The arrival of the Internet and digital communication tools have led to the emergence of new "paper-less organizations" that operate without an office, schedule, or even employees. With the Internet, we can now work remotely, synchronously and asynchronously, without having to move from one place to another. We no longer need to live in the same place, or meet face to face, in order to collaborate on a common endeavour. We simply need to connect to the Internet network to find colleagues, partners or customers. The hope was that this global communication network would eventually lead to increased participation and greater opportunities for people all over the world to participate to the global economy. And to some extent, it did.

Yet, the shift was only a partial one. Over the past 20 years, we have progressively moved away from the traditional model of centralized organizations, where large operators (often with a dominant position) were responsible for providing a service to a group of passive consumers. Today, we are witnessing the emergence of new organizational structures in the digital domain, which are much more distributed in nature. These so-called "crowdsourced organizations" are responsible for aggregating the resources of multiple people to provide a service to a much more active group of consumers. Indeed, if we look at the modus operandi of today's internet giants -such as Google, Facebook, Twitter, Uber, or Airbnb -we notice that they all have one thing in common: they rely on the contributions of users as a means to generate value within their own platforms. The problem with this model is that, in most cases, the value produced by the crowd is not equally re-distributed among all those who have contributed to the value creation. Bost of the profits are instead captured by the large intermediaries who operate these platforms. But the most revolutionary aspect of blockchain technology is that it is also a means for individuals to coordinate common activities, to interact directly with one another, and to govern themselves in a more secure and decentralized manner. Indeed, modern blockchain-based networks make it possible for people not only to transact value between each other, but also to execute software in a secure and decentralized manner. With a blockchain, software applications no longer need to be deployed on a centralized server: they can be run on a peer-to-peer network that is not controlled by any single party. These blockchain-based applications can be used to coordinate the activities of a large number of individuals, who can organize themselves without the help of a third party.

There are already a few of such applications that have been deployed on a blockchain. For instance, Steemit, Sapien, and Akasha are distributed social networks and media platforms that operate without a centralized authority. Instead of the content being stored on a centralized server, operated by a centralized organization that can control and manage the content that is displayed to the public, these platforms stores content on a decentralized network, using blockchain technology to coordinate individuals and manage the content they contribute to the platforms through a set of code-based protocols and rules.

Similarly, OpenBazaar is a decentralized marketplace, like eBay or Amazon, that operates independently of any intermediary operator. The platform relies the Bitcoin blockchain to ensure that buyers and sellers interact directly with one another, without passing through any centralized middleman. Anyone is free to offer a product for sale on the platform at a given price. Once a buyer agrees to the price for that product, an escrow account is created on the blockchain requiring two out of three people (i.e., the buyer, the seller, and a potential third-party arbitrator) to agree for the funds to be released (a so-called multisignature account). Once the buyer has sent the payment to the escrow account, the seller ships the product, and after receiving the product, the buyer releases the funds from the escrow account. Only if an issue arises will the system require the intervention of a third party (e.g., an arbitrator) to decide whether to release the payment to the seller or whether to return the money to the buyer.

There also have been some attempts at creating generic infrastructure for decentralized organizations, such as DAOstack and Aragon, which provide the basic building blocks for creating decentralized crowdsourcing organisations, administered without a centralized operator. These organizations are governed by the code deployed on a blockchain-based infrastructure, which is designed to govern peer-to-peer interactions between multiple actors.

Blockchain technology thus facilitates the emergence of new forms of decentralized organizations, which have no director or CEO, nor any sort of hierarchical structure. These organizations are administered, collectively, by all individuals interacting on a blockchain. As such, it is important not to confuse them with the traditional model of "crowdsourcing," where people contribute to a platform but do not benefit from the success of that platform. To the contrary, blockchain technologies can support a much more cooperative form of crowdsourcing -sometimes referred to as "platform cooperativism"-where users qualify both as contributors and shareholders of the platforms to which they contribute. And since there is no intermediary operator, the value produced within these platforms can be more equally redistributed among those who have contributed to the value creation.

With this new opportunity for increased "cooperativism," we're moving toward a true sharing or collaborative economy -one that is not controlled by a few large intermediary operators, but that is governed by and for the people. Blockchain technology makes it possible to replace the model of top-down hierarchical organizations with a system of distributed, bottom-up cooperation. Ultimately, this shift could change the way wealth is distributed in the first place, enabling people to cooperate toward the creation of a common good, while ensuring that everyone will be duly compensated for their efforts and contributions.

While most of these blockchain-based organizations have been-thus far-developed mostly to facilitate the coordination of individuals in the digital world, the possibilities provided by these new organisational structures can also be found in the physical world. Cities, municipalities and local communities can leverage the power of blockchain technology in order to increase transparency and accountability in many sectors of activities, while providing new opportunities for anyone to engage and participate in the local economy. Indeed, blockchain technology is currently being explored as a way to support local energy microgrids with a peer-to-peer exchange between neighbors (see, e.g. Grid Singularity) or to provide more transparency in the food supply chain (with projects such as Provenance, for instance) by recording information in the form of immutable cryptographic records on a distributed ledger.

Digital technologies create many new opportunities to increase the capacity of local production within communities, neighborhoods and cities with the use of urban farming technologies (aquaponics, aeroponics, synthetic biology), solar panels or wind turbines, and digital fabrication tools (from personal 3d printers to flexible factories). These technologies could contribute to great operational efficiencies by reducing the costs of production and unleashing new business opportunities for manufacturers worldwide. Yet, the question of governance remains a critical issue that still needs to be properly addressed. Indeed, in the physical world, commons-pool resources are subject to the "tragedy of the commons": without a proper governance structure or incentivization scheme, people are likely to free-ride, leading to an over-exploitation and/or undercontribution to these common-pool resources. In order to increase the chances that these new technologies contribute to the flourishing of a healthy ecosystem of local production, we need to identity the proper incentivization mechanisms that will encourage people to contribute ressources, without being subject to the scrutiny of a centralized authority.

