Pushpendra Singh
email: spushp@gmail.com

Comments on the Representations of Instantaneous Frequency using the Hilbert Transform, Direct Quadrature and Hilbert Quadrature

Keywords: Analytic signal, Hilbert transform, Direct Quadrature, Hilbert Quadrature, Instantaneous frequency

Using the two parameters amplitude and frequency of a signal, four different types of representations can be obtained. The first two representations of a signal with (i) constant amplitudes and constant frequencies (CA-CF) by the Fourier theory, (ii) variable amplitudes and variable frequencies (VA-VF) using AM-FM model are well established. In this study, we present methods to obtain other two representations: (iii) constant amplitudes and variable frequencies (CA-VF) representation (FM model), and (iv) variable amplitudes and constant frequencies (VA-CF) representation (AM model) for time-frequency analysis of a signal. Moreover, contrary to perception available in literature, we show that the direct quadrature (DQ), which is a CA-VF type representation, cannot produce full sine wave form cosine wave and vice versa, thus not suitable for instantaneous frequency (IF) computation. In order to overcome the limitations of DQ, a novel Hilbert Quadrature (HQ) representation of a signal is proposed. Practically, some noise is always present in the signal under analysis and in such cases we demonstrate that the analytic signal representation obtained by suprressing the negative frequecnies in the Fourier method, which is equivalent to the Hilbert transform, is the most robust and suitable for the computation of IF.

Introduction

There are various nonlinear and nonstationary signal representation, decomposition and analysis methods such as empirical mode decomposition (EMD) based algorithms [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and non-stationary time series analysis[END_REF][START_REF] Wu | Ensemble Empirical Mode Decomposition: a noise-assisted data analysis method[END_REF][START_REF] Rehman | Multivariate empirical mode decomposition[END_REF][START_REF] Singh | The Hilbert spectrum and the Energy Preserving Empirical Mode Decomposition[END_REF][START_REF] Singh | Some studies on nonpolynomial interpolation and error analysis[END_REF][START_REF] Singh | Nonpolynomial spline based empirical mode decomposition[END_REF], wavelet based synchrosqueezed wavelet transforms (SSWT) [START_REF] Daubechies | Synchrosqueezed Wavelet Transforms: an Empirical Mode Decomposition-like Tool[END_REF], Wiener filter based variational mode decomposition (VMD) [START_REF] Dragomiretskiy | Variational Mode Decomposition[END_REF], eigenvalue decomposition (EVD) [START_REF] Jain | An iterative approach for decomposition of multi-component nonstationary signals based on eigenvalue decomposition of the Hankel matrix[END_REF], empirical wavelet transform (EWT) [START_REF] Gilles | Empirical Wavelet Transform[END_REF], the Fourier theory and zero-phase filtering based methods [START_REF] Singh | Some studies on a generalized Fourier expansion for nonlinear and nonstationary time series analysis[END_REF][START_REF] Singh | The Fourier decomposition method for nonlinear and non-stationary time series analysis[END_REF][START_REF] Singh | Fourier-based Feature Extraction for Classification of EEG Signals Using EEG Rhythms[END_REF].

The time and frequency domain representations are two classical representations of a signal. The Time-Frequency Distributions (TFDs) provide localized signal information in both the time and frequency domain. The TFD provides insight into the complex structure of a signal consisting of several components. There exist many types of time-frequency analysis methods such as Gabor transform, short-time Fourier transform (STFT), wavelet transforms and Wigner-Ville distribution.

The [START_REF] Carson | Variable frequency electric circuit theory with application to the theory of frequency modulation[END_REF] proposed the concept of time-varying frequency [START_REF] Carson | Variable frequency electric circuit theory with application to the theory of frequency modulation[END_REF], with application to the frequency modulation (FM) scheme, as a generalization of the definition of constant frequency. The time-varying frequency is the basis of time-frequency representation required for the nonstationary signals and nonlinear systems analysis. The instantaneous frequency (IF) is the basis of the TFD or time-frequency-energy (TFE) representation and analysis of a signal. The IF is a practically important parameter of a signal which can reveal the underlying process and provides explanation for physical phenomenon in many applications such as acoustic, vibration, meteorological and atmospheric applications [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and non-stationary time series analysis[END_REF], seismic and speech signal analysis [START_REF] Singh | The Fourier decomposition method for nonlinear and non-stationary time series analysis[END_REF], radar, sonar, structural engineering, solar physics, health monitoring, communications, biomedical and medical applications [START_REF] Cummings | Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand[END_REF], mechanical systems analysis [START_REF] Guo | Application of EMD method to friction signal processing[END_REF][START_REF] Lei | A review on empirical mode decomposition in fault diagnosis of rotating machinery[END_REF], cosmological gravity wave and financial market data analysis.

The Fourier series is a well-known and most important representation of a periodic function in the mathematics, science and engineering for spectral analysis of a physical phenomena. The discretetime Fourier series (DTFS) represents a N -periodic discrete-time signal x[n] as linear combinations of harmonically related complex exponential (or sinusoidal) functions, i.e.,

x[n] = N -1 k=0 a k exp(jkω 0 n) = N -1 k=0 |a k | exp(jkω 0 n + jφ k ) for n = 0, 1, . . . , N -1, (1)
with

a k = 1 N N -1 n=0 x[n] exp(-jkω 0 n) for k = 0, 1, . . . , N -1, (2)
where fundamental frequency

ω 0 = 2π N , φ k = tan -1 (Img{a k }/Re{a k }), Img{a k }
and Re{a k } denote imaginary and real part of a k , respectively. Unless otherwise specified, the phase is always computed by the function, atan2(Img{a k }, Re{a k }), which produces the result in the range (-π, π] and also avoids the problems of division by zero. It is to be noticed that, in the representation [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and non-stationary time series analysis[END_REF], each term has (a) constant but different frequency (kω 0 ) value which is integer multiple of fundamental frequency ω 0 , (b) constant amplitude (|a k |) value which may or my not be different from the amplitude value of other terms, and (c) linearly varying phase (kω 0 n + φ k ). If signal x[n] is real valued function in [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and non-stationary time series analysis[END_REF], then it can be written as

x[n] = N -1 k=0 |a k | cos(kω 0 n + φ k ) and N -1 k=0 |a k | sin(kω 0 n + φ k ) = 0 for n = 0, 1, . . . , N -1, (3)
where amplitudes |a k | and frequencies kω 0 are constants for each k.

Using the two parameters amplitude and frequency of a signal, four different types of representations of a signal can be obtained: (i) constant amplitudes and constant frequencies by the Fourier theory, (ii) variable amplitudes and variable frequencies using AM-FM model, (iii) constant amplitudes and variable frequencies (FM model), and (iv) variable amplitudes and constant frequencies representation (AM model) to perform time-frequency analysis of a signal. First two representations are well established and, in this work, we present other two representations of a signal.

Four representations of a signal

In this section, we discuss all four representations of a signal for time-frequency analysis.

Constant amplitudes and constant frequencies (CA-CF) representation

The Fourier representations, (1) and (3), are viewed as superposition of sinusoidal functions with constant amplitudes and constant frequencies.

Variable amplitudes and variable frequencies (VA-VF) representation

The representation of a signal x[n] with variable amplitudes and variable frequencies (AM-FM model) can easily be obtained by any signal decomposition algorithms such as the EMD algorithm [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and non-stationary time series analysis[END_REF], Fourier decomposition method (FDM) or filter mode decomposition (FMD) [START_REF] Singh | Breaking the Limits -Redefining the Instantaneous Frequency[END_REF][START_REF] Singh | Some studies on a generalized Fourier expansion for nonlinear and nonstationary time series analysis[END_REF][START_REF] Singh | The Fourier decomposition method for nonlinear and non-stationary time series analysis[END_REF][START_REF] Singh | The Taylor's nonpolynomial series approximation[END_REF][START_REF] Singh | Some studies on multidimensional Fourier theory for Hilbert transform, analytic signal and space-time series analysis[END_REF][START_REF] Singh | LINOEP vectors, spiral of Theodorus, and nonlinear time-invariant system models of mode decomposition[END_REF]. The signal x[n] is modelled as

x[n] = M i=1 y i [n] + c 0 = M i=1 a i [n] cos(φ i [n]) + c 0 , (4)
where c 0 is a constant, each term

y i [n] = a i [n] cos(φ i [n]) is a mono-component (or narrow-band) function such that a i [n] ≥ 0 and φ i [n] is a monotonically increasing function, i.e. φ i [n + 1] ≥ φ i [n],
for all n. In order to obtain this signal model, we use the FDM/FMD (Algorithm 2) that generates a set of Linearly Independent Non Orthogonal yet Energy Preserving (LINOEP) vectors [START_REF] Singh | The Hilbert spectrum and the Energy Preserving Empirical Mode Decomposition[END_REF][START_REF] Singh | The Linearly Independent Non Orthogonal yet Energy Preserving (LINOEP) vectors[END_REF]. An orthogonal set of vectors can also be obtained by the discrete Fourier transform (DFT) based zerophase filter-bank as outlined in [START_REF] Singh | Breaking the Limits -Redefining the Instantaneous Frequency[END_REF]. The analytic representation of x[n], by excluding constant c 0 , can be written as

z[n] = M i=1 (y i [n] + j ŷi [n]) = M i=1 a i [n] exp(jφ i [n]), (5)
where

a i [n] = y 2 i [n] + ŷ2 i [n], φ i [n] = tan -1 (ŷ i [n]/y i [n]
), and imaginary part of z[n] is the Hilbert transform (HT) of real part which is defined as HT{y

i [n]} = ŷi [n] = y i [n] * 1-cos(πn)

πn

, where * denotes convolution operation.

The instantaneous frequency (IF) is the time derivative of the instantaneous phase that is obtained by the Gabor's analytic signal representation [START_REF] Gabor | Theory of communication[END_REF]. The IF is well-defined only when time derivative of phase is positive, if this derivative is negative then IF does not provide any physical significance [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and non-stationary time series analysis[END_REF][START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 1: Fundamentals[END_REF][START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 2: Algorithms and Applications[END_REF]. The IF for an analytic signal is, traditionally, defined as [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 2: Algorithms and Applications[END_REF][START_REF] Ville | Theorie et application de la notion de signal analytic[END_REF]

ω i [n] = φ id [n], (6)
which can be negative for some time instants. Recently, author in [START_REF] Singh | Breaking the Limits -Redefining the Instantaneous Frequency[END_REF] has obtained an important enhancement in the definition of the IF where it is redefined in such that IF is valid for all types of signals such as monocomponent and multicomponent, narrowband and wideband, stationary and nonstationary, linear and nonlinear signals. In order to obtain positive IF for all time, it is redefined in [START_REF] Singh | Breaking the Limits -Redefining the Instantaneous Frequency[END_REF] as

ω i [n] = φ id [n] if φ id [n] ≥ 0, φ id [n] + π otherwise, (7)
which is mathematically validated by the fact that

φ id [n] = tan -1 (ŷ i [n]/y i [n]) and φ id [n] + knπ = tan -1 (ŷ i [n]/y i [n]), ∀k, n ∈ Z (due to periodicity of tangent function, i.e. tan(φ id [n]) = tan(φ id [n] + knπ)).
In the above expressions ( 6) and ( 7), an approximation of differentiation in discrete-time, φ id [n], can be evaluated by forward finite difference (FFD) or backward finite difference (BFD) or central finite difference (CFD) defined as [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 2: Algorithms and Applications[END_REF] φ id

[n] = φ i [n + 1] -φ i [n] , (FFD) φ id [n] = φ i [n] -φ i [n -1] , (BFD) φ id [n] = φ i [n + 1] -φ i [n -1] /2, (CFD)      (8)
where φ i [n] is unwrapped phase (to obtain correct phase, subtract 2π if consecutive phase difference is > 2π and add 2π if consecutive phase difference is < 2π as shown in Algorithm 1) which ensure that all appropriate multiples of 2π have been included in phase angle. Sometimes, due to quantization and finite precision arithmetic operation in digital signal processing (DSP), phase unwrapping Algorithm 1A as well as 'unwrap' function of MATLAB do not generate correct output. Thus, in order to obtain correct phase, we propose phase unwrapping Algorithm 1B where some tolerance value is used. One such example where proposed method is generating correct output and others fail is

x[n] = A cos(πn) and x[n] = A sin(πn). Clearly, here, phase φ[n] = atan2(x[n], x[n]) = πn and frequency ω[n] = π. Outputs with 'unwrap(atan2(x[n], x[n]
))' function of MATLAB which fails to obtain correct phase/frequency, and proposed algorithm that produces the desired results are shown in Figure 1.

