Robust Statistics Based Noise Variance Estimation ...

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Robust Statistics Based Noise Variance Estimation: Application to Wideband Interception of Non-Cooperative Communications François-Xavier Socheleau, Dominique Pastor and Abdeldjalil A¨ıssa-El-Bey Institut Telecom; Telecom Bretagne; UMR CNRS 3192 Lab-STICC, Université européenne de Bretagne Technopôle Brest Iroise-CS 83818, 29238 Brest Cedex, France Email: {fx.socheleau, dominique.pastor, abdeldjalil.aissaelbey}@telecom-bretagne.eu

Abstract Based on recent results that link sparsity hypotheses and robust statistics, a new noise variance estimator for application to communication electronic support is derived in this contribution. Numerical simulations indicate that the proposed estimator clearly outperforms the median absolute deviation measure. They also highlight the benefits of this new estimator for CFAR detection and show that it can be implemented in systems with high spectral scanning rate. The Matlab code corresponding to the proposed estimator is available at http://perso.telecom-bretagne.eu/pastor/software.

Index Terms Communication Electronic Support, COMINT, noise variance, robust statistics, sparsity.

I. I NTRODUCTION Communication Electronic Support (CES) refers to measures taken to gather information intercepted from radio-frequency emissions of non-cooperative communication systems [1]. Current CES systems are based on HF, VHF or UHF acquisitions, which are usually wideband in order to maximise the probability of intercepting the radiated emissions. In fact, such systems are designed to totally or partially March 18, 2010

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intercept military communication systems that embed electronic counter-countermeasures (ECCM) such as frequency hopping for instance [2]–[4]. Typically, the acquisition bandwidth of CES systems is around a few hundreds of kHz in the HF range and between 20 and 40 MHz for the VHF and the UHF ranges [5]– [8]. Signals resulting from these wideband interceptions are usually sparse in the time-frequency domain since they are composed, in most cases, of either a noisy mixture of few narrowband transmissions or intermittent wideband transmissions. Figure 1 shows an example of a 20 MHz interception in the low military VHF range (30-88 MHz). The resulting signal is a mixture of narrow band (between 10 to 25 kHz) fixed frequency communications such as those described in [9] and military frequency hopping communications.

Fig. 1.

Example of a 20 MHz interception in the low military VHF range: two frequency hopping transmitters as well as

various fixed frequency communications are present in the wideband CES signal with a global time-frequency activity rate lower than 10%.

Knowledge of the noise power is of prime importance for CES signal processing. For instance, being in a non-cooperative context and therefore having little or no prior information on the intercepted signals, the detection of non-cooperative transmissions is usually performed using “constant false alarm” like detectors that require prior knowledge of the noise power. This knowledge can also improve the performance March 18, 2010

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of blind modulation recognition or blind demodulation algorithms that follow the detection. The noise variance is often unknown and must be estimated to process the observations. Since very little is generally known about the signals, it is often relevant to estimate the noise standard deviation via a robust estimator such as the median absolute deviation (MAD) estimator [10], [11]. The present paper, as a continuation of [12], addresses the problem of estimating the noise variance in CES applications where the number or the amplitudes of the outliers are too large for the MAD estimator to perform well. We suggest estimating the noise variance thanks to an algorithm derived from a theoretical result involving sparse hypotheses of the same type as those given in [13] and [14]. In our context, the short term Fourier transform can be seen as a sparse transform in the sense that it makes possible to represent a wideband CES signal by coefficients that are mostly small except a few ones whose amplitudes are large (cf. figure 1). This contribution is organised as follows. The noise variance estimator is derived in section II. This estimator is then applied to wideband CES interception in section III. Section III-B specifies the way it is implemented for this application and performance measurements are given in section III-C. Finally, conclusions and perspectives are presented in section IV. Additionally, the Matlab code corresponding to the proposed robust estimator is available at http://perso.telecom-bretagne.eu/pastor/software. II. N OISE

VARIANCE ESTIMATION

On the basis of a heuristic approach, the next section presents the theoretical result [12, Theorem 1] on which our estimator for the noise standard deviation is based. This theoretical result and the particular case of interest addressed in section II-B rely on sparsity assumptions. Our estimator is then described in section II-C. A. Theoretical background The random vectors and variables introduced below are assumed to be defined on the same probability space (Ω, B, P). As usual, N stands for the set of all natural numbers. Let Y = (Yk )k∈N be a sequence of independent observations that are d-dimensional random vectors such that Yk = εk Θk + Xk for k ∈ N where, given k ∈ N: Θk stands for some possible random signal with unknown distribution; ǫk is a random variable valued in {0, 1} that models the possible occurrence of Θk ; Xk stands for some centred

d-dimensional Gaussian distributed real random vector with covariance matrix σ02 Id where σ0 6= 0, Id

being the d × d identity matrix. We assume that εk , Θk and Xk are independent for every natural number

