Luciano Da Fontoura Costa
email: luciano@ifsc.usp.br

Convolution!

Introduction

Convolution... or, should we shout, convolution! This seemingly intricate operation is, at the same time, surprisingly important in physics, engineering, statistics, mathematics, and many other theoretical and applied areas.

Given two functions, convolution combines them in a specific (linear and equitable) way, facilitating several key mathematical operations, ranging from integrodifferential operations to pattern recognition -passing through interpolation, system modeling, and filteringto name but a few more immediate applications.

It is important to notice convolution is not a function, but a binary operation in the sense that it receives two functions as input, producing a third function as output, as illustrated in Figure 1. The Dirac's delta 'function' acts as the identity element, in the sense that, when convolved with another function g(t), yields that same function. However, there The convolution between two real/complex functions g(t) and h(t) is a third real/complex function w(t) that can often be understood as a combination of the two input functions. Convolution is commutative, so there is no need to distinguish between its two inputs.

is no guaranteed inverse function of g(t), so that the binary operation of convolution does not formally constitute a group (e.g. [START_REF] Da | Group theory: A primer. Researchgate[END_REF]). Yet, convolution is a bilinear operation, in the sense that when one of the functions is kept fixed, the convolution acts linearly on the other function, and vice-versa.

Despite its importance, convolution can seem intricate at times. This probably stems from its own definition, in particular the variable transformation τ -t, the fact that convolution corresponds to an infinite integral, the involvement of two functions, and the fact that complex functions are often adopted. Interestingly, even if only as a coincidence, the own related term convoluted is traditionally taken to be a synonymous of intricate.

Better understanding and application of mathematical concepts and methods can greatly benefit from respec-tive intuitive and conceptual assimilations. Fortunately, there are several intuitive interpretations that can help one to develop a more conceptual and complete understanding of the type of operations performed by the convolution between two functions. In this work, we resource to several approaches so as to contribute to that finality. More specifically, we start by presenting an alternative, but equivalent, definition of convolution that allow the variable shifting required by this operation to be performed without involving the additive inverse of that variable, as typically implied by the more traditional definition of convolution. In addition, we also discuss convolution as the 'blending' or 'mixing' between the two involved functions, and also as 'matching' between those functions. The convolution is also approached considering the frequency domain, as allowed by the convolution theorem and the Fourier transform (which are also briefly presented). This approach allows one to realize that the convolution between two sinusoidal functions if null except for the case in which the two function have exactly the same frequency.

The consideration of several properties and applications of this important binary operation, including linear systems modeling and filtering, can provide further insights and consolidation of a more integrated and complete conceptual understanding of convolution.

This constitute the main objective of the current text.

Convolution

The convolution between two functions g(t) and h(t), both mapping from real values t into complex images, is traditionally defined as

[g * h] (τ ) = ∞ -∞ g(t)h(τ -t)dt (1)
If we make the variable substitution u = τ -t, we have du = -dt, from which follows that

[g * h] (τ ) = ∞ -∞ g(τ -u)h(u)du (2)
implying that convolution is commutative, i.e. g(t) * h(t) = h(t) * g(t).

Though this definition has been frequently considered, it involves a combination of the value τ with -t, which is perhaps not very intuitive, as the addition of τ in the argument of h(), i.e. h(τ -t), implies this function being displaced to the right, and not to the left as it would happen otherwise if t were not negated, i.e. h(t + τ ).

A more intuitive expression for the convolution can be obtained by making the variable substitution u = -t, so that

g(t) * h(t) = [g * h] (τ ) = ∞ -∞ g(-t)h(t + τ )dt (3)
Now, we have that the variable τ , which is external to the integral, is added to t instead of -t. This is achieved at the expense of reflecting g(t) into g(-t), but this is more intuitive since this reflection does not involve shiftings by τ .

The above alternative definition of convolution, henceforth adopted in this work, also allows a simple graphical explanation of the operation implemented by the convolution between the two original functions g(t) and h(t), which is illustrated in Figure 2 for an even g(t) (i.e. g(t) = g(-t)). First, we rotate function g(t) around the vertical axes, which has no effect in this particular case as g(t) is an even function. Then, for each value of -∞ < τ < ∞, a shifted version of h(t), namely h(t+τ ) is obtained and multiplied by g(-t), yielding a new function, shown in gray, whose area A corresponds to [g * h] (τ ) for each specific value of τ .

The more general situation in which g(t) is not even can be immediately addressed by rotating this function around the vertical axis as the first step.

