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OPTIMAL DETECTION AND ESTIMATION IN FMCW RADAR Andrzej Wojtkiewicz∗ , Rafał Rytel-Andrianik# The paper is mainly concerned with detection of an accelerating target echo by FMCW radar. The processed signal is modeled as a polynomial phase signal, contaminated either by white or colored additive gaussian noise. Obtained tests, optimal in the Neyman-Pearson sense, are linked with the Maximum Likelihood estimation procedures dealt with in earlier papers. An example shows importance of proper suiting the test to noise statistics.
1
I NTRODUCTION
We are concerned with the Frequency Modulated Continuous Wave (FMCW) radar that transmits constant amplitude wave str (t) = Atr cos(ψ(t)) with sawtooth frequency modulation. The instantaneous frequency, within one modulation period of length T , rises linearly: 1 dψ(t) = fc + αt 2π dt
t ∈< 0, T >
(1)
where fc is a carrier frequency and α determines modulation slope. The transmitted signal is scattered by the target, and considerably attenuated returns to the radar after delay time τ(t) = 2R(t)/c (c and R(t) stand for light velocity and moving target range, respectively). The received and transmitted signals are mixed (homodyne reception) forming complex valued video signal (which consists of inphase and quadrature components as real and imaginary parts, respectively)[1]. We record the video signal from K consecutive modulation periods (i.e. observation time lasts KT ): s(t, k) = Aexp( jφ(t, k))
(2a)
φ(t, k) = ψ(t) − ψ(t − τ(t)) = 2πτ(t + kT )( fc + αt − 0.5ατ(t + kT ))
t ∈< τmax , T >, k = 1, ..., K (2b)
where t denotes time ”inside” k-th modulation period, and τmax = 2Rmax /c is maximal signal delay corresponding to maximal radar range. Adopting quadratic target movement model R(t) = R0 + v0t + 0.5at 2
(3)
the equation (2b) describing video signal phase becomes quite complicated. Fortunately some reasonable simplifications are possible [2], and equation (2b) can be approximated as: φ(t, k) ≈ φ0 + ωRt + av k + aa k2
(4a)
where ωR = 2π
2R0 α c
av = 2π
2v0 fc T c
aa = 2π
2a 1 2 fc T c 2
(4b)
In real-life radar the video signal is corrupted by noise. Analysis of recorded radar signals (not included here) revealed that an additive noise plays the major role. That is, we observe y(t, k) = s(t, k) + n(t, k) (when the target is present) or y(t, k) = n(t, k) (when the target is absent), where n(t, k) represents noise signal. In the paper we give solution to the following problem: Given noisy recorded video signal y(t, k) decide in an optimal manner whether target with specified parameters is present or absent. By optimal we mean a method which maximizes probability of detection, when the probability of false alarm is fixed. That is we use NeymanPearson theory. ∗ #
Institute of Electronic Systems, Warsaw University of Technology, Nowowiejska 15/19, 00–665 Warsaw, Poland, email: ∗
[email protected] #
[email protected] 1
2
S TATEMENT OF THE DETECTION PROBLEM
The detection problem described in the previous section, can be stated formally as a binary hypothesis testing problem [3]. The two hypotheses are: H0 : y(m∆, k) = n(m∆, k) H1 : y(m∆, k) = s(m∆, k) + n(m∆, k)
m = Mmax , ..., M Mmax = k = 1, ..., K
τmax ∆
M=
T ∆
(5)
where y(m∆, k) is the recorded noisy video signal sampled with time period ∆; n(m∆, k) is a sampled complex, gaussian (either white ot colored) noise, and s(m∆, k) represents sampled video signal given by (2a). Target parameters (R0 , v0 , a) are assumed to be known and signal s(m∆, k) would be deterministic if it were not for the fact that the amplitude A, and the constant phase φ0 are not known. Here, we assume that amplitude is an unknown deterministic constant, and constant phase is a random variable with pdf uniform on (−π, π). Thus, this is the case of a known signal with nuisance parameters, buried in additive noise [3]. The optimal 1) Neyman-Pearson test consists of comparing likelihood ratio p(y|H p(y|H0 ) with preset threshold [3]. If the threshold is exceeded, then hypothesis H1 is decided to be true, otherwise we choose hypothesis H0 .
