Gilles Chabriel
email: gilles.chabriel@im2np.fr

Ghislain Gassier
email: ghislain.gassier@gmail.com

Jean Barrère
email: jean.barrere@im2np.fr

Francoise Briolle
email: francoise.briolle@gmail.com

Claude Jauffret
email: claude.jauffret@im2np.fr

A Unifying Approach for Disturbance Cancellation and Target Detection in Passive Radar Using OFDM

come

D R A F T

I. Introduction

A Passive bistatic radar (PBR) exploits illuminators of opportunity to detect moving targets from their Doppler-shifted frequency echoes (see e.g. [START_REF] Forster | Passive radar for target tracking and classification[END_REF], [START_REF] Griffiths | Passive coherent location radar systems. part 1: performance prediction[END_REF], [START_REF] Kulpa | Recent developments in passive radars[END_REF]). We focus here on the very widespread digital video broadcasting -terrestrial (DVB-T) transmitters. These emissions are interesting from a signal processing point of view: the bandwidth (7.6 MHz in Europe) allows the target position to be theoretically estimated with a good resolution (20 m at best); the underlying orthogonal frequency division multiplexing (OFDM) allows the broadcast signal detection through a multipath channel [4], and an antenna dedicated to the reference signal can then be avoided [5].

A main difficulty concerns the presence of the very powerful line-of-sight (LOS) path and its rooted replicas. In the surveillance signal, this strong continuous zero Doppler contribution (ZDC) may mask the echoes of the smallest radar cross-section (RCS) targets in the range-frequency plane of the cross-ambiguity function (CAF). More precisely, since the integration time is necessarily limited 1 , the dominant paths induce both secondary lobes (essentially along the Doppler axis) and a random pedestal abusively called clutter floor, whose magnitude can be higher than the peaks of interest. Since, for moving targets, the observation time must remain reasonably low, a ZDC rejection preprocessing is often performed [START_REF] Colone | A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar[END_REF], [START_REF] Meller | Cheap Cancellation of Strong Echoes for Digital Passive and Noise Radars[END_REF], [START_REF] Poullin | Passive Radar using COFDM (DAB or DVB-T) Broadcasters as Opportunistic Illuminators[END_REF], [START_REF] Chabriel | Passive Covert Radars using CP-OFDM signals A new efficient method to extract targets echos[END_REF]. Classically, a weighted set of time-delayed versions of the broadcast signal is fitted, in the least-squares (LS) sense, to the surveillance signal [START_REF] Colone | A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar[END_REF], [START_REF] Meller | Cheap Cancellation of Strong Echoes for Digital Passive and Noise Radars[END_REF] then subtracted. The main drawbacks in the aforementioned LS-based ZDC rejection are a quite heavy computational and memory loads. To decrease the memory load, a batch implementation is also proposed in [START_REF] Colone | A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar[END_REF]: the ZDC is estimated using consecutive smaller signal portions instead of the whole signal duration. Unfortunately, in this case, remaining correlations with low Doppler sources conduct to a detection loss on both sides near the null Doppler axis. Moreover, for DVB-T signals, the periodical insertion of boosted pilot symbols, used for synchronization and channel equalization, gives rise to multiple ghost-peaks "spamming" the CAF output [START_REF] Palmer | DVB-T Passive Radar Signal Processing[END_REF], [START_REF] Harms | Understanding the signal structure in DVB-T signals for passive radar detection[END_REF]. An alternative method using the OFDM structure and reducing both the memory load and the computational complexity, was proposed by D. Poullin in [START_REF] Poullin | Passive Radar using COFDM (DAB or DVB-T) Broadcasters as Opportunistic Illuminators[END_REF]. The surveillance signal is projected orthogonally to the frequency subspace corresponding to the ZDC. The projector is cleverly approximated by a set of inverse covariance matrices evaluated for each OFDM subcarrier. Let us also mention [START_REF] Chabriel | Passive Covert Radars using CP-OFDM signals A new efficient method to extract targets echos[END_REF] where a low complexity ZDC rejection method based on the CP-OFDM structure of the surveillance signal is described. Note that in all the previous approaches, despite the ZDC mitigation, a stronger target may mask a weaker one as well. Such dominant echoes must then specifically D R A F T be removed by a multistage elimination process as it is proposed in [START_REF] Colone | A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar[END_REF].

Beside these ZDC mitigation studies, some authors suggested to perform a CAF using so-called "mismatched" versions of the illumination signal as a new reference: first, Glende [START_REF] Glende | PCL signal processing for sidelobe reduction in case of periodical illuminator signals[END_REF] proposed, for systems working with analog TV, to use a modulus equalized (a reciprocal filtered) version of the transmitted signal. For DVB-T emissions, Palmer et al. [START_REF] Palmer | DVB-T Passive Radar Signal Processing[END_REF] use some mismatched reference signal to reduce the false alarms due to pilot data. Then, they extended their approach [START_REF] Searle | Evaluation of the ambiguity function for passive radar with ofdm transmissions[END_REF] by using the reciprocal filter associated to a circular correlator. Using a particular sampling along the Doppler dimension (note that such a sampling was already proposed by Berger et al. [START_REF] Berger | Signal Processing for Passive Radar Using OFDM Waveforms[END_REF] for CAF on DAB signals), they obtain a well-performing detector without the need of a ZDC mitigation pre-stage. We will denote this method by the inverse filter approach (IFA).

As main contributions, we propose in this paper, a new interpretation of IFA called "channel detector" (CHAD) and a theoretical study of its performance. The principle of this new approach is to estimate the so-called Doppler channel due to paths with a same Doppler shift; the corresponding impulse response is directly used to detect the underlying echoes. This detector is shown to be theoretically robust to clutter and we prove that its random pedestal is generally lower than that of the classical CAF.

The paper is organized as follows: Section II presents the notion of time-varying Doppler channel associated with the frequency ν characterized by its frequency response H ν or equivalently by the impulse response h ν .

CHAD approach is presented in Section III: an estimation of |h ν | 2 is used to detect the set of echoes with the same Doppler-shifted frequency ν. Theoretical performance comparisons between CHAD and CAF are presented. The validity of the mathematical derivations is illustrated by numerical experiments on simulated and experimental data (Section VI and Section VII).

II. Notion of Doppler Channel

A typical passive bistatic radar scenario is depicted in Fig. 1, where a transmitter illuminates a scene composed of different static scatterers due to topography, buildings . . . , and one or several moving reflectors of interest, namely the targets. A wide spatial-aperture receive antenna collects the radiations from the different paths. Let us consider a passband high frequency broadcast signal of opportunity and let us denote by s e its baseband version. The observation time is here assumed to be sufficiently small so that the Doppler dilatation/compression effect can be neglected. Due to the multipath propagation channel, the baseband received signal y(t) then corresponds to a noisy linear combination of time-delayed replicas of s e (t), with frequency shifts depending on reflector velocities. Denoting by I I , the index set {0, 1, . . . , I -1}, the received signal can then be classically modeled as:

y(t) = i∈I I α i s e (t -τ i -t 0 ) e 2πfp i t +η(t), (1)
where:

• i is the reflector/path index varying from 1 to I -1; i = 0 being reserved for the LOS path, • t 0 is the LOS path propagation delay (from the transmitter to the radar antenna), • τ i is the relative propagation delay such that τ i + t 0 corresponds to the propagation delay of the path i (i.e. from the transmitter to the receive antenna, via the i th reflector), • f p i is the Doppler-shifted frequency corresponding to the path i, • α i are factors depending on positions, RCS of the different reflectors, . . . • η(t) corresponds to the additive white Gaussian noise (AWGN) channel.

The LOS path being indexed by 0, one has necessarily τ 0 = 0 and f p 0 = 0.

Unlike conventional active radar systems, the emitted signal s e is here not directly available. Nevertheless, with a DVB-T emission, one can perfectly recover the transmitted data and consequently synthesize a baseband temporal version of the LOS received signal. This synthetic signal, referred as the "reference signal", is denoted by s in the sequel. The surveillance signal can now be expressed as a function of s:

y(t) = i∈I I α i s (t -τ i ) e 2πfp i t +η(t).
(

Assuming that F different Doppler-shifted frequencies are observed, we gather the paths with the same Dopplershifted frequency, say f j , and define the corresponding index sets

J fj = {i ∈ I I |f p i = f j }, j ∈ I F .
It can then be considered that the reference signal is received through F distinct linear time-varying channels, called subsequently Doppler channels and the observation (2) is rewritten as

y(t) = j∈I F (h fj * s)(t) e 2πfj t +η(t) (3)
where the impulse response h fj (t):

h fj (t) i∈J f j α i δ(t -τ i ) (4)
characterizes the time-invariant part of the j th Doppler channel. The particular channel h 0 (corresponding to the null Doppler frequency f 0 = 0), will be called the static channel.

III. CHAD (Doppler CHAnnel Detection)

A. Recall of DVB-T Signal Main Features

The DVB-T analog signal is a sequence of so-called CP-OFDM symbols, of duration T s . Each symbol is composed of two parts: the useful part of duration T u and a redundant part of duration T cp , called the cyclic prefix (CP). The useful part of a symbol m is composed of K orthogonal subcarriers f k = k Tu carrying a complex data d m k . Most of the subcarriers are modulated by information data d m k which can only take Q complex discrete values depending on the Q-ary QAM modulation used. Some of them are modulated by a sequence of known data: the pilot data. Finally, a set of low and high subcarrier frequencies are not modulated ensuring that the overall bandwidth B stays within K/T u . We denote by E K = {K m , . . . , K M } ⊂ I K , the index subset of the effectively modulated subcarriers2 . The cardinality of E K is denoted #E K .