Observe that the convolution operation tends to increase the extension along t (non-zero values) of the resulting function with respect to the two original inputs. Actually, it can be shown that this extension corresponds to the sum of the extensions of the g(t) and h(t) in the case these extensions are bound.

This example illustrates why convolution is sometimes understood as a blend of its two input functions g(t) and h(t), in the sense that [g * h] (τ ) inherits characteristics from both original functions.

3 The Dirac Delta 'Function'

Formally speaking, the Dirac delta 'function' δ(t) is not a function as its valued is not defined at t = 0. According to the theory of distributions (e.g. [START_REF] Van Dijk | Distribution Theory[END_REF]), this function corresponds to a functional that takes an original, 'wellbehaved' function g(t) into the scalar corresponding to its value at zero, i.e. g(0).

Consider the rectangular function with area 1 given as

r(t) = c = 1 2a , for x ∈ [-a, a] 0, otherwise (4)
The Dirac delta 'function' can be informally understood as the following limit of r(t)

δ(t) = lim a→∞ r(t) (5)
Since r(t) has unit area for any a > 0, it follows that

∞ -∞ δ(t)dt = 1 (6)
We will informally understand that

δ(t) g(t) = δ(t) g(0) (7)
and we obtain

∞ -∞ δ(t)g(t)dt = g(0) (8)
which is sometimes understood as the sampling property of the Dirac delta 'function'.

This result allows us to easily obtain the convolution of a function g(t) with the Dirac delta, i.e.

∞ -∞ δ(-t)g(t + τ )dt = ∞ -∞ δ(-t)g(τ )dt = g(τ ) (9)
Interestingly, this results in the original function g(t), and we can understand the Dirac delta as a kind of identity element of the convolution. Figure 3 illustrates this result graphically. A relatively frequent situation involves convolving a function g(t) with a sum of Dirac deltas, such as

S(t) = ∞ k=-∞ δ(t -k ∆t) ( 10
)
where ∆t is the spacing between the infinite sequence of Dirac deltas.

The effect of convolving S(t) with a function g(t) can be easily understood by taking into account that the convolution is a bilinear operation, from which follows that

[S * g] (τ ) = ∞ k=-∞ g(t -k ∆t) (11)
In other words, this convolution involves adding g(t) at each of the positions k∆t at which the Dirac deltas are located.

Convolution as 'Blending' of Two Functions

Convolution can often be understood as performing some kind of 'blending' or 'mixing' between the two involved functions g(t) and h(t), in the sense that the resulting function 'inherits' features from both g(t) and h(t).

In order to understand this type of interpretation, let's start by observing that the convolution of any function g(t) with the constant function h(t) = c yields another constant function, namely z(t) = c ∞ -∞ g(t)dt (i.e. the area of g(t)), as a result.

Let's also start by considering the sinc function as being defined as:

sinc(t) = sin(πt) πt , for t = 0 1, for t = 0 (12)
Observe that this is the normalized version (area equal to 1) of the sinc function, otherwise defined as sin(t)/t. The sinc is an example of a function localized along the time domain, in the sense that its 'mass' is concentrated at some point R (in this case 0), as can be inferred considering the mean and standard deviation of the sinc function. Informally speaking, a localized function will have most of its area ('mass') concentrated around the interval having a few standard deviations as width and centered at R. Being localized also implies that the function tends to decrease to the left and righthand sides R.

Let's also consider the rectangle function with width L as being defined as:

R L (t) = 1, for -L 2 < t ≤ L 2 0, otherwise (13)
If L is much larger than the standard deviation of the sinc (or any other localized function), we have that, except the regions near the transitions of R L (t), the convolution between the sinc (g(t)) and the rectangle function (h(t)) can be well approximated by a convolution between the sinc and a constant function, so that the it results approximately to the sinc area ∞ -∞ g(t)dt, which we know by definition to be equal to 1. So, convolving the sinc over the rectangle function will have almost no effect in the middle portion of the latter function, which will remain almost unaltered.

However, the convolution will produce different results around each of the left and right transitions of the rectangle functions. Figure 4 illustrates the convolution between a sinc (a) and a wider rectangle function (b). The resulting function (c) can be understood as a 'blend' or 'mix' of the two original functions, inheriting features from both of them.

Convolution as 'Matching' of Two Functions

When two localized functions have similar sizes, such as in Figures 5(a) and (b), the convolution between them tends to provide a function (c) that peaks at the time position corresponding to the best 'matching' or 'alignment' between the two original functions. Usually, a better 'matching' can be obtained by using the correlation (see Section 12) instead of convolution, since this binary operation does not involve mirroring of any of the two original functions. The 'matching' feature of the convolution is related to the concept of internal product between two real functions g(t) and h(t), which is typically defined as:

g(t), h(t) = ∞ -∞ g(t)h(t)dt (14)
Observe that, in the case of complex functions g(t) and h(t), the internal product typically involves the conjugation of h(t).