The time-frequency distribution (TFD) of a signal is obtained by 3D plot of {n, ω i [n], a 2 i [n]} for i = 1, 2, . . . , M . Moreover, using the IF [START_REF] Singh | The Linearly Independent Non Orthogonal yet Energy Preserving (LINOEP) vectors[END_REF], we can obtain TFD of any signal directly without decomposition, which is not possible by using the traditional IF (6) because it becomes negative in some instants if signal under analysis is not monocomponent.

Constant amplitudes and variable frequencies (CA-VF) representation

A representation of a signal with constant amplitudes and variable frequencies (FM model) can be obtained from AM-FM model by the algorithm 'the normalization scheme: an empirical AM and Frequency with proposed algorithm Algorithm 1: Algorithm 1A: A MATLAB code for the phase unwrapping [START_REF] Gdeisat | One-Dimensional Phase Unwrapping Problem[END_REF]. Algorithm 1B: Proposed MATLAB code for the phase unwrapping where some tolerance value is used.

% Algorithm 1A for i=2:length(Phw) difference = Phw(i)-Phw(i-
[n] = y i [n]/A i ∈ [-1, 1]
, ∀n, then (4) can be written by excluding constant c 0 as

x[n] = M i=1 A i p i [n]. (9)
One can easily construct a complex signal, z DQ [n], with constant amplitudes and variable frequencies, by direct quadrature (DQ) representation as

z DQ [n] = M i=1 A i p i [n] + j 1 -p 2 i [n] = M i=1 A i exp(jϕ i [n]), (10)
where

p i [n] = cos(ϕ i [n]) ⇔ ϕ i [n] = cos -1 (p i [n]) = tan -1 1 -p 2 i [n]/p i [n] ∈ [0, π] and sin(ϕ i [n]) = 1 -p 2 i [n] ≥ 0. However, we need sin(ϕ i [n]) = ± 1 -p 2 i [n]
to obtain correct phase in the range (-π, π]. Hence, contrary to perception available in literature such as [START_REF] Huang | On inatantaneous frequency[END_REF], DQ representation cannot produce full sine wave form cosine wave and vice versa, because there is always ambiguity in selecting correct sign as explained above.

Thus, in order to obtain correct positive or negative sign, we use sign of pi [n], which is the HT of

p i [n]
, in imaginary part of [START_REF] Jain | An iterative approach for decomposition of multi-component nonstationary signals based on eigenvalue decomposition of the Hankel matrix[END_REF] and construct a complex Hilbert quadrature (HQ) signal, z[n], as

z[n] = M i=1 A i p i [n] + j 1 -p 2 i [n] sgn(p i [n]) = M i=1 A i exp(j φi [n]), ( 11
)
where sgn is a sign function that we define as

sgn(p i [n]) = 1 if pi [n] ≥ 0, -1 if pi [n] < 0. ( 12
)
As the imaginary part of ( 11) is obtained by the HT and DQ representation, so we term it as the HQ of the real part. The TFD of a signal can be obtained by 3D plot of {n, ωi

[n] = A 1 -cos 2 (φ[n]) = A| sin(φ[n])| (with period π).
However, the Fourier spectrum of the signal and its magnitude is different. Let us consider both cases: (a) First case when

A 1 cos(θ[n]) = A cos(φ[n]) and A 1 sin(θ[n]) = |A sin(φ[n])|, which implies A 1 = A and θ[n] = tan -1 | sin(φ[n])| cos(φ[n]) = φ[n] if 0 ≤ φ[n] ≤ π, -φ[n] if -π ≤ φ[n] < 0, ( 13
)
as tan -1 | sin(φ[n])| cos(φ[n]) ∈ [0, π]. (b) Second case when A 1 cos(θ[n]) = |A cos(φ[n])| and A 1 sin(θ[n]) = |A sin(φ[n])|, which implies A 1 = A and θ[n] = tan -1 | sin(φ[n])| | cos(φ[n])| = tan -1 (| tan(φ[n])|) = φ[n] if 0 ≤ φ[n] ≤ π/2, -φ[n] if -π/2 ≤ φ[n] < 0, ( 14
) as tan -1 (| tan(φ[n])|) ∈ [0, π/2].
In order to obtain positive IF for all time, here, we redefined it for FM signal representation by DQ as

ω[n] = φ d [n] if φ d [n] ≥ 0, -φ d [n] otherwise, ( 15
)
and justification of this definition can be seen in Example 1. From ( 13) and ( 14), one can easily observe that the phase unwrapping is not required as absolute consecutive phase difference is always less than or equal to π.

Variable amplitudes and constant frequencies (VA-CF) representation

The VA-CF representation a of signal can be considered as an AM model. Any discrete-time signal x[n] can be written as a sum of weighted Kronecker delta functions, i.e.,

x[n] = ∞ l=-∞ x[l]δ[n -l]. (16)
Thus, the representation ( 16) can be viewed as a way to describe a signal as linear combinations of shifted delta functions, much the way the Fourier series describes a periodic signal as linear combinations of harmonically related complex exponential functions.

Using the analytic representation of unit sample sequence, we obtain variable amplitudes and constant frequency representation as follows. The unit sample sequence is defined as x[n] = δ[n-n 0 ] = 1 at n = n 0 and zero otherwise. Using the discrete-time Fourier transform (DTFT)

X(ω) = ∞ n=-∞ x[n] exp(-jωn), (17)
Algorithm 2: The FDM/FMD algorithm [START_REF] Singh | Breaking the Limits -Redefining the Instantaneous Frequency[END_REF] to obtain LINOEP vectors c i from decomposition of a signal x such that x = c 0 + M i=1 y i and y i ⊥ M l=i+1 y l . Use Algorithm 2 A (Algorithm 1 B) to obtain {y 1 , • • • , y M } in order of highest to lowest (lowest to highest) frequency components. %Algorithm 2 A c 0 = mean(x);

x 1 = x -c 0 ; for i = 1 to M -1 do s i = ZP HP F i (x i , f ci ); r i = x i -s i ; α i = s i ,r i r i ,r i ; y i = s i -α i r i ; ỹi+1 = (1 + α i )r i ; x i+1 = ỹi+1 ; y M = ỹM ; %Algorithm 2 B c 0 = mean(x); x 1 = x -c 0 ; for i = 1 to M -1 do s i = ZP LP F i (x i , f ci ); r i = x i -s i ; α i = r i ,s i s i ,s i ; y i = (1 + α i )s i ; ỹi+1 = r i -α i s i ; x i+1 = ỹi+1 ; y M = ỹM ;
one can obtain the DTFT of unit sample sequence as X(ω) = exp(-jωn 0 ) ⇒ |X(ω)| = 1. Signal is represented by inverse DTFT (IDTFT)

x[n] = 1 2π π -π X(ω) exp(jωn) dω, (18)
therefore, unit sample sequence can be written as

x[n] = 1 2π π -π exp(jω(n -n 0 )) dω.
This implies all frequency components (-π ≤ ω ≤ π radians/sample or -Fs 2 ≤ f ≤ Fs 2 Hz) are equally present in unit sample sequence. Thus, it is concentrated in time and spread over all frequencies, hence, following the uncertainty principle.

By using the relation z

[n] = 1 π π 0 X(ω) exp(jωn) dω, we obtain the analytic representation of δ[n -n 0 ] as z[n] = sin(π(n -n 0 )) + j[1 -cos(π(n -n 0 ))] π(n -n 0 ) = a[n] exp(jφ[n]), (19)
where a

[n] = sin( π 2 (n-n 0 )) π 2 (n-n 0 )) , φ[n] = π 2 (n -n 0 ) and real part of z[n] is delta function δ[n -n 0 ] = sin(π(n -n 0 )) π(n -n 0 ) = a[n] cos(φ[n]). (20)
The IF in analytic representation of delta function ( 19) can be obtained by [START_REF] Daubechies | Synchrosqueezed Wavelet Transforms: an Empirical Mode Decomposition-like Tool[END_REF] as ω[n] = π 2 , which corresponds to one-fourth of the sampling frequency, i.e. Fs 4 Hz. Using ( 19) and ( 20), we rewrite (16) as

x[n] = ∞ l=-∞ x[l] sin( π 2 (n -l)) π 2 (n -l)) cos π 2 (n -l) = ∞ l=-∞ a[l, n] cos (ω c (n -l)) , (21)
where

a[l, n] = x[l] sin( π 2 (n-l)) π 2 (n-l))
and ω c = π 2 . Hence, the representation (21) can be viewed as a way to describe a discrete-time signal as linear combinations of shifted cosine functions of constant frequency π 2 radians. Thus, in the representation [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 1: Fundamentals[END_REF], each term has a constant frequency (ω c = π 2 ) value, variable amplitude a[l, n] value, and linearly varying phase (ω c (n -l)).

Simulation results

In this section, we perform simulations and compute numerical results to demonstrate the efficacy of the different representations. 13), IF estimated (middle) by ( 6) and IF estimated (bottom) by ( 15) with initial phase φ = 7π. Spikes present in the estimation of IF are due to non-zero initial phase.

Example 1:

First, we consider a signal with linearly increasing phase x(t) = A cos(2πf 0 t + φ) with f 0 = 10, F s = 8000Hz, t = nT s = n/F s ∈ [0, 1]s, and A = 3. Figure 2 (with φ = 7π) and Figure 3 (with φ = 0) are the plots of phase estimated by [START_REF] Singh | Some studies on a generalized Fourier expansion for nonlinear and nonstationary time series analysis[END_REF], IF estimated by ( 6) which yields both positive and negative values, IF estimated by [START_REF] Singh | The Taylor's nonpolynomial series approximation[END_REF] which yields only positive values. The instantaneous phase is limited to [0, π] and spikes in frequency are located at the time instants which corresponds to phase multiple of π radians in original unwrapped phase.

Similarly, Figure 4 (with φ = 7π) and Figure 5 (with φ = 0) are the plots of phase estimated by [START_REF] Singh | The Fourier decomposition method for nonlinear and non-stationary time series analysis[END_REF], IF estimated by ( 6) which yields both positive and negative values, IF estimated by [START_REF] Singh | The Taylor's nonpolynomial series approximation[END_REF] which yields only positive values. The instantaneous phase is limited to [0, π/2] and spikes in frequency are located at the time instants which corresponds to phase multiple of π/2 radians in original unwrapped phase.

