The paper deals with the problem of development of the e ective time delay di erence estimator of beat signals, obtained by processing the frequency modulated signals. Such beat signals can be formed by the heterodyne scheme of receiver of linear frequency modulated signals with internal coherence. The described problem is actual in many radar applications such as radio direction nding. The paper is carried out the analysis of Cramer-Rao lower bound for delay as signal parameter and its comparison to frequency and initial phase. On the base of the considered approach the estimator of delay di erence of beat signals is proposed in the class of smooth parabolic nite di erence estimators. Investigation of the proposed method is carried out. Advantages of the developed estimator are shown in comparison with other techniques in such class in the areas of variance and in uence of the parasitic signals on the bias.
The problem of time delay di erence estimation of received signals is actual in
many applications of radar, sonar systems and also in ultrasonic, laser and other
measurement tasks [
The time delay estimation methods can be divided in two groups in the
frequency and time domain. Methods of estimation in frequency domain consist
in the calculation of di erence values of phase spectrum, which correspond to the
searched signals. Many authors notice the drawbacks of such methods which are
connected with in uence of the side lobes [
The time domain delay di erence estimation methods can be separated as
follows. Methods based on the zero- or threshold- crossings detection. Methods
based on calculation of signal model parameters. Methods that use a prior
information about phase of signal [
One of the most e ective ways for increasing the accuracy of time delay
measurements in the short range radar problem is using the continuous waves
with complex modulation. Such signals allow one to provide signal to noise ratio
(SNR) higher than pulsed systems with equal power [
In radar problems signals with linear frequency modulated continuous waves
(FMCW) are often used. One of the main advantages of those one is straight
dependence of processed frequency on delay of signals received from each aim.
The beat tone obtained by correlation scheme has the following expression [
The beat tone contains information about the time delay of the received
signal in its frequency and phase. As a rule, only a frequency is measured [
The aim of this paper is investigation of possibility of using the phase information of beat signals for increasing the accuracy of small (less than one period) time delay di erence estimation. 2
The cross correlation based method for the time delay estimation is widely used
in practice. In one of it implementations, it consists in phase di erence
estimation by using of maximum of the corss-correlation function [
X s2(t)s1(t).qX s22(t) X s21(t) ; (3) where (0) is the normalized cross-correlation coe cient; is conjugation.
The advantage of estimator (3) is the relatively low computational
complexity. However, it is shown that the estimator (3) is not asymptotically e ective
and has a bias in general case [
The Author of paper [
'(t) = !^t + ^; where '(t) is the approximation line; !^ is the estimated frequency; ^ is the estimated initial phase.
By using the least-square method (LSM ), the initial phase of (4) can be
found as [
N (N 1)=2
X n= (N 1)=2 arg s(n); (4) (5) (6) (7) (8) where N is the sample length; arg is the operator of obtaining phase of a complex number; s(n) is the analyzed sample.
In the considered problem s(n) = s2(n)s1(n), where s2(n); s1(n) are complex signals, the delay di erence between which is measured.
It is noted that solution (5) has the variance which coincides with the CramerRao low bound (CRLB).
The Authors of [
The model of real-valued harmonic signals of FMCW radars and correlation scheme of processing (2) can be described as follow:
s1(t) = A1 cos(!1t + 1); s2(t) = A2 cos(!2t + 2); where !1; !2 are the frequencies of analyzed beat tones; A1; A2; 1; 2 are the corresponding amplitudes and initial phases.
The result of multiplication of the signals s1 and s2 (s(n) = s2(n)s1(n)) without noises N (t) in anaclitic form can be written as where s(n) is the processed signal; ! is the beat frequency di erence; is the initial phase di erence; !; are the frequency and initial phase sum respectively; k is the delay of the k-th signals; W (n) is the weight coe cient which is given as
f n Taking into account the noise term N (t) in case of the single harmonic signal (p = 1) (9) can be given as var[ ]! = (
Tmf )2( 2fs )2 12SNN3R 1 3SN R 1 2 f 2N ; where var[ ]! is the variance of delay estimation by frequency as parameter of the signal. In (13), it is supposed that the sample length coincides with one period of modulation (Tm), and, hence, Tm = N=fs.