The cyclic prefix is obtained by copying the last part of the symbol and attaching it in front of the useful part. Its duration T cp is chosen to be greater than the echo delays, to avoid intersymbol (ISI) and intercarrier (ICI) interferences [4]. The temporal expression of any baseband emitted CP-OFDM symbol m is then:

s m (t ) = 1 K k∈E K d m k e 2π k Tu t , ( 5
)
where t is a local time variable such as -T cp ≤ t < T u .

During the observation time, the entire baseband emitted signal forms a sequence of M symbols we will denote by

s(t) = s 0 (t ), . . . , s M -1 (t ) = s m (t ) m∈I M ( 6
)
where t is now the global time variable 0 ≤ t < M T s .

Assuming an error-free DVB-T demodulation, the data d m k , m ∈ I M , k ∈ E K , are assumed known throughout the remainder of the paper.

B. Detection Principle

Since each f j -Dopplerized target is revealed by a peak (ideally a Dirac pulse, see ( 4)) in the impulse response h fj corresponding to its Doppler channel, the main idea is to estimate such an impulse response from the sampled received signal y[k] and to use it as a detection function. Introducing the under test Doppler frequency ν, the estimation of h ν needs the underlying Doppler channel to be stationarized. We then define the corresponding Doppler compensated observation:

y ν (t) y(t) e -2πνt . ( 7
)
Using ( 3), y ν (t) becomes

y ν (t) = j∈I F (h fj * s)(t) e 2π(fj -ν)t +η ν (t), (8)
where η ν (t) is the frequency-shifted noise contribution.

Introducing the index set J ν = i ∈ I I /f p i = ν , let us now rewrite [START_REF] Poullin | Passive Radar using COFDM (DAB or DVB-T) Broadcasters as Opportunistic Illuminators[END_REF] as

y ν (t) = ȳν (t) + ỹν (t) + η ν (t), (9)
where the so-called static part ȳν (t) of y ν (t), if it exists (i.e. if J ν = ∅), gathers the contributions of all paths having a null Doppler shift after compensation (the set of paths with f p i = ν before compensation), while the dynamic part ỹν collects the remaining paths. One has

ȳν (t) = h ν * s (t) (10)
where (ideally)

h ν (t) = i∈Jν α i δ (t -τ i ) . ( 11
)
Let us assume that t = 0 corresponds to the beginning of the useful part of an OFDM symbol coming up the direct path3 . The compensated observed signal can then be cut out into blocks of duration T s :

y ν (t) = y m ν (t ) m∈I M , 0 ≤ t < T s .
Thanks to the CP-OFDM structure of the signal and assuming that the delays τ i , i ∈ J ν are less than the CP duration 4 , it can be shown from ( 4), ( 5) and ( 10) that the static part of each block yields

D R A F T ȳm ν (t ) = k∈E K H ν [k]d m k e 2π k Tu t , 0 ≤ t < T u , ( 12
)
where the #E K complex coefficients H ν [k] correspond to the frequency response of the ν-Doppler channel at frequencies k Tu :

H ν [k] = i∈Jν α i exp -2π k Tu τ i , k ∈ E K . ( 13
)
Let us sample the compensated received signal at the data symbol period T u /K. Using t = n T u /K, we can write for each block

y m ν [n ], m ∈ I M 5 : y m ν [n ] = k∈E K H ν [k]d m k e 2π kn K +ỹ m ν [n ] + η m ν [n ], ( 14
)
n ∈ I K .
Applying a K-points discrete Fourier transform (DFT) on the previous compensated received block, one obtains the following spectral representation:

Y m ν [k] =H ν [k]d m k + Ỹ m ν [k] + Ξ m ν [k], k ∈ E K , ( 15
) with Ỹ m ν DFT{ỹ m ν } and Ξ m ν DFT{η m ν }. It is proved (Subsection A of the Appendix) that ∀k ∈ E K and ∀ν, lim M →+∞ 1 M m∈I M Ỹ m ν [k] d m k = 0 ( 16
)
and lim

M →+∞ 1 M m∈I M Ξ m ν [k] d m k = 0. ( 17
) Hence, Ĥν [k] 1 M m∈I M Y m ν [k] d m k , k ∈ E K ( 18
)
is an asymptotically unbiased estimator of the ν-Doppler channel frequency response H ν . Thereafter, an estimation of the corresponding impulse response ĥν is obtained by a simple Inverse Discrete Fourier transform (IDFT)

ĥν [n] = 1 K n∈I K Ĥν [k] e j2π kn K . ( 19
)
Finally, a parametric sweep over the Doppler frequency shift ν, allows to define the so-called channel detector CHAD through the following range-frequency function:

D(τ, ν) | ĥν [n]| 2 , with τ = nT u /K. ( 20
)
Note that, since a DVB-T signal is bandlimited, the coefficients Ĥν [k], k ∈ I K \E K are non-observable and have then to be replaced by zero. This leads to (squared) sinc-shaped detection peaks (instead of Dirac pulses), limiting the resolution along the range axis.

Finally, for the Doppler-shifted frequencies ν we are interested in 6 , the OFDM symbol duration T u given by the DVB-T standard can be considered small enough to ensure that the phase rotation within one OFDM remains almost constant: e j2πνn Tu K ≈ 1 for all n ∈ I K [START_REF] Moscardini | Batches algorithm for passive radar: a theoretical analysis[END_REF], [START_REF] Howland | Bistatic Radar: Emerging Technology[END_REF]. Hence, it is reasonable to make the following approximation simplifying the computation of each compensated signal block

y m ν [n] ≈ y m [n] e -2πνmTs , ∀n ∈ I K . ( 21
)
Based on the previous derivations, the detection scheme practical main steps can now be detailed as follows:

1) From the different blocks of the sampled surveillance signal, form the M following vectors, ∀m ∈ I M :

y m = y m [0], . . . , y m [n], . . . , y m [K -1] T .
2) Compute the K-points discrete Fourier transforms of each vector y m :

Y m = Y m [0], . . . , Y m [k], . . . , Y m [K -1] T , with ∀m ∈ I M , ∀k ∈ I K : Y m [k] = 1 K n∈I K y m [n] e -j2π kn K .
3) Compute the M channel frequency responses corresponding to the different blocks

H m = H m [0], . . . , H m [k], . . . , H m [K -1] T ,
with ∀m ∈ I M :

H m [k] = Y m [k]/d m k , ∀k ∈ E K , H m [k] = 0, ∀k ∈ I K \E K .
4) Choose a value ν for the Doppler compensation and compute the associated channel frequency response using the following linear combination (see [START_REF] Karr | Probability[END_REF] and ( 21)

) Ĥν = 1 M m∈I M e -j2πνmTs H m . ( 22

C. Performance Analysis

In this subsection, we compare for the two studied detectors (CAF and CHAD), the detection peak mean shape, and the floor mean levels for clutter and noise. The study will be conducted considering d m k as a random variable and neglecting the effect of the pilot deterministic sequences inserted in the data stream.

Let us consider that the surveillance antenna is impinged by a single DVB-T signal i whose coordinates are (n i Tu K , f p i ) in the delay-Doppler plane. This signal should 6 -500 Hz < ν < 500 Hz be seen as a potential source of disturbance (secondary lobes and a clutter floor) for the detection of a weaker target. Assuming that the bistatic propagation delay n i Tu K is less than the guard interval duration, an m th noise-free, powernormalized, received sampled block yields, ∀n ∈ I K

y m [n ] = s m [n -n i ] e 2πfp i (n -n i )Tu K +mTs , ( 23
)
where the unit-variance reference signal s is assumed to be stationary block m by block m i.e. E{|d m k | 2 } does not depend on m. Now since in the DVB-T standard, one has

E{|d m k | 2 } = 1, it comes that ∀m s m [n ] = 1 √ #E K k∈E K d m k e 2π n k K , ( 24
)
ensuring the unit-variance assumption.

1) CAF based detector: the cross-ambiguity function A is here defined as the squared modulus of the crosscorrelation function X :

A = |X [n, ν]| 2 , ( 25
)
where

X [n, ν] = 1 M K m∈I M n ∈I K y m [n ](s m [n -n]) * e -2πνmTs , (26)
is computed from the symbol useful parts (M K samples) with a constant phase rotation within a symbol (see [START_REF] Berger | Signal Processing for Passive Radar Using OFDM Waveforms[END_REF]).

The two parameters under test (n Tu K , ν) are respectively the bistatic propagation delay (bistatic range) and the Doppler-shifted frequency.