Through large number of simulations and empirically we found that the spikes in the estimated frequency are function of initial phase, sampling frequency and amplitude estimated by maxima of signal in DQ method.

Example 2:

Next, we consider the frequency estimation of a sinusoidal signal

x[n] = A cos(2πf 0 nT s + φ), (22)
with 13), IF estimated (middle) by ( 6) and IF estimated (bottom) by [START_REF] Singh | The Taylor's nonpolynomial series approximation[END_REF] with phase φ = 0 (bottom top 3 figures). Spikes are not present in the estimation of IF with zero initial phase. 14), IF estimated (middle) by [START_REF] Singh | Nonpolynomial spline based empirical mode decomposition[END_REF] and IF estimated (bottom) by ( 15) with (a) phase φ = 7π. Spikes present in the estimation of IF (bottom) are due to non-zero initial phase.

A = 1, f 0 = 0.101, T s = 1/F s = 1, φ = π/7, n = 0, 1, . . . , N -

Example 3:

Finally, we consider a unit sample sequence x[n] = δ[n -n 0 ] and obtain complex signal signal z DQ [START_REF] Jain | An iterative approach for decomposition of multi-component nonstationary signals based on eigenvalue decomposition of the Hankel matrix[END_REF], where a[n] = 1 and phase

[n] = δ[n -n 0 ] + j(1 -δ[n -n 0 ]) = a[n] cos(ϕ[n]) using DQ representation
ϕ[n] = tan -1 1 -δ[n -n 0 ] δ[n -n 0 ] = π/2 if n = n 0 , 0 if n = n 0 . (23)
Using FFD [START_REF] Daubechies | Synchrosqueezed Wavelet Transforms: an Empirical Mode Decomposition-like Tool[END_REF], its frequency is given by (blue solid line) and its envelope (red dashed line), (d) signal (blue solid line) and its envelope (red dashed line) obtained by the proposed method [START_REF] Gilles | Empirical Wavelet Transform[END_REF].

ω[n] = ϕ[n + 1] -ϕ[n] =            0 if n ≤ n 0 -2, -π/2 if n = n 0 -1, π/2 if n = n 0 , 0 if n ≥ n 0 + 1, (24) 0 0
which takes both positive and negative values. As negative frequency do not provide any physical significance, it can be easily converted to positive one by using the proposed IF definition [START_REF] Singh | The Taylor's nonpolynomial series approximation[END_REF]. This unit sample sequence is analyzed by the analytic signal representation z[n] [START_REF] Carson | Variable frequency electric circuit theory with application to the theory of frequency modulation[END_REF], where it has only positive frequency value. Figure 8 shows the plots of real, imaginary part and absolute value of analytic signal with n 0 = 199, sampling frequency F s = 100 Hz, length N = 400. Theoretically, this clearly indicate that most of the energy of signal δ[n -n 0 ] is concentrated at time t = 1.99 sec. (n 0 = 199) and frequency f = 25 Hz (one-fourth of the sampling frequency, i.e., ω = π/2). Figure 9 shows the time-frequency-energy (TFE) distribution of unit sample sequence δ[n -n 0 ] from the analytic signal representation using the FDM, ensemble empirical mode decomposition (EEMD) and continuous wavelet transform (CWT) methods. This clearly indicate that the TFE distribution obtained by the FDM method is same as theoretical estimation [START_REF] Carson | Variable frequency electric circuit theory with application to the theory of frequency modulation[END_REF] and [START_REF] Cummings | Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand[END_REF], whereas there is energy spread over a range of frequencies and lack of accuracy in TFE distribution obtained by the EEMD and CWT methods.

Conclusion

The representations of a signal with constant amplitudes and constant frequencies using the Fourier theory, variable amplitudes and variable frequencies using AM-FM model are well established. In this study, we presented the novel methods to obtain constant amplitudes and variable frequencies, variable amplitudes and constant frequencies representation of a signal. Contrary to perception available in literature, we demonstrated that the direct quadrature (DQ) cannot produce full sine wave form cosine wave and vice versa, thus cannot be used for the instantaneous frequemcy (IF) computation. Therefore, to overcome the limitation of DQ, a novel Hilbert Quadrature (HQ) representation of a signal is proposed. We have also proposed a phase unwrapping algorithm with some tolerance which overcomes the problems of quantization and finite precision arithmetic operation errors introduced in widely-used phase unwrapping algorithm. Practically, real-life data is always noisy and we demontstrated that the IF estimation from such data by the analyisc signal representation obtained by suprressing the negative frequecnies in the Fourier method, which is equivalent to the Hilbert transform, is the most robust, stable and suitable for time-frequency analysis.

Figure 1 :

 1 Figure 1: The phase and frequency plots, obtained form Example x[n] = A cos(πn) and x[n] = A sin(πn) with A = 1, (a) phase plot by the MATLAB function 'unwrap(atan2(x[n], x[n]))' which is not able to produce correct phase and by the proposed Algorithm 1B which produces correct monomonically increasing phase, (b) frequency plot by the MATLAB function which generates both positive and negative IFs, and (c) frequency plot by the proposed Algorithm 1B which produces only correct positive IFs.

  [n], A 2 i } for i = 1, 2, . . . , M . Discussion: It is interesting to note that the DQ of a signal p[n] = A cos(φ[n]) (with period 2π) and its magnitude, i.e. |p[n]| = |A cos(φ[n])| which has period π, is same without any change in phase φ[n] values, which can be written as q

Figure 2 :

 2 Figure 2: Plots, obtained form Example 1, of phase estimated (top) by (13), IF estimated (middle) by (6) and IF estimated (bottom) by (15) with initial phase φ = 7π. Spikes present in the estimation of IF are due to non-zero initial phase.

1 ,

 1 N = 512. There are four sub-plots in Figure6(top to bottom) (a) cosine wave x[n] obtained by[START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 2: Algorithms and Applications[END_REF], (b) sine wave obtained by DQ[START_REF] Jain | An iterative approach for decomposition of multi-component nonstationary signals based on eigenvalue decomposition of the Hankel matrix[END_REF], (c) Hilbert transform of x[n] (blue solid line) and its envelope (red dashed line) obtained by (5), (d) signal (blue solid line) and its envelope (red dashed line) obtained by the proposed HQ[START_REF] Gilles | Empirical Wavelet Transform[END_REF].

  Figure 7 shows the IF plot of signal x[n] (22) obtained by Direct Quadrature (DQ) (10), Hilbert transform (HT) (5) and Hilbert Quadrature (HQ) (11) (a) without noise (top figure) (b) with noise SNR=60dB (middle figure) and (c) with noise SNR=30dB (bottom figure). These plots clearly demonstrate that, in the absence of noise, there are unnecessary fluctuations in IF computed by DQ over the entire range, some fluctuations at both ends of the signal in IF estimated using the HT, whereas IF obtained by proposed DQ is accurate and stable over entire range of the signal except at the beginning of the data. It is also evident that when signal is corrupted by the white noise, accuracy of the IF computed by HT is better, more robust and stable as compared to the HQ and DQ methods.

FrequencyFigure 3 :

 3 Figure 3: Plots, obtained form Example 1, of phase estimated (top) by (13), IF estimated (middle) by (6) and IF estimated (bottom) by[START_REF] Singh | The Taylor's nonpolynomial series approximation[END_REF] with phase φ = 0 (bottom top 3 figures). Spikes are not present in the estimation of IF with zero initial phase.

FrequencyFigure 4 :

 4 Figure 4: Plots, obtained form Example 1, of phase estimated (top) by (14), IF estimated (middle) by (6) and IF estimated (bottom) by (15) with (a) phase φ = 7π. Spikes present in the estimation of IF (bottom) are due to non-zero initial phase.

Figure 5 :Figure 6 :

 56 Figure 5: Plots, obtained form Example 1, of phase estimated (top) by (14), IF estimated (middle) by (6) and IF estimated (bottom) by[START_REF] Singh | The Taylor's nonpolynomial series approximation[END_REF] with phase φ = 0. there are are Spikes in the estimation of IF with zero initial phase.

Figure 7 :

 7 Figure 7: IF of signal x[n] (22) obtained by Direct Quadrature (10), Hilbert transform (5) and Hilbert Quadrature (11) (a) without noise (top figure) (b) with SNR=60dB (middle figure) and (c) with SNR=30dB (bottom figure).

Figure 8 :

 8 Figure 8: The analytic representation of δ[n -n 0 ] (with, n 0 = 199, sampling frequency Fs = 100 Hz, length N = 400) (a) Real part of z[n] (top), (b) Imaginary part of z[n] (middle), and (c) absolute value of z[n] (bottom).

Figure 9 :

 9 Figure 9: The TFE analysis of unit sample sequence δ[n -n 0 ] (with, n 0 = 199, sampling frequency Fs = 100 Hz, length N = 400) (a) FDM (top), (b) EEMD (middle), and (c) CWT (bottom).


Elie Amani

Karim Djouani

Anish Kurien

Jean-Rémi De Boer

Willy Vigneau

Lionel Ries

GPS Multipath Detection in the Frequency Domain

Keywords: GPS, multipath detection, FFT-based detector, tracking error, scalar tracking loop, vector tracking loop

Multipath is among the major sources of errors in precise positioning using GPS and continues to be extensively studied. Two Fast Fourier Transform (FFT)-based detectors are presented in this paper as GPS multipath detection techniques. The detectors are formulated as binary hypothesis tests under the assumption that the multipath exists for a sufficient time frame that allows its detection based on the quadrature arm of the coherent Early-minus-Late discriminator (Q EmL ) for a scalar tracking loop (STL) or on the quadrature (Q EmL ) and/or in-phase arm (I EmL ) for a vector tracking loop (VTL), using an observation window of N samples. Performance analysis of the suggested detectors is done on multipath signal data acquired from the multipath environment simulator developed by the German Aerospace Centre (DLR) as well as on multipath data from real GPS signals. Application of the detection tests to correlator outputs of scalar and vector tracking loops shows that they may be used to exclude multipath contaminated satellites from the navigation solution. These detection techniques can be extended to other Global Navigation Satellite Systems (GNSS) such as GLONASS, Galileo and Beidou.