The interpretation of this important binary operation can be more conceptually approached by considering its discrete version, i.e. considering finite versions of g(t) and h(t) sampled along time and within a time window, giving rise to two respective real vectors g and h with N elements each. In this case, the internal product is given as:

g i h i = | g|| h|cos(θ) ( 15
)
where θ is the smallest angle between the two vectors. It follows from this definition that, while keeping | g| and | h| vixed, the values of the internal product will depend on θ, reaching its highest value for θ = 0, minimal (negative) value for θ = π, and null value for θ = π/4. So, the internal product between g and h can be conceptually understood as a measurement of the similarity between the angles of these vectors (remember that we chose to keep their magnitudes constant). This interpretation can be somewhat extended to the internal product between the two original functions g(t) and h(t) from which the vectors g and h were derived. Now, we can go back to the convolution definition(Eq. 3) and observe that it actually corresponds to the internal product between g(-t) and h(τ + t). As such, we can conceptually interprete that, for functions with fixed magnitude, the convolution at a given value of relative displacement τ provides some measurement of the 'similarity between g(-t) and h(t). That is, ultimately, the reason why the convolution can be understood as a 'matching' between the two involved real functions. It also explains why, usually, a better 'matching' can be provided by the correlation, which does not invert one of the functions along time, therefore corresponding more directly to the internal product between g(t) and h(t). In case these functions are even, there will be no difference in inferring the 'matching' between them by using either correlation or convolution.

Observe that special care needs to be taken regarding the magnitudes of the involved functions, which were assumed constant in the above discussion. In case the magnitudes can vary, we need to take their direct effect on the 'matching' intensity.

The Fourier Transform

The Fourier transform is intrinsically related to convolution and the Dirac delta 'function', and as such it allows further insights to be obtained regarding the meaning of convolution. In this section we briefly present the continuous Fourier transform in one dimension (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF]).

Let g(t) be a complex (or real) function of t. Under certain circumstances (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF]), its Fourier transform exists and can be calculated as

F {g(t)} = G(f ) = ∞ -∞ g(t)exp {-i2πf t} dt (16)
It is also possible to adopt ω = 2πf .

Observe that this transform maps the original function g(t) in the 'time' domain into a new complex function G(f ) in the 'frequency' domain. The original function g(t) can be recovered (under some conditions) from G(f ) as

F -1 {G(f )} = g(t) = ∞ -∞ G(f )exp {i2πf t} df (17)
When both the above direct and inverse Fourier transforms exist, we have a respective Fourier transform pair.

g(t) ←→ G(f ) (18)
A first interesting property of the Fourier transform, which follows easily from its definition, is that this transform is linear, i.e. F {a g(t)

+ b h(t)} = a F {g(t)} + b F {h(t)}.
Let's calculate the Fourier transform of δ(t).

F {δ(t)} = ∞ -∞ δ(t)exp {-i2πf t} dt = = ∞ -∞ δ(t)exp {-i2πf (0)} dt = ∞ -∞ δ(t)dt = 1 (19)
It can be shown that the inverse Fourier transform of 1 exists and corresponds to δ(t), so we can write our first Fourier transform pair as

δ(t) ↔ 1 (20)
It can be verified that the Fourier transform of a purely real and even function g(t) is real and even. Similarly, the Fourier transform of a purely imaginary and odd function is imaginary and odd.

The Fourier transform has a particularly interesting, though not so often applied, property known as symmetry (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF]). If we have the Fourier pair g(t) ↔ G(f ), it follows that

G(t) ←→ g(-f ) (21)
This property allows us to immediately derive several new Fourier transform pairs from already known results. For instance, if we apply this property to the pair δ(t) ↔ 1, we conclude that

1 ↔ δ(f ) (22)
Let's now see what happens when we shift a function g(t) to the left by t 0 , i.e. g(t -t 0 ). From the Fourier transform definition, we have that

F {g(t -t 0 )} = ∞ -∞ g(t -t 0 )exp {-i2πf t} dt (23)
now, we make the variable transformation u = t -t 0 , which implies