Using (23), the expression (26) yields

X [n, ν] = γ M K m∈I M e 2π(fp i -ν)mTs n ∈I K s m [n -n i ](s m [n -n]) * e 2πfp i n Tu K , ( 27
) with γ = e -2πfp i n i Tu K .
Finally, replacing the expression of s m using (24), it comes

X [n, ν] = γ M K#E K m∈I M e 2π(fp i -ν)mTs k,k ∈E 2 K d m k (d m k ) * e 2π(nk -n i k) K s kk (f p i ), ( 28
)
where

s kk (f p i ) n ∈I K e 2π(k-k +fp i Tu )n K = sin π(k -k + f p i T u ) sin π(k -k + f p i T u )/K . ( 29
)
Note that s kk (f p i ) does not depend on k. Now, observing a CAF detection output, for example in Figure 5, one can clearly distinguish, in the range-Doppler plane, both a non-stationary part and a stationary part. The non-stationary part (a squared sinc-like function) corresponds to the detection peak shape while the stationary part is a random variable we shall refer as the clutter floor. This behavior can be explained using the following decomposition:

A[n, ν] = |X e [n, ν] + X d [n, ν]| 2 , ( 30
)
with

X e [n, ν] = X [n, ν] when k = k in (28), ( 31
)
and

X d [n, ν] = X [n, ν] when k = k . ( 32
)
Let us study the behavior of E{A[n, ν]}. First, the random variables d m k and d m k being zero-mean and uncorrelated for all k = k [START_REF] Wei | Convergence of the Complex Envelope of Bandlimited OFDM Signals[END_REF], it can easily be shown that X d [n, ν] is centered for all (n, ν). Second, it is proved in Appendix B that X e and X d are uncorrelated. Then, introducing the zero-mean random function Y X e -E{X e } + X d , it comes

E{A[n, ν]} = E{X e } 2 + Var{Y}. ( 33
)
As we will see subsequently, only the first right-hand side term of (33) varies with (n, ν): it corresponds to the peak shape; the second (constant) term matches the clutter floor level. a) Peak shape: the CAF peak average shape is given by (see Appendix C for detailed derivations):

|E {X e [n, ν]}| 2 = sin 2 πfp i Tu sin 2 πfp i Tu/K (M K) 2 (#E K ) 2 sin 2 π(f p i -ν)M T s sin 2 π(f p i -ν)T s sin 2 π(n -n i )#E K /K sin 2 π(n -n i )/K . ( 34
)
As expected and due to the limited bandwidth, the peak shape corresponds to a squared sinc-like function (a Dirichlet kernel) limiting the resolution along the range axis. Similar secondary lobes are observed along the Doppler axis, depending on the number M of OFDM symbols used.

Replacing n by n i and ν by f p i in the latter equation and considering the approximation

sin 2 πfp i Tu sin 2 πfp i Tu/K ≈ K 2 , one obtains the detection peak mean level E X e [n i , f p i ] 2 ≈ 1 M 2 (#E K ) 2 M 2 (#E K ) 2 i.e. 0dB. ( 35
)
b) Clutter floor mean level: we have

Var{Y} = Var{X e } + Var{X d }.
The calculations of the variances of X e and X d are detailed respectively in Appendix D and Appendix E. We get

Var{X e [n, ν]} = sin 2 πfp i Tu sin 2 πfp i Tu/K M K 2 #E K E |d m k | 4 -1 (36) D R A F T (Note that for QAM signals, E |d m k | 4 never exceeds 1.

381), and

Var X d = 1 M K 2 (#E K ) 2 k k =k |s kk (f p i )| 2 . ( 37
)
It follows that Var{Y} does not depend on (n, ν).

Note that Var X d < Var X e and the variable |X e [n, ν] -E{X e [n, ν]} 2 remains the main source of clutter.

c) Noise floor mean level: let us consider now that the received signal is white noise only:

y m [n ] = η m [n ].
One can then write using (26)

X [n, ν] = 1 M K √ #E K m∈I M e -2πνmTs k∈E K (d m k ) * e 2πnk K n ∈I K η m [n ] e -2πn k K . ( 38
)
Since the two independent random variables d m k and η m [n ] are assumed to be i.i.d. sequences, computing the modulus squared then the mathematical expectation, the noise floor mean level yields using

E{|d m k | 2 } = 1 E{A[n, ν]} = 1 M 2 K 2 m∈I M n ∈I K E{|η m [n ]| 2 } (39) = E{|η m [n ]| 2 } M K . ( 40
)
2) CHAD: applying a DFT on a received symbol

y m , Y m [k ] n ∈I K y m [n ] e -2πn k K
, and dividing by

√ #E K d m k , k ∈ E K ,
one obtains after some derivations (see the Appendix F) the expression of the frequency response of the propagation channel corresponding to the reception of the m th symbol:

H m [k ] Y m [k ] √ #E K d m k = γ #E K e 2πfp i (mTs) k∈E K d m k d m k e -2πn i k K s kk (f p i ), (41)
where γ = e -2πfp i n i Tu K . Applying now the Doppler correction e -2πν(mTs) for each symbol m and averaging over the M available symbols, one has

H ν [k ] 1 M m∈I M e -2πν(mTs) H m [k ] = (42) γ M #E K m∈I M e 2π(fp i -ν)mTs k∈E K d m k d m k e -2πn i k K s kk (f p i ). ( 43
)
The corresponding impulse response is obtained by IDFT:

h ν [n] = γ M K#E K m∈I M e 2π(fp i -ν)mTs k ∈E K e 2πnk K k∈E K d m k d m k e -2πn i k K s kk (f p i ). ( 44
)
Let us consider the detection function

D[n, ν] = |h ν [n]| 2 ,
with the following decomposition:

h ν [n] = h e ν [n] + h d ν [n] ( 45
)
where h e ν gathers the terms in (44) with k = k and where h d ν stands for the remaining terms (k = k ). Since h e ν is deterministic (it does not depends on data d m k ) and h d ν is a centered random variable (here E{

d m k d m k } = 0), it comes E{D} = |h e ν | 2 + E |h d ν | 2 , ( 46
h e ν [n] = γ M K#E K m∈I M e 2π(fp i -ν)mTs k∈E K e 2π(n-n i )k K s kk (f p i ). ( 47
)
Since s kk (f p i ) depends no more on k, it comes

h e ν [n] = (48) γ M K#E K e π(fp i -ν)(M -1)Ts e π(n-ni) K M +Km -1 K e πfp i Tu K-1 K sin π(f p i -ν)M T s sin π(f p i -ν)T s sin π(n -n i )#E K /K sin π(n -n i )/K sin πf p i T u sin πf p i T u /K .
Using |γ| = 1, we straightforwardly obtain the CHAD detection peak shape as the squared modulus of the latter expression:

|h e ν [n]| 2 = sin 2 πfp i Tu sin 2 πfp i Tu/K M 2 K 2 (#E K ) 2 sin 2 π(n -n i )#E K /K sin 2 π(n -n i )/K sin 2 π(f p i -ν)M T s sin 2 π(f p i -ν)T s . ( 49
)
which appear identical to that obtained with classical CAF.

b) Clutter floor mean level: as for CAF, the clutter forms a stochastic pedestal. The difference is that, here, only the terms where k = k contribute to the clutter floor.

Using (44), one finds:

E h d ν [n] 2 = 1 (M K#E K ) 2 m,m ∈I 2 M e 2π(fp i -ν)(m-m )Ts (50) k2,k 2 ∈E 2 K e 2πn(k 2 -k 2 ) K k1 =k2,k 1 =k 2 ∈E 2 K E d m k1 d m k2 (d m k 1 ) * (d m k 2 ) * e 2πn i (k 1 -k 1 ) K s k1k2 (f p i )s * k 1 k 2 (f p i ). Since E {d m k } = 0 and E 1 d m k = 0, then E d m k 1 d m k 2 (d m k 1 ) * (d m k 2
) * = 0 for m = m . Additionally, it can be shown that for QAM signals E

d m k (d m k ) *
= 0, hence only the terms with k 1 = k 1 and k 2 = k 2 are not null in the previous expression and the clutter mean level yields

Var h d ν [n] = E 1 |d m k | 2 M (K#E K ) 2 k k =k s 2 kk (f p i ). ( 51
H ν [k] = 1 M m∈I M Y m [k] √ #E K d m k e -2πνmTs (52)
= 1 M √ #E K m∈I M 1 d m k e -2πνmTs n ∈I K η m [n ] e -2πn k K .
Applying a IDFT, we get the corresponding impulse response

h ν [n] = 1 M K √ #E K k∈E K m∈I M 1 d m k e -2πνmTs n ∈I K η m [n ] e 2π(n-n )k K . ( 53
)
The two independent random variables d m k and η m [n ] being zero mean and temporally white, variance of h ν [n] yields

E{|h ν [n]| 2 } = 1 (M K) 2 #E K m∈I M k∈E K E 1 |d m k | 2 n ∈I K E{|η m [n ]| 2 } (54) = ρ E{|η m [n ]| 2 } M K , ( 55
)
where

ρ = 1 #E K k∈E K E 1 |d m k | 2 ≈ E 1 |d m k | 2 .
This latter expression indicates that the noise floor mean level depends on the QAM type. On can note that for 4-QAM, CAF and CHAD exhibits the same noise floor level (ρ = 1). For other types of QAM, the noise floor is unfortunately higher, since one has ρ = 1.8 for 16-QAM (+2.5dB), and ρ = 2.6 for 64-QAM (+4.1dB).

D. Performance comparison

From the clutter floor theoretical mean level of each studied detector (37) for CAF and (51) for CHAD, we will evaluate the improvement brought by CHAD compared to CAF. Let us consider the two zero mean complex variables: Y = X e -E{X e }+X d (for CAF) and Z h d ν (for CHAD), and the clutter floor attenuation coefficient defined by

c Z/Y (f p i ) Var{Z}
Var{Y} . The attenuation c Z/Y (f p i ), depends on the absolute value of the Doppler-shifted frequency f p i of the disturbance source (s kk f p i is an even function) and is parametrized by the number of subcarriers and the QAM type. One has after some developments:

c Z/Y (f p i ) = Var{h d ν } Var{X e } + Var{X d } = (56) E 1 |d m k | 2 k k =k s 2 kk (f p i ) #E K sin 2 πfp i Tu sin 2 πfp i Tu/K E |d m k | 4 -1 + k k =k s 2 kk (f p i )
. One can see that, using CHAD, the clutter floor level remains lower than that obtained with a CAF for low frequencies (up to about 300Hz for the 8K mode). In particular and very interestingly, since s kk (0) = 0, the clutter floor induced by the set of zero-Doppler paths reduces to zero. Var{Z} |Z| 2 are then chi-squared distributed with 2 degrees of freedom (χ 2 2 ) and we get the two following results

Prob |Y| 2 > 9.21 Var{Y} 2 = 1%, ( 57
)
Prob |Z| 2 > 9.21 Var{Z} 2 = 1%, ( 58
)
giving a 1% confidence interval for the clutter floor level of each detector.