I. INTRODUCTION

Multipath remains one of the major sources of errors in precise positioning using the Global Positioning System (GPS) and all other Global Navigation Satellite Systems (GNSS). GPS pseudorange measurements can be severely compromised by multipath signals because these signals destroy the correlation function shape used for time delay estimation. Recently, several signal processing techniques have been developed to deal with errors induced by multipath signals in a GPS receiver. Three categories of techniques can be distinguished. The first category relies on detection of multipath using statistical detection methods [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]. No mitigation is performed but the satellite whose line-of-sight (LOS) signal is absent or severely affected by secondary paths is excluded from the navigation solution. Examples of methods that fall under this category are the Early Late Phase (ELP)-based detection [START_REF] Mubarak | Exclusion of Multipath-Affected Satellites Using Early Late Phase[END_REF] [3], the ANOVA-based detection [START_REF] Brenneman | An ANOVA-Based GPS Multipath Detection Algorithm Using Multi-Channel Software Receivers[END_REF], detection based on analysis of SNR fluctuation [START_REF] Fang | A GNSS Satellite Selection Method Based on SNR Fluctuation in Multipath Environments[END_REF], detection based on code minus carrier delta-range measurement [START_REF] Beitler | CMCD: Multipath Detection for Mobile GNSS Receivers[END_REF]. The ANOVA method requires a multiantenna receiver (physical or logical antenna array). With the SNR-based method, the fluctuations occur with a periodicity of 1.5 to 20 minutes for GPS satellites, which requires the multipath detection test to be set over long periods of time (10 minutes in [START_REF] Fang | A GNSS Satellite Selection Method Based on SNR Fluctuation in Multipath Environments[END_REF]) therefore delaying multipath detection. The second category of techniques modifies the receiver tracking loop to make it resistant to multipath signals. Under this category fall the Narrow Correlator [START_REF] Dierendonck | Theory and Performance of Narrow Correlator Spacing in a GPS Receiver[END_REF], the Edge Correlator [START_REF] Veitsel | The Mitigation of Multipath Errors by Strobe Correlators in GPS/GLONASS Receiver[END_REF], the Strobe Correlator [START_REF] Veitsel | The Mitigation of Multipath Errors by Strobe Correlators in GPS/GLONASS Receiver[END_REF], the High Resolution Correlator (HRC) [START_REF] Mcgraw | Practical GPS Carrier Phase Multipath Mitigation using High Resolution Correlator Techniques[END_REF], the Gated Correlator [START_REF] Mcgraw | GNSS Multipath Mitigation using Gated and High Resolution Correlator Concepts[END_REF], the Multipath Elimination Technology (MET) [START_REF] Townsend | A Practical Approach to the Reduction of Pseudorange Multipath Errors in a L1 GPS Receiver[END_REF], and the A-Posteriori Multipath Estimation (APME) Technique [START_REF] Sleewaegen | Mitigating short-delay multipath: a promising new technique[END_REF]. The third category relies on joint detection and estimation of the line-ofsight (LOS) and/or multipath signal parameters (amplitude, delay and phase) using statistical estimation methods [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]. It includes the Multipath Estimating Delay Lock Loop (MEDLL) [START_REF] Van Nee | The Multipath Estimating Delay Lock Loop[END_REF], the Modified RAKE Delay Lock Loop (MRDLL) [START_REF] Cahn | Multipath Corrections for a GPS Receiver[END_REF], the Multipath Mitigation Technology (MMT) [START_REF] Weil | Multipath Mitigation using Modernized GPS Signals: How Good Can it Get?[END_REF], the Vision Correlator (VC) [START_REF] Fenton | The Theory and Performance of NovAtel Inc.'s Vision Correlator[END_REF], the Fast Iterative Maximum-Likelihood Algorithm (FIMLA) [START_REF] Sahmoudi | Multipath Mitigation Techniques Using Maximum-Likelihood Principle[END_REF], Deconvolution Approaches [START_REF] Kumar | Deconvolution Approach to Carrier and Code Multipath Error Elimination in High Precision GPS[END_REF] [START_REF] Lohan | A Deconvolution Algorithm for Estimating Jointly the Line-Of-Sight Code Delay and Carrier Phase of GNSS Signals[END_REF], and Frequency Domain Processing [START_REF] Yang | Frequency-Domain Characterization of GPS Multipath for Estimation and Mitigation[END_REF] [START_REF] Yang | Multipath-Desensitized Delay Estimation with GPS Signal Channel Transfer Function Filtering[END_REF].

The techniques suggested by this paper are multipath detection techniques. They are suitable for a mono or multi antenna receiver and the time delay before multipath detection is significantly small (between 30 ms and 1 sec) compared to the technique in [START_REF] Fang | A GNSS Satellite Selection Method Based on SNR Fluctuation in Multipath Environments[END_REF]. No multipath mitigation is performed. Instead, these detection techniques may be used to exclude multipath affected satellites from calculation of the position, velocity, and time (PVT) solution. In fact, multipath estimation techniques, especially those based on frequency domain processing, may present a high computational burden compared to multipath detection techniques. And the fact that many satellite constellations can now be jointly used to obtain the PVT solution in one GNSS receiver, excluding multipath contaminated satellites instead of spending computational resources to mitigate multipath can sometimes be a more handy approach. The detectors avoid the additional computational burden of multipath parameters estimation but still incur the computational cost of Fast Fourier Transform (FFT) calculation. The detection tests are proposed for both scalar and vector tracking loops (STL and VTL). The detectors indications can also be used to switch between STL and VTL tracking modes. The detectors are based on correlator outputs, meaning that the test metrics are defined using correlator outputs. For a scalar tracking loop (STL) utilizing a coherent Early-minus-Late (EmL) discriminator, the EmL correlation output is directly used in the detectors metrics. Indeed, for a STL, depending on the relative phase of the multipath signal, the presence of multipath increases the signal power on the quadrature arm of the EmL correlation point (Q EmL ) compared to the power that is usually observed in the absence of multipath. This increase in signal power is used to detect multipath with the specification of a proper threshold. For a vector tracking loop (VTL) using the coherent EmL discriminator, both the in-phase and quadrature arms of the EmL correlation outputs are utilized in the detectors metrics. In fact, for a VTL, the increase in signal power due to the presence of multipath may occur on the in-phase arm (I EmL ) and/or on the quadrature arm (Q EmL ) of the EmL correlation output depending on the delay and phase of the multipath signal. The tracking scheme used in this paper is a vector delay frequency locked loop (VDFLL) assisted by a scalar phase locked loop (PLL), with the possibility to switch to a scalar delay locked loop (DLL) and a frequency-assisted phase locked loop (FPLL).

The rest of the paper is organized as follows. Section II presents the multipath model that was used to theoretically assess the performance of the proposed detection techniques before applying them to DLR multipath data and real GPS multipath signal data. Section III explains the link between correlator outputs and multipath detection. In section IV, the detection approach used for each detector is described in detail. Experimental results are provided in section V and section VI gives concluding remarks.

II. MULTIPATH MODEL AND CORRELATOR OUTPUTS

If a specular multipath model with a finite number of multipath signals is considered, the signal entering the code and phase loop, neglecting the low rate data, is expressed as:

 ) ( ) cos( ) ( ) cos( ) ( ) ( 1 0 0 0 t w t t C A t t C A t x L l l l l                    
where L is the number of multipath signals, 0 A and l A are the line-of-sight (LOS) and th l multipath amplitudes respectively,

) (t C is the spreading code, 0  , l  , 0  , l
 are the time and phase delays induced by the transmission from satellite to receiver for the LOS and th l multipath signals respectively, and  is the nominal GPS L1, L2 or L5 pulsation and ) (t w is the zero-mean additive white Gaussian noise with variance 2  .The signal at the output of the prompt correlator at an instant of time is therefore:

 P L l l l l P w j R A j R A U            ] ) ( exp[ ) ( ) exp( ) ( 1 0        
where R is the correlation function,   is the error between the LOS signal delay and the estimated code replica delay,   is the error between the LOS carrier phase and the estimated carrier replica phase, l  is the delay of the th l multipath with respect to the LOS, l  is the phase shift of the th l multipath with respect to the LOS, and P w is the postcorrelation noise. For the rest of the paper a one-path specular multipath model is used. In fact, it can be assumed that all reflected signals have the same frequency and combine into a single effective multipath. In that case, the signal entering the code and phase loops can be expressed as:

 ) ( ) cos( ) ( ) cos( ) ( ) ( 0 0 0 t w t t C A t t C A t x M M M                  
where the subscript M refers to the multipath resulting from a vector sum of all multipath signals in presence. The multipaths with a different frequency would contribute to an increase in noise power. Thus, the in-phase and quadrature outputs of the prompt correlator in the presence of a specular multipath, at an instant of time, are:

 P I M M M P w R A R A I , 0 ) cos( ) ( ) cos( ) (                 a  P Q M M M P w R A R A Q , 0 ) sin( ) ( ) sin( ) (                 b
Similarly, the Early and Late in-phase and quadrature correlator outputs are given by:

 E I M M M E w d R A d R A I , 0 ) cos( ) ( ) cos( ) (                   a  E Q M M M E w d R A d R A Q , 0 ) sin( ) ( ) sin( ) (                   b  L I M M M L w d R A d R A I , 0 ) cos( ) ( ) cos( ) (                   a  L Q M M M L w d R A d R A Q , 0 ) sin( ) ( ) sin( ) (                   b
where d is half the Early-Late chip spacing

d 2    1 0    .
The EmL in-phase and quadrature outputs are obtained by subtracting Eq.from Eq. 

     EmL I M M M M EmL w d R d R A d R d R A I , 0 ) cos( ) ( ) ( ) cos( ) ( ) (                            a      EmL Q M M M M EmL w d R d R A d R d R A Q , 0 ) sin( ) ( ) ( ) sin( ) ( ) (                             b
For a locked VTL, it can be assumed that

0   , which implies that 0 ) ( ) (       d R d R  
as well. The EmL inphase and quadrature outputs in this case are:

  EmL I M M M M EmL w d R d R A I , ) cos( ) ( ) (              a   EmL Q M M M M EmL w d R d R A Q , ) sin( ) ( ) (               b
While for a STL, the quadrature arm of the coherent EmL discriminator output (7b) is used in the detectors metrics that are described in section IV; for a VTL, the modulus of the complex output of the coherent EmL discriminator is used, i.e. the in-phase and quadrature EmL outputs (8a and 8b) are taken into account.

         EmL M M M M L E L E EmL EmL w j d R d R A Q Q j I I jQ I EmL                ) ( exp ) ( ) (     a    2 2 EmL EmL Q I EmL    b

III. CORRELATOR OUTPUTS AND MULTIPATH DETECTION

A careful analysis of GPS correlator outputs in the absence and in the presence of multipath can lead to the design of multipath detection techniques.

In the absence of multipath, the in-phase prompt correlator output carries the LOS signal power. In the presence of a multipath (MP) signal, the prompt correlator output is composed of the sum of the LOS and MP signals and the STL locking point is adjusted to this sum. As the tracking loop constantly seeks to bring the quadrature prompt power to zero, the quadrature prompt output will have part of the LOS + MP signal power only for a short transient time following MP arrival then will get back to zero, unless the MP signal is in phase or opposition of phase with the LOS signal.

The situation in the presence of multipath is different however for early and late correlator outputs and consequently for in-phase EmL and quadrature EmL outputs as can be observed in Fig. 1 for the STL and in Fig. 2 for the VTL. For the STL, in the presence of multipath, the signal energy in the quadrature arm of the EmL correlator output during transient and steady-state times following multipath arrival is significantly higher than in the absence of multipath, unless the MP signal is in phase or opposition of phase with the LOS signal. In the absence of multipath, only noise is observed on the quadrature EmL output.