F {g(t -t 0 )} = ∞ -∞ g(u)e -i2πf (u+t0) du = = e -i2πf t0 ∞ -∞ g(u)e -i2πf u du = e -i2πf t0 G(f ) (24)
It can be shown that this time shifting property also holds with respect to the inverse Fourier transform, so that we obtain the pair

g(t -t 0 ) ←→ e -i2πf t0 G(f ) (25)
The frequency shifting property can be similarly derived as

e i2πtt0 G(t) ←→ g(f -f 0 ) (26)
At this point, we can obtain several interesting Fourier pairs involving sinusoidals and complex exponentials. We start with the derivation of the Fourier pair for e i2πf t . By combining the fact that 1 ↔ δ(f ) with the frequency shift property, it follows that

e i2πf0t ←→ δ(f -f 0 ) (27)
Given that cos(2πf 0 t) = 1/2 e i2πf0t + e -i2πf0t , we immediately find that

cos(2πf 0 t) ←→ 1 2 [δ(f + f 0 ) + δ(f -f 0 )] (28)
Analogously, it follows from sin(2πf

0 t) = 1/(2i) e i2πf0t -e -i2πf0t that sin(2πf 0 t) ←→ 1 2i [-δ(f + f 0 ) + δ(f -f 0 )] (29) hence sin(2πf 0 t) ←→ i 2 [δ(f + f 0 ) -δ(f -f 0 )] (30)
We also have, from the time shifting property, that

δ(t -t 0 ) ←→ e -2πf t0 (31)
So, if we imagine that the original function g(t) is composed by a (possibly infinite) sum of properly time shifted Dirac deltas δ(t -t 0 ), we obtain in the frequency domain a respective sum of above complex exponentials (involving sines and cosines). When f is discretized, function g(t) becomes periodic and the inverse Fourier transform becomes related to the Fourier series.

This fact explains why the sine (and cosine) functions are particularly important for the Fourier transform, defining its respective basis, which is used to express the function g(t). The recover of g(t) is performed by the inverse Fourier transform in terms of typically infinite linear combinations of sines and cosines, which we will call harmonic components.

Table 1 lists some commonly used pairs of Fourier transforms.

The Convolution Theorem

One of the strong connections between convolution and the Fourier transform is established through the convolution theorem, which states that

g(t) * h(t) ←→ G(f )H(f ) (32)
We also have that

g(t) h(t) ←→ G(f ) * H(f ) (33)
So, the Fourier transform allows convolutions in the time domain to be performed in terms of function pointwise multiplications in the frequency domain, and vice versa. This can often allow not only substantial savings of computational expenses, but also contribute to a more complete conceptual understanding of the convolution, as developed in the following section.

Table 2 lists some other commonly used Fourier transform properties.

Convolution in the Frequency Domain

We are now in a position to consider convolution between any pair of sinusoidal functions, such as two cosines. It is relatively difficult to infer the result of the convolution of these two functions by considering their distribution along time, e.g. by using graphical constructions such as that described in Section 2.

Figure 6 shows two such functions, with respective frequencies f 1 = 1kHz and f 2 = 2kHz, as well as their respective Fourier transforms (recall that, as a consequence of evenness of the cosine functions, we have purely real Fourier transform results). We have from the convolution theorem that this convolution between two cosines can be obtained as the inverse Fourier transform of the product of their respective transforms. As a consequence of f 1 = f 2 , we have that this product in the frequency domain results in the null function (0 everywhere), so that the respective inverse Fourier is also null. We conclude that the convolution between two cosines (or sines) will only be non-zero in case they both have the same frequency.

This interesting result immediately extends any two functions defined by finite sums of sines and/or cosines (i.e. periodic functions, related to the Fourier series).

Figure 7 illustrates a situation with respect to two functions containing only one (f 1 = 1kHz) and two (f 1 = 1kHz and f 2 = 2kHz) harmonic components, respectively.

Now, the components respective to f 1 = 1kHz in both functions align in the frequency domain, resulting in a non-null convolution outcome that contains only this frequency, though with changed amplitude.

So, the consideration of the convolution action in the frequency domain, as allowed by the Fourier transform, can contribute to our better understanding of the nature and effects of the convolution operation of periodic func- tions, including sines and cosines. 9 Linear, Time Invariant Systems Let's now consider an interesting application of the convolution as a powerful means to accurately model linear, time invariant dynamical systems. Figure 8 illustrates the typical graphic representation of a dynamic system, involving respective input x(t) and output y(t). The respective Fourier transforms X(f ) and Y (f ) are also shown. This system will be linear iff S(a x 1 (t) + b x 2 (t)) = a S(x 1 (t)) + b S(x 2 (t)).

We also consider that system S is time-invariant, i.e. if y(t) = S(x(t)) then y(t-t 0 ) = S(x(t-t 0 )) for any t 0 ∈ R. More informally, this means that the properties of S do not change along time.