Enabling local processes of production to reduce the impact of the current industrial globalisation is crucial, but enabling mechanisms to incentivise, accelerate and scale this process is fundamental and urgent. This is where blockchain technology could come at hand by creating an open platform and decentralized incentivization scheme that can be articulated between multiple stakeholders. Local communities have been experimenting with local currencies for a long time, but because of the limited scope, these have never managed to reach a global audience. For instance, with a blockchain, multiple cities around the world could incentivize local communities to contribute to the commons and engage into productive and collaborative activities by rewarding these practices with a global social impact currency. This would enable local communities to coordinate, on a global level, in order to promote a paradigm shift in terms of recycling, reuse, relocalisation of supply chains, and other practices that reduce the impact of the linear economy.

The opportunities are huge, and yet, nothing should be taken for granted. The decentralized potential of blockchain technology does not necessarily mean that it will in fact be used in a decentralized manner. Just as the internet has evolved from a highly decentralized infrastructure into an increasingly centralized system controlled by only a few large online operators, there is always the risk that big giants will eventually form in the blockchain space. If we, as a society, really value the concept of a true sharing economy, where disparate groups of individuals can coordinate themselves and cooperate on a peer-to-peer basis, and those producing value are fairly rewarded for their efforts, it behooves us all to engage and experiment with this emerging technology, in order to explore the new opportunities it provides and deploy large, successful, community-driven applications that enable us to achieve the promises of a true collaborative economy in the context of new productive cities.

  Blockchain : a global infrastructure for distributed governance and local manufacturing. 1 By Primavera De Filippi in Diez, T. (Ed.) The Mass Distribution of Almost Everything. Institute for Advanced Architecture of Catalonia, Spain

  How can these communities govern themselves without falling into the same centralized paradigm that has become so widespread in the context of the sharing economy? Recently, a new technology has emerged that could contribute an answer to that question. Blockchain technology-the technology that underpins Bitcoin-facilitates the exchange of value in a secure and decentralized manner, without the need for an intermediary. As such, it enabled the emergence of virtual currencies and other distributed ledger technologies that are prone to disrupt existing intermediaries in the financial sector, and beyond.

	technologies and tools inside new fabrication spaces 2 . Today, millions of people are
	connected to the Internet using open source software, and employ digital fabrication tools
	(including 3d printing) to build the largest distributed design and manufacturing ecosystem
	in the world. This opens a new potential for a more equal redistribution of production
	means, both digitally and physically. And yet, as it happened with the Internet a few
	decades ago, the value generated by makers is likely to be captured by the upcoming
	manufacturing and distribution giants.
	Conversely, in the physical domain, the development of new open source hardware and
	software tools during the last decades have been fostering new modes of learning,
	designing, manufacturing and collaborating that actually promote individual participation
	into an open ecosystem of value creation and re-distribution. Inspired by the Open Source
	movement in software, maker communities have been building new hardware-based

Edits from the article What Blockchain Means for the Sharing Economy?: https://hbr.org/2017/03/whatblockchain-means-for-the-sharing-economy

See, for instance, the Fab Lab Network, which consists of over 1000 consolidated fab labs around the world, all sharing a common inventory and core values around collaborative and sustainable production, educational programs, and ongoing projects at a global scale.


Sylvain Meignen
email: sylvain.meignen@imag.fr

Duong-Hung Pham
email: duong-hung.pham@imag.fr

D-H Pham

On Multicomponent Signal Denoising and Mode Reconstruction from Short-Time Fourier Transform

Keywords: Time-frequency, signal denoising, short-time Fourier transform, downsampling, reassignment, AM/FM multicomponent signals. I

In this paper, we investigate the reconstruction and denoising of multicomponent signals from their downsampled short-time Fourier transform. We first recall signal reconstruction techniques based on short-time Fourier transforms and how to use them for signal denoising. We then investigate how to adapt the proposed denoising technique to the context of multicomponent signals. Comparisons with recently developed techniques based on synhrosqueezing conclude the paper.

or STFT contexts [START_REF] Daubechies | Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool[END_REF] [3] [START_REF] Wu | Adaptive analysis of complex data sets[END_REF], several extensions were recently proposed to better take into account the modulation either by demodulating the signal [START_REF] Wang | Matching demodulation transform and synchrosqueezing transform in time-frequency analysis[END_REF], by taking into account phase and amplitude modulations in the synchrosqueezing operators [START_REF] Oberlin | Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations[END_REF] [START_REF] Pham | High-order synchrosqueezing transform for multicomponent signals analysiswith an application to gravitational-wave signal[END_REF] or both [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF].

Despite some theoretical analyses have been carried out [START_REF] Yang | Statistical analysis of synchrosqueezed transforms[END_REF], the impact of noise on synchrosqueezing operators still needs to be further studied, in particular in heavy noise situations. From a practical point of view, different techniques have been developed to handle the noise in the SST context, among which a very promising one, inspired by multitaper approaches [START_REF] Xiao | Multitaper time-frequency reassignment for nonstationary spectrum estimation and chirp enhancement[END_REF], is based on averaging the SSTs obtained with different kernels [START_REF] Daubechies | concentration of frequency and time via multitapered synchrosqueezed transform[END_REF]. Another issue related to the use of SST is that while the latter is sharper than the TF representation it is based on, mode reconstruction cannot be carried out from SST downsampled in time. On the contrary, such a reconstruction can be achieved from a downsampled STFT: in some sense, SST, by sharpening the frequency representation, annihilates the time redundancy of the TF representation it is based on. Our goal is thus to show that, using the time redundancy of STFT, the reconstruction of the modes of a noisy MCS can be performed with a better accuracy and using less STFT coefficients than when STFT based SST is considered.

For that purpose, in Section II, we recall the different signal reconstruction techniques from shifted downsampled STFT, and then study how to use these for signal denoising in Section III, assuming the noise is Gaussian white. Exploiting these findings, we introduce, in Section IV, a novel algorithm, based on downsampled STFT, for the retrieval of the modes of a MCS, of which we study, in the next section, the sensitivity to its parameters. We finally carry out a comparison with two other methods based on SST and recently introduced in [START_REF] Oberlin | Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations[END_REF] [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], on both simulated and real data.