Figure 1 represents the Q EmL output with a reduced number of oscillations in order to show the oscillatory nature of the EmL Q output within the envelope. Instead of 1575 cycles per C/A code chip for GPS L1, 25 cycles per C/A code chip are considered. The figure shows that the Q EmL output is zero for some multipath phase values even in the presence of multipath. More specifically, the Q EmL output is zero when the multipath signal is in phase (

  360 0 k M   
) or opposition of phase

(   360 180 k M   
) with the LOS signal, with k being an integer. Except for those phase values, the Q EmL output in the presence of multipath oscillates along the different multipath delay values between a maximum and a minimum which depend on the multipath to LOS amplitude ratio  and on the Early-Late correlator chip spacing  . These multipath phase values (

  360 0 k M    and   360 180 k M    ) correspond to multipath delay values 1 / L c M f nR   and 1 / ) 5 . 0 ( L c M f R n   
, with n an integer, if multipath phase is related to multipath delay using

C M L M R f / 2 1    
, i.e. if it is assumed that the multipath phase is only due to the differential path delay. In general, at the moment of reflection or diffraction, the multipath signal undergoes a relative phase M  that can be modelled using the differential path delay and reflector and antenna parameters [START_REF] Ray | Mitigation of GPS Code and Carrier Phase Multipath Effects Using a Multi-Antenna System[END_REF] or else assumed random [START_REF] Heiries | Optimisation d'une chaîne de réception pour signaux de radionavigation par satellite à porteuse à double décalage (BOC)[END_REF]. For the VTL, both the EmL I and EmL Q outputs increase in signal power in the presence of multipath as shown in Fig. 2 and when EmL Q is at zero, EmL I is not and vice versa. This means that for all multipath phase or delay values, the absolute value

2 2 EmL EmL Q I EmL  
increases in amplitude in the presence of multipath.

IV. FFT TEST FOR MULTIPATH DETECTION

A. Detection Test on Scalar Tracking Loop (STL)

A binary hypothesis test can be defined as follows for an observation window of N samples and of initial index n 0:

H 0 : Q EmL (n) = w Q,EmL H 1 : Q EmL (n) = s(n) + w Q,EmL , with n ϵ {1, …, N } + n 0
In other terms, under hypothesis H 0 , no signal is present on the EmL Q output, only noise is observed which implies that no multipath signal is present; whereas under hypothesis H 1 a signal s(n) plus noise are observed, which implies the presence of multipath. The Q EmL post-correlation noise term w Q,EmL is zero-mean white Gaussian with variance 2  .The problem is to decide for H 1 or H 0 from Q EmL , i.e. to decide whether s(n) is present or not. Two frequency-domain MP detectors are suggested.

1) STL-MP Detector I

First, the standard Fast Fourier Transform (FFT) detector [START_REF] Kay | Fundamentals of statistical signal processing[END_REF] [START_REF] Chan | Detection of stochastic signals in the frequency domain[END_REF], used in the sonar and radar systems, which detects the presence of a signal in the frequency domain is applied to GPS MP detection. This detector computes the periodogram

 N mn j N n n n EmL EmL EmL e n Q m Q where m Q N m  2 1 2 0 0 ) ( ) ( ) ( 1 ) (           ) (m Q EmL is the discrete Fourier transform of ) (n Q EmL
and is implemented via an N -point FFT algorithm. The detector subsequently chooses the largest value of ) (m  , normalizes it with the maximum likelihood estimate (MLE) of noise variance 2   and then compares the result against a threshold  . The detection test to decide for H 0 or H 1 is therefore formulated as:

          0 1 2 2 ˆ) ( max ˆ) ( max H H m or m       where       0 1 1 2 0 2 1 1 ˆn n Q N Q and Q n n Q N N n EmL N n EmL          
Detector I is optimum if s(n) is a sinusoid. This sinusoid may have unknown amplitude, phase and frequency, but optimality requires the frequency to be a bin frequency, i.e. to be equal to

N m  2
, with m being an integer [START_REF] Chan | Detection of stochastic signals in the frequency domain[END_REF]. Optimum detection entails that for a given probability of false alarm (PFA), the detector gives the maximum probability of detection (PD).

The detection threshold is

     1 2 1 exp 2 1               N PFA cdf      
with cdf being the cumulative distribution function of the standard normal distribution, 1  cdf the inverse cumulative distribution function, and PFA the probability of false alarm. The values of PFA and N have a great impact on the performance of Detector I, and there is a trade-off between obtaining a high detection capability or a low false alarm rate. Increasing the value of N and/or PFA improves the detection capability of Detector I in theory. However, in practice, increasing the value of N delays the instant, after system initialisation, when the multipath is detected, and increases the complexity of the N -point FFT computation. On the other hand, increasing PFA may result in many false detections and this may not be beneficial if the objective is to exclude the multipath contaminated satellites from the navigation solution. One might wind up with fewer satellites than needed to compute the PVT solution. If N is assumed sufficiently large ( 32  N

), the probability of detection (PD) is given by [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]:

                                      SNR PFA cdf cdf SNR PFA cdf cdf PD 2 1 2 1 2 1 1  
where SNR is the post-correlation signal to noise ratio of the Q EmL output and is given by This PD depends on the chosen PFA, on the amplitude A of the signal s(n) on the Q EmL output, and on the noise power on that output. This noise power is given by

  r K n   1 2 2   [7]where S n f N 0 2  
is the thermal noise power ( 0 N and S f are noise spectral density and baseband sampling frequency respectively), K is the number of correlation points, and

                           ) sin( ) ( ) ( ) sin( ) ( ) ( 0 M M M d R d R d R d R A A           where ) ( sin 0 fT c K C A   
with C the LOS power before correlation,

T the coherent integration time (

S Tf K 
) and f  the error between the LOS carrier frequency and the estimated carrier replica frequency. The post-correlation SNR at the Q EmL output has thus the following expression linking it to the C/N 0 : 2) STL-MP Detector II The second MP detector computes an FFT on blocs of 1024 samples at the frequency of correlators. In practice, this 1024-point FFT is a very complex calculation, but for the simplicity of algorithm description, it is chosen. The obtained frequency resolution is ~1 Hz for a coherent integration time of 1 ms. A power spectral density (PSD) estimator is then derived from the FFT bloc output for |f|  500 Hz. A periodogram can be used as a PSD estimator. This PSD estimator allows the search for s(n) signal frequency in the zone of interest, by measuring the signal power using a sliding window of length 3. More specifically, this PSD estimator makes it possible:  To estimate the noise power by summing samples of the spectral density for |f| > 200 Hz.  To estimate the power of the signal by summing samples of the spectral density for |f| < 200 Hz.  To estimate a signal to noise power ratio and compare it against a threshold  .

       2 2 0 2 2 ) sin( ) ( ) ( ) sin( ) ( ) ( 2 sin 2 2                           M M M d R d R d R d R d fT c NT N C NA SNR             
Detector II algorithm can therefore be summarized as follows:  Compute 1024-point FFT on blocs of 1024 samples at the frequency of correlators

 1024 ) ( ) ( 2 1 0 0        N where e n Q m Q N mn j N n n n EmL EmL     Derive a periodogram (PSD estimator) for |f|  500 Hz.
Letting m denote the frequency f, is the detection threshold. The probability of detection (PD) is given by [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]:

 2 ) ( 1 ) ( m Q N m EmL          Estimate the noise power           
                  PFA N CDF PD 1 2 ln 2 1

 

where CDF is the cumulative distribution function of the noncentral chi-squared distribution with 2 degrees of freedom and non-centrality parameter

B. Detection Test on Vector Tracking Loop (VTL)

A binary hypothesis test can be defined as follows for an observation window of N samples and of initial index n 0:

H 0 : EmL (n) = w EmL H 1 : EmL (n) = s(n) + w EmL , with n ϵ {1, …, N } + n 0
The detection test to decide for H 0 or H 1 is formulated as:

       0
 2 ) ( 1 ) ( m EmL N m         V. RESULTS

A. DLR Multipath Environment Simulator Data

The DLR multipath environment simulator has been developed by The German Aerospace Center (German: Deutsches Zentrum für Luft-und Raumfahrt), abbreviated DLR. This DLR simulator generates signals that have the characteristics of GNSS signals collected in semi-urban and urban environments for a fixed-point, a moving person or a moving vehicle. These synthesized signals have the specific purpose of providing a tool for multipath study and analysis. They are manipulated then fed to the input of the code and carrier tracking loops in Matlab. These synthesized signals with the variety of multipath environment scenarios that they represent are used to evaluate the validity of the proposed detection techniques. Some scenarios involve a vehicle moving parallel to the buildings in an urban environment and a vehicle moving in an unconstrained (open sky) environment. These are analysed in Figures 5 to 14. Figures 5,7, 9, 11 and 13 depict the parameters (amplitude, phase and delay) of the different signal paths (LOS and/or multipath) for the scenarios and channels under study. Amplitude values are normalized, with the LOS amplitude having a maximum value of 1 (0 dB). The LOS amplitude is represented in blue in all figures depicting parameters throughout the paper. The LOS phase and delay parameters are represented either in blue or cyan colour. Both Detectors I and II perform successfully as illustrated by figures 6, 8, 10, 12 and 14. Figure 5 through the displayed amplitude, phase and delay parameters shows the presence of many multipath signals apart from the LOS signal for a channel (labelled Channel 1) tracking a satellite in STL mode for a duration of 60 seconds. Figure 6 shows the detection tests results for that channel. For simplicity of performance comparison between the two detectors, a sliding window of N=1024 samples and PFA values of 10 -4 (Detector I) and 10 -2 (Detector II) are used to calculate the test metric values and set the detection thresholds. The test metric values go above the defined thresholds meaning that the presence of multipath is detected for both Detectors I and II. The signal power observed on the Q Eml arm whose magnitude is higher than the signal power that would be observed in the absence of multipath (see Fig. 12 for comparison) also confirms the presence of multipath. In fact, in the absence of multipath, only noise is observed on the Q EmL arm. It is important to mention that for faster MP detection (after system initialisation) and lower computation complexity, smaller values of N such as N = 32, 64, 128, 256 can be used for Detector I, some without compromising the detection capability in low SNR scenarios. N=1024 is used with Detector II for the sake of obtaining a frequency resolution of ~1 Hz, but again smaller values of N may be used with some risk of reduced detector performance.

Two other scenarios from the DLR multipath environment simulator are defined to allow multipath to appear first around the 30 th tracking second (see Figures 7 and8 for STL -Channel 7 and Figures 13 and14 for VTL -Channel 1) and then around the 20 th tracking second (see Figures 9 and10 for STL -Channel 9). The detectors perform as expected. The test metric values start below threshold for the 30 (respectively 20) tracking seconds where only the LOS signal is present then go above threshold around the 30 th (respectively 20 th ) second and the rest of the time. This is in accordance with the signal power increase on the Q EmL arm (STL case) and the increase in the absolute value of EmL (VTL case) which also occur around the 30 th (respectively 20 th ) tracking second. Figures 11 and12 are a case of a moving car in an unconstrained environment (open sky) and the tracking mode is STL. The test metric values remain below the thresholds for both Detectors I and II, meaning that no multipath is detected as expected. The signal power on the Q EmL arm remains minimal (made of noise only), which confirms the absence of multipath.

Figures 15 through 20 illustrate detection test results for a pedestrian moving in an urban environment. Figures 15,16 and 17 show the case of a channel that is severely affected by multipath. At time intervals where the LOS signal power is so weak compared to multipath signals power or where the LOS is absent, the detection tests in STL tracking mode fail to detect the presence of multipath as attested by observing Fig. 15 and Fig. [START_REF] Weil | Multipath Mitigation using Modernized GPS Signals: How Good Can it Get?[END_REF]. In fact the STL loses track, as can be confirmed by the delay and phase lock indicators in Fig. 21, which renders the detection tests useless. Figure 17 shows that the VTL manages to keep tracking the signal on the same severely affected channel in Fig. 15 even at the time intervals where the LOS signal is absent or has very weak power. Detector II test on the VTL even manages to succeed in detecting multipath where it couldn't with the STL tracking mode. This result is consistent with the VTL tracking robustness in multipath environments compared to the STL. The same phenomenon is observed on Fig. 18, 19 and 20. Again a severely affected channel as shown by the parameters displayed in Fig. 18 is studied. The detection tests performed on the STL fail to detect multipath as shown in Fig. 19. The channel is severely affected to the point of having the STL diverge, and even lose lock, making the detection tests unusable. Once more the VTL manages to keep tracking the signal on the channel and to detect multipath as illustrated in Fig. 20.