Let's also define the impulse response of the system S as the function y 0 (t) obtained when we input a Dirac delta x(t) = δ(t) (the impulse) into S (also shown in Figure 8).

Under these circumstances, it can be shown that, for any input function x(t), we have that

y(t) = x(t) * y 0 (t) (34)
So, once the system has been probed with the impulse δ(t), yielding the respective impulse response y o (t), its respective operation considering any other input can be provided by the above convolution. In other words, all dynamical properties of S are captured into y 0 (t), which therefore assimilates every possible information about the inner workings of S.

This interesting -and useful -property, which may initially sound like 'magic, can be easily understood in an intuitive way in the frequency domain. First we express the expression in Equation 34 in the frequency domain

Y (f ) = X(f ) Y 0 (f ) (35)
Now, remember that the Fourier transform of the Dirac delta impulse is the constant function h(t) = 1. In other words, this function will equally stimulate every possible frequency inside the system S, as it modifies each of these frequencies in its specific way, yielding the resulting frequency response, or transfer function Y 0 (f ).

So, we can understand the system operation as being defined by the Y 0 (f ) and, as a consequence, the response to any other input function x(t) can now be obtained as the inverse Fourier transform of the signal X(f ) Y 0 (f ).

Linear Filters

The above developed framework to model linear, timeinvariant systems can be used to introduce the concept of linear filters. Basically, we consider the filtering function as corresponding to the transfer function above, which is kept fixed during the system operation which will, therefore, act on the signal X(f ) modifying it as specified by the filter function Y o (f ),.

Linear filters are frequently used in a wide range of applications, such as removing additive noise, change the tone of sound signals (bass/treble), communications, emphasizing signal transitions, etc.

Basically, the operation of the filter Y 0 (f ) is specified by the distribution of its values along f . Figure 9 It is important to keep in mind that the terms 'high' and 'low' typically refer to the absolute value of the involved frequencies. Observe that Fourier transforms normally exted along both sides of the frequency axis.

Therefore, low-pass filters, allowing the lower frequencies to pass while attenuating or eliminating the higher frequencies, involve a filter function having higher (positive or negative) values near the frequency axis origin. High-pass filters have an opposite nature. Band and rejection-pass filters allow only an interval of frequencies to pass, reducing or blocking the other frequencies.

Deconvolution

We have seen how a function x(t) can be modified by convolving it with another function y(t) (e.g. a filter), resulting a function z(t) or, in the frequency domain

Z(f ) = X(f )Y (f ).
It is often necessary to consider how x(t) can be recovered from z(t). Provided Z(f ) do not assume any zero value at any frequency f , we can use the convolution theorem to obtain

X(f ) = Z(f ) Y (f ) (36)
This deterministic procedure is called deconvolution, as the means of reversing the convolution effects.

In practice, we often do not known Y (f ), which therefore has to be estimated somehow. In the case of experimental signals, we often will also have some involved noise signal (t), such as in the following case of additive noise z(t) = x(t) * y(t) + (t) (37)

The procedure of determining y(t) in these circumstances is the subject of interest of the area of statistical signal processing, involving methods such as the Wiener and Kalman filters (e.g. [START_REF] Brown | Introduction to Random Signals and Applied Kalman[END_REF]).

Correlation

Correlation corresponds to another interesting operation similar to the convolution that also finds several applications, such as in statistics (self-and cross-correlations) as well as in pattern matching or recognition. The (cross-)correlation between two complex functions g(t) and h(t) is typically defined as

g(t) ⊗ h(t) = ∞ -∞ g(t) * h(t -τ )dt ( 38
)
where * means complex conjugation. The alternative convolution integral adopted in the present work allows a direct comparison between convolution and correlation. Comparing Equations 3 and 38, we observe that convolution and correlation are very similar except for the negation of t in g(t) and the the fact that h(t) is shifted in opposite direction.

Correlation has a property analogous to the convolution theorem:

g(t) ⊗ h(t) ←→ G * (f ) H(f ) (39)
This is often called the correlation theorem. Given a signal g(t), its autocorrelation can be obtained in the frequency domain as

G * (f ) G(f ) = |G(f )| 2 (40)
which corresponds to the power spectrum of g(t).

The energy in g(t) can be understood as corresponding to

E g = ∞ -∞ |g(t)| 2 dt (41)
Parseval's theorem indicates that the energy of the original signal g(t) is preserved in the respective frequency representation G(f ), i.e.