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k . Basically, Y models a sequence of independent observations where random signals are either present

or absent in independent and additive white Gaussian noise modelled by the sequence X = (Xk )k∈N . For any τ ∈ [0, ∞) and any q ∈ N, consider the sample moments q

1X kYk kγ I(kYk k ≤ σ0 τ ), γ ∈ [0, ∞). µγ (q, τ ) = q

(1)

k=1

where k · k stands for the standard Euclidean norm in Rd , I(A) denotes the indicator function of a given event A: I(A) assigns 1 to any element of A and 0 to any element of the complementary set Ω \ A of A. Assume, for a while, that the random vectors Θk and, thus, the random vectors Yk , are independent

and identically distributed (iid). According to the strong law of large numbers and writing, as usual, (a-s) for “almost surely”, µγ (q, τ ) tends to E [kYk kγ I(kYk k ≤ σ0 τ )] (a-s) when q tends to ∞. Therefore, let r and s be two non-negative real numbers such that 0 ≤ s < r, we can write that µr (q, τ ) E [kYk kr I(kYk k ≤ σ0 τ )] ≈ , µs (q, τ ) E [kYk ks I(kYk k ≤ σ0 τ )]

(2)

in the sense specified by the strong law of large numbers. In Eqs. (1) and (2), the real value σ0 τ plays the role of a threshold height. In this respect, let us define the thresholding test with threshold height σ0 τ as the statistical test whose decision is that εk equals 1 if kYk k > σ0 τ and whose decision is that εk

equals 0, otherwise. Suppose that the norms of the signals Θk are all significantly large in comparison to the noise standard deviation. We can then expect the existence of a threshold height σ0 τ such that the probability of error of the thresholding test with threshold height σ0 τ is small for making a decision about the value of any εk . For this threshold, the probability that kYk k 6 σ0 τ conditioned to εk = 0, that is, the probability that kXk k 6 σ0 τ , should be large. Similarly, the probability that kYk k 6 σ0 τ conditioned to εk = 1, that is, the probability that kΘk + Xk k 6 σ0 τ , should be small. Now, we have E [kYk kγ I(kYk k ≤ σ0 τ )] = E [kΘk + Xk kγ I(kΘk + Xk k ≤ σ0 τ )] P[εk = 1] + E [kXk kγ I(kXk k ≤ σ0 τ )] P[εk = 0]

(3)

and E [kΘk + Xk kγ I(kΘk + Xk k ≤ σ0 τ )] 6 (σ0 τ )γ P[kΘk + Xk k ≤ σ0 τ ]. Therefore, when kΘk k is large enough, it can be expected that the first term on the right hand side of Eq. (3) becomes negligible in comparison to the second so that the following approximation E [kYk kγ I(kYk k ≤ σ0 τ )] ≈ E [kXk kγ I(kXk k ≤ σ0 τ )] P[εk = 0]

(4)

holds true for any γ ∈ [0, ∞). By combining the latter approximation with Eq. (2), it follows that, when q and the norms of the signals are large enough, µr (q, τ )/µs (q, τ ) should tend, in a certain sense to

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specify, to E [kXk kr I(kXk k ≤ σ0 τ )] /E [kXk ks I(kXk k ≤ σ0 τ )] = σ0r−s Υr (τ )/Υs (τ ) (5) Z τ 2 tγ+d−1 e−t /2 dt. Eq. (5) derives from where, for any τ ∈ [0, ∞) and any γ ∈ [0, ∞), Υγ (τ ) = 0

the standard change-of-variable formula [15, Theorem 16.13] by noticing that the distribution of kXk k

follows from that of kXk k2 , which has the centred chi-2 distribution with d degrees of freedom. The

difficulty in combining Eqs. (2) and (4) is that the former involves an almost sure convergence when q tends to infinity whereas the latter involves a convergence when the norms of the signals are large

enough. Therefore, the resulting convergence criterion, according to which µr (q, τ )/µs (q, τ ) tends to σ0r−s Υr (τ )/Υs (τ ) when q and the norms of the signals are large enough, cannot be trivial. In fact and

surprisingly enough, this convergence can be established even when the signals are not iid, if we introduce the notion of minimum amplitude of signals, the notion of thresholding function and make two additional assumptions. The minimum amplitude of the sequence Θ = (Θk )k∈N is defined as the supremum ̺ of the set of those ρ ∈ [0, ∞] such that, for every natural number k , kΘk k is larger than or equal to ρ (a-s): ̺ = sup {ρ ∈ [0, ∞] : ∀k ∈ N, kΘk k ≥ ρ (a-s)} .