2.1 The White noise case In this section we assume that n(m∆, k) in (5) is a gaussian circular white noise with variance σ2 . Computation of likelihood ratio reveals (we omit derivations) that the optimal test is independent of unknown amplitude A (i.e. the test is uniformly most powerful) and has the following form: H1
D(R0 , v0 , a) ≷ γ
(6)
H0
where K
D(R0 , v0 , a) = | ∑
M
∑
y(m∆, k)e− jφ(m∆,k) |2
(7)
k=1 m=Mmax
The threshold γ is selected as to meet required probability of false alarm: solved to obtain, that: γ = − ln(Pf a )σ2 MK
R∞ γ
fD(R,v,a)|H0 (t) dt = Pf a which can be (8)
Most often, target parameters are not known a priori and many binary hypothesis tests must by performed, each of them checking for existence of an object with different parameters. Computation of D(R0 , v0 , a) for all combinations of (R0 , v0 , a) can be time consuming when product MN is large - disadvantage which inhibits using (7) with exact video signal phase φ(m∆, k), given by (2b), in real-time systems. To overcome this difficulty we have to resort to using approximate model of the phase, given by (4). Then 2
− jωR m∆ e− j(av k+aa k ) |2 D(R0 , v0 , a) = | ∑Kk=1 ∑M m=Mmax y(m∆, k)e 2 = | ∑Kk=1 Y (θR , k)e− j(av k+aa k ) |2
(9)
where so called range transform Y (θR , k) is a Discrete Fourier Transform (DFT) of y(m∆, k) for k-th modulation period M
Y (θR , k) =
∑
y(m∆, k)e− jωR m∆
(10)
m=Mmax
and the normed radian frequency θR = ωR ∆. Now, (9) can be computed using fast FFT algorithm. Because DFT is a linear transform, the following equation holds: Y (θR , k) = S(θR , k) + N(θR , k)
(11)
where range spectrum S(θR , k) and white gaussian noise N(θR , k) are DFTs of s(m∆, k) and n(m∆, k), respectively. Expression (9) means that computation of D(R0 , v0 , a) consists of computing K range transforms for all modulation periods k = 1, ..., K, and then computing chirp transforms of Y (θR , k) for all ranges θR . 2
When the target is already detected, often, we want to estimate its parameters. It turns out [2] that Maximum Likelihood (ML) estimates of (R0 , v0 , a) are given by maximizing (9), i.e.: (Rˆ 0 , vˆ0 , a) ˆ = arg max D(R0 , v0 , a). (R0 ,v0 ,a)
Thus the same quantity D(R0 , v0 , a) is used both for detection and estimation. If we assume zero target acceleration, then aa = 0, chirp transform in (9) simplifies to DFT of Y (θR , k), and D(R0 , v0 , 0) is a two-dimensional DFT of the video signal y(m∆, k). Like in the accelerating target case, D(R0 , v0 , a) can be used for estimation - now, as a = 0, maximization is two-dimensional [1]. Furthermore, when the target is motionless, then additionally av = 0 and so D(R0 , 0, 0) = | ∑Kk=1 Y (θR , k)|2 , thus in the optimal test we compare with the threshold magnitude of an averaged range spectrum at hypothesized range θR . This stems from the fact that the video signal is then harmonic with frequency corresponding to target range [1]. In [4] it was shown that even for fast-moving targets, video signal within on modulation period can be considered harmonic. It motivates computing range spectrum Y (θR , k) as in (10) also when target is moving.