E g = ∞ -∞ |g(t)| 2 dt = ∞ -∞ |G(f )| 2 df = E G (42)
13 Concluding Remarks

Convolution plays a particularly general and important role in many scientific and technological areas, from physics to dynamic systems. Yet, given its relatively intricate complex integral definition, it is sometimes not so well understood intuitive or conceptually. In this present text, we attempted at providing a smooth introduction to convolution, possibly leading to a better and more comprehensive understanding of this important concept.

In order to do so, we resorted to an alternative integral definition of the convolution where the time variable appears without negation, allowing a potentially simpler and more intuitive understanding and graphical representation. In addition, the convolution was discussed as a means to 'blend' or 'mix' the two original function (usually when these are localized along time and one of them is wider than the other), and also as a means to identify the time position where the two original function best 'match' or 'alignt' one another (typically when the two functions are localized and have similar widths). We also considered the close relationship between convolution and the Fourier transform and the convolution theorem, which allowed us to understand that convolution between functions containing a finite number of discrete harmonic components often yield the null function as result.

The developed concepts also allowed a straightforward presentation of the important concepts as the impulse response and filters in linear, time-invariant dynamic systems. The concepts of deconvolution and correlation were also briefly introduced.

In a universe permeated by 'waves' of complexity (e.g. [START_REF] Da | Quantifying complexity. Researchgate[END_REF]), convolution represents a relatively simple concept paving the way to effectively mixing waves in the most diverse range of possible applications.

Acknowledgments.

Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) for sponsorship. This work has benefited from FAPESP grant 15/22308-2.

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Table 1: Some Fourier transform pairs g(t) ↔ G(f ). H(t) is the heavyside step function, sng(t) = 2H(t) -1 is the sign function, R L is the rectangle function with width L, and ∆t = 1/f 0 ∈ R.

g(t)

G(f )

1 δ(f ) δ(t) 1
e i2πf0t δ(f -f 0 ) e -a|t| 2a a 2 +(2πf ) 2 cos(2πf 0 t) 1 2 [δ(f + f 0 ) + δ(f -f 0 )] sin(2πf 0 t) i 2 [δ(f + f 0 ) -δ(f -f 0 )] H(t) 1 2 δ(f ) + 1 i2πf R L (t)
L sinc(L t) sgn(t)

1 i2πf e -at 2 √ π a e -π 2 f 2 /a ∞ n=-∞ δ(t -n∆t) 1 ∆t ∞ n=-∞ δ(t -n ∆t )
G(t) ←→ g(-f ) g(-t) ←→ G(-f ) e i2πf0t ←→ G(f -f 0 ) g(t -t 0 ) ←→ e -i2πt0f G(f ) g(a t) ←→ 1 |a| G( f a ) g(t) * h(t) ←→ G(f )H(f ) g(t)h(t) ←→ G(f ) * H(f ) g(t) ⊗ h(t) ←→ G * (f )H(f ) d a dt a g(t) ←→ (j2πf ) a G(f ) t -∞ g(τ )dτ ←→ G(f ) (j2πf ) + 1 2 G(0)δ(f ) ∞ -∞ |g(t)| 2 dt = ∞ -∞ |G(f )| 2 df

Figure 1 :

 1 Figure1: The convolution between two real/complex functions g(t) and h(t) is a third real/complex function w(t) that can often be understood as a combination of the two input functions. Convolution is commutative, so there is no need to distinguish between its two inputs.

Figure 2 :

 2 Figure 2: The convolution between two input functions g(t) and h(t) in the specific case of g(t) being even.

Figure 3

 3 Figure 3: .

Figure 4 :

 4 Figure 4: The convolution between a localized function (a) and a wider function (b) tends to yield a new function (c) that can be understood as a 'blend' or 'mixing' of the two original functions, in the sense of inheriting features from both of them.

Figure 5 :

 5 Figure 5: The convolution between two localized functions with similar widths (a,b) tends to yield a new function (c) peaking at the time position corresponding to the best 'matching' or 'alignment' between the two original functions.

Figure 6 :

 6 Figure 6: A function g(t) corresponding to the sum of two cosine functions with distinct respective frequencies f 1 = 1kHz and f 2 = 2kHz and its Fourier transform.

Figure 7 :

 7 Figure 7: Two functions defined by sums of cosines, sharing one of the respective frequencies, and the respective Fourier transforms.

Figure 8 :

 8 Figure 8: A linear, time invariant dynamic system, including its input and output represented in the time and frequency domain. The impulse response corresponds to y 0 (t), while Y 0 (f ) is often called frequency response or transfer function.

  illustrates four basic types of filters, namely: (a) low-pass, (b) highpass, (c) band-pass and (d) band-rejection (or band-stop).

