmit it towards potential targets. Transmitter, antenna, and receiver are linked using a circulator. After reflection on an obstacle, the returned wave is demodulated. The obtained signal is sampled at transmitted code frequency. The signal processing is assured by the Digital Processor Bloc. Its main function is the correlation (CORR) calculation between the reference code c(i) and the received signal y(i). The maximum of this correlation is detected, and the corresponding time position is noted k. Knowing the speed of the wave (c = 3.108 m/s), the distance from the target can be deduced using the following formula: D=k
c 2f
(1)
Where, D is the distance to the target (in m) and f is the radar code frequency (in Hz). In this paper we focus on the case of one target, and the received signal can be given as: y(i) = Ac(i − k) + n(i)
(2)
where, A is the attenuation coefficient due to channel propagation and k is the time delay. At the output of the correlator we have:
−1 1 NX c(i − i0 )y(i) N i=0 = A.Rcc (i0 − k) + Rcn (i0 )
Rcy (i0 ) =
(3)
This signal is the sum of the autocorrelation function Rcc (i0 ) of the signal c(i) attenuated with factor A, and delayed in the time by k samples, and Rcn (i0 ) the cross correlation between c(i) and n(i). The use of the correlation receiver is therefore particularly adapted to the detection of signals submerged in a white gaussian noise [Jor86, Run53]. So it’s interesting to use a radar waveform c(i) which gives a good autocorrelation function (a peak as narrow as possible and of weak lateral lobes). The BPRS (Binary Pseudo-Random Sequences) can be used [MV01, FRÉ03] because they offer good performances for the detection and are easy to generate.
3
Algorithms based on HOS
In order to improve the performances of radar detection, a bibliographic study has been done to found new algorithms. These algorithms must be compatible with real time use, must consume less memory resources and must reduce processing time. Several algorithms based on Higher Order Statitistics (HOS) were proposed in [HC90, Tug91, Tug93, C.L88] for estimation of the delay between two signals provided by two different sensors. It has been demonstrated that the cross correlation between two sensors signals gives less effective results when the noises of the two sensors are correlated. In our case we have only one sensor that gives the signal modelled by eq.2. From this signal we want to estimate the delay k of the signal submerged in a gaussian noise . One of the key motivations behind the use of HOS in signal processing problems and for our case is their ability to suppress additive gaussian noise. This ability of noise suppression is due to the fact that the nth order cumulants of a gaussian signal are equal to zero for n > 2. In the next section we describe the principals interesting algorithms.
3.1 Tugnait Algorithm of order 3 The Tugnait algorithm of order 3 is effective for signals having the following proprieties [Tug93] : 1. The signal c(i) is zero mean, stationary and non gaussian such that either its bispectrum or trispectrum is nonvanishing. 2. The noise n(i) is stationary, zero mean and gaussian. The equivalent expression, in the case of one sensor, of this algorithm used in the study [Tug93] is the following: PN −1
Cccc (j)Cccy (j + i0 ) J3,org (i0 ) = q P i=0 PN −1 2 −1 2 [ N i=0 Cccc (j)].[ i=0 Cccy (j)] P
(4)
−1 Where, Cccy (j) = N i=0 c(i)c(i + n)y(i + j), is the third-order cumulant with n as strictly nonvanishing whatever j.
This algorithm is based on the assertion that the noise being gaussian has zero third order cumulant. By multilinearity of cumulant operators, we have:
Cccy (j + i0 ) =
N −1 X
c(i)c(i + n)y(i + j + i0 )
(5)
i=0 N −1 X
A c(i)c(i + n)c(i + j + i0 − k) N i=0 = A.Cccc (j + i0 − k) =
Cccc (j) and Cccy (j + i0 ) are therefore equal to the shift of i0 − k. The correlation between these two cumulants is therefore maximum when i0 = k so the value of the estimator J3,org (i0 ) is maximum at this instant. The denominator of 4 is constant for every received signal y(i). It gives no additional information about the delay estimation. The new algorithm named T U G3 algorithm takes only account the numerator: JT U G3 (i0 ) =
N −1 X
(6)
Cccc (j)Cccy (j + i0 )
i=0
3.2
The Proposed Algorithms
Note that in eq.6 the first cumulant uses only the reference code c(i). In order to exploit a maximum the information given by the received signal y(i), we propose new algorithms called T OCE1 (Third Order Cumulant Estimator 1) and T OCE2 which consist in calculating the correlation between two third-order cumulants respectively: JT OCE1 (i0 ) =
ÃN −1 X
!2
Cycc (j)Cycy (j + i0 )
(7)
i=0
And JT OCE2 (i0 ) =
ÃN −1 X
!2
Cyyc (j)Cyyy (j + i0 )
(8)
i=0
P
P
N −1 −1 Where, Cycc (j) = N i=0 c(i)c(i + n)y(i + j) are the i=0 c(i)c(i + n)y(i + j), and Cyyc (j) = third-order cumulants, with n as strictly nonvanishing whatever j. We can demonstrate, like T U G3 algorithm, that theses algorithms have peaks on i0 = k. In the following section we compare theses algorithms with the below presented algorithms CORR and T U G3.