II. SIGNAL RECONSTRUCTION FROM DOWNSAMPLED SHORT-TIME FOURIER TRANSFORM

For a signal f in l 2 (Z), and a discrete window g also in l 2 (Z), the STFT downsampled by a factor of R (a positive integer) and shifted by an integer parameter p ( 0 ≤ p < R) is defined for each ω by:

V g f (mR + p, ω) = n∈Z f [n]g[n -mR -p]e -i2πω(n-mR-p) . (1)
If one assumes the signal is of length L and supported on {0, • • • , L -1} , the filter g supported on

{-M, • • • , M } such that L g := 2M + 1 ≤ N ≤ L/2
, where N is the number of frequency bins, one

January 25, 2018 DRAFT may write:

V g f (mR + p, k N ) = n∈Z f [n]g[n -mR -p]e -i2π k(n-mR-p) N = M n=-M f [mR + p + n]g[n]e -i2π kn N = 2M n=0 f [mR + p + n -M ]g[n -M ]e -i2π k(n-M ) N .
Since g is null on {M + 1, • • • , N -1 -M }, STFT can be rewritten as:

V g f (mR + p, k N )e -i2π kM N = N -1 n=0 f [mR + p + n -M ]g[n -M ]e -i2π kn N .
Using the properties of the discrete Fourier transform (DFT), one obtains, for any n ∈ {0, • • • , N -1}:

f [mR + p + n -M ]g[n -M ] = 1 N N -1 k=0 V g f (mR + p, k N )e i2π k(n-M ) N . ( 2
)
When g is non-zero on {-p, • • • , R -1 -p}, one gets, for any q ∈ {0, • • • , R -1}:

f [mR + q] = 1 g[q -p]N N -1 k=0 V g f (mR + p, k N )e i2π k(q-p) N . (3)
This reconstruction technique will be referred to as the first reconstruction technique (RT1) in the sequel.

Since p varies in {0, • • • , R -1}, if one wants to use the same filter g whatever p, the former has to be non-zero on

{-R + 1, • • • , R -1}.
Alternatively, using (2) and assuming m∈Z g[n -mR] 2 = 1 for all n, we may write:

f [n] = N -1 k=0 m∈Z V g f (mR + p, k N )g[n -mR -p] e i2π k(n-mR-P ) N N . (4)
The proof of ( 4) is available in Section Appendix. Furthermore, when such a filter is used, one also has energy conservation, namely:

n∈Z f [n] 2 = m∈Z 1 N N -1 k=0 |V g f (mR + p, k N )| 2 , (5)
the proof being also available in Section Appendix. Note that the reconstruction can still be carried out by making the appropriate renormalization as follows: It is important to remark that such a reconstruction formula was already studied in [START_REF] Griffin | Signal estimation from modified short-time Fcourier transform[END_REF] but for infinite signals and assuming a continuous frequency. Here, the signal is finite and the frequencies are discretized. Also, in our context the number of frequency bins is much smaller than the signal length for the method to be tractable for long signals.

f [n] = N -1 k=0 m∈Z V g f (mR + p, k N )g[n -mR -p] e i2π k(n-mR-p) N N m∈Z g[n -mR -p] 2 , (6)
Alternatively, assuming this time m∈Z g[n -mR] = 1 for all n, one similarly has:

f [n] = N -1 k=0 m∈Z V g f (mR + p, k N ) e i2π k(n-mR-p) N N , (7)
the proof being available in Section Appendix. Note that, as soon as m∈Z g[n -mR] = 0 for all n, one can still reconstruct f through:

f [n] = N -1 k=0 m∈Z V g f (mR + p, k N ) e i2π k(n-mR-p) N N m∈Z g[n -mR -p] . (8)
This reconstruction formula is referred to as third reconstruction technique (RT3) in the sequel.

Remark 2. Typical filters g that can be used in RT2 and RT3 are those satisfying the constant overlap add property [START_REF] Allen | Short term spectral analysis, synthesis, and modification by discrete Fourier transform[END_REF] with hop-size R (COLA(R)): m∈Z g[n -mR] = 1, for all n.

For example, the rectangular window defined on {0,

• • • , R-1} is COLA(R).
Other popular examples of COLA(R) windows is the Bartlett window:

w[n] = w r [n] 1 -|n| R , for n ∈ {-R, • • • , R}
and where w r is the rectangular window on {-R, • • • , R}, the Hann window defined, for n ∈ {-R, • • • , R}, by:

w[n] = w r [n] 1 2 + 1 2 cos( πn R ) , or the Hamming window, w[n] = w r [n] 0.54 + (1 -0.54) cos( πn R ) , for this time n ∈ {-R, • • • , R -1}
, and putting w[R] = 0. One has however to bear in mind that COLA(R) condition is only sufficient for signal reconstruction from downsampled STFT.

III. SIGNAL DENOISING FROM DOWNSAMPLED STFT

Let us consider a discrete-time noisy signal f = f + σΦ where Φ is a unit variance, zero mean white Gaussian noise. When the signal samples are complex numbers, the noise is also complex, and Φ is such that V g Φ (m, k N ) is also Gaussian with zero mean and satisfies:

Var V g Φ (m, k N ) = Var V g Φ (m, k N ) = σ 2 g 2 2 , (9)
where {X} (resp. {X}) denotes the real (resp. imaginary) part of complex number X, and Var the variance. From this, we deduce that

|V g Φ (m, k N )| 2 ) k,m is χ 2 distributed with 2 degrees of freedom. Because of (9), the expectation of |V g Φ (m, k N )| 2 is 2σ 2 g 2 2 .
Now, the probability of false alarm is less

than 1% if |V g Φ (m, k N )| 2 is larger than 9 × 2σ 2 g 2 2 .
Therefore, assuming the variance of the noise σ 2 is known, we define the following thresholding strategy:

V g f (m, k N ) =    V g f (m, k N ), if |V g f (m, k N )| ≥ 3 √ 2σ g 2 0 otherwise, ( 10
)
where g 2 is the l 2 norm of filter g. Denoised signals are then obtained using one of the three reconstruction formulae replacing V g f by V g f . For the sake of simplicity, we will also denote these techniques by RT1, RT2 and RT3, and f r a reconstructed signal using one of these methods. To assess the quality of the reconstruction procedure using a shifted downsampled STFT, we remark that a good reconstruction technique should be very slightly sensitive to the shift parameter p. To measure the quality of reconstruction we consider the output SNR corresponding to

SN R(f, f r ) = 20 log 10 f 2 f -f r 2 . ( 11
)
Pointing out that the higher the SNR the lower the l 2 error f -f r 2 we remark that, for RT1, we may write:

R-1 q=0 |f [mR + q] -f r [mR + q]| 2 = R-1 q=0 1 g[q -p] 2 N 2 | N -1 k=0 V g f (mR + p, k N ) -V g f (mR + p, k N e i2π k(q-p) N | 2 . ( 12
)
If we define

K(p, q) := | N -1 k=0 V g f (mR + p, k N ) -V g f (mR + p, k N ) e i2π k(q-p) N | 2
, and assume it is independent of p and q, then the l 2 error is minimal when

R-1 q=0 1 g[q-p
] 2 is minimal for some p. For low-pass, positive, and symmetric windows (like Hamming or Bartlett windows, for instance), this corresponds to considering the central taps of filter g, i.e. p = R/2 . Regarding method RT2, energy conservation enables us to write:

q∈Z |f [q] -f r [q]| 2 = m∈Z 1 N N -1 k=0 |V g f (mR + p, k N ) -V g f (mR + p, k N )| 2 , (13)
which is independent of p, provided K(p, p) is independent of p.

To illustrate this, we consider a linear chirp defined for t in {0, ..., L-1} by f (t) = e 2iπ(0.05t+(0.10/L)t 2 ) , with L = 4096. The maximal reduced frequency is 0.25, and the Shannon-Nyquist sampling theory tells us that one needs at least L/2 samples to have perfect reconstruction, meaning the sampling period is half the critical one (i.e. the signal is oversampled by a factor of 2). Then, we compute the STFT of the signal using the Hamming window with length L g = 63 and consider a number N = 256 of frequency bins, and R = 32 (the largest downsampling factor compatible with the three reconstruction techniques since M = 32). For p = 0, we obtain the representation of For RT1, having numerically checked that K(p, q) is almost independent of p and q, we notice that, as expected, best reconstruction results are obtained when p = R/2 . On the contrary, we check that RT2 is insensitive to p, as a consequence of energy conservation. The behavior of RT3 is very similar to that of RT2 but is more difficult to analyze. As we are interested in reconstructing the signal from its January 25, 2018 DRAFT downsampled STFT for any given shift parameter, RT1 should not be used. However, such a reconstruction should be borne in mind since it is used in the synchrosqueezing techniques discussed at the end of this paper.

It is also worth mentioning that when the SNR is equal to 10 dB, and no downsampling is applied, the output SNRs for the linear chirp and with either of the three methods RT1, RT2 and RT3 are 23.39, 23.27 and 22.87 dB respectively: looking at Figure 1 B, we notice that the denoising performance are not sensitive to downsampling factor since similar results are obtained using downsampled STFT in RT2, RT3 and also for RT1, provided the shift parameter is optimally chosen for the latter.

This behavior is also observable when applying the reconstruction techniques to a more general signal, as the MCS whose downsampled STFT is depicted in Figure 1 C, for which the output SNR is displayed in Figure 1 D. In this example, the reduced frequencies are 400 L + 30 L cos(3π t L ) and 1000 L + 60 L cos(3π t L ) with L = 4096, meaning that the maximal reduced frequency equals 0.38: the signal is over-sampled by a factor less than 2 (so with R = 32, the downsampling factor compared with the critical Nyquist sampling is even bigger than in the previous example). As previously, to use downsampled STFT instead of the full transform does not result in a significant performance loss. In the following section, we are going to exploit these findings to propose a novel algorithm for the reconstruction and denoising of the modes of a MCS.

IV. MODE RECONTRUCTION FROM DOWNSAMPLED STFT OF NOISY MCS

In the following, we study more in details MCSs f defined by:

f [n] = K l=1 f l [n], with f l [n] = A l [n]e i2πφl[n] , (14)
where A l [n] > 0 and φ l [n] > 0. We also assume the f l s, called modes or components, are separated with resolution ∆ and with bounded modulation M , i.e. for all times n,

∀ 1 ≤ l ≤ K -1, φ l+1 [n] -φ l [n] > 2∆ ∀ 1 ≤ l ≤ K, |φ l [n]| ≤ M. ( 15
)
Mode reconstruction in the TF plane is carried out by integrating STFT along the frequency axis and in the vicinity of so-called ridges, approximating the φ l s, called instantaneous frequencies (IF) of the modes. Therefore, we first explain how ridges are detected and then detail how mode retrieval is performed. In what follows, f will be a noisy version of f as introduced in Section III.

January 25, 2018 DRAFT

A. Ridge Extraction from Noisy Downsampled STFT

We here recall how to compute an estimate of the ridges (n, φ l [n]) l . Assuming knowledge of the number K of modes, we are going to use an adaption of the algorithm detailled in [START_REF] Daubechies | Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool[END_REF] or [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF], which was originally proposed in [START_REF] Carmona | Characterization of signals by the ridges of their wavelet transforms[END_REF]. This computes a local minimum of the functional :

E f (ψ 1 , • • • , ψ K ) = K l=1 L-1 n=0 -|V g f (n, ψ l [n])| 2 + λψ l [n] 2 dt + βψ l [n] 2 , (16)
so that ψ l approximates φ l . However, equation ( 16) does not offer any algorithmic means to compute the ridges and does not take into account the downsampling factor. It was shown in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF] through a numerical study, that to use regularization terms in the energy functional resulted in less accurate ridge estimation.