Figure 9 :

 9 Figure 9: Four types of commonly considered filters: (a) low-pass, (b) high-pass, (c) band-pass and (d) band-rejection (or band-stop).

Table 2 :

 2 Some properties of the Fourier transform, assuming g(t) ↔

G(f ), h(t) ↔ H(f ), and a, b ∈ C. a g(t) + b h(t) ←→ a G(f ) + b H(f )


Natacha Cyrulnik
email: natacha.cyrulnik@univ-amu.fr

The politics of the body or shared intimacy in documentary cinema La politique du corps ou l'intimité partagée dans le cinéma documentaire

Keywords: documentary movie, territory, community, sharing Mots clés corps, documentaire, territoire, communauté, partage

To speak of the body in a relationship with cinema is to enter into a form of intimacy through the representation that is made of this body. Introducing it into a documentary system means affirming the apprehension of a reality anchored in a place at the same time as the various protagonists of the film are positioning themselves. Based on the interactions between filmmaker, filmmakers and spectators, this film genre is based on an educational approach insofar as it allows to acquire knowledge on how to live in a territory. The experience of fifteen years of making workshop and documentary films in the so-called "cities" in the South of France, offers a unique situation to build a representation of this way of being in a space particularly stigmatized by the media. A process of collaboration, (co-)creation or participation develops at the same time as the bodies set up. These are affirmed at the same time as man positions himself within society, thus testifying to a policy of bodies.

Résumé

Parler du corps dans un rapport au cinéma, c'est entrer dans une forme d'intimité à travers la représentation qui est faite de ce corps. L'introduire dans un dispositif documentaire, c'est affirmer l'appréhension d'une réalité ancrée dans un lieu en même temps que les différents protagonistes du film se positionnent. A partir des interactions entre filmeur, filmés et spectateurs, ce genre cinématographique s'appuie sur une approche pédagogique dans la mesure où il permet d'acquérir une connaissance sur la manière d'habiter un territoire. L'expérience de quinze années de réalisation de films d'ateliers et de documentaires dans ce que l'on appelle les « cités » dans le Sud de la France, propose une situation singulière pour construire une représentation de cette manière d'être dans un espace particulièrement stigmatisé médiatiquement. Un processus de collaboration, (co-)création ou de participation se développe en même temps que les corps mettent en place. Ceux -ci s'affirment en même temps que l'homme se positionne au sein de la société, témoignant ainsi d'une politique des corps.

1 Natacha Cyrulnik is a documentary filmmaker, qualified to conduct research at the University of Aix-Marseille, and responsible for the "production and filmaker professions" programme in the SATIS (Sciences, Arts and Techniques of Image and Sound) department in Aubagne. She participated in the creation of the UMR PRISM (Perception, Representations, Image, Sound, Music) as head of axis 2 "Creations, practices and artistic explorations". His work focuses generally on the representation of territories through documentary film, film education, and the processes of (co-)creation.

To speak of the body in a relationship with cinema is to enter into a form of intimacy through the representation that is made of this body. And defining intimacy can only be done by thinking about one's connection to the other. It is therefore necessarily shared ! The question then is to know to what extent this intimacy is shared with others ? This somewhat provocative assertion has the advantage of linking the private sphere almost necessarily with the public sphere in the cinema. It is in this logic that Eric Maigret and Eric Macé defined the term "mediacultures2 " (2005). Then, at the crossroads of media images and the culture in which we live, the understanding of images would become a civic duty insofar as it contributes to the definition of our society. It is in any case in line with this logic, and even according to this claim, that artistic education workshops for cinema (Cyrulnik, 2016a), as well as documentaries, have been implemented in the french suburbs called « cités » for about fifteen years. Devices such as "image couriers" have highlighted it since their creation, insofar as its ancestor "a summer at the cinema" already wrote in these brochures in 1999 that the aim was to "send young directors to the red-light districts"! The ability to read images, or at least to have a critical mind about those consumed on a daily basis, is becoming a social issue.

A striking example of the change in the investment of young people3 in the suburbs and the position of their bodies in relation to a camera is the emergence of reality TV show in France in 2003. While I had started to produce artistic workshops in cinema with the young people of the Cité Berthe in La Seyne-sur-mer as early as 1999, the young people were moving much further in front of the camera in order to become rich and famous by the simple fact of being filmed as suggested by these new shows in France. This district is scary at first sight because it opened the 8pm news on television with burning cars and other stone throwing. It represented in a way the caricature of these territories that the Anglo-Saxon media call "no-go-zones" and that are stigmatized in France under the terms of "ghettos" or "lawless zone", among others (Avenel, 2005). These television representations of social housing in France have become almost systematic on television for the particular territories that are « cités » in France. These are often large neighbourhoods, mostly bounded by traffic lanes that surround the area geographically; where immigrants who came to help rebuild France were initially welcomed in housing that was well equipped at the time but often became poorly maintained with a socially and economically precarious population; and with marked behavioural, verbal and clothing codes, thus creating a distinct culture and a sense of marginalization for the inhabitants of the cities compared to the rest of French society. Thus, building new representations by making films with these inhabitants helps to better understand this relationship to the world and the place of bodies in this space.