4
Comparison Performances
The objective of this simulation is to study the performances of the algorithms discussed in the last part. The considered channel is AW GN (Additive White Gaussian Noise) and the input Signal-to-Noise Ratio (SN R) was varied between −30 dB and 0 dB. The results of simulations give the mean values of 1000 realizations. For every realization N samples
40 CORR, TUG3 TOCE1 TOCE2
30
Dynamic (dB)
20
10
0
−10
−20
−30 −30
−25
−20
−15 −10 Input SNR (dB)
−5
0
Figure 2: Dynamic as function of the Input SNR values. of the noise are generated and added of N samples of a BP RS sequence delayed of kwon k samples. In our simulations, the code length is N = 1023 and the delay is k = 100. The output signal is then processed and is exploited according to two criterions: 1. Dynamic: It’s calculated according to the following expression: ¯ ¯ ¯ ¯ ys (i)i=k ¯ ¯ D = 20.log ¯ ¯ ¯ max(ys (i))i6=k ¯
2. OutputSN R : It’s calculated as follows: Ã
SN Rout
|ys (i)i=k |2 = 10.log σy2n
!
Where, ys (i) is the output of considered algorithm when its input is y(i) and yn (i) is the output of considered algorithm when its input is n(i). The expression of the InputSN R is: Ã
A2 SN Rin = 10.log σn2
!
4.1 The dynamic: The dynamic represents the ratio of considered algorithm at the instant i = k and the maximum of the algorithm in the instants i 6= k. The simulation results are presented in Figure 2. The interesting part of the curves is the part giving positive values of dynamic. This part corresponds to the InputSN R > −20 dB for CORR, T U G3 algorithms, and to the
120 CORR, TUG3 TOCE1 TOCE2
100
Output SNR (dB)
80
60
40
20
0
−20 −30
−25
−20
−15 −10 Input SNR (dB)
−5
0
Figure 3: Output SNR as function of Input SNR values. InputSN R > −16 dB for T OCE1 and T OCE2 algorithms. We notice that, in the band −20 dB to 12 dB the CORR and T U G3 are better than T OCE1 and T OCE2 algorithms. For the Input SN R > −12 dB these proposed algorithms are better.
5
The Output SNR:
To evaluate the performances of the studied algorithms, we must calculate the output Signal to Noise Ratio which traduces the difference between the peak level and the noise part in the computed detection function. The Figure 3 presents the simulation results. For this criterion, the T OCE1 is the most effective in comparison with the others algorithms. At Input SN R = 0 dB we have an output SN R = 118 dB. The CORR and T U G3 algorithms have the same OutputSN R and it’s the close preferment. For the InputSN R > −15 dB the T OCE2 algorithm became in the second position after the T OCE1 algorithm. According to these results the T OCE1 algorithm is the best for the output SN R criterion An interesting study has been developed to adapt these HOS algorithms in order to be implemented on FPGA plate-form. The T OCE1 algorithm seems to be more interesting for easy implementation.
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Conclusion
The principle function of coded anti-collision radar is to estimate the distance of a detected target. New approach of digital receiver for this radar has been presented. This receiver is based on time delay estimation algorithm between the transmission and the reception. From the estimated delay, the distance is deduced according to a simple equation. The correlation remains the classical algorithm which can be used. This algorithm is the optimal because it
maximizes the outputSN R. The more efficient signal waveforms for this detector are the Binary Pseudo Random Sequences (BP RS) since they present good autocorrelation functions. Others algorithms based on the HOS are presented. The principle raison to use the HOS is that they can suppress the additive gaussian noise. We have proposed new algorithms called T OCE1 and TOCE2. These algorithms have been compared to CORR and T U G3 algorithms. The output SN R and the Dynamic are the criterions of this comparison. By analysing the simulation results, we show that the CORR algorithm is the worst for the two criterions. For Dynamic, we have seen that the CORR algorithm is the optimal in the band between −20 dB and −12 dB. The T OCE1 algorithm is the best in the band between −12 dB and 0 dB. For the output SN R, the T OCE1 algorithm is optimal. Future studies will be focused on HOS algorithms implementation on FPGA device and the validation of the T OCE1 algorithm in real conditions.
References [A.M96] A.Menhaj. Études de systèmes anti-collision basés sur les techniques radar pour véhicules routiers. PhD thesis, University of Valenciennes (In French), December 1996. [C.L88] R. Pan C.L.Nikias. Time delay estimation in unknown gaussian spatially correlated noise. IEEE transaction on acoustic, speech and signal processing, pages 1706–1714, November 1988. [FRÉ03] Bruno FRÉMONT. tude et réalisation d’un système radar coopératif destiné aux systèmes de transport guidés. PhD thesis, University of Valenciennes (In French), November 2003. [HC90] H.Chiang and C.L.Nikias. A new method for adaptative time delay estimation for non-gaussian signals. IEEE transaction on acoustics speech and signal processing, vol. 38, n◦ 2, pages 209–219, Febrary 1990. [Jor86]
J.R Jordan. Correlation algorithms, circuit and measurement applications. IEE Proc. G. Circuit Device and systems, pages 58–74, 1986.
[MV01] M.Saint-Venant. Radar anticollision à corrélation : Etude et réalisation. PhD thesis, University of Valenciennes (In French), December 2001. [Run53] P. Rundnick. The detection of weak signals by correlation methods. J. Appl. Phys, pages 128–131, 1953. [Tug91] J. K. Tugnait. On time delay estimation with unknow spatially correlated gaussian noise using fourth-order cumulants and cross cumulants. IEEE transaction on signal processing, vol. 39, n◦ 6, pages 1258–1267, June 1991. [Tug93] J. K. Tugnait. Time delay estimation with unknow spatially correlated gaussian noise. IEEE, Transaction on signal processing, vol. 42, n◦ 2, pages 549–558, Febrary 1993.