Thus, to detect the ridge of a monocomponent signal and in the downsampled STFT context, ( 16) reduces to:

ψ[mR + p] = argmax 0≤k≤N -1 |V g f (mR + p, k N )|, (17)
making m vary. The performance of this IF estimator is available in [START_REF] Stankovic | Performance of spectrogram as if estimator[END_REF].

Algorithm 1 (Ridge extraction from downsampled STFT) Pick p ∈ {0, • • • , R -1} Define Z p,R = {mR + p, m ∈ L-1 R } ∆ f = R × M × N for l = 1 to K do 1. Pick t m = mR + p ∈ Z p,R . 2. p m,l := argmax k |V g f (t m , k N )|. 3. I m,l := [p m,l -∆ f , p m,l + ∆ f ] {0, ..., N -1}. 4. p m-1,l := argmax k∈Im,l |V g f (t m-1 , k N )|, iterate backward. 5. p m+1,l := argmax k∈Im,l |V g f (t m+1 , k N )|, iterate forward. 6. For t m ∈ Z p,R , set V g f (t m , k/N ) := 0, for k ∈ [p m,l -∆N, p m,l + ∆N ] Sort (p m,l ) l in increasing order to obtain (p m,l ) l (ψ l [t m ]) m = ( pm,l N ) m
To extract the ridges of a MCS from its downsampled STFT, we generalize [START_REF] Donoho | Ideal spatial adaptation via wavelet shrinkage[END_REF], using a procedure summarized in Algorithm 1. The ridges are extracted one by one as follows. One first picks a random time January 25, 2018 DRAFT and finds the local maximum of the modulus of STFT at that time (steps 1 and 2, this point is supposed to be on a ridge), then, since mode modulation is bounded by M , one finds neighboring points on the ridge using the fact that the maximum displacement of a point on a ridge between two successive locations is ±M × R (which means ±M × R × N in terms of indices, steps 3-5), and finally iterates backward and forward (steps 4 and 5). One then removes the detected ridge and corresponding TF neighborhood (step 6, the ridges are associated with K sufficiently separated local maxima at each time instant), and proceeds with the remaining ridges.

Note that to improve the quality of ridge detection, for each mode, steps 1 to 5 are iterated and the retained ridge corresponds to the most energetic one after these iterations. This algorithm is similar to that proposed in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF] for ridge extraction, except that we take into account the downsampling factor R in the search for p m-1,l and p m+1,l from p m,l .

B. Mode Reconstruction

Based on RT2 and the signal denoising procedure from downsampled STFT introduced in Section III, we define a novel technique for the retrieval of the modes of a MCS (to consider RT3 for mode reconstruction instead would not change the results that much). Let us focus on the reconstruction of the l-th mode, assuming ψ l is known. For that purpose, we recall that a first order approximation of STFT close to the lth ridge and in the continuous time setting, reads [START_REF] Oberlin | The Fourier-based synchrosqueezing transform[END_REF]:

V g f ,c (t, ω) ≈ A l (t)ĝ(ω -φ l (t)), (18)
where

V g f,c (t, ω) = R f (u)g(u-t)e -i2πω(u-t) du (assuming g is a real window) and f (ω) = R f (u)e -i2πωu du. If the window g is the Gaussian window σ -1/2 s e -π x 2 σ 2 s then ĝ(ξ) = σ 1/2 s e -πσ 2 s ξ 2 .
Using such a window, considering the approximation of φ l by ψ l , and adapting [START_REF] Baraniuk | Measuring time-frequency information content using the renyi entropies[END_REF] to our discrete time and frequency settings, we get at time t m := mR + p:

V g f (t m , k N ) ≈ |V g f (t m , ψ l [t m ])|e -πσ 2 s ( k N -ψl[tm]) 2 , for k N sufficiently close to ψ l [t m ].
If, as previously, the noise variance is σ 2 , the coefficients associated with mode l one selects at each time t m , following the procedure detailled in Section III, correspond to the indices k satisfying:

|V g f (t m , ψ l [t m ])|e -πσ 2 s ( k N -ψl[tm]) 2 ≥ 3 √ 2σ g 2 ,
namely the indices inside the interval J l,m :=

[N ψ l [t m ] -Γ l,m , N ψ l [t m ] + Γ l,m ], with Γ l,m = N σ s - 1 π log 3 √ 2σ g 2 |V g f (t m , ψ k [t m ])|
.

Knowing interval J l,m , mode f l is reconstructed through:

f r,l [n] := m∈Z k∈Jl,m V g f (t m , k N )g[n -t m ] e i2π k(n-tm ) N N . (19)
This method will be denoted by M 1 in the sequel. Doing so, one selects the portion of signal above noise level in the vicinity of each ridge. However, such a simple expression for J l,m exists only because one assumes the window is Gaussian. Also, the width of interval J l,m only depends on the magnitude of STFT at local maxima location, but not on the frequency modulation of the corresponding modes.

Therefore, an alternative to compute the frequency interval for mode reconstruction is to define η

[1]
l,m and η [START_REF] Daubechies | Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool[END_REF] l,m : η

[1] l,m = argmax k k N < ψ l [t m ], |V g f (t m , k N )| < 3 √ 2σ g 2 η [2] l,m = argmin k k N > ψ l [t m ], |V g f (t m , k N )| < 3 √ 2σ g 2 ,
and then set :

J l,m := N ψ l [t m ] -η [1] l,m , N ψ l [t m ] + η [2] l,m . (20)
In such a case, the interval is no longer symmetric with respect to the maximum amplitude location and adapts to frequency modulation. This method will be denoted by M 2 in the sequel. The comparison between methods M 1 and M 2 will be carried out in the Section Numerical Applications.