As an example, Figure 3 presents, for a 64-QAM -8K, 90 symbols emission, the clutter floor 1% confidence intervals (in dB) for both CAF and CHAD.

E. Short study of the case: τ i > T cp

Without any loss of generality, we consider a single delayed static contribution, such that T cp < τ i < T u ( i.e 0 < n i < K -1). It follows an overlapping of symbol m + 1 on the m th received block. For a noise-free case, one has from Equation ( 14)

y m [n ] = k ∈E K H 0 [k ]d m-1 k e j2π k n K n = 0, . . . , n i -1 (59) + k ∈E K H 0 [k ]d m k e j2π k n K n = n i , . . . , K -1.
After DFT and data division, we obtain

Y m [k] d m k = 1 K K-1 n =0 y m [n ] d m k e -j2π kn K (60) = 1 K ni-1 n =0 k ∈E K H 0 [k ] d m-1 k d m k e j2π (k -k)n K + 1 K K-1 n =ni k ∈E K H 0 [k ] d m k d m k e j2π (k -k)n K ,
whose the expectation value is

E Y m [k] d m k = 1 K ni-1 n =0 k ∈E K H 0 [k ] E d m-1 k d m k e j2π (k -k)n K (61) + 1 K K-1 n =ni k ∈E K H 0 [k ] E d m k d m k e j2π (k -k)n K = 1 K K-1 n =ni H 0 [k] = K -n i K H 0 [k].
Thus, when τ i varies from T cp to T s , the corresponding channel frequency response linearly decreases. When τ i > T s , it remains null. This result is still valid for any Doppler echo.

IV. ZDC rejection from static channel estimation

Whether with CHAD or CAF, the static echo secondary lobes remain and their strong amplitude can still compromise the detection of the weakest targets. This disturbing zero-Doppler contribution has then to be removed and we therefore propose two ways.

A. ZDC rejection preprocessing

A classical strategy is to implement the ZDC rejection as a preprocessing basically applied on the received signal y[k] (see e.g. [START_REF] Colone | A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar[END_REF], [START_REF] Poullin | Passive Radar using COFDM (DAB or DVB-T) Broadcasters as Opportunistic Illuminators[END_REF], [START_REF] Chabriel | Passive Covert Radars using CP-OFDM signals A new efficient method to extract targets echos[END_REF]). As in [START_REF] Chabriel | Passive Covert Radars using CP-OFDM signals A new efficient method to extract targets echos[END_REF], it can be done here using the estimation Ĥ0 [k] of the zero-Doppler channel frequency response (obtained using [START_REF] Karr | Probability[END_REF] with ν = 0). Then, we directly get the estimate of the spectral representation of the dynamic part, computing for each block m ∈ I M :

Ỹ m [k] Y m [k] -Ĥ0 [k]d m k , k ∈ E K . ( 62
)
This ZDC-free signal can be used to detect most of the formerly hidden targets, implementing a classical CAF or a CHAD detector.

Note that, similarly to the approach in [START_REF] Poullin | Passive Radar using COFDM (DAB or DVB-T) Broadcasters as Opportunistic Illuminators[END_REF], the computational cost (the number of complex multiplications) of such a preprocessing is only O(M K log 2 (K)) while the least mean squares (LMS) method [START_REF] Colone | A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar[END_REF] exhibits a complexity having a cubic growth with K (O(M K 3 )).

B. Online ZDC rejection

The CHAD approach allows to directly implement the ZDC rejection during the estimation of the Doppler channel frequency response. From (62), it can be done replacing (22) by

Ĥν = 1 M m (H m -Ĥ0 ) e -2πνmTs , (63)
where Ĥ0 = 1 M m H m corresponds to the static channel vector estimation and where Ĥν stands now for the ZDCfree stationarized Doppler channel. After some derivations, it can be rewritten as

Ĥν = 1 M m e -2πνmTs - 1 -e -2πνM Ts M (1 -e -2πνTs ) H m , ( 64
)
unifying in one and the same step the ZDC rejection and the detector construction while not increasing the computational complexity.

V. CHAD algorithm

Based on the previous derivations, the resulting channel detector algorithm (CHAD) is given in Table I. Note that since 1

d m k = (d m k ) * |d m k | 2 ,
we retrieve here the approach in [START_REF] Searle | Evaluation of the ambiguity function for passive radar with ofdm transmissions[END_REF] where the reference signal is mismatched by dividing by the squared modulus of each datum d m k (modulus equalization). As outlined in section I, Searle et al. [START_REF] Searle | Evaluation of the ambiguity function for passive radar with ofdm transmissions[END_REF] propose to ignore the secondary lobes by computing the "mismatched reference signal" CAF with a Doppler shift resolution such that "any Fourier bin exists at a zero of any other bin", that is to say using ∆ ν = 1 M Ts . It is very interesting because ZDC sidelobes (artificially) disappear and ZDC mitigation may appear now useless. However, it follows a detection loss and consequently we preferred to keep in most of the numerical results, both an oversampling on the Doppler dimension and a ZDC cancellation. Note that as suggested in [START_REF] Berger | Signal Processing for Passive Radar Using OFDM Waveforms[END_REF], [START_REF] Searle | Evaluation of the ambiguity function for passive radar with ofdm transmissions[END_REF], an ad hoc Doppler shift range allows the straightforward estimate of H ν by FFT.

VI. Numerical Simulations

We consider a simulation scenario where four moving targets are illuminated by a 8K, 64-QAM, DVB-T emission with T cp = T u /16. The target echoes Tg 1 , Tg 2 ,Tg 3 and Tg 4 are received with respectively a 40 dB, 60 dB, 65 dB and 90 dB power attenuation (relatively to the LOS emission). Their coordinates in the range-frequency plane are respectively: (3 km, 30 Hz), (6 km, 90 Hz), (7 km, -150 Hz) and (12 km, -100 Hz). Additionally, we simulate the presence of about 30 static echoes with a minimal attenuation of 30 dB. These echoes are randomly distributed in a bistatic range less than 16 km and have consequently bistatic propagation delays less than the guard interval. A 50 dB attenuation AWGN channel is considered. Finally, the Doppler shift step ∆ ν is set to 1 Hz. The stochastic pedestal level induced by the LOS corresponding peak does not exceed (1% confidence interval) about -55 dB (see Figure 3 for a null Dopplershifted frequency). Consequently, and as it can be seen in the figure, only the most powerful target Tg 1 remains visible while the other three Doppler echoes are hidden by the LOS clutter level.

Three ZDC rejection preprocessings have been implemented and tested before the CAF detection. Figure 5 shows a CAF output obtained using the approach proposed in subsection IV-A . The stochastic floor level induced by the new dominant echo Tg 1 (30 Hz) reduces to -95 dB, corresponding to the theoretical value obtained by combining its power attenuation (-40 dB) and the CAF clutter level for a unitpower source (-55 dB, see the zoomed view in Figure 3). The targets Tg 2 and Tg 3 are now detected, but Tg 4 outcrops the clutter level and is then not clearly visible. A false detection peak betraying the presence of pilot signals, appears (ghost target on the figure). Note that this ambiguity peak could be reduced using a reference signal synthesized with a pilot level reduction [START_REF] Palmer | DVB-T Passive Radar Signal Processing[END_REF], [START_REF] Harms | Understanding the signal structure in DVB-T signals for passive radar detection[END_REF].

Very close results are obtained by the other two preprocessings we tested: the LMS-based method [START_REF] Colone | A Multistage Processing Algorithm for Disturbance Removal and Target Detection in Passive Bistatic Radar[END_REF] and the zero-Doppler orthogonal projection [START_REF] Poullin | Passive Radar using COFDM (DAB or DVB-T) Broadcasters as Opportunistic Illuminators[END_REF]. For the LMS-based method, both a high computational load (O(M K 3 )), and a high memory load are necessary since a M K × T cp = M K × K/16 dimensional matrix has to be pseudo-inverted. Since the stochastic pedestal induced by the static emissions is now null, the three most powerful targets are clearly detected but the weakest target remains hidden by the secondary lobes of the LOS detection peak. Finally, one can see that, since CHAD detection peak is data-independent (see eq. ( 49)), the ghost peak appears no more.

VII. Experimental Data

We present here a detection result obtained with an acquisition system located near the city of Rennes in France. The target is a ferryboat illuminated by the DVB-T transmitter "Bécherel" whose modulation parameters are: 8K mode, 64-QAM and T cp = T u /32. In the rangefrequency plane, the target bistatic coordinates are about (46 km, -67 Hz).

The figure 8 presents the detection maps obtained without ZDC rejection with both CAF (left) and CHAD (right) using M = 190 OFDM symbols and ∆ ν = 1 Hz. The ferryboat location conducts to a bistatic delay higher than six times the CP duration and consequently out of the τ i < T cp assumption. It follows (see III-E) a detection loss depending on the amount of data outside the guard interval i.e. here about 10 log 10 ((32 -6) 2 /32 2 ) ≈ -1.8 dB, see eq. ( 61). One can see that despite this slight loss of detection, the clutter level being greatly attenuated (about 30 dB), the target is clearly visible (lower right corner) with CHAD while classical CAF fails. Note that the clutter reduction level obtained with CHAD is apparently lower than in the previous simulations. This can be explained by a higher noise floor which is reached here, limiting the performance.