B. Real GPS Signals Data

In this section, the performance of one of the proposed multipath detection techniques, Detector I, is also evaluated on real GPS signals collected in urban, semi-urban, and foliage environments, for a STL. Digitized signals from the radio frequency (RF) front-end of different GPS receivers are used. A receiver with a high gain antenna provides signals with high carrier to noise ratio (C/N 0 ). A high C/N 0 results in a high post-correlation SNR which increases the probability of multipath detection (PD) and theoretically the performance of Detector I as demonstrated in section IV, Fig. 3. The receivers are calibrated to generate digitized signals at different Intermediate Frequencies (IF). Signals that are digitized with different sampling frequencies (f s ) are used. The acquisition, code and carrier tracking algorithms, as well as the proposed detection algorithm are implemented using these signals in Matlab. The coherent integration time used during tracking for all signals is 1ms. This means that if a sliding window of N=64 samples is used to set the detection test, 64ms time delay results before detection just after system initialisation. Figures 22 to 24 are a case of a moving vehicle in a heavy foliage environment, where the detection test threshold is set with PFA=10 -1 , PFA=10 -3 and PFA=10 -5 . The tracking mode is scalar (STL). A tracking duration of 20 seconds is studied. This is a case of a weak signal (low C/N 0 ). The probability of multipath detection with a low PFA is low for a low postcorrelation SNR. Figure 24 represents the same channel as Fig. 22 but compares the signal power on the quadrature arm of EmL with that of the in-phase prompt arm. The figures show that PFAs of 10 -3 or 10 -5 (which are appropriate PFA values to avoid false detections) fail to detect multipath on channels 1 and 2 which are tracking the GPS satellites SV1 and SV20 respectively. With a PFA of 0.1 however, multipath is detected on both channels. A PFA of 0.1 is not recommended because this results in several false detections. It appears however that for a very weak signal, only a high PFA achieves multipath detection for a STL. Depending on whether the number of visible satellites at a given time is enough to compute a navigation solution, a high PFA can be chosen in order to increase the chance of excluding all multipath affected satellites, and a lower PFA may be chosen to more likely exclude only the ones that are severely affected. Figures 25 to 27 show scenarios of a moving vehicle in a semi-urban environment, and the test is set with PFA=10 -1 , PFA=10 -3 and PFA=10 -5 . The C/N 0 is high enough to get multipath detections with PFA=10 -3 even with PFA=10 -5 in some cases on channels 4 and 7 which are tracking satellites SV22 and SV3 respectively. that when the multipath signal arrives aligned with the LOS signal, i.e. in phase (relative phase of 0° + k360°) or opposition of phase (relative phase of 180° + k360°), the Q EmL output remains blind to multipath presence. This means that for those multipath phase values, the Q EmL signal power does not change according to the presence or absence of multipath and only the post correlation noise is observed on the Q EmL arm. This results in missed detection if the multipath remains in phase or opposition of phase for the duration of the N-sample size window used to set the detection test. This blindness to multipath that is in phase or opposition of phase with the LOS is however not observed in the VTL case. In VTL, when Q EmL is blind to multipath I EmL is not and vice versa. As both I EmL and Q EmL are used in the VTL detection metric, the detectors performance is not affected. These detectors can be utilized as tools to exclude multipath contaminated satellites from the calculation of the PVT solution and to switch between STL and VTL tracking modes depending on accuracy and availability needs for a receiver embedded with both tracking modes. Also, the tests can be applied with minor adjustments to other GNSS systems such as Galileo, GLONASS and Beidou.

Fig. 1 .

 1 Fig. 1. I EmL and Q EmL amplitudes vs. Multipath delay for STL in lock (reduced oscillations).

Fig. 2 .

 2 Fig. 2. I EmL , Q EmL and |EmL| amplitudes vs. Multipath delay for VTL in lock (reduced oscillations).



  of correlation between the Early and Late outputs. Taking, in Eq. (the amplitude A=s(n) is given by

Figure 3

 3 Figure 3 illustrates the probability of detection (PD) given different PFA values for different values of SNR It can be

Fig. 3 .

 3 Fig. 3. Probability of multipath detection versus PFA for different SNRs (Detector I)



  Decide for H 0 or H 1 using the following test:

.Fig. 4 .

 4 Fig. 4. Probability of multipath detection versus PFA for different SNRs (Detector II)

Fig. 5 .

 5 Fig. 5. Moving vehicle -urban environment, STL, Chan1.

Fig. 6 .

 6 Fig. 6. Moving vehicle -urban environment, STL, Chan1, PFA=10 -4 (Detector I), PFA=10 -2 (Detector II).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Moving vehicle -urban environment, STL, Chan7.

Fig. 9 .

 9 Fig. 9. Moving vehicle -urban environment, STL, Chan9.

Fig. 10 .

 10 Fig. 10. Moving vehicle -urban environment, STL, Chan9, PFA=10 -4(Detector I), PFA=10 -2 (Detector II).

Fig. 11 .

 11 Fig. 11. Moving vehicle -open sky environment, STL, Chan1.

Fig. 12 .

 12 Fig. 12. Moving vehicle -open sky environment, STL, Chan1, PFA=10 -4 (Detector I), PFA=10 -2 (Detector II).

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Moving vehicle -urban environment, VTL, Chan1.

Fig. 15 .

 15 Fig. 15. Moving pedestrian -urban environment, Chan4.

Fig. 16 .Fig. 17 .Fig. 18 .

 161718 Fig. 16. Moving pedestrian -urban environment, STL, Chan4, PFA=10 -4 (Detector I), PFA=10 -2 (Detector II)

Fig. 19 .Fig. 20 .Fig. 21 .

 192021 Fig. 19. Moving pedestrian -urban environment, STL, Chan5, PFA=10 -4 (Detector I), PFA=10 -2 (Detector II)

  the signals synthesized by the DLR multipath simulator, it is possible to control at what instant during tracking the multipath signals can be superimposed on the LOS signal. But with real signals, this is not possible as the provided digitized signal is a mixture of the LOS and multipath signals already. The signals were just collected in environments that are prone to generate multipath contaminated signals.

Fig. 22 .Fig. 23 .Fig. 24 .

 222324 Fig. 22. Moving vehicle -foliage environment, STL, SV1, low C/N 0 , IF = 0 MHz, f s = 26 MHz, Detector I.

Fig. 25 .

 25 Fig. 25. Moving vehicle -semi-urban environment, STL, SV22, high C/N 0 , IF = 9.548 MHz, f s = 38.192 MHz, Detector I.

Fig. 26 .

 26 Fig. 26. Moving vehicle -semi-urban environment, STL, SV3, high C/N 0 , IF = 9.548 MHz, f s = 38.192 MHz, Detector I.

Fig. 28 .

 28 Fig. 28. Fixed point -urban environment, STL, SV6, high C/N 0 , IF = 1.25 MHz, f s = 5 MHz, Detector I.

Fig. 31 .Figure 28

 3128 Fig. 31. Fixed point -urban environment, STL, SV26, high C/N 0 , IF = 1.25 MHz, f s = 5 MHz, Lock Indicators


M Sahli

A Cablé

K Chetehouna

S Hamamda

N Gascoin

S Revo

Preparation and Characterization of Polytetrafluoroethylene (PTFE)/Thermally Expanded Graphite (TEG) Nanocomposites

Keywords: Nanocomposites, Physical properties, Thermal properties

In this study, TEG-reinforced polytetrafluoroethylene (PTFE) nanocomposites were prepared.

The structural properties of the PTFE/TEG composites were then investigated using X-ray diffraction (XRD) and infrared spectroscopy (FT-IR). Subsequently, the thermal stability, thermal resistance of the composites were studied through differential scanning calorimetry (DSC) and thermo-gravimetric analysis (TGA). The XRD results allowed to report a high degree of dispersion of the TEG inside of the PTFE matrix. Moreover, milling at high temperature was found to enhance the dispersion of the TEG inside of the polymer matrix. Furthermore, the results revealed an increase of the glass-transition temperature when using an increased concentration of TEG. This result confirms that the degradation of the TEG/PTFE nanocomposite occurs at a higher temperature with a greater TEG loading. These finds give an indication of the potential of TEG to improve the thermal properties and durability of PTFE, for instance with application in the field of aeronautics.

Introduction

Polymer nanocomposites are multiphase materials obtained by dispersing a filling agent with dimension in the order of nanometers into a polymer matrix. Various materials are commonly used as nanofillers, such as clay, silica, carbon-based materials. The resulting polymer nanocomposite displays enhanced properties, for instance in terms of rigidity, strain resistance, electrical conductivity, etc. In this context, a significant increase of interest has been observed in the last decade in the field of black carbon, carbon nanotubes and expanded graphite (EG) used as nanofillers of polymer materials [1] [START_REF] El Achaby | Mechanical, thermal, and rheological properties of graphene-based polypropylene nanocomposites prepared by melt mixing[END_REF]. The present study focuses on EG as nanofiller, and in particular EG obtained by thermal treatment, ie. Thermally Expanded Graphite (TEG). The latter has several advantages, including availability at relatively low cost and ease of production [START_REF] Yakovlev | Thermally expanded graphite: synthesis, properties, and prospects for use[END_REF]. Numerous properties can be enhanced by adding EG as nanofiller in a polymer matrix. These improvements are well documented in the literature in the context of various applications. [START_REF] Gupta | A comparative study on superhydrophobic sponges and their application as fluid channel for continuous separation of oils and organic solvents from water[END_REF][START_REF] Lutfullin | Petroleum products and water sorption by expanded graphite enhanced with magnetic iron phases[END_REF][START_REF] Ding | A new magnetic expanded graphite for removal of oil leakage[END_REF][START_REF] Dedov | Kinetics of sorption of petroleum products from the surface of water by thermally expanded graphite[END_REF] have reported an improvement of the oil absorption properties by using EG mixed with magnetic sorbents. This is particularly relevant in the context of environmental issues, for instance oil spill remediation, or water purification [START_REF] Yakovlev | Use of thermally expanded graphite in water-purification and water-treatment systems[END_REF]. The potential of EG has also been extensively studied in the field of energy storage in buildings, in particular regarding Phase Change Materials and thermal conductivity of building materials [START_REF] Sobolciak | Thermal characterization of phase change materials based on linear low-density polyethylene, paraffin wax and expanded graphite[END_REF][START_REF] Tian | Preparation of binary eutectic chloride/expanded graphite as high-temperature thermal energy storage materials[END_REF][START_REF] Yu | Preparation and properties of stearic acid/expanded graphite composite phase change material for low-temperature solar thermal application[END_REF][START_REF] Mochane | The effect of expanded graphite on the flammability and thermal conductivity properties of phase change material based on PP/wax blends[END_REF][START_REF] Fan | Effects of various carbon nanofillers on the thermal conductivity and energy storage properties of paraffin-based nanocomposite phase change materials[END_REF]. The electrical conductivity of polymers is generally low. Adding nanofillers in an amount that exceeds the percolation threshold allows to improve their electrical and thermal conductive properties for application in the field of electronics [START_REF] Meng | In situ intercalation polymerization approach to polyamide-6/graphite nanoflakes for enhanced thermal conductivity[END_REF][START_REF] Kuester | Processing and characterization of conductive composites based on poly (styrene-b-ethylene-ran-butylene-b-styrene)(SEBS) and carbon additives: A comparative study of expanded graphite and carbon black[END_REF][START_REF] Huang | Fe 3 C@ carbon nanocapsules/expanded graphite as anode materials for lithium ion batteries[END_REF][START_REF] Wen | Expanded graphite as superior anode for sodium-ion batteries[END_REF][START_REF] Wang | Preparation of expanded graphite/poly (phenylene sulfide) composites with high thermal and electrical conductivity by rotating solid-state premixing and melt processing[END_REF]. In this context, the potential of EG to improve the performance of Fuel cells has been reported by [START_REF] Dursun | Expanded graphite-epoxy-flexible silica composite bipolar plates for PEM fuel cells[END_REF][START_REF] Du | Highly conductive epoxy/graphite polymer composite bipolar plates in proton exchange membrane (PEM) fuel cells[END_REF][START_REF] Show | Stainless steel bipolar plate coated with carbon nanotube (CNT)/polytetrafluoroethylene (PTFE) composite film for proton exchange membrane fuel cell (PEMFC)[END_REF].