V. PARAMETERS ESTIMATION

A. Estimation of the Threshold for Denoising and Mode Reconstruction

To perform coefficients selection for STFT denoising prior to mode reconstruction, one needs to estimate γ := σ g 2 from the STFT of f . This is done by considering the median absolute deviation approach [START_REF] Sachs | Applied Statistics: A Handbook of Techniques[END_REF], which we adapt to our discrete STFT context. Indeed, as shown in [START_REF] Donoho | Ideal spatial adaptation via wavelet shrinkage[END_REF] but for wavelet decompositions, since there are very few STFT coefficients associated with the signal and since the real part of STFT of the noise is a zero mean Gaussian variable, γ can be estimated through:

γp = median | V g f (mR + p, k N ) m,k | 0.6745 . ( 21
)
Computing the estimation γp given by formula (21) on the linear chirp associated with Figure 1 A, when the SNR equals 10 dB, we get an estimate fluctuating around 1.22 (the ground truth being 1.14), the amplitude of the variations are less than 1% when both the shift parameter p and the downsampling factor R vary (tested values for R were 8, 16, 32 and 64, while 0 ≤ p ≤ R -1). Similar conclusions could be obtained with another SNR. Finally, the same type of results are derived with more complicated signals like the one associated with Figure 1 C.

B. Optimal Window Length Determination

An important parameter in all these TF methods is the length of the analysis window. Our approach is to use, as in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF] [18], the Rényi entropy to determine an optimal window length. For the Gaussian window, this corresponds to the determination of an optimal window parameter σ s , and an optimal L g for the Hamming window. In our framework, the optimal window length is the one leading to the most concentrated representation, i.e. associated with the lowest Rényi entropy defined by:

H(σ) = 1 1 -α log 2   N -1 k=0 L-1 n=0   |V g f (n, k N )| V g f 1   α   . (22)
Here, we measure the Rényi entropy on the full STFT, but the conclusions would be the same on a downsampled STFT. Parameter α is set to 3 but another value would lead to similar results. In Figure 2 A and B, we display the Rényi entropy of the STFT of the signal associated with Figure 1 C, for various input SNRs, and with respect to the filter length parameter, for the Gaussian and Hamming windows respectively. Looking at these figures, and as already noticed in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], the minimum entropy clearly puts forward an optimal window length which varies very little with the noise level. In these simulations, we took N = 512 and then the optimal window parameter σ s equals 0.15 and the optimal L g equals 161 for the Hamming window.

VI. NUMERICAL APPLICATIONS

In this section, we first investigate the quality of mode reconstruction with respect to the type of reconstruction method, i.e. M 1 or M 2 , type of filter, downsampling factor and shift parameter. Then, we compare the proposed mode reconstruction methods with other ones based on synchrosqueezing transform [6] [8]. In this regard, a particular emphasis is put on the amount of information needed to perform mode reconstruction (the MATLAB code to reproduce the different figures of this paper is available at https://github.com/meignen/STFT MCS DENOISING).

A. Sensitivity of Mode Reconstruction to Filter Choice, Downsampling Factor and Shift Parameter

We here study the sensitivity of mode reconstruction techniques M 1 and M 2 , to the choice of downsampling factor, shift parameter, and filter g. Our experiments are again based on noisy versions of the signal associated with Figure 1 C, with N = 512, and the optimal filter is chosen as explained in the previous subsection (we only consider Hamming or Gaussian filter in our simulations). Taken into account the optimal length associated with each of these filters, the largest downsampling factor one can consider is R = 128, but to take R ≤ 64 is sufficiently informative on the behaviors of the proposed techniques.

To measure the performance of the different reconstruction techniques we compute the output SNRs associated with the reconstruction of each mode, i.e. SN R(f l , f r,l ), where the output SNR is defined in [START_REF] Daubechies | concentration of frequency and time via multitapered synchrosqueezed transform[END_REF] and f r,l in [START_REF] Behera | Theoretical analysis of the second-order synchrosqueezing transform[END_REF].

As for the whole signal reconstruction procedure studied in section III, the output SNR, when considering mode reconstruction, is almost insensitive to the shift parameter. Therefore, for the sake of simplicity, we omit this parameter in the simulations and consider p = 0. The results depicted in Figure 3 A, for an input SNR equal to 0 dB or 10dB, show that the quality of reconstruction for f 1 worsens when the downsampling factor increases, whatever the type of reconstruction method used, but the performance loss is not significant at least for M 2 . On that example, M 1 behaves slightly better than M 2 but the discrepancy between the results provided by the two methods remains within a range of 1 dB. Finally, to use the Hamming or the Gaussian window makes very little difference. If one carries out the same study on mode f 2 , which is much more modulated than f 1 , the importance of taking into account the modulation in the reconstruction procedure transpires. The gain obtained by using M 2 rather than M 1 is at least 2 dB at high noise level, and the benefits of using the former technique is even greater when the noise level is low. Based on these results, we will use M 2 in the comparison with synchrosqueezing techniques that follows.

B. Comparison with Synchrosqueezing Transform Based Methods

Recently, a technique extending STFT-based synchrosqueezing transform (FSST) to MCSs containing strongly frequency modulated modes was proposed in [START_REF] Oberlin | Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations[END_REF] [19], and was called second order synchrosqueezing (VSST) (original description of STFT-based synchrosqueezing transform can be found in [3] [20]). Such a method was also used in a demodulation algorithm, still for the purpose of mode retrieval [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF]. After having recalled the principle of VSST and of the algorithm based on VSST and demodulation for mode retrieval [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], we are going to carry out comparisons between these techniques and M 2 .

The principle of VSST is to reassign the TF representation given by STFT, along the frequency axis, using the following complex estimate of the second order derivative of the phase of f , defined as follows:

q f (n, k N ) = 1 2iπ V g f (n, k N )V g f (n, k N ) -(V g f (n, k N )) 2 V tg f (n, k N )V g f (n, k N ) -V tg f (n, k N )V g f (n, k N ) . January 25, 2018 DRAFT Introducing ω f (n, k N ) = ∂tV g f (n, k N ) 2iπV g f (n, k N ) and t f (n, k N ) = n - ∂ηV g f (n, k N ) 2iπV g f (n, k N )
, enables the definition of an IF estimate of f as [START_REF] Behera | Theoretical analysis of the second-order synchrosqueezing transform[END_REF]:

ω(2) f (n, k N ) =          {ω f (n, k N ) + q f (n, k N )(n -t f (n, k N ))} if ∂ t t f (n, k N ) = 0 {ω f (n, k N )
} otherwise, which is proved to be exact for a noise-free Gaussian modulated linear chirp [START_REF] Behera | Theoretical analysis of the second-order synchrosqueezing transform[END_REF]. VSST then consists of reassigning the STFT of f through:

T V g f (n, k N ) = 1 g[0] |V g f (n, q N )| ≥ γ round(N ω(2) f (n, q N )) = k V g f (n, q N ), (23)
where round(X) denotes the nearest integer to X and γ is the threshold introduced in Section III. Note that (23) can be viewed as a local version of RT1 (with R = 1) , the coefficient being summed up at their reallocation point.