The last figure 9 presents the detection maps obtained by each detector after a ZDC rejection. The results are close to each other but one can observe that, as expected, CHAD exhibits a noise floor level slightly higher than the classical CAF.

VIII. Conclusion and Perspectives

In this paper, we introduce the notion of "Doppler channel". We show how its associated impulse response can be efficiently computed for DVB-T emissions and used as a new performing target detection tool, we named CHAD (channel detector). We theoretically compare the performance of CHAD to that of classical CAF. In particular, we demonstrate that, very interestingly, for low Doppler-shifted contributions, the induced clutter level remains significantly inferior to that of CAF. This level even reaches zero for zero-Doppler contributions. CHAD actually appears to be equivalent to a CAF based on the mismatched reference signal proposed in [START_REF] Glende | PCL signal processing for sidelobe reduction in case of periodical illuminator signals[END_REF], [START_REF] Searle | Evaluation of the ambiguity function for passive radar with ofdm transmissions[END_REF]. However, this new interpretation in terms of Doppler channels opens a much wider range of possibilities. For example, one can now directly merge the ZDC rejection with the detector without increasing the computational load. In the Doppler channel domain, the detection problem becomes a simple harmonic analysis problem and one can imagine that we might improve the performance, using e.g. high-resolution spectrum-estimation methods. Finally, a CHAD extension to spatial array-processing is still possible.

Appendix

This appendix provides details of the longest derivations referred to throughout the text.

A. Estimation of the static channel

With no loss of generality, the mathematical derivations will be conducted with a null Doppler compensation (ν = 0) and a single Doppler path i (f p i = 0). From ( 15), we then have,

∀k ∈ I K 1 M m∈I M Y m 0 [k] d m k =H 0 [k] + 1 M m∈I M Ỹ m 0 [k] d m k + 1 M m∈I M Ξ m 0 [k] d m k .
To simplify the notations, the subscript 0 will be omitted in the sequel.

1) Study of 1 M m∈I M Ỹ m [k] d m k
: this sum has to be considered as a source of disturbance (in addition to the noise) for the channel H estimation. Its contribution to the LOSsynchronized received signal yields ỹ(t) = α i s (t -τ i ) e 2πfp i t , where the relative propagation delay τ i is assumed to be less than T cp .

The received signal being sampled and cut off according to the OFDM structure, the corresponding m th sampled received block ỹm [n ] can then be written as ỹm

[n ] = α i s m n T u K -τ i -mT s e 2πfp i (n Tu K -τi-mTs) , n ∈ I K . with s n T u K -τ i -mT s = 1 K k∈E K d m k e 2π k Tu (n Tu K -τi-mTs) . It comes ỹm [n ] = α i k∈E K e -2π kτ i Tu d m k e 2π kn K e 2πfp i (n Tu K -mTs) , n ∈ I K ,
where α i = αi K e -2πfp i τi .

The Doppler-shifted frequencies f p i we deal with, are assumed sufficiently low so that the phase rotation D R A F T e 2πfp i n Tu K can be considered as constant e 2πfp i n Tu K ≈ 1, ∀n ∈ I K . We can then write ỹm

[n ] = α i k∈E K e -2π kτ i Tu d m k e 2π kn K e -2πfp i mTs , n ∈ I K . Now applying a K-points DFT, one obtains Ỹ m [k ] = α i k∈E K e -2π kτ i Tu d m k n ∈I K e 2π (k-k )n K e -2πfp i mTs , k ∈ E K . ( 65
) Since n ∈I K e 2π (k-k )n K is null except for k = k , then Ỹ m [k ] simplifies to Ỹ m [k ] = α i d m k e -2πfp i mTs , k ∈ E K , ( 66
) with α i = Kα i e -2π k τ i Tu . 1 M m∈I M Ỹ m [k ] d m k = α i M m∈I M e -2πfp i mTs , = α i M 1 -exp -2πf p i M T s 1 -exp -2πf p i T s , ∝ 1
p i )| 2 (M K#E K ) 2 m∈I M k∈E K E |d m k | 4 -E |d m k | 2 2 = sin 2 πfp i Tu sin 2 πfp i Tu/K M K 2 #E K E |d m k | 4 -1 .

E. CAF clutter floor level -equation (37)

Using (28) with k = k , one has:

Var X d = 1 (M K#E K ) 2
Var X d = 1 M K 2 (#E K ) 2 k∈E K k ∈E K , k =k s 2 kk (f p i ).
F. Equation ( 41) We recognize in the latter summation, the expression of the function s kk (f p i ) defined in (29). It comes

H m [k ] Y m [k ] √ #E K d m k = n ,
H m [k ] = γ #E K e 2πfp i (mTs) k∈E K d m k d m k e -2πn i k K s kk (f p i ),
where γ = e -2πfp i n i Tu K .

1 A

 1 Unifying Approach for Disturbance Cancellation and Target Detection in Passive Radar using OFDM G. Gassier, G. Chabriel, J. Barrère, F. Briolle and C. Jauffret Abstract-This paper addresses the problem of mobile target detection in multipath scenarios with a passive radar using DVB-T transmitters of opportunity. For such emissions, it has been shown the interest in implementing "mismatched" correlators, reducing both the zero Doppler contribution (ZDC) masking effects and the false alarm rate. A very efficient mismatched reference signal is obtained with the reciprocal filter (or inverse filter) which consist in a modulus frequential equalization of the transmitted signal. We propose here to revisit the reciprocal filter based correlator and to reinterpret it as a so-called Doppler channel detector (CHAD). This new interpretation allows a direct rejection of the ZDC, unifying in one and the same step the main disturbance mitigation and the detector construction. We provide a statistical theoretical study of the performance and a comparison with the matched correlator i.e. the classical cross-ambiguity function (CAF). We demonstrate that CHAD has a random pedestal (a clutter floor level) significantly lower than that of the classical CAF for low Doppler frequency shifts. Numerical experiments on simulated and real data as well validate the mathematical derivations. Index Terms-Passive bistatic radar (PBR), passive covert radar (PCR), passive coherent location (PCL), digital video broadcasting-terrestrial (DVB-T), orthogonal frequency division multiplexing (OFDM), crossambiguity function (CAF), mismatched filter, reciprocal filter, clutter.

Fig. 1 .

 1 Fig. 1. Passive bistatic radar typical scene

) 5 )

 5 Compute the detection function corresponding to the Doppler shiftν: D(τ, ν) = | ĥν [n]| 2, where ĥν [n] is obtained by IDFT of vector Ĥν (see[START_REF] Adali | Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety[END_REF]) and where τ = nT u /K. 6) Repeat steps 4) and 5) for different values of parameter ν.

) were the deterministic function |h e ν | 2

 2 corresponds to the detection peak and the mean value of the random variable |h d ν | 2 corresponds to the clutter floor mean level. a) Peak shape: let us study the deterministic part corresponding to the terms where k = k in (44)

  ) c) Noise floor mean level: as for CAF, let us consider that the received signal is white noise only: y m [n ] = η m [n ]. One obtains the following Doppler channel frequency response

Figure 2

 2 Figure 2 presents plots of the attenuation c Z/Y according to the disturbing path Doppler shift f p i , for 64-QAM with the two DVB-T K-modes.

Fig. 2 .

 2 Fig. 2. CHAD/CAF clutter floor attenuation coefficient c Z/Y (dB).

2 I 2 2 I 2 . The two variables 2 Var{Y}

 22222 The variables Y (CAF) and Z (CHAD) are obtained by summing a large number (M K) of independent data d m k . Hence, invoking the central limit theorem, it can be considered that these variables are normally distributed. Moreover, since Re{d m k } and Im{d m k } are independent random variables with the same variance, one has for the two following real random vectors: (Re{Y}, Im{Y}) ∼ N 0, Var{Y} and (Re{Z}, Im{Z}) ∼ N 0, Var{Z} |Y| 2 and 2

Fig. 3 .

 3 Fig. 3. Clutter floor level for a unit-power source, 90 symbols, 64-QAM, 8K (1% confidence interval in dB).

1 T

 1 0) Inputs d m = d m 0 , . . . , d m k , . . . , d m K-, m ∈ I M y m = y m [1], . . . , y m [n ], . . . , y m [K -1] T , m ∈ I M 1) Compute H m = DFT[y m ] 1 d m where stands for the element-wise product 2) Estimation of the Doppler channel impulse responses for l = -L/2, . . . , L/2 a. ν[l] = l∆ν b. Hν = 1 M m H m wm, wm = e -2πνmTs for a straightforward detection e -2πνmTs -1-e -2πνM Ts M (1-e -2πνTs ) detection including ZDC rejection c. hν = IDFT[Hν ] end 3) Echo detection Construct the K × L detection map D such that D[n, l] = |h ν[l] [n]| 2 .

Figure 4

 4 Figure 4 presents a CAF output computed over 90 consecutive OFDM symbols (i.e. a 85 ms duration and 90 × 8192 = 737280 samples) from the original received sequence (y[n]) n∈I N .

Fig. 4 .

 4 Fig. 4. CAF using the original received signal.

Fig. 5 .

 5 Fig. 5. CAF after a ZDC rejection prior stage.

Figure 6

 6 Figure 6 presents the detection map obtained using CHAD (90 symbols) without ZDC rejection (22).

Fig. 6 .

 6 Fig. 6. CHAD output D without ZDC rejection.

Figure 7

 7 Figure 7 corresponds to the detection map obtained with CHAD using an H 0 compensation (64) in order to remove the LOS secondary lobes. The clutter level induced by the most powerful remaining target Tg 1 is -113.25 dB (obtained by combining the -40 dB power attenuation and the -73.25 dB CHAD theoretical clutter level of Figure 3), that is to say less than the noise floor (-50 -10 log 10 (90 × 8192) ≈ -108 dB) which becomes the main hiding effect.