Finally, [START_REF] Kang | Thermal, impact and toughness behaviors of expanded graphite/graphite oxide-filled epoxy composites[END_REF][START_REF] Gantayat | Expanded graphite as a filler for epoxy matrix composites to improve their thermal, mechanical and electrical properties[END_REF][START_REF] Revo | Structure, tribotechnical, and thermophysical characteristics of the fluoroplastic carbonnanotubes material[END_REF][START_REF] Haghighi | Thermal performance of poly (ethylene disulfide)/expanded graphite nanocomposites[END_REF] have studied the potential of EG to improve the durability and mechanical properties of polymer materials, while [START_REF] Gogoi | Jatropha curcas oil based alkyd/epoxy resin/expanded graphite (EG) reinforced bio-composite: Evaluation of the thermal, mechanical and flame retardancy properties[END_REF] [START_REF] Dittrich | Carbon black, multiwall carbon nanotubes, expanded graphite and functionalized graphene flame retarded polypropylene nanocomposites[END_REF] have observed an increase of flame retardation properties. Hence, numerous studies have been carried out regarding the preparation and characterization of polymer nanocomposites. They differ by the studied materials (type of polymer and nanofiller), the preparation method for the nanocomposites, and the characterization techniques involved. Commonly studied polymers include PET [START_REF] Paszkiewicz | Influence of expanded graphite (EG) and graphene oxide (GO) on physical properties of PET based nanocomposites[END_REF], Epoxy [22][23], PEDS [START_REF] Haghighi | Thermal performance of poly (ethylene disulfide)/expanded graphite nanocomposites[END_REF] and PTFE [START_REF] Revo | Structure, tribotechnical, and thermophysical characteristics of the fluoroplastic carbonnanotubes material[END_REF][START_REF] Kandanur | Effect of activated carbon and various other nanoparticle fillers on PTFE wear[END_REF][START_REF] Sujuan | Tribological properties of PTFE and PTFE composites at different temperatures[END_REF][START_REF] Yan | Preparation and mechanical properties of PTFE/nano-EG composites reinforced with nanoparticles[END_REF][START_REF] Burris | Polytetrafluoroethylene matrix nanocomposites for tribological applications[END_REF][START_REF] Lee | Tribological behavior of PTFE nanocomposite films reinforced with carbon nanoparticles[END_REF][START_REF] Xiang | On the tribological properties of PTFE filled with alumina nanoparticles and graphite[END_REF]. In the present study, TEG was used as reinforcing nanofiller in a fluoroplastic (PTFE) polymer matrix, whose general formula is: -[-CF2-CF2-] n . PTFE, owing to its' qualities of low friction and non-reactivity, is used in a wide array of applications in the industry -from wiring in aerospace, machinery lubricant, to non-stick coating for cookware. PTFE and TEG were chosen to study the possibility of increasing the functional characteristics of PTFE, including enhancing its kinetic characteristics, elastic limit, wear resistance, etc. In fact, various studies have been carried out regarding nanocomposites based on PTFE and carbon-based nanofillers [START_REF] Revo | Structure, tribotechnical, and thermophysical characteristics of the fluoroplastic carbonnanotubes material[END_REF][START_REF] Kandanur | Effect of activated carbon and various other nanoparticle fillers on PTFE wear[END_REF][START_REF] Sujuan | Tribological properties of PTFE and PTFE composites at different temperatures[END_REF][START_REF] Yan | Preparation and mechanical properties of PTFE/nano-EG composites reinforced with nanoparticles[END_REF][START_REF] Burris | Polytetrafluoroethylene matrix nanocomposites for tribological applications[END_REF][START_REF] Lee | Tribological behavior of PTFE nanocomposite films reinforced with carbon nanoparticles[END_REF][START_REF] Xiang | On the tribological properties of PTFE filled with alumina nanoparticles and graphite[END_REF][START_REF] Avramenko | Effect of structural and morphological features of a nanocarbon component on electrophysical properties of fluoroplastic composite materials[END_REF][START_REF] Dovbeshko | Nanostructure of exfoliated graphite[END_REF]. However, these studies focused mainly on the potential of carbon nanofillers to improve the mechanical and tribological properties of PTFE. They concluded towards an enhanced durability and increased resistance to wear of the obtained nanocomposites compared to raw PTFE. In this context, the aim of the present work was to further assess the potential of TEG as filling agent for PTFE. In particular, the focus was put on the impact of TEG on the structural and on the thermal properties and degradation process of the resulting PTFE/TEG nanocomposites. The influence of the conditions when preparing the nanocomposites was as well considered. In the present document, the procedure for preparing the TEG and PTFE/TEG nanocomposites is first presented, followed by a description of the characterization methods employed and associated results.

Experimental procedure

2.1.Preparation of TEG

The raw material used for the preparation of TEG was natural graphite, which was oxidized by sulfuric acid. The latter enables to optimize the expansion volume of the graphite after thermal treatment [START_REF] Kim | High latent heat storage and high thermal conductive phase change materials using exfoliated graphite nanoplatelets[END_REF]. Graphite oxide was then exposed to thermal treatment in the apparatus (Fig. 1), at a temperature of ~1000°C during 1 minute, which gives the best results in terms of expansion volume according to [START_REF] Zhang | Composite reactive block for heat transformer system and improvement of system performance[END_REF]. An apparatus for the production of TEG has been developed in the context of the present study. The apparatus is presented on Figure 1 ; it allows the preparation of TEG at a rate of 1.5 kg/h. A gas flow rate valve is used to control the combustion and maintain the combustion chamber at the desired temperature. The developed apparatus has numerous advantages: its design is technologically relatively simple and it can be easily carried around. This makes it highly relevant for instance for applications requiring onsite production of TEG, such as oil spill remediation. The aspect of the TEG obtained with the apparatus and used for the samples is depicted on Figure 2, as well as the microstructure of the TEG obtained by scanning electron microscope.

2.2.Preparation of PTFE/TEG nanocomposites

The PTFE/TEG nanocomposite samples were prepared by manual milling. An analytical balance was used to weight the PTFE and TEG, which were then sieved and mixed together by vigorous manual milling in a mortar during approximately one hour. Three different mass fractions of TEG were considered for the samples: 2.5wt %, 5wt %, and 10 wt%. These values were chosen such as to obtain a composite material in the percolation threshold. In addition, two different thermal pre-treatment were considered, ie. milling at ambient temperature (around 23°C), and milling at a temperature of 160°C . For the samples milled at a temperature of 160°C, the PTFE and TEG were first mixed in a heated crucible until the mixture reached the temperature of 160°C (sample temperature measured by IR thermometry). Then, the mixture was milled in a mortar in a similar manner than for the samples obtained at ambient temperature. Thus, a total of 6 samples were obtained and characterized. The aspect of the PTFE used for the samples is depicted on Figure 2.

2.3.Characterization methods

The following methods have been used in order to study the morphological and thermal properties of the samples. X-ray diffraction (XRD) of the composites was carried out using a XPERT-PRO diffractometer system, with CuKα radiation at 45kV and 40mA in order to assess the structural properties of the PTFE/TEG nanocomposites. In particular, XRD allowed to follow the impact of the addition of TEG on the crystallinity of the obtained nanocomposite material. The measurements were carried out at a temperature of 25°C, with a copper anode.

The sample was examined within the interval 2θ = (20…70)°. The Fourier Transform Infra-Red (FT-IR) spectroscopy measurements were conducted using a Jasco FT/130 IR-6300 (Jasco Analytical Instruments, Easton, MD, USA) instrument to further study the structure of the material. The investigation was carried out in the frequency range of (600…4000) cm -1 , with a resolution of 4cm -1 . The scanning speed was 2mm /sec. The ATR method was used, with a TGS detector. Thermogravimetric analysis (TGA) and Differential Scanning Calorimetry (DSC) measurements were performed in order to study the thermal properties of the samples. They were performed simultaneously for all samples, with a Jupiter STA 449 F3 calorimeter by NETZSCH (NETZSCH, Selb, Germany). An Al 2 O 3 crucible was used. The samples were studied in the temperature range (24…320)°C, at a heating rate of 10K/min. All TGA-DSC measurements were carried out under an Argon atmosphere with flow rate of 20.0ml/min. The mass of the studied samples were measured as follows -for the samples milled at T=23°C: 33.95mg (10wt % TEG) 29.10mg (5wt % TEG) [START_REF] Avramenko | Effect of structural and morphological features of a nanocarbon component on electrophysical properties of fluoroplastic composite materials[END_REF].86 (2.5wt % TEG). For the samples milled at T=160°C: 25.10mg (10wt % TEG) 30.05mg (5wt % TEG) 27.22mg

(2.5wt % TEG).

Results and discussion

Structural characterization

XRD

The structural properties of the PTFE/TEG nanocomposite was first investigated using XRD.