The reconstruction of f l is subsequently performed by means of the following formula:

f l [n] ≈ |k-round(N φ l [n])|<d T V g f (n, k N ). (24)
We shall remark that d is an integer that enables to compensate for reassignment and ridge estimation errors. It can be increased up to ∆ and, in the absence of noise, and, if one uses a large d, one ends up with a reconstruction close to RT1 (except that one performs the reconstruction mode by mode) so that output SNR is better. In noisy situations, this appears to be numerically also the case since most of the noise has been removed by the thresholding procedure. In practice φ l is unknown and is replaced by its approximation ψ l computed on T V g f , using the same approach as the one developed in Algorithm 1 but with R = 1.

Alternatively, one can use a demodulation algorithm prior to reconstruction as proposed in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], by using the ridge estimate given by (ψ l ) l=1,••• ,K , and reconstruct the modes through Algorithm 2 (in which b 0 is a constant and Ψ l a primitive of ψ l null at 0). This method was shown, in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], to behave better in terms of mode retrieval.

At first glance, since with VSST techniques, the information is reassigned in the vicinity of the ridge associated with one mode, for a given time, mode reconstruction involves much less coefficients than with STFT [START_REF] Oberlin | Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations[END_REF]. Nevertheless, we have just proven that we could still perform mode reconstruction from a downsampled version of STFT which is not the case with VSST. So, even though VSST leads to a much January 25, 2018 DRAFT Algorithm 2 (mode reconstruction using demodulation and VSST)

Compute ridge estimates (ψ 1 , • • • , ψ K ) from VSST of f for l = 1 to K do 1. Compute fD,l [n] = f [n]e -i2π(Ψl[n]-b0n)
2. Extract ridge ψ D,l from VSST of fD,l using single ridge detection in [b 0 -∆, b 0 + ∆].

3. Reconstruct f l through:

f l [n] ≈ |k-N ψD,l(t)|<d T V g fD,l
(n, k N ) e i2π(Ψl[n]-b0n) .

sharper representation than STFT, if one is interested in signal reconstruction using the least amount of information as possible, we are going to show that the reconstruction based on STFT performs better than those based on VSST. To quantify this, we compute the mode reconstruction performance for the signal associated with Figure 1 C. We investigate the behavior of the just recalled VSST techniques (either based on direct reconstruction or using demodulation beforehand) with respect to parameter d, along with mode reconstruction based on method M 2 , using the optimized Hamming filter and when the downsampling parameter R varies.

The output SNRs are displayed in Figure 4 A-D, and correspond to input SNRs 0 or 10 dB (the results are averaged over 5 realizations, Algo 2 denoting the reconstruction technique associated with Algorithm 2). As the reconstruction with M 2 does not depend on d, in these figures, the mode reconstruction results

January 25, 2018 DRAFT with this technique are displayed as constant functions. As already noticed, M 2 is not much sensitive to the downsampling factor when the latter goes from R = 16 to R = 64, for both f 1 and f 2 .

Analyzing the results regarding VSST techniques, and as reported in [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF], to demodulate the signal as proposed in Algorithm 2, results in better mode reconstruction than when using VSST directly. Now comparing M 2 with VSST based techniques, M 2 leads better results when only the information on the ridge is considered for VSST (d = 0), but when d increases, typically d = 5, mode reconstruction performed with Algorithm 2 seems to perform similarly to M 2 in most cases.

However, there is one key issue which is hidden in the just computed output SNRs, which is the number of coefficients used to perform the reconstruction with each method. Indeed, for a fair comparison, we compute the number of coefficients required by the different reconstruction procedures. If the reconstruction procedure is based on Algorithm 2 or on VSST, the total number of coefficients to be stored is N V SST = K(2d + 1)L. On the contrary, the number of coefficients used by M 2 corresponds to:

N M2 = K l=1 m # J l,m Z , (25)
where J l,m is the interval defined in [START_REF] Thakur | Synchrosqueezing-based recovery of instantaneous frequency from nonuniform samples[END_REF] and # {X} the cardinal of set X. Note that the range for m depends on the downsampling factor R. In Figure 5, we depict for input SNRs equal to 0 or 10 dB, the output SNRs associated with mode reconstruction with respect to the number of coefficients kept, i.e. N V SST or N M2 , divided by signal length L. Doing so, we highlight the very different nature of the mode reconstruction techniques based on Algorithm 2 and M 2 . Indeed, whatever the type of modes considered, for a given output SNR, mode retrieval is performed using much less information using the latter technique than the former. But more than that: the latter technique, while denoising the modes, can also compress the whole signal, since the number of coefficients kept over the signal length L is lesser than one when R ≥ 64. On the contrary, VSST based denoising techniques always require more than L coefficients. In this example, the amount of information needed to get the same output SNR is about 15 times larger with the technique associated with Algorithm 2 than with M 2 . Note finally that for the former technique the results are not sensitive to the modulation (f 1 reconstruction versus that of f 2 ), which is expected because this algorithm demodulates each mode before proceeding with reconstruction ( see step 1 of Algorithm 2).