Fig. 7 .

 7 Fig. 7. CHAD output D including H 0 compensation.

Fig. 8 .

 8 Fig. 8. CAF (left) and CHAD (right) on real data without ZDC rejection.

Fig. 9 .

 9 Fig. 9. CAF (left) and CHAD (right) on real data with ZDC rejection.

M:.. 2 = 0 1 K 2 = 2 = 2 m,m ∈I 2 Me 2 K E |d m k | 2 -

 201222222 sin πf p i M T s sin πf p i T s .Now, since | sin πf p i M T s | ≤ 1 , it comes lim from the strong law of large numbers, one has (see[START_REF] Karr | Probability[END_REF] pp. 151-152)The zero-mean random variables Ξ m [k] and 1 of the cross-correlation between X e and X d One has using (28) both with k = k and k = k :E X e (X d ) * = E s k1k1 (f p i ) M K#E K m1∈I M e 2π(fp i -ν)m1Ts k1∈E K |d m1 k1 | 2 e 2π(n-n i )k 1 K 1 M K#E K m2∈I M e -2π(fp i -ν)m2Ts k2,k 2 =k2 Since E |d m1 k1 | 2 (d m2 k2 ) * d m2 k for any triplet (k 1 , k 2 , k 2 ) such that k 2 = k 2 , we get E X e (X d ) * = 0,and the random variables X e and X d are then uncorrelated.C. CAF peak shape -equation (34)Since s kk (f p i ) does not depend on k, it comes using (28) with k = k :E {X e [n, ν]} = γs kk (f p i ) M K#E K m∈I M e 2π(fp i -ν)mTs K M k=Km e 2π(n-n i )k K = γs kk (f p i ) M K#E K m∈I M e 2π(fp i -ν)mTs e 2π(n-n i )Km K K M -Km k=0 e 2π(n-n i )k K = γs kk (f p i ) M K#E K e π(fp i -ν)(M -1)Ts e π(n-ni) K M +Kmsin π(f p i -ν)M T s sin π(f p i -ν)T s sin π(n -n i )#E K /K sin π(n -n i )/K .Taking the squared modulus, then using (29) and |γ| = 1, we finally obtain the expression of the detection peak mean shape|E {X e [n, ν]}| 2 = sin 2 πfp i Tu sin 2 πfp i Tu/K (M K#E K ) 2 sin 2 π(f p i -ν)M T s sin 2 π(f p i -ν)T s sin 2 π(n -n i )#E K /K sin 2 π(n -n i )/K .D. CAF clutter floor mean level -equation (36)One has, using (28) with k = k :Var{X e } = E X e -E{X e } E s kk (f p i ) M K#E K m∈I M e 2π(fp i -ν)mTs k∈E K (|d m k | 2 -E{|d m k | 2 }) e 2π(n-n i )k K |s kk (f p i )| 2 (M K#E K ) 2π(fp i -ν)(m-m )Ts k,k ∈E E{|d m k | 2 } |d m k | 2 -E{|d m k | 2 } e 2π(n-ni) (k-k )K This expression is clearly null for m = m or k = k and one finally obtains Var{X e [n, ν]} = |s kk (f

k 2 =k2∈E 2 K d m k 2 1 ) * (d m k2 ) * d m k 2 }

 2212 m k 1 ) * e 2π(nk 1 -n i k 1 ) K s k1k 1 (f p i ) m ∈I M e -2π(fp i -ν)m Ts k2,(d m k2 ) * e -2π(nk 2 -n i k 2 ) K (s k2k 2 (f p i )) * = 0 for m = m and since a QAM signal is second-order circular (E{(d m k ) 2 } = 0)[START_REF] Adali | Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety[END_REF], only the terms with k 1 = k 2 and k 1 = k 2 remain. Hence, it comes using (Var {d m k }) 2 = 1:

  k∈I K ×E K

TABLE I CHAD

 I

(Channel Detector)

Note that in the DVB-T standard, #E K /K ≈ 0.83.

CP/pilot-aided standard synchronization.

This assumption may seem restrictive but it corresponds anyway to bistatic distances up to 70km (8K mode, Tcp = Tu/4). Moreover, we will see that this assumption can be reasonably overridden.

In practice, y m ν [n ] is built from the sampled version of the received signal, that is to say y m ν [n ] = y m [n ] e -2πν(n +m) Tu K .

Ghislain Gassier received his Master degree in applied mathematics (Statistics -Data Processing) from the Aix Marseille Université (AMU), France, in 2010, and his Ph.D. degree from the Université de Toulon (UTLN), France, in 2016. He has been working as a teaching assistant for the French Ministry of Defence (French Air Force Academy -Ecole de l'Air) and did his research activities both in the Morpho-Analysis in Signal Team (MAS'Air) of the Research Center of the French Air Force (CReA), Salon-De-Provence, and in the Signals and Systems Team of IM2NP Laboratory, Toulon, France.

Gilles


Jean Marc Menaud
email: jean-marc.menaud@imt-atlantique.fr

Jonathan Pastor
email: jonathan.pastor@imt-atlantique.fr

SeDuCe: Toward a testbed for research on thermal and power management in datacenters

With the advent of Cloud Computing, the size of datacenters is ever increasing and the management of servers and their power consumption and heat production have become challenges. The management of the heat produced by servers has been experimentally less explored than the management of their power consumption. It can be partly explained by the lack of a public testbed that provides reliable access to both thermal and power metrics of server rooms.

In this paper we propose SeDuCe, a testbed that targets research on power and thermal management of servers, by providing public access to precise data about the power consumption and the thermal dissipation of 48 servers integrated in Grid'5000 as the new ecotype cluster. We present the chosen software and hardware architecture for the first version of the SeDuCe testbed, highlighting its current limitation and proposing some improvements that will increase its relevance.

INTRODUCTION

The advent of web sites with a global audience and the democratization of Cloud Computing have led to the construction of datacenters all over the world. Datacenters are facilities that concentrate from a few servers up to hundreds of thousands of servers hosted in rooms specially designed to provide energy and cooling for the servers. These facilities are widely used for applications such as the hosting of web services or High Perfomance Computing.

In recent years, the size of datacenters is ever increasing, which leads to new challenges such as designing fault tolerant software to manage at large scale the servers and energy management of server rooms. On the latter challenge, many research efforts have been conducted [START_REF] Hermenier | Entropy: a consolidation manager for clusters[END_REF] [START_REF] Feller | Energy-aware ant colony based workload placement in clouds[END_REF], most of them focusing on the implementation of on demand power management systems, such as DVFS [START_REF] Von Laszewski | Poweraware scheduling of virtual machines in dvfs-enabled clusters[END_REF] [10] and VOVO [START_REF] Pinheiro | Dynamic cluster reconfiguration for power and performance[END_REF] [START_REF] Jeffrey S Chase | Managing energy and server resources in hosting centers[END_REF]. Some work has been made to extend existing scientific testbeds with power monitoring of experiments : for example Kwapi [START_REF] Clouet | A unified monitoring framework for energy consumption and network traffic[END_REF] enables researchers to track the power consumption of their experiments conducted on Grid'5000.

On the other hand, the thermal management of servers has been less explored, a large part of the existing work considering only simulations [START_REF] Sun | Spatio-temporal thermal-aware scheduling for homogeneous high-performance computing datacenters[END_REF]. This can be explained, partly, by the difficulty of conducting experiments involving thermal monitoring of servers : to ensure that data recorded experimentally is valid, experimentions must be conducted on a testbed that contains many temperature sensors, not only positionned on cooling systems, but also at the front and the back of each server of racks.

In addition, such a testbed must enable reproducible experimentations, by providing its users with a full control on experimental conditions of their experiments and by exposing its data in a non misleading way, via a well documented API (Application Programming Interface).

Finally, as power management and temperature management of servers are related problems [START_REF] Moore | Making Scheduling" Cool": Temperature-Aware Workload Placement in Data Centers[END_REF], there is a need for a testbed that enables users to access to both thermal and power data of servers.

As far as we know, there is no public testbed that enables researchers to work on both power and thermal aspects of servers functionning. The objective of the SeDuCe project is to propose such a testbed : SeDuCe testbed enables its users to use, in the context of the ecotype cluster integrated in the Grid'5000 infrastructure [START_REF] Bolze | Grid'5000: A large scale and highly reconfigurable experimental grid testbed[END_REF], 48 servers located in 5 airtight racks with a dedicated central cooling system (CCS) positioned inside one of the rack. In parallel of conducting the experiment by leveraging the tools provided by Grid'5000, users can get access to thermal and power data of the testbed via a web portal and a user-friendly API. Stability of experimental conditions is guaranteed by hosting the testbed in a dedicated room equipped with a secondary cooling system (SCS) that enables a precise thermo-regulation of the environment outside the cluster. As resources of the testbed are made publicly available via the Grid'5000 infrastructure, all its users will be able to perform reproducible research on thermal and power management of servers.