(6)

A thresholding function is any non-decreasing continuous and positive real function δ : [0, ∞) → (0, ∞) such that δ(ρ) = Cρ + γ(ρ) where 0 < C < 1, γ(ρ) is positive for sufficiently large values of ρ

and lim γ(ρ) = 0. ρ→∞

The first additional assumption is the existence of some p ∈ [0, 1) such that P[εk = 1] 6 p for every k ∈ N (A1). This assumption can be regarded as a weak assumption of sparsity because it imposes that

the signals are not always present without imposing small probabilities of occurrence for the signals. The second assumption is the existence of some positive real number ν such that supk∈N E[kΘk kν ] is finite (A2). This second assumption means that the ν th moments of the signal norms are finite and upperbounded. Note that the probability distributions and the probabilities of occurrence of the signals remain unspecified. With these assumptions and the material introduced so far, it follows from [12, Theorem 1] that, given two non-negative real numbers r and s such that 0 ≤ s < r ≤ ν/2 and a thresholding function δ , σ0 is the unique positive real number σ such that, for every β0 ∈ (0, 1] and uniformly in β ∈ [β0 , 1],

lim lim sup ∆q (σ, βδ(̺/σ))

=0 ̺→∞

q→∞

(7)

∞

where, for any pair (σ, τ ) of positive real numbers,

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q X kYk kr I(kYk k ≤ στ ) k=1 Υ (τ ) r r−s . ∆q (σ, τ ) = q −σ Υs (τ ) X kYk ks I(kYk k ≤ στ ) k=1

We recall that, given a sequence (uq )q∈N of real numbers, the upper limit lim sup uq of this sequence q→∞

is the limit, when q tends to ∞ of the non-increasing sequence sup{uk : k > q}. In Eq. (7), the L∞ norm k · k∞ is applied to the random variable ∆q (σ, βδ(̺/σ)). We remind the reader that, given a complex random variable Z , kZk∞ is called the essential supremum of Z and is defined by kZk∞ = inf {ρ ∈ [0, ∞) : |Z| 6 ρ (a-s)} so that |Z| 6 ρ (a-s) if and only if ρ > kZk∞ . It is in the sense of

the convergence criterion specified by Eq. (7) when β = 1 that µr (q, δ(̺/σ0 ))/µs (q, δ(̺/σ0 )) can be said to tend to σ0r−s Υr (τ )/Υs (τ ) when q and ̺ are large enough. Even though the existence of some convergence can be guessed thanks to the heuristic approach proposed above, the specific form of the convergence criterion of Eq. (7) is not intuitive. Given q independent observations Y1 , . . . , Yq and under the assumption that the minimum amplitude of the signals is ̺, Eq. (7) suggests estimating σ0 by σ˘0 = argmin σ

sup

∆q (σ, βℓ δ(̺/σ)),

ℓ∈{1,...,L}

where L ∈ N and βℓ = ℓ/L for every ℓ ∈ {1, . . . , L}. Details about the derivation of this discrete cost are given in [12]. Following the terminology of [12], the estimate σ˘0 is hereafter called the noise standard deviation Essential Supremum Estimate (ESE) associated with the thresholding function δ . This name follows from the crucial role played by the essential supremum in Eq. (7) and [12, Theorem 1]. B. The case of signals with finitely upper-bounded energies and less present than absent Let us consider now the case described by the following two assumptions. First, assume that (A1) is satisfied with p = 1/2 so that the signals are less present than absent (A1’). Second, assume that (A2) is satisfied with ν = 2 so that the energies of the signals are finitely upper-bounded (A2’). These assumptions are of immediate and practical interest regarding the problem addressed in this paper. For every given ρ ∈ [0, ∞), let ξ(ρ) be the unique positive solution for x in the equation 0 F1 (d/2; ρ

2 2

x /4) = eρ

2

/2

,

(8)

where 0 F1 is the generalised hypergeometric function [16, p. 275]. This map ξ can be proved (see [12]) to be a thresholding function in the sense given above, with C = 1/2. When signals are less present than absent, ξ is particularly relevant for the following reason. As mentioned above, the approximation March 18, 2010