The CRLB for estimation of the delay by the initial phase is given by [
It can be supposed that signal (10) has two random parameters A0 and .
The Fisher matrix in this case has rank 2. Each diagonal element of the Fisher
Matrix has the following expression [
J =
n=0 ] =
A20 NX1 2 n=0
W 2(n); where J is the diagonal element of the Fisher matrix. The Cramer-Rao low bound for estimation of parameter of signals in from (10) is given by where var[ ] is the variance of delay estimation by delay as parameter.
The CRLB for estimation of delay by frequency is given by [
1 2 f0 ) 2 12SN R 1
N 3SN R 1 2f02N ; where var[ ] is the variance of delay estimation by frequency.
From comparing expressions (12)-(14) the following relations can be given var[ ] 12 var[ ]
12 (f0 +
f =2)2 f 2 0 (f0 +
f =2)2 f 2 var[ ]!: (15)
Relations (15) show that the proposed approach of using delay as parameter gives the minimal CRLB. For instance, if f0 = 2 f , then expression (15) gives: var[ ] = 18:75var[ ] = 75var[ ]!.
The results mentioned above show the advantages of using full information of the phase of beat tone (that equivalent to use the delay as parameter) for time delay estimation. The main drawback of this approach is restriction, which is connected with the phase uncertainty if its value crosses bound.
Results described above lead to the supposition that it is actual to further investigation of the e ective estimator design for the problem of time delay di erence measurement of beat signals of FMCW radar systems. 3
The beat tone (10) can be transformed in a Fourier sequence with the follow coe cients
Cl(n) = exp iW (n) l; where l is the value of grid of discreet transform, k = 0; : : : ; N of sample.
The Fourier transform by coe cients (16) is given by 1 are numbers (16) (17) (18) (19) where S(l) is the specter by l grid. It is obvious that the maximum of (17) corresponds to delay.
It can be supposed that the analogue of the Nyquist theorem for (17) can be expressed as where N 1 is the maximum value of the grid l ; max is the maximum allowable delay. Condition (18) is equal to the restriction of the phase in range 2 . Thus, N 1
2 max; N 1
1=(2f0): S(l) = S( l) =
N 1 X s(n)Cl(n) = n=0
N 1
X s(n)e iW (n) l ;
n=0
The width of peak in transform (17) is proportional to 1= f . As it shown in [
p p X A0 exp (i[W (n) k + Ik(n)]) = X jsk(n)jei arg sk(n); k=1 k=1 (20) where Ik(n) are the phase noises sk(n). The maximum of transform (17) for signal (20) corresponds to the null of its derivative
N 1 j l= k = Im[ X n=0 k=0
p W (n) X jsk(n)je(i arg sk(n)e iW (n) l ] = 0; (21) where Im[] is the imaginary part of complex expression.
Follow the approach proposed in [
Analysis of equation (22) for p = 1 gives the following solution where ^ is the estimated delay di erence.
In general case, estimator (23) can be expressed as ^ =
PnN=01 W js(n)j arg s(n)
PnN=01 W 2js(n)j
; ^ = [W T AW ] 1W T A ; (22) (23) (24) (25) where W = [W (0); : : : ; W (N 1)]T is the coe cient vector; A is the vector of amplitudes of the sample; is the vector of arguments of the sample; = [arg(s(0)); : : : ; arg(s(N 1))]T ; A = diag[js(0)j; : : : ; js(N 1)j].
In a special case when js(n)j = const; A =const diag[1; : : : ; 1] the estimator (24) is given as
^ = [W T W ] 1W T : 4
Investigation of the estimator properties
The proposed estimator (24) corresponds to expression of the Gauss-Markov
theorem, thus this estimator is a linear, unbiased and asymptotically e ective
[
Figure 1 shows the relation of standard deviation (STD) in logarithmic scale to SNR for proposed estimator (prop. ) and compared ones (Tret' (3); FK' (4) and MaxCorr (3)) . Also in Fig. 1 the results are shown for the maximum likelihood (ML) estimator and CRLB (12); (13) and (14). The ML was performed as the procedure of calculation of frequency di erence which corresponds to the maximum of spectrums. The spectrum calculation was carried out by the Fast Fourier transform (FFT) with zero padding to 222 sample size.