2 TESTBED DESIGN 2.1 Ecotype : a Grid'5000 cluster dedicated to the study of power and thermal management of servers

We have built the ecotype cluster, which is composed of 48 servers and has been designed for research related to power and thermal management in datacenters. The ecotype cluster is integrated in the Grid'5000 infrastructure and is made available to any of its users : they can reserve servers of the ecotype cluster and conduct experiments on them by using the usual Grid'5000 tools. Meanwhile they can access in real time to information regarding the temperature of the servers involved in their experiments, and get the power consumption of any parts of the testbed (servers, switches, cooling systems, ...). Each server of the ecotype cluster is a DELL PowerEdge R630 that contains a pair of Intel Xeon E5-2630L v4 CPUs (10 cores, 20 threads per CPU), 128GB of RAM, and 400GB SSD. The CPUs have been designed to have a power consumption that is lower than other CPUs of the XEON 26XX serie, with a thermal design power (TDP) of 55W. Each server is connected via two 10GBe links to the Grid'5000 production network, and via a single 1GBe link to the Grid'5000 management network. Additionally, each server is certified to work in environments where temperature can be up to 35°C.

The cluster is composed of 5 Racks (Z1, Z2, Z3, Z4, Z5) and, as shown on Figure 1, one rack (Z4) is in charge of cooling the cooling the cluster by hosting a dedicated central cooling system (CCS), while remaining racks are computing racks and are dedicated to hosting servers.

As depicted by Figure 1, computing racks host 12 servers, following two layouts of server positions : one layout where servers are organised in a concentrated way with vertical space between servers (Z1 and Z2), and a second layout where servers are spaced at 1U intervals (Z4 and Z5).

We have deliberately chosen to use these two layouts : they will enable to study the impact of a layout on the temperature and the power consumption of servers.

In addition to servers, the cluster contains three network switches that are in charge of connecting servers to production and management networks of the Grid'5000 testbed. Three racks (Z2, Z4, Z5) are hosting each a network switch.

The cluster has also been designed to leverage "Schneider Electric IN-ROW" racks that create an inside airtight environment for servers, and guarantee that the environment outside the cluster has a limited impact on temperatures inside the racks. The temperature inside the cluster is regulated by central cooling system (CCS), connected to a dedicated management network, which implements an SNMP service that enables to remote control the cooling and to access its operating data.

In addition, the temperature outside the cluster is regulated by a secondary cooling system (SCS) which is mounted from the ceiling of the server room : the SCS is in charge of maintaining a constant temperature in the server room.

Finally, we have installed Airflow management panels between servers, in order to separate the air of the hot aisle from the air of the cold aisle.

Power Monitoring

Each element composing the cluster (servers, network switches, fans, condensators, ...) is electrically monitored, and its power consumption is recorded every second in a database.

Electrical plugs of servers and network switches are connected to power distribution units (PDUs), which in turn share power consumption of servers and network switches via a dedicated service network. Each computing racks contains two PDUs, and each server of a computing rack has two electrical plugs. As depicted in Figure 2, the electrical plugs of a server are connected to two different PDUs, In a similar way, energy consumption of the CCS can be fetched by requesting its SNMP service. This SNMP service is able to provide an overall power consumption of the CCS and also the power consumption of each internal part such as the condensator or the fans. The SCS does not implement any built-in networking access, and thus cannot share its metrics with any component over a network. To solve this problem, we instrumented several parts of the SCS by using a Fluksometer 1 : a Fluksometer is a connected device that can monitor several electrical metrics (power consumption, voltage, amperage, ...) and expose their values over a network via a web-service.

Finally, we have added an additional system that tracks overall power consumption of both servers and the CCS. This additional system is based on the Socomec G50 metering board 2 , and enables to check that the aforementioned source of power consumption data are sound. These additional metrics are by using the modbus protocol.

Temperature Monitoring

In order to track the thermal behavior of the ecotype cluster, each server is monitored by a pair of temperature sensors : one sensor is positioned at the front of the server (in the cold aisle) and another sensor is positioned at the back of the server (in the hot aisle).

As depicted by Figure 2, each temperature sensor is a member of a bus (based on the 1wire protocol) connected to a Scanner (implemented by using a wifi arduino) in charge of gathering data from the temperature sensors of the bus. As the front and the back of each server is monitored by temperature sensors, computing racks have in total two Scanners and two buses : a front bus for monitoring the cold aisle and a back bus dedicated to the hot aisle. Scanners fetch temperatures from their sensors at a frequency of one reading per sensor every second.

For a first version of the SeDuCe testbed, we have chosen to use the DS18B20 sensor, a temperature sensor produced by "Maxim Integrated" 3 that costs approximately 3$. According to the specifications provided by its constructor, this sensor is able to provide a temperature reading every 750ms with a precision of 0.5°C between -10 °C and 85 °C.

The key feature of the DS18B20 sensor is that it is able to work as part of an 1wire bus. In the context of the SeDuCe infrastructure, 12 DS18B20 sensors are connected together to form an 1wire bus, and a Scanner, an nodeMCU arduino with built-in wifi capabilities, fetches periodically their temperature readings. The current version of the firmware used by Scanners scans an 1wire bus every seconds, and then pushes temperature data to a Temperature Registerer service, as illustrated in Figure 3.

We also developed a contextualisation tool to generate firmwares for the Scanners. It leverages the PlatformIO framework to program a Scanner that pushes data to a web-service. Using this contextualisation tool is simple : a developer needs to define a program template in a language close to C language and marks some parts of code with special tags to indicate that these parts need to be contextualized with additional information, such as initializing a variable with the ID of a Scanner device or with the address of a remote web-service (such as the one that will receive temperature records). The contextualisation tool takes this program and a context as input parameters, analyses the template program, and completes parts that requires contextualisation with information provided in the context, which results in valid C language source file. Then, the firmware is compiled and automatically uploaded to Scanners via 1 https://www.flukso.net/about 2 https://www.socomec.com/range-communication-interfaces_en.html?product= /diris-g_en.html 3 https://datasheets.maximintegrated.com/en/ds/DS18B20.pdf their serial ports. By leveraging this contextualisation tool, we can remotely configure Scanners and update their firmware. To help users to easily access power and thermal metrics generated by the SeDuCe testbed, we developed a platform that exposes publicly two components : a web portal 4 and a documented API 5(Application Programming Interface).

Seduce portal

As illustrated by Figure 3, the web portal and the API fetch data from a time series database (TSDB) based on InfluxDB6 . InfluxDB enables to store a large quantity of immutable time series data in a scalable way. In the background, InfluxDB creates aggregates of data by grouping periodically data from a same series. These aggregated sets of data enable the web portal to promptly load data used for visualization.

Two kind of components are in charge of inserting data in the database : the Power consumption crawlers and the Temperature Registerer. Power consumption crawlers are programs that are in charge of polling data from PDUs, Socomecs, Flukso, the CCS and the SCS. In turn, this data is inserted in the database. On the other hand, the Temperature Registerer is a web service that receives temperature data pushed from nodeMCU arduino devices, and inserts it in the database.

The web portal and the API are both written in Python and leverage the "Flask" micro web framework. The API component makes an extensive use of the Swagger framework 7 which automatises the generation of complete REST web services and their documentations from a single description file (written in JSON or YAML). This choice has enabled us to focus on the definition and the implementation of the API, by reducing the quantity of required boilerplate code.

All the components depicted in Figure 3 are implemented as micro-services. Our system is able to register 200 metrics per seconds with minimal hardware requirements (it is currently hosted on a single computer). In the case we add more sensors to our testbed, it is likely that the existing components would be sufficient.

SeDuCe: Toward a testbed for research on thermal and power management in datacenters e-Energy '18, June 12-15, 2018, Karlsruhe, Germany

In the case that one of the component would not be able to cope with the additional workload, it would be easy to setup an high availability approach by using a load balancer such as NGINX that would forward requests to a pool of instances of the component.

EVALUATING THE DS18B20 SENSOR 3.1 Experimental setup

In this section we evaluate the behaviour of the DS18B20 sensor in order to evaluate its fitness to the thermal measurement of the dissipated by servers. The performance evaluation of the DS18B20 can be made regarding two criteria : the precision the temperature it reports, and its response time. The precision of a sensor corresponds to the difference between the reported temperature and the actual temperature (usually it is estimated by a calibrated reference thermometer). On the other hand, the response time of a sensor corresponds to the time taken by the temperature reported by the sensor to converge to the reference temperature.

The evaluation of these two parameters requires an experimental environment and the definition of an experimental protocol.

To ensure that the validation of the DS18B20 sensor is based on reliable data, we used an Hart Scientific 7103 microbath which contained a silicon oil maintained at a specified homogeneous temperature. The temperature inside the microbath was controlled by two thermometers (a PT100 probe and a Hart Scientific 1502A). We conducted two experiments : a first experiment to evaluate the precision of the temperatures reported by DS18B20 sensors, and a second experiment to evaluate the reaction time of DS18B20 sensors. To evaluate the precision of DS18B20 sensors, we plunged 5 sensors into a silicon oil configured to be at a precise target temperature. Once the temperature of the silicon oil is stabilized, we measured the temperatures reported by the 5 DS18B20 sensors during 20 minutes. The target temperature was initially set at 10 °C, and incrementally increased by steps of 10 °C.

Sensor precision

Table 1 summarizes the temperatures reported by DS18B20 sensors at each step of temperature.

First, it is noticeable that for each targeted temperature, the average temperature reported by a DS18B20 sensor is in a range of 1°C band around the target temperature. While this value is higher than the 0.5°C advertised by the manufacturer, it should be relativised : when the target temperature is set to 20°C the difference between the worst sensor and the reference temperature is less than 5%, and at 50°C it becomes lower than 2%. These results should also be put in the perspective of the price range of DS18B20 sensors (approximately 3$ per unit).

Second, it appears that for a given sensor, the difference between the reported temperature and the reference temperature is stable for any temperature steps. Thus, by comparing several temperature records of a DS18B20 sensor with those of a reference thermometer (such as the one we used to stabilize the silicon oil), it would be possible to modelise the difference between the temperature reported by the sensor and the reference temperature. This would enable the implementation of a calibration mechanism that will be in charge of correcting the difference between the reported temperature and the reference temperature.