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of Eq. (4) is expected for a threshold height σ0 τ such that the probability of error of the thresholding test with threshold height σ0 τ is small. Since it is assumed that kΘk k > ̺ (a-s) and P[εk = 1] 6 1/2, it follows from [17, Theorem VII-1] that, for deciding on the value of εk , the thresholding test with threshold height σ0 ξ(̺/σ0 ) has a probability of error less than or equal to V (̺/σ0 ), a rapidly decreasing function of ̺/σ0 . Therefore, with τ = ξ(̺/σ0 ), the approximation of Eq. (4) should be accurate enough to estimate σ0 via the ESE. Since we assume that ν = 2, we can choose s = 0 and r = 1. Other values for these parameters have not yet been considered. By choosing the thresholding function ξ , an appropriate search interval [σmin , σmax ] can be specified for the computation of the ESE. The detailed computation of this search

interval can be found in [12, Section 3.2]. Under assumptions (A1’) and (A2’), it follows from the foregoing that the ESE associated with the thresholding function ξ can be computed according to the subsequent formula:

σ˘0 =

argmin σ∈[σmin ,σmax ]

 q   X     kYk kI(kYk k ≤ βℓ σξ(̺/σ))      Υ (β ξ(̺/σ)) 1 ℓ k=1 sup −σ . q X Υ0 (βℓ ξ(̺/σ))  ℓ∈{1,...,L}        I(kYk k ≤ βℓ σξ(̺/σ))  

(9)

k=1

A minimisation routine for scalar bounded non-linear functions can be used for the computation of the ESE. The experimental described below were obtained by using the MATLAB routine fminbnd.m for this minimisation. This routine is based on parabolic interpolation (see [18]).

C. The Complex and the Modified Complex Essential Supremum Estimates We now focus on the case of practical relevance where the observations, and thus, the signals and noise, are two-dimensional random vectors, or, equivalently, complex random variables. As above, the observations are assumed to be independent and not necessarily identically distributed. Such observations can be the complex values provided by the standard I and Q decomposition encountered in most receivers in radar, sonar and telecommunication systems. These complex observations can also be the outcome of a Discrete Fourier Transform (DFT). In section III, these complex values are those provided by the DFT of the signal specified by Eq. (14). In continuation of section II-B, we assume that the signals are less present than absent (A1’) and have finite energies (A2’) and . In the two-dimensional case, that is, when d = 2, the expression of ξ simplifies. According to [19, Eq. 9.6.47, p. 377], I0 (x) = 0 F1 (1; x2 /4) for every x ∈ [0, ∞), where I0 is the zeroth-order modified Bessel ρ function of the first kind, so that ξ(ρ) = I−1 0 (e

March 18, 2010

2

/2 )/ρ

for any ρ ∈ [0, ∞). In the two-dimensional case, the DRAFT

8

expression of Υ0 simplifies as well since Υ0 (τ ) = 1−exp(−τ 2 /2) for τ ∈ [0, ∞). The ESE empirical bias and empirical Mean-Square Error (MSE) are computed in [12, Section 4] for two-dimensional random signals that are independent with probabilities of presence less than or equal to one half and that are uniformly distributed on a circle centred at the origin with known radius ̺. The two components of each of these signals can be regarded as the in-phase and quadrature components of some modulated sinusoidal carrier with amplitude ̺ and phase uniformly distributed in [0, 2π]. Such signals are met in many signal processing applications. The values then obtained for the empirical bias and the empirical MSE when ̺ ∈ {0, 0.25, 0.5, 0.75, . . . , 5} and any probability of presence in {0.1, 0.2, 0.3, 0.4, 0.5} suggest that the

asymptotic conditions in Eq. (7) can be relaxed in practice. This is the reason why, in [20], we set ̺ to √ 0 (the most trivial lower-bound for the norms of the signals) and, by taking into account that ξ(0) = 2 √ (in the general d-dimensional case, ξ(0) = d, see [17]), we estimate σ0 by  q  X  √     kYk kI(kYk k ≤ βℓ σ 2) √      2) Υ (β 1 ℓ k=1 √ . σ f0 = argmin sup −σ (10) q X  √ Υ (β 2) σ∈[σmin ,σmax ] ℓ∈{1,...,L}  0   ℓ     I(kYk k ≤ βℓ σ 2)   k=1

where [σmin , σmax ] is the same search interval as in Eq. (9). The estimate σ f0 is called the Complex Essential Supremum Estimate (C-ESE).