The values that are shown in Fig. 1 were calculated for signals (7) with delays 1 and 1.0001. The beat signals was simulated for the follow con guration of FMCW: initial frequency 100; frequency deviation 50; period of modulation 0.05; sample frequency 10000 (samples size 500 points). All values above are given in relative units. It should be noted that the selected con guration does not interfere with the generality of carried investigation.
The obtained results (see Fig. 1) shows that the proposed estimator has
the threshold value of STD in range 6 - 10 dB, which is coincides with the
theoretical estimations [
Investigation of the in uence of a parasitic signal in the analyzed sample on the bias of estimation was performed for proposed estimators (24) and (25) and compared ones. The following model of superposition of fundamental and parasitic beat tones was used: s1(n) = A1 exp (i[W (n) 1]) + A2 exp (i[W (n)( 1 + 21]); (27) s2(n) = A1 exp (i[W (n)( 1 + 1]) + A2 exp (i[W (n)( 1 + 1 + 21 + 21)]); where 21 = 2 1; 21 = 2 1; 1; 2 are the delays for the rst fundamental and parasitic signals respectively (for s1(n)); 1; 2 are the delay di erences for second and rst fundamental and parasitic signals respectively (for s2(n)).
In Figure 2 relations are shown for the relative bias in % to the di erence of delays ( 21) normalized on the 1. Obtained values were calculated for the proposed estimators prop. 1 (24) and prop. 1a (25) and compared ones for A2 = 0:1A1 and 12 = 0:00015. The conditions of numerical experiment are similar to those ones that was used for obtaining Fig. 1. The Figure 3 is shown the relations for the most accurate (unbiased) estimators from g. 2 for in range of 0:2%.
The Analysis of the performed results (see Fig. 2) show the advantages of the proposed estimator prop. a (25). It is also should be noted that ML method has the biggest bias with compared ones. The results of estimators: prop. , FK and MaxCorr have similar bias.
The real part of the considered model of signals (27) can be transformed in the following manner: s = A1 cos ('1(t)) + A2 cos ('2(t)) = A cos ('1(t) + b); (28) in % to delay di erence 12 normalized on the 1 for (29) (30)
in % to delay di erence 12 normalized on the 1 for q where A = (A1 + A2 cos '(t))2 + A22 sin2 '(t) and b = arcsin (A2 sin '(t)=A).
If A1 A2 and b is su ciently small than it can be made the assumption that sin b b then b A2 sin '(t)=A1. In this case, the analytic form of the (28) is given as s = s2(n)s1(n) = js(n)j exp (i[W (n) (31)
It can be supposed that for the most part of applications 21W (n) 1, and, hence, 21 1 and s = 21W (n) cos (
The results of the bias relation is the frequency deviation that is normalized
by the initial frequency are shown in the Fig. 4. The presented results are
obtained for (33), int the integral replacement (see 34) and calculated biases
for prop a and prop . The relation f =f0 changes in range [
: (34) The paper propose a new approach for time delay di erence estimation of beat signals, which are formed by heterodyne scheme of FMCW signals processing. The method is based on the idea of using delay as a parameter of received beat tones. The Cramer-Rao low bound analysis for this case shows the advantages compared to the traditional parameters: frequency and initial phase. For instance, if f0 = 2 f , then the variance of considered method is by 19 times smaller then for phase and in 75 times smaller then for frequency as parameters.
Based on the proposed method, a new estimator has been designed in class of the smooth parabolic nite di erence estimators. It was shown that the considered estimator is a linear, consistent, asymptotically e ective, and, also corresponds to the Gauss-Markov theorem. However, the proposed approach has restrictions, which are connected with cyclic nature of phase. The SNR threshold is about 6 dB.
Advantages of the developed estimator are shown in comparison with other techniques in such class in the in uence of the parasitic signals on the bias of obtained value. The analysis of bias expression shows its dependence on FMCW signals con guration and on time delay di erence between the fundamental and parasitic beat tones. The increasing of the last parameter value leads to the bias reduction as square of its value.
Results of the carried investigation shows the advantages of the proposed approach, particulary, of estimator prop. a for the tasks of the time delay di erence measurement of beat signals in comparison with the considered traditional techniques in the areas of the variance and in uence of the parasitic signals on the bias.