Sensor reactivity

To evaluate the reaction time of DS18B20 sensors, we plunged a single DS18B20 sensor from the ambient air at a temperature of 23°C into a silicon oil at a temperature of 30°C. We measured the time taken by the temperature reported by the DS18B20 sensor to converge to the temperature reported by the reference thermometer. Figure 4 plots these measurements.

The sensors takes about 10 seconds to stabilize at the temperature reported by the reference thermometer. This delay can be explained by the fact the DS18B20 sensor uses a "TO-92" semiconductor package which leverages plastic to protect its internal components from the external environment. This plastic shield enables the sensor to be used in a wide range of environments. However, the drawback is that it creates a thermal inertia, as the heat has to first penetrate the plastic shield before reaching the electronic part of the sensor.

This result indicates that DS18B20 are not suitable for configurations where temperature can suddenly change as the reactivity of the DS18B20 sensor is low. However, it appears to be appropriate for cases where the temperature is temperature is already stabilized and will only change smoothly, such as about a few °C over tens of seconds, such as in the hot aisle of the cluster.

EXPERIMENTATION

To illustrate the potential of the SeDuCe platform, we conducted an experiment that mixes energetic and thermal monitoring The goal of the experiment was to verify that the data produced by the SeDuCe testbed is reliable, by designing an experiment that will use both the thermal and the power data produced by the testbed. These data would be used to reproduce a scientifically validated observation, such as the impact of idle servers on the power consumption and the temperature of a server room.

Such experiment has been conducted in the past [START_REF] Moore | Making Scheduling" Cool": Temperature-Aware Workload Placement in Data Centers[END_REF], however as far as we know there is no public testbed that would enable researchers to reproduce this result : by reproducing this result on the SeDuCe testbed, we think that it would demonstrate the soundness of our approach and the usefulness of our testbed.

Description of the experiment

To illustrate the scientific relevance of our testbed, we wanted to reproduce the observations made by third party publication [START_REF] Moore | Making Scheduling" Cool": Temperature-Aware Workload Placement in Data Centers[END_REF]. In [START_REF] Moore | Making Scheduling" Cool": Temperature-Aware Workload Placement in Data Centers[END_REF], authors have highlighted an interesting fact : in a datacenter idle servers (i.e. servers that are turned on while not being used to execute any workload) have a significant impact on power consumption and heat production. We decided to try to reproduce this observation.

For this experiment, servers of the ecotype cluster are divided in three sets of servers:

• Active servers: servers with an even number (ecotype-2, 4, ..., ecotype-48) were executing a benchmark that generates a CPU intensive workload. • Idle servers: a defined quantity (0, 6, 12, 18, 24 servers) of servers with an odd number (ecotype-1, ecotype-3, ..., ecotype-47) was remaining idle. • Turned-off servers: remaining servers were electrically turned off. and during one hour we recorded the power consumption of the CCS and the average temperature in the hot aisle of the ecotype cluster.

The CPU intensive workload was based on the "sysbench" tool : the goal was to stress CPUs of each servers, resulting in an important power consumption and a bigger dissipation of heat.

To guarantee the statistical significance of the measurements, each experimental configuration was repeated 5 times, leading to a total number of 25 experiments.

We executed two sets of experiments : one with the SCS turnedon (Figure 5) and the other while the SCS was turned off (Figure 6). The objective of turning off the SCS was to identify the impact of the SCS over the CCS.

Results

Figure 5 plots the cumulated power consumption of the CCS and the average temperature in the hot aisle of the cluster with the SCS enabled.

First, it is noticeable that as the number of idle nodes increases, both the energy consumed by the SCS and the temperature in the hot aisle of the rack increase. This can be explained by the fact that an idle node consumes some energy and produces some heat, which increases the workload of the CCS.

The second element highlighted by Figure 5 is that the impact of idle nodes is not linear : the red line representing the CCS consumption follows an exponential pattern and the blue line representing the average temperature in the hot aisle follows a sigmoid pattern. The exponential pattern of the power consumption of the CCS can be explained by the fact that the heat produced by a given server has an impact on the temperature of surrounding servers, thus increasing their needs in terms of cooling. Consequently, to cool the additional heat produced by idle servers, the CCS has to work more frequently. On the other hand, the sigmoid pattern of the average temperature in the hot aisle is explained by the fact that the CCS has its own monitoring of the temperature inside racks, and when temperatures at the back of servers reach a threshold, the CCS works more frequently and intensively to make the average temperature back under a specific threshold, thus attenuating the increase of the temperature in the hot aisle.

Figure 6 plots the cumulated power consumption of the CCS and the average temperature in the hot aisle of the cluster while the SCS is disabled. This figure highlights that the power consumption of the CCS is lower when the SCS is disabled. This can be explained by the fact that the SCS was configured to maintain a temperature of 19 °C in the outside room, which is close to the maximum temperatures in the cold aisle : as the SCS does not cool down enough the outside air, by means of thermal conduction, it warms the temperature inside the racks. As a consequence, it increases the needs in term of cooling inside the cluster, leading to an higher power consumption of the CCS.

This experimental campaign has shown that idle servers have an important impact on the power consumption of cooling systems and overall racks temperature, thus it confirms the observation made in this publication [START_REF] Moore | Making Scheduling" Cool": Temperature-Aware Workload Placement in Data Centers[END_REF].

FUTURE WORK

We are considering the replacement of DS18B20 sensors by more reactive and precise sensors. While being suitable for monitoring temperatures in the hot aisle of our cluster, we have noticed that DS18B20 sensors are not fitting for configurations where temperature can suddenly change in short period of time, as it happens in the cold aisle. Figure 7 compares a DS18B20 sensor and a thermocouple based sensor, both positioned in the cold aisle at the front the "ecotype-6" server, and recording temperatures during a typical instance of the CCS workload. At t=0s, the CCSs just finished cooling the cold aisle, thus the temperature has reached its lowest level and is going to increase. The CCS restarts to cool the cluster at t=300s. It is noticeable that the temperature measured by the thermocouple based sensor drops significantly after few seconds, while the DS18B20 sensor measures a temperature that is falling gradually over a hundred of seconds. Even worse, we can see at t=420s that the minimal temperature measured by the DS18B20 sensor does not correspond to the minimal temperature reported by the reference thermometer, but rather to the intersection between its decreasing measured temperature and the increasing reference temperature. We explain this observation by the thermal design of the DS18B20 : as stated in Section 3.3 it leverages the "TO-92" semiconductor package which leads to a significant thermal inertia in air environment.

The thermocouple sensor seems more fitted for measuring the temperature of the cold aisle of the cluster : the MAX31850 is thermocouple amplifier compatible with the 1wire protocol, which is compatible with the software architecture we have developed. Thermocouples based sensors would lead to a significant improvement in terms of precision and thermal reactivity : we are considering replacing DS18B20 sensors by sensors based on MAX31850K.

CONCLUSION

In this article we have presented our initial work on building the Se-DuCe testbed, which targets research related to power and thermal management in datacenters. We have described the architecture of the testbed, which is built on buses of sensors, storage of power and thermal metrics in a time series oriented database (InfluxDB) and an user friendly web portal and a documented API. We have also detailed the components used in this first version of the SeDuCe testbed, and their limitations such as the lack of reactivity of the DS18B20 sensor, a low cost temperature sensor that shows its limits in situation where temperature changes suddendly. However, this model of sensor remains relevant in a large number of situations.

Future work on this first version of the SeDuCe testbed will mainly focus on evaluating more reactive temperature sensors. We are thinking to replace all or a part of the existing sensors by thermocouple based sensors, whose main advantages are that they are more reactive and more precise while remaining affordable.

Figure 1 :Figure 2 :

 12 Figure 1: Layout of the ecotype cluster (front view)

Figure 3 :

 3 Figure 3: Architecture of the SeDuCe portal

Figure 4 :

 4 Figure 4: Reactivity of a DS18B20 plunged in a silicon oil at 30°C temperature

Figure 5 :

 5 Figure 5: Central cooling consumption and average temperature in the hot aisle (SCS enabled)

Figure 6 :

 6 Figure 6: Central cooling consumption and average temperature in the hot aisle (SCS disabled)

Figure 7 :

 7 Figure 7: Comparison between DS18B20 and thermocouples

Table 1 :

 1 Comparison of 5 DS18B20 sensors (mean temperature ± standard deviation)

	Target	Sensors Sensor 1	Sensor 3	Sensor 3	Sensor 4	Sensor 5
	10 °C 10.86 ± 0.05 10.24 ± 0.03 10.06 ± 0.06 10.91 ± 0.05 10.13 ± 0.02
	20 °C 20.75 ± 0.00 20.00 ± 0.02 19.98 ± 0.05 20.86 ± 0.04 20.00 ± 0.00
	30 °C 30.74 ± 0.04 29.93 ± 0.07 29.91 ± 0.06 30.82 ± 0.06 29.97 ± 0.06
	40 °C 40.72 ± 0.05 39.81 ± 0.06 39.87 ± 0.05 40.77 ± 0.06 39.92 ± 0.07
	50 °C 50.72 ± 0.05 49.75 ± 0.07 49.85 ± 0.06 50.75 ± 0.07 49.89 ± 0.08
	60 °C 60.76 ± 0.03 59.76 ± 0.05 59.88 ± 0.03 60.79 ± 0.06 59.93 ± 0.06
	70 °C 70.83 ± 0.06 69.79 ± 0.06 69.92 ± 0.06 70.86 ± 0.03 69.99 ± 0.04

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