Although the C-ESE is computed under non-asymptotic conditions, its empirical bias and empirical MSE, presented in [20] for modulated sinusoidal carriers with amplitude ̺ and phase uniformly distributed in [0, 2π], show that σ f0 is already a reasonably good estimate of σ0 . A new estimate based on the C-ESE

is then proposed in [20]. Hereafter called the Modified C-ESE (MC-ESE) and denoted by σ c0 , this new estimate is computed on the basis of σ f0 by setting v uX q √ u u kYk k2 I(kYk k ≤ σ f0 2) u u σ c0 = λu k=1 q u X √ t I(kY k ≤ σ f 2) k

(11)

0

k=1

where λ is some constant to choose. This proposition results from the following intuitive approach, whose mathematical justification is still an open issue. When β = 1 and under the assumptions considered in this section, Eq. (7) applied to the thresholding function ξ means that, when the amplitudes of the signals

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are larger than or equal to some sufficiently large value ̺ and the sample size is large enough, we have

σ0r−s ≈

q X

Υs (ξ(̺/σ0 )) k=1 q Υr (ξ(̺/σ0 )) X k=1

kYk kr I(kYk k ≤ σξ(̺/σ0 ))

(12) s

kYk k I(kYk k ≤ σξ(̺/σ0 ))

Therefore, the computation of the C-ESE by minimising the discrete cost (10) amounts to considering that the foregoing approximation remains valid when r = 1, s = 0 and ̺ = 0. Assume now that the signals are such that supk∈N kΘk k4 < ∞, which remains a reasonable assumption, although stronger than the “finite energy” assumption. Then, Eq. (7) holds true with r = 2 and s = 0 so that Eq. (12) leads now to σ02 ≈

q X

kYk k2 I(kYk k ≤ σξ(̺/σ0 ))

Υ0 (ξ(̺/σ0 )) k=1 q Υ2 (ξ(̺/σ0 )) X k=1

(13) I(kYk k ≤ σξ(̺/σ0 ))

The expression of the MC-ESE is then obtained by assuming that the foregoing approximation remains q √ √ valid for ̺ = 0, which suggests that the constant in Eq. (11) should be λ = Υ0 ( 2)/Υ2 ( 2) = 1.0937.

Designed for dealing with signals whose prior probabilities of presence are less than or equal to one

half, MC-ESE can be regarded as an alternative to the Median Absolute Deviation (MAD) estimator, which performs poorly when the number or the amplitudes of the outliers are too large (see section III-C). In what follows, we show the potentiality of MC-ESE for applications in CES interception. III. A PPLICATION

TO WIDEBAND

CES

SIGNALS

A. Signal model Considering a single sensor CES system and assuming that the wideband intercepted signal is a mixture of Ns sources, the discrete-time baseband equivalent signal at reception can be expressed as y(m) =

Ns LX i −1 X i=1 ℓ=0

hi (ℓ)zi (m − ℓ) + x(m)

(14)

where Ns is the number of sources, zi denotes the i-th source signal and {hi (ℓ)}ℓ=0,··· ,Li −1 is the baseband equivalent discrete-time channel impulse response between the i-th source and the receiver, x is the additive and independent complex white Gaussian noise with variance σ0 and, as usual, we write x(m) ∼ CN 0, σ02 .

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B. MC-ESE implementation The MC-ESE noise variance estimator is very well adapted to non-cooperative interception since it is based on very few priors. The only limitation is that assumption (A1’) requires that the wideband CES signal of interest has a time-frequency activity rate less than 1/2. Experience shows that, on average, this activity rate is proportional to the ratio fc /Bw , where fc and Bw are the central frequency and the bandwidth of the intercepted signal, respectively. In most cases, wideband CES signals are sufficiently sparse to verify assumption (A1’). This assumption is easily verified in the case of fixed-frequency or frequency-hopping (FH) communications (see Figure 1 for instance) but can as well be valid for signals using direct sequence (DS) spread spectrum. A relevant example is the JTIDS/MIDS [21] NATO standard that combines both FH and DS spread spectrum. To date, this standard is regarded as one or even the best communication system with respect to ECCM techniques. It has a DS spreading factor of 32/5 leading to a 5 MHz instantaneous bandwidth and a pulse duration of 6.4 µs for an inter-pulse gap of 6.6 µs. These features result to an average time-frequency activity rate of 12% over a 20 MHz interception bandwidth. To apply the MC-ESE algorithm, let us consider that at least M samples of the signal y are available at reception. Split this set of observations into K disjoint frames of N samples each such that M = KN . Apply an N -DFT to each frame. We then obtain a matrix [Yk,n ]k∈{1,...,K},n∈{0,...,N −1} of complex values where k is the frame index and n the DFT bin number N −1 1 X Yk,n = √ y[kN + m]e−2iπnm . N m=0