C. Application to Real Data

In this section, we investigate the behavior of our new method for the denoising of a bat echolocation call, whose noise-free STFT is depicted in Figure 6 A. As one does not know the ground truth in terms of the modes to be extracted, one cannot compute output SNR for each mode ; the only thing one can do is to compute a global output SNR, as defined in [START_REF] Daubechies | concentration of frequency and time via multitapered synchrosqueezed transform[END_REF]. In this regard, we are going to check that we do not lose much information by considering mode extraction as in M 2 . Another difference between this signal and the one associated with We again analyze the quality of denoising with respect to the number of coefficient kept over the length of the signal, the results depicted in Figure 6 B and C, respectively correspond to a noise level of 0 and 10 dB. The number K of modes is set to 3 or 4 in M 2 and also in Algorithm 2 . We first remark that while the latter technique is sensitive to the number of modes (N V sst is linear with respect to K), M 2 is not. Indeed, if one considers an irrelevant mode number with M 2 , the corresponding coefficients will be mostly below the noise level and will thus not be considered. Another interesting fact is that, in such a context, to extract the mode for denoising purpose may be useless, because the coefficients above the noise threshold are almost all located in the vicinity of the different ridges: given a downsampling factor, the number of selected coefficients is almost the same with M 2 or with the denoising procedure introduced in Section III (denoted by "No ridge" in Figures 6 B andC). Finally, we also remark that when the downsampling factor is large, the denoising performance becomes worse when extracting the ridges in the denoising process: if one wants to denoise the signal while compressing it, in such a situation, ridge extraction should be precluded.

VII. CONCLUSION

In this paper, we first recalled different signal reconstruction procedure using downsampled versions of

APPENDIX

If g is such that, for any n, m∈Z g[n -mR] 2 = 1, from (2) and putting q = n + mR + p -M , we may write:

f [q]g[q -mR -p] = 1 N N -1 k=0 V g f (mR + p, k N
)e i2π k(q-mR-p) N .

From this we deduce:

f [q] m∈Z g[q -mR -p] 2 = m∈Z 1 N N -1 k=0 V g f (mR + p, k N )e i2π k(q-mR-p) N g[q -mR -p] ⇔ f [q] = m∈Z 1 N N -1 k=0
V g f (mR + p, k N )e i2π k(q-mR-p) N g[q -mR -p] m∈Z g[q -mR -p] 2 .

If m∈Z g[n -mR] 2 = 1, we get (4), and if it is only non-zero for any n, (6) follows.

Regarding energy conservation, from (2) we may write:

m∈Z N -1+mR+p-M q=mR+p-M f [q] 2 g[q -mR -p] 2 = m∈Z N -1+mR+p-M q=mR+p-M | 1 N N -1 k=0 V g f (mR + p, k N )e i2π k(q-mR-p) N | 2 ⇔ m∈Z q∈Z f [q] 2 g[q -mR -p] 2 = m∈Z 1 N N -1 k=0 |V g f (mR + p, k N )| 2 ⇔ q∈Z f [q] 2 = m∈Z 1 N N -1 k=0 |V g f (mR + p, k N )| 2 .
The proof of ( 7) is similar to that of (4). Indeed, we have:

f [q] m∈Z g[q -mR -p] = m∈Z 1 N N -1 k=0 V g f (mR + p, k N )e i2π k(q-mR-p) N ⇔ f [q] = m∈Z 1 N N -1 k=0
V g f (mR + p, k N )e i2π k(q-mR-p) N m∈Z g[q -mR -p] .

If m∈Z g[n -mR] = 1 for any n, then one obtains [START_REF] Pham | High-order synchrosqueezing transform for multicomponent signals analysiswith an application to gravitational-wave signal[END_REF] and otherwise if the sum is only non-zero [START_REF] Meignen | On demodulation, ridge detection and synchrosqueezing for multicomponent signals[END_REF] follows.

  provided m∈Z g[n -mR] 2 = 0 for all n. This type of reconstruction technique is denoted second reconstruction technique (RT2) in the sequel. January 25, 2018 DRAFT Remark 1.

DFig. 1 .

 1 Fig. 1. A: STFT of a linear chirp; B: output SNR corresponding to the reconstruction of signal associated with STFT displayed in A, when the shift parameter varies, for different input SNR and for RT1, RT2, and RT3; C: STFT of a MCS; D: output SNR corresponding to the reconstruction of signal associated with STFT displayed in C, when the shift parameter varies, for different input SNRs and for RT1, RT2, and RT3.

Figure 1 A

 1 , corresponding to a 256 × 128 matrix and, changing p does not change the matrix size. So, if one compares with the critical sampling, requiring 2048 values for f , the downsampling factor actually equals 16. Then, for different noise levels, shift parameter in {0, • • • , R -1} and input SNRs, we display in Figure 1 B, the output SNR corresponding to the reconstruction of the linear chirp using either RT1, RT2, or RT3.

Fig. 2 .

 2 Fig. 2. A: Computation of Rényi entropy with respect to σs when the Gaussian window is used to compute STFT, for different noise level ; B: same as A but when the Hamming window is used instead

DFig. 4 .

 4 Fig. 4. A: output SNR associated with the reconstruction of mode f1 of the signal associated with Figure 1 C, using different VSST techniques and with respect to parameter d, or M2 with optimized Hamming filter and different downsampling factors R (SNR 0 dB); B: same as A but for mode f2; C: same as A but with SNR 10 dB; D: same as B but with SNR 10 dB.

Fig. 5 .

 5 Fig. 5. Output SNR associated with the reconstruction of the signal associated with Figure 1 C with respect to the number of coefficients kept divided by the signal length (input SNR 0 or 10 dB)

Figure 1 CFig. 6 .

 16 Fig. 6. A: STFT of a bat echolocation call, B: output SNR associated with the reconstruction of the signal associated with A with respect to the number of coefficients kept divided by the signal length (input SNR 0 dB), the sensitivity to the number K of modes is also tested; C: same as B but with an input SNR of 10 dB.

  short-time Fourier transform. From this, we have derived a new algorithm for signal denoising which we have then adapted to the denoising of multicomponent signals and mode extraction. We then showed that the proposed new technique compares favorably, on simulated and real data, with the most recent denoising techniques based on synchrosqueezing in terms of the number of coefficients used in the reconstruction procedure. Future work should involve the generalization of the proposed denoising approach to non Gaussian noises.

January 25, 2018 DRAFT