The XRD pattern of the PTFE/TEG samples is shown on Figure 3, while the corresponding measured values are presented in Table 1 andTable 2. Two broad peaks can be observed at 2Ө around 26.5° and 54.6°. These peaks correspond to the (002) and (004) peaks, which are respectively the X-ray diffraction of carbon in the PTFE matrix, and also in TEG. The XRD patterns reveal that a high dispersion of the TEG nanofillers within the PTFE matrix was achieved. In fact, the peak at 2Ө around 54.6° (004) is barely noticeable and its relative intensity is low for all samples (see Table 1 andTable 2). This behavior gives an indication that milling allows an appropriate dispersion of TEG, and is thus an acceptable method for the preparation of PTFE/TEG nanocomposites. The effect of milling temperature on the TEG dispersion is noticeable as well. For a given TEG concentration, the relative intensity of the (004) peak diminished when milling at T=160°C compared to milling at T=23°C; see Table 1 and Table 2. This trend was observed for all samples. Thus, increasing the milling temperature improved the dispersion of TEG within the PTFE matrix. It can also be observed from the intensity of the (002) peaks that the crystallinity rate was enhanced when the TEG concentration increased, and that it was particularly high for the 10% TEG sample. Furthermore, it is possible to determine the size of the crystals which compose the sample by applying the Scherrer relation to the different X-ray diffraction peaks [START_REF] Patterson | The Scherrer formula for X-ray particle size determination[END_REF]. The width of the (002) peaks is quite important for all samples, which is an indication of the presence of crystals of small size in the samples (here in the range of 47 to 55 nm) according to Scherrer's equation [START_REF] Patterson | The Scherrer formula for X-ray particle size determination[END_REF], see Table 1 and Table 2. Table 1. XRD results for the samples milled at T=23°C Table 2. XRD results for the samples milled at T=160°C

FTIR

The structure of the samples was further investigated using Fourier Transform Infra-Red spectroscopy. The obtained results are presented on Figure 4. The dominant IR bands obtained were 1123 cm -1 , 966 cm -1 , 1194 cm -1 , 585 cm -1 , 661 cm -1 , 1283 cm -1 and 2358 cm -1 .

Based on the general formula of PTFE: -[-CF2-CF2-] n and on the literature, the following assignments have been made: CF 3 symmetric stretching (1123 cm -1 ) [START_REF] Mihály | FTIR and FT-Raman spectroscopic study on polymer based high pressure digestion vessels[END_REF], CH out-of-plane deformation of trans double bonds (966 cm -1 ) [START_REF] Kim | High latent heat storage and high thermal conductive phase change materials using exfoliated graphite nanoplatelets[END_REF], CF 2 stretching (1194 cm -1 ) [START_REF] Mihály | FTIR and FT-Raman spectroscopic study on polymer based high pressure digestion vessels[END_REF][42] [START_REF] Liang | Infrared spectra of high polymers. III. Polytetrafluoroethylene and polychlorotrifluoroethylene[END_REF],

CF stretching (1283 cm -1 ) [START_REF] Liang | Infrared spectra of high polymers. III. Polytetrafluoroethylene and polychlorotrifluoroethylene[END_REF], CF 2 wagging (585 and 661 cm -1 ) [START_REF] Liang | Infrared spectra of high polymers. III. Polytetrafluoroethylene and polychlorotrifluoroethylene[END_REF], and CO 2 (2358 cm -1 ) [START_REF] Shi | Preparation of pectin-ZnO nanocomposite[END_REF].

It can be observed that the intensity of the peaks increased with an increase of TEG concentration. Similar peaks were observed for the different TEG concentrations considered, at the exception of 2358 cm -1 (CO 2 ), which was absent for both the 2.5wt% and 5wt% TEG samples milled at ambient temperature, and which can certainly be assigned to atmospheric CO 2 . When increasing the milling temperature to T=160°C, the same peaks were observed but with increased intensity, except for the 2.5wt% TEG sample. Furthermore, low intensity IR bands appeared in the range (1500-1750 cm -1 ) and (3500-4000 cm -1 ) which were not present for the samples milled at ambient temperature. These peaks may correspond to byproducts created as a result of the increased milling temperature, and corresponding among others to CH 2 , CH 3 deformation (1458 cm -1 ) [START_REF] Mihály | FTIR and FT-Raman spectroscopic study on polymer based high pressure digestion vessels[END_REF]. This latter observation is congruent with the low fusion temperature of PTFE.

3.2.Thermal characterization

DSC

The effect of the inclusion of TEG on the thermal behavior of the nanocomposites during heating was examined by DSC. The DSC thermograms are presented on Figure 5, both for the samples milled at T=23°C and at T=160°C.

For the samples milled at ambient temperature, the curves corresponding to the different TEG loading display a similar behavior. From ambient temperature to 100°C, the DSC of the 5wt% TEG nanocomposite sample is superior to the two other samples. At about 100°C, a change of behavior occurs between the curves associated with the 5wt% TEG and 10wt% TEG samples.

From approximately 100°C onwards, the heat flux of the 10wt% TEG nanocomposite is significantly higher compared to those of the other two samples. The exothermal is particularly important with the increase of the concentration at 320°C. The I 2.5 /I 10 and I 5 /I 10 ratios are respectively equal to 30% and 70%. The 2.5wt% TEG sample displays a DSC inferior to the other two samples over the whole temperature range. Moreover, the concentration of TEG was found to impact the value of the glass-transition temperature. In fact, for the samples milled at T=23°C, a glass-transition temperature of 195°C was observed for 2.5wt% and 205°C for 5wt% TEG, while it reached approximately 208°C for 10wt% TEG. A similar trend was observed for the samples milled at T=160°C. Thus, the addition of TEG is responsible for an increase of the glass-transition temperature. This indicates that the degradation of the PTFE/TEG nanocomposite occurs at a relatively higher temperature with a greater TEG loading, and confirms the potential of TEG to improve the thermal properties of PTFE.

When the samples undergo milling at 160°C during one hour, the concentration of TEG induces as well an increase of the DSC curves and glass-transition temperature. However, a change of behavior is observed for the heat flux curves. In particular, a steep increase of the heat flux is observed for the 10wt% TEG sample above the curves of the other two samples, and the intersection between the curves corresponding to 5wt% and 10wt% which was observed at 100°C for the sample milled at ambient temperature totally disappeared. In this case, the I 2.5 /I 10 and I 5 /I 10 ratios hold the value of 7% and 50%, respectively. In addition, by comparing the DSC results for the samples milled at T=23°C and at T=160°C, it can be observed that a higher milling temperature seems to slightly reduce the glass-transition temperature, with values of approximately 190°C for 2.5wt% TEG, 195°C for 5wt% TEG and 200°C for 10wt% TEG. This result is an indication that milling at a lower temperature would be preferable in order to ensure the best resistance to heat for the TEG/PTFE nanocomposites.

TGA

The degradation kinetics of the PTFE/TEG nanocomposites were then studied using Thermogravimetric Analysis. The mass reduction versus time obtained from the TGA measurements are presented on Figure 6. The highest temperature for which the samples were tested was 320°C. The reason for that is that total calcination of the sample occurred at a higher temperature, as found by the authors in a previous work [START_REF] Revo | Structure, tribotechnical, and thermophysical characteristics of the fluoroplastic carbonnanotubes material[END_REF]. In the latter work, a DSC analysis of carbon-nanotubes enriched PTFE displayed an intense peak around 330°C, and became then unusable. Thus, it was not possible to study the residual weight of the different samples in the present study.

For the PTFE/TEG samples obtained by milling at ambient temperature, a very low weight loss was observed for all samples (less than 2% in all cases), even up to a temperature of 320°C. For the sample with 2.5 wt% TEG, a plateau can be observed which indicates an almost constant sample weight. The measured weight loss increased along the TEG concentration for the other samples. For the PTFE/TEG samples milled at 160°C, the measured weight loss was similarly relatively low over the whole temperature range (less than 2.5% in all cases). Regarding the impact of TEG loading on the weight loss, the tendency is however inverted compared to the samples milled at ambient temperature. In fact, the curves obtained for 5 wt% and 10wt% TEG are practically identical; which is an indication that similar mechanisms are involved. However, for the sample containing 2.5 wt% TEG, a decreasing TGA curve can be observed over the whole temperature range, with a significant decrease compared to the other two samples beginning from 100°C onwards. The higher thermal conductivity observed from the TGA measurements for the samples obtained by milling at T=160°C compared to the samples milled at T=23°C can be explained by the greater dispersion of the EG nanofiller inside of the polymer matrix which was already observed during the structural analysis. Thus, the impact of the TEG concentration on the thermal properties of the PTFE/TEG nanocomposite could be clearly observed as well as the influence of the temperature during milling. For the samples milled at a temperature of 160°C, the addition of TEG allowed to reduce the thermal degradation of the polymer nanocomposite. This phenomenon was further emphasized by an increase of the concentration of nanofiller. Finally, a concentration of 2.5 wt% TEG appeared to give the best thermal performance for the samples milled at ambient temperature. Figure 6. Weight loss versus temperature for the PTFE/TEG nanocomposites (heating rate of 10°C/min) for 2.5 wt%, 5 wt % and 10 wt % TEG; samples obtained by milling at T=23°C and by milling at T=160°C.

Conclusion

A mobile apparatus was conceived for the production of Thermally-Expanded Graphite (TEG), and TEG-reinforced polytetrafluoroethylene (PTFE) nanocomposites were prepared by manual milling. The obtained samples were then characterized in terms of structural and thermal properties. The XRD results revealed that an acceptable dispersion of the TEG nanofiller inside of the polymer matrix was obtained, and that increasing the temperature during milling enhanced the TEG dispersion compared to milling at ambient temperature. Furthermore, the DSC measurements gave indications of a relationship between the glasstransition temperature and the concentration of TEG in the samples; an increased concentration of TEG leading to a higher glass-transition temperature. However, a higher milling temperature seemed to slightly reduce the glass-transition temperature, which indicates that milling at a lower temperature would allow to ensure the best behavior regarding thermal degradation for the PTFE/TEG nanocomposites. The influence of the TEG loading on the thermal properties of the PTFE/TEG nanocomposite was further studied with TGA. The thermal performance was found to vary depending on the temperature during milling. For the samples milled at a temperature of 160°C, the addition of TEG allows to reduce the thermal degradation of the polymer nanocomposite, and even more so when the concentration of nanofiller increases. However, a concentration of 2.5 wt% TEG and milling at ambient temperature gave the best thermal performance for the samples. Thus, the milling temperature should be chosen accordingly to the desired application for the PTFE/TEG nanocomposites. The present work could be further extended in order to assess the potential of TEG to improve the properties of PTFE, for instance in terms of mechanical and electrical properties.

Acknowledgments

The present work has been carried out in the context of the PNR research project, funded by the Algerian General Directorate for Scientific Research and Technological Development (DG-RSDT), who is gratefully acknowledged.

Figure 1 .

 1 Figure 1. View of the apparatus for direct production of TEG at a rate of 1.5 kg/h. 1: Combustion chamber. 2: Container for graphite supply to the combustion chamber. 3: Raw graphite transport system. 4: Gas flow rate control valve. 5: TEG outlet. 6: Gas bottle. 7: Power supply.

Figure 2 .

 2 Figure 2. Aspect of the TEG produced with the experimental device: macroscopic view (left), microstructure (middle) obtained by scanning electron microscope; pure PTFE (right).

Figure 3 .

 3 Figure 3. XRD pattern of the PTFE/TEG nanocomposite at different percentage of TEG mass fraction for the samples milled at T=23°C (ambient temperature) and at T=160°C.

Figure 4 .

 4 Figure 4. Fourier Transform Infra-Red spectroscopy of PTFE/TEG nanocomposites with different TEG mass fraction (2.5 wt%, 5 wt% and 10 wt%) for the samples obtained by milling at T=23°C and T=160°C.

Figure 5 .

 5 Figure 5. Differential scanning calorimetric diagrams of PTFE/TEG nanocomposites with different TEG mass fraction (2.5 wt%, 5 wt% and 10 wt%) for the samples obtained by milling at T=23°C and at T=160°C. Heating rate of 10°C/min.

[ 1 ]

 1 Schadler, L. S., Brinson, L. C., & Sawyer, W. G. (2007). Polymer nanocomposites: a small part of the story. Jom, 59(3), 53-60.