(15)

For each frame k and each bin n, we assume the random presence of a CES signal frequency component denoted by Θk,n . We therefore have Yk,n = εk,n Θk,n + Xk,n . Similarly to the previous section, εk,n is valued in {0, 1} and indicates whether the CES signal frequency component Θk,n is present or absent in the nth bin of the k th frame. Since noise is white and Gaussian with standard deviation σ0 , the complex random variables Xk,n are mutually independent and identically distributed with Xk,n ∼ CN 0, σ02 . In order to deal with q observations that can reasonably be considered as mutually independent (see

section II-A), we randomly rearrange our M = KN observations and, then, split the resulting sequence into subsets of q observations each. We compute a MC-ESE of σ02 on each of these subsets and average the MC-ESEs thus obtained. Each MC-ESE is computed according to the specifications given in section II-C with L = q .

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C. Simulations In order to validate the noise variance estimator derived in this contribution, we hereafter consider the scenario of a CES payload, embedded in an aircraft, monitoring the VHF range. Each source i is convolved with a time-variant propagation channel hi modeled as a Rice fading channel commonly used in ground-air communication simulations [22]. The channel parameters are detailed in the following table. TABLE I C HANNEL MODEL

Scenario Number of echo path

En-Route 2

Maximum delay τmax (µs)

[6,100]

Rice factor KRice (dB)

[10,20]

Doppler (Hz)

60

Note that for each hi , τmax and KRice are randomly and uniformly chosen within the ranges given by the table. We simulate a 20 MHz analysis bandwidth CES system with a 25.6 MHz sampling frequency. A 1024 point DFT is performed at reception (see Eq. (15)). This corresponds to a 25 kHz channelisation, which is very well suitable for VHF interception. The source allocation is assumed to be iid in the timefrequency plane. The number of sources Ns is set to ⌈N × P[εk,n = 1]⌉ where ⌈·⌉ denotes the integer ceiling operation. The Signal-to-Noise Ratio (SNR) is defined as ! E |Θk,n |2 SNR(dB) = 10log10 , σ02 which corresponds to the average SNR of the active time-frequency slots Yk,n . As far as the MC-ESE parameters are concerned for application to wideband CES signals, λ is set to 1.0937 as recommended in section II-C and q = N = 1024. All the results presented hereafter are averaged over 500 Monte-Carlo runs. Figure 2 highlights the impact of the signal duration on the noise variance estimation performance for an average realistic scenario (25% activity rate and a SNR of 10 dB). The estimation Normalised Mean Square Error (NMSE) of the C-ESE and the Modified C-ESE (MC-ESE) are compared. This scenario shows the benefits of the MC-ESE since it outperforms the C-ESE by up to 20 dB. Moreover, the accuracy of the MC-ESE estimate improves as the signal duration increases whereas the NMSE of the C-ESE stays flat. It can be noticed that the MC-ESE performance would be good enough (NMSE< −20 dB) for most March 18, 2010

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CES applications with available signal portions of around one millisecond only. This has the great benefit of allowing the MC-ESE algorithm to be implemented in CES systems with very high spectral scanning rate. Typical CES systems scan around 3 GHz per second with a 20 or 40 MHz interception bandwidth, which only allows a few milliseconds to acquire and process the signal. With respect to the foregoing results, the sequel focuses on the MC-ESE only.

−5

NMSE (dB)

−10

−15

−20

−25 C−ESE MC−ESE −30 0

0.5

1

1.5

2

2.5

3

3.5

4

Signal duration (ms)

Fig. 2.

Impact of the signal duration on the C-ESE and MC-ESE performance (P[εk,n = 1] = 0.25, SNR = 10 dB).