2.2
The Colored noise case
It happens often that at a given range θR , power spectrum of the noise N(θR , k) in (11) is not uniform. For example if ground clutter dominates then it is concentrated around zero frequency, or if echos from moving mist dominate then the noise spectrum is concentrated around frequency corresponding to the mean wind velocity. The complex noise N(θR , k) is gaussian, with zero mean and the autocorrelation matrix (assumed to be known): RR (k1 , k2 ) = E[N(θR , k1 )N ∗ (θR , k2 )]
(12) H1
−1 2 The optimal Neyman-Pearson test is now given by: D0c (R0 , v0 , a) ≷ γ0c where D0c (R0 , v0 , a) = |sH 0 RR YR | and the H0
2
vectors are defined as s0 = [s0 (1), ..., s0 (K)]T , YR = [Y (θR , 1), ...,Y (θR , K)]T and s0 (k) = e j(av k+aa k ) . As before the −1 threshold is set as to attain desired Pf a , resulting in γ0c = − ln(Pf a )(sH 0 RR s0 ). Vector s0 depends on target parameters (v0 , a), and so depends the threshold γ0c . To make the threshold in test −1 constant we can divide both sides of the test by (sH 0 RR s0 ) to obtain a new test (the expression that we divide by is −1 positive, as matrices RR and RR are positive definite), then: H1
Dc (R0 , v0 , a) ≷ γc
(13a)
H0
where Dc (R0 , v0 , a) =
R−1YR |2 D0c (R0 , v0 , a) |sH = 0H R−1 −1 H s0 RR s0 s0 RR s0
(13b)
γc = − ln(Pf a )
(13c)
Like previously, the quantity Dc (R0 , v0 , a) used in detection, can be used to obtain ML estimates of target parameters, as (Rˆ 0 , vˆ0 , a) ˆ = arg max Dc (R0 , v0 , a) [5]. It is also worth noting, that in a white noise case the inverse (R0 ,v0 ,a)
autocorrelation matrix RR −1 is a unity matrix and test (13a) reduces to (6). It is possible to speed up computing Dc (R0 , v0 , a) by employing DFT. To do so we must approximate autocorrelation matrix RR by circulant autocorrelation matrix RRc what is equivalent to sampling signal spectrum. Circulant matrices are diagonalised by DFT matrix Fm,n = exp( j2π(m − 1)(n − 1)/K) [6]: RRc = K1 FH ΛR F, where ΛR = diag([λ1 , · · · , λK ]) is a diagonal matrix with elements given by DFT of the first column of RRc (and represents power spectrum of sampled noise): λk =
K
∑ rc (m)e− j(m−1)(k−1)/K
k = 1, · · · , K
(14)
m=1
Thus we can rewrite (13b) as Dc (R0 , v0 , a) =
2 |(Fy)H Λ−1 R (Fs0 )| −1 H (Fs0 ) ΛR (Fs0 )
(15)
Denominator of the above equation is responsible for keeping threshold constant. Numerator can be considered as a weighted dot product of DFTs of modeled signal and recorded signal. It means that this is a kind of weighted matched filter with weights being reciprocals of noise power in summed spectrum lags. 3
3
C OMPUTER SIMULATIONS
In this section we present an example, showing how important it is to match detection algorithm properly to to the noise statistics. The noise was modeled as AR(1) process with the complex pole at √12 · e jπ/4 . Radar and target parameters were assumed as follows: carrier frequency fc =10GHz, modulation period T =1ms, target radial mean velocity v0 =-5m/s, target radial acceleration a=10m/s2 .
(b)
0
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0 a
20
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(a)
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28 20 26 24
18
dB
dB
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a [m/s2]
Figure 1: (a) coefficient D(R0 = const, v0 , a) - suited for white noise (b) coefficient Dc (R0 = const, v0 , a) - suited for actual colored noise Values of coefficients D(R0 , v0 , a) (which is compared with a threshold in the test (6) suited to the white noise) and Dc (R0 , v0 , a) (which corresponds to the test (13a) matched to proper noise autocorrelation matrix) for different hypothesized parameters (v0 , a), at fixed range R0 are depicted on figure 1. The figure shows that in the optimal method the noise has been whitened, and so the target is much easier to discriminate. So big improvement has been achieved mainly due to a large difference in velocities of target and clutter - if they traveled with the same velocity (for example a man waking in a moving mist of the same velocity) it would be much more difficult to discriminate them.
R EFERENCES [1] A.Wojtkiewicz, J.Misiurewicz, M.Nał˛ecz, K.J˛edrzejewski, K.Kulpa, Two-dimensional signal processing in FMCW radars, Proc. XX KKTOiUE, Kołobrzeg, Poland, 1997, pp 475-480. [2] A.Wojtkiewicz, R.Rytel-Andrianik A new estimation method for target range, velocity and acceleration in FMCW radar, Proc. Int. Conf. on Signals and Electronic Systems Ustro´n’2000, Poland. [3] H.L. Van Trees, Detection, Estimation and Modulation Theory - Part I, New York: Willey, 1968. [4] R.Rytel-Andrianik, Analiza dokładno´sci pomiaru odległo´sci poruszajacego ˛ si˛e obiektu, wykrywanego przez radar FMCW, Proc. KST, Bydgoszcz’2001 Poland. [5] A.Wojtkiewicz, R.Rytel-Andrianik, Estimation method for velocity and acceleration of a target detected in clutter by FMCW radar, Proc. PCH Workshop, Budapest’2001, Hungary. [6] T.K.Moon W.C.Stirling, Mathematical Methods and Algorithms for Signal Processing, Prentice Hall, 1999.
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