Figure 3 compares the Normalised Mean Square Error of the MC-ESE to that of the well known MAD estimator, for various SNR on a 250 µs signal. As expected, the performance of each of the two non parametric estimators under consideration strongly depends on P[εk,n = 1]. However, MC-ESE clearly outperforms the MAD estimator for P[εk,n = 1] & 5% as the latter is not resistant to large numbers of outliers or to large outlier amplitudes. Moreover, it can be seen that the estimation performance tends to deteriorate as the SNR increases. This observation mainly results from the artificial increase of P[εk,n = 1] due to multipath propagation. In fact, VHF monitoring from an aircraft can lead to channel maximum delays τmax longer than a 1024 point DFT frame (e.g 100µs compared to 1024/25.6 = 40µs). It is also worth noticing the seemingly existence of an optimum for the NMSE of the MC-ESE. This was not expected theoretically and deserves some attention in forthcoming work. As detection is usually the first and one of the most critical operation in CES, the proposed estimator is

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15 10

MAD, SNR=15dB MAD, SNR=5dB MC−ESE, SNR=15dB MC−ESE, SNR=5dB

5

NMSE (dB)

0 −5 −10 −15 −20 −25 −30 −35 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

P[ε =1] k,n

Fig. 3.

MAD and MC-ESE performance comparison (signal duration = 0.25 ms).

indirectly evaluated in figures 5, 4 and 6 through the performance of a constant false alarm rate detector. These figures compare the true detection rate (Pdet ) for various theoretical false alarm rate (Pf a ) when the noise variance is perfectly known and when it is estimated. The decision on detection is made by comparing |Yk,n |2 to a positive threshold that aims at guaranteeing a specified false alarm rate. Given that

the noise is complex-valued and Gaussian, 2|Yk,n |2 /σ02 follows a chi-square distribution with 2 degrees of freedom when εk,n = 0. Therefore, when σ0 is known, the detector decides that εk,n equals 1 if |Yk,n |2 > −σ02 ln(Pf a ) and that εk,n equals 0, otherwise. It is usual to summarise this decision-making

on the value of εk,n by writing |Yk,n |2

εk,n =1 > < εk,n =0

− σ02 ln(Pf a ).

(16)

When σ0 is estimated by MC-ESE, we replace σ0 by its estimate σ c0 in Eq. (16). Figures 5, 4 and 6 confirm the benefit of the MC-ESE algorithm for CES application. In fact, for an activity rate of 25% (Figure

5) the difference between the detection rate achieved by the ideal detector of Eq. (16) and the detection rate obtained by the detector based on σ c0 is negligible whatever the value of the SNR. This emphasizes the results of Figure 3 where the NMSE ranges from -18 to -24 dB for P[εk,n = 1] = 0.25. Also, it can be seen on Figure 4 and Figure 6 that for an activity rate of 10% and 40% respectively, the detection rate difference is more important but still acceptable with regards to CES system requirements. The fact March 18, 2010

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that the MC-ESE detection rate is above (resp. below) the ideal rate indicates that the noise variance is under-estimated (resp. over-estimated). This is in agreement with Table III-C where the effective false alarm rate is measured for different theoretical Pf a and activity rates. This table shows a lower effective false alarm rate than the theoretical one for activity rates above 30%. This, once again, indicates an overestimation of the noise variance. It can also be noticed on this table that the detector is approximatively CFAR for 0.1 < P[εk,n = 1] < 0.3 and that the noise variance is under-estimated for activity rates lower than 10%.

1 0.9 0.8 0.7

P

det

0.6 0.5 0.4 0.3

SNR=15dB, MC−ESE 2 SNR=15dB, perfect knowledge of σ0

0.2

SNR=10dB, MC−ESE SNR=10dB, perfect knowledge of σ20

0.1


0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P

fa

Fig. 4. Comparison of the receiver operating characteristics when the noise variance is perfectly known and when it is estimated by MC-ESE (P[εk,n = 1] = 0.1, signal duration = 0.25 ms).

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1 0.9 0.8 0.7

P

det

0.6 0.5 0.4 0.3


0.2


0.1


0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P

fa


1 0.9 0.8 0.7

P

det

0.6 0.5 0.4 0.3


0.2


0.1


0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P

fa


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TABLE II E FFECTIVE FALSE ALARM RATE VS

THEORETICAL

Pf a

FOR DIFFERENT

P[εk,n = 1], SNR=10 D B.

Pf a = 10

−3

P[εk,n = 1] =

10−2

10−1

0

3.5 10−3

2.2 10−2

1.5 10−1

0.1

2.1 10−3

1.5 10−2

1.2 10−1

0.2

1.1 10−3

1.2 10−2

1.0 10−1

0.3

8.2 10−4

8.5 10−3

9.3 10−2

0.4

4.6 10−4

5.8 10−3

7.5 10−2

0.5

2.8 10−4

3.1 10−3

6.4 10−2

IV. C ONCLUSION

AND PERSPECTIVES

On the basis of recent results in robust statistics, this contribution has presented a noise variance estimator, the MC-ESE, applied to wideband CES interception. This estimator requires very little prior knowledge about the signal. It only relies on the sparsity hypothesis that CES signals, in a given timefrequency zone, have their probability of presence upper-bounded by one half. Performance evaluation indicates that MC-ESE outperforms the standard MAD estimator and is suitable for CES systems with high spectral scanning rate. However, simulations have also shown that in scenarios with long delay spread propagation channels and high SNRs, the estimator accuracy is slightly degraded. Forthcoming studies could then evaluate the opportunity to mitigate this degradation by the use of multiple sensors. From a more general point of view, further theoretical and experimental studies should address the following points. To begin with, the behaviour of MC-ESE (bias, consistency etc.) must be better understood since the asymptotic Eq. (7) and [12, Theorem 1] do not explain the good performance measurements of this estimator at low SNRs. In particular, figure 3 suggests the possible existence of an optimal upper bound for the signal probability of presence. The influence of parameters L, r and s is also a topic to consider in forthcoming work and the use of multiple values for these parameters could be thinkable for the computation of the estimate. Finally, the following point should be addressed. Choosing ̺ = 0 for the lower bound of the signal amplitudes makes it possible to avoid any prior knowledge on the signal for the computation of the MC-ESE. However, if the value of ̺ could be measured on some preliminary data basis, this prior knowledge on the signal could prove helpful to extend MC-ESE to the case of signals with probabilities of presence above one half. This case is of practical interest and theoretically conceivable according to Eq. (7) and [12, Theorem 1]. Despite these open questions, MC-ESE offers March 18, 2010

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new perspectives for robust noise variance estimation and has already a strong operational interest for wideband CES interception. R EFERENCES [1] R. Poisel, Introduction to Communication Electronic Warfare Systems, Artech House Publishers, 2002. [2] S.G. di Pasquale P. Lagarde, “The PR4G VHF ECCM system: extensive tactical communications for the battlefield,” in MILCOM, 1992. [3] “Sincgars,” http://en.wikipedia.org/wiki/SINCGARS. [4] “Tadiran,” http://www.tadcomm.com/. [5] “PAC COMINT analysis station (France),” Jane’s Military Communications, Surveillance and signal analysis, Aug. 2008. [6] “Israel Aerospace Industrie Ltd,” http://www.iai.co.il. [7] “Sozvezdie,” http://www.sozvezdie.su/. [8] “Telemus,” http://www.telemus.com/. [9] “Tetrapol,” http://en.wikipedia.org/wiki/TETRAPOL. [10] D. Donoho and I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol. 81, no. 3, pp. 425–455, Sept. 1994. [11] P. J. Rousseeuw and C. Croux, “Alternatives to the Median Absolute Deviation,” Journal of the American Statistical Association, vol. 88, no. 424, pp. 1273–1283, Dec. 1993. [12] D. Pastor, “A theoretical result for processing signals that have unknown distributions and priors in white Gaussian noise,” Computational statistics and data analysis, vol. 52, pp. 3167–3186, 2008. [13] D. Pastor A. Atto and G. Mercier, “Detection thresholds for non-parametric estimation,” Signal, Image and Video processing, 2008. [14] D. Pastor and A. Atto, “Sparsity from binary hypothesis testing and application to non-parametric estimation,” in Proc. European Signal Processing Conference, Aug. 2008. [15] P. Billingsley, Probability and Measure, third edition, Wiley, 1995. [16] N. N. Lebedev, Special Functions and their Applications, Prentice-Hall, Englewood Cliffs, 1965. [17] D. Pastor, R. Gay, and A. Gronenboom, “A sharp upper bound for the probability of error of likelihood ratio test for detecting signals in white gaussian noise,” IEEE Trans. Inf. Theory, vol. 48, no. 1, pp. 228–238, Jan. 2002. [18] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical recipes in C, The Art of Scientific Computing, second edition., University Press, Cambridge., 1992. [19] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Ninth printing., Dover Publications Inc., New York., 1972. [20] D. Pastor and A. Amehraye, “Algorithms and Applications for Estimating the Standard Deviation of AWGN when Observations are not Signal-Free,” Journal of Computers, vol. 2, no. 7, Sept. 2007. [21] C. Golliday, “Data link communications in tactical air command and control systems,” IEEE J. Sel. Areas Commun., vol. 3, no. 5, pp. 779–794, 1985. [22] E. Haas, “Aeronautical Channel Modeling,” IEEE Trans. Veh. Technol., vol. 51, no. 2, pp. 254–264, Mar. 2002.

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