So I realized with the population of the Cité Berthe of La Seyne-sur-mer (and other cities also in France and abroad 4 ) workshops of artistic practice in cinema during ten years. I then travelled around the South region of France as head of the artistic education centre for two years to see all the workshop practices with a camera, and then wrote a book about all these cinematographic experiences. I also extended these experiences in the suburbs through a documentary series entitled " Living in the territory ", which includes nine films in total on four cities with these kind of suburbs in the South of France. The idea was to produce documentaries (one per year in some social housing) to try to identify the urban and human evolutions of these spaces so stigmatized by the media in order to nuance their representations, whether social, urban, filmic, of oneself or of a community. The documentary, which comes etymologically from "documentum", which can be translated as "example, model, lesson, teaching, demonstration", thus testifies here to a form of pedagogical extension of the first workshop situations insofar as it allows new knowledge of this situation to be acquired. Knowledge and understanding through this film genre [START_REF] Niney | L'épreuve du réel à l'écran. Essai sur le principe de réalité documentaire[END_REF]2002) makes it possible to understand these particular territories in order to nuance the caricatured television comments from which the inhabitants say they suffer. These films made in the suburbs involve interactions between the different protagonists who are the filmmaker, the filmed and the spectator [START_REF] Cyrulnik | Interactions entre les différents protagonistes d'un documentaire et affirmations d'identités territoriales[END_REF]. From their exchanges, a community is created, both during the filming and during the debate at the end of the screening, where the spectators reformulate together what they have experienced (Cyrulnik, 2015a). Thus, through these exchanges, cinema encourages a way of asserting itself individually within a community that federates. These different situations of interaction position man in a creative situation that generates both a playful approach by playing at creating a representation, and a political assertion by engaging with others. This is the hypothesis that we are making and that we will develop on the basis of these fifteen years of cinematographic experience in French « cités ».

Cinema, community and identity

The simple fact of walking through the streets of a city with a camera already appeals to its inhabitants. After going beyond the relationship with the police, money, and reality TV which she immediately suggests, a reflection on what we could do with it challenges suburban youth. The presence of a single director with a camera quickly raises questions about the relationship to the images consumed daily. These become a first reference on which to build others. Little by little the camera comes to suggest new ways of representing this particular territory that is the suburb. A more personal register is set up, an complicity. A thought is articulated and a narrative is composed [START_REF] Paillé | L'analyse qualitative en sciences humaines et sociales[END_REF] : new representations take shape [START_REF] Cyrulnik | Rénovations urbaines mises en récit pour de nouvelles représentations des cités[END_REF].

With the new technologies the relationship to images has changed. Mobile phones shake up this apprehension of something personal that is made public, and vice versa. This object is used in an intimate register with its own selfies, musics, photos, etc. that reflect a personality. At the same time, it is a constant link to the outside world for communication. This duality reveals once again the need to be able to capture images of everyday life. But this relationship to intimacy already existed before these new technologies. These exacerbate it. The need to be able to read and/or construct images becomes a form of political commitment. To film is to acquire a critical and civic mind at the same time. It is also about bringing a new community together and finding its place in it. The positioning of the bodies in relation to the camera or screen suggests civic engagement [START_REF] Cyrulnik | Représenter le territoire, filmer la cité[END_REF]. In making these films, we are at the crossroads of an individual and collective affirmation at the same time. As we have seen since the introduction, the whole process of artistic education in cinema is built on the desire to promote free will: for participants, as well as for those responsible for these situations or the inhabitants. However, this need to be able to read these images with hindsight is more a civic duty than an apprehension of a true artistic dimension that would promote personal development and would be a priority. In this world of "mediacultures", "[i]t is up to the deliverer [media education] to build at the same time the motivation of the learner, and the conviction that he is right to educate himself in this way; [...] that [this education] is primarily aimed at developing learners distance from the media themselves so that teaching does not sink into bric-a-brac or whatever. Building critical thinking skills in the specialty is a major objective" (Porcher, 2006: 80). In addition to Louis Porcher's comments on media education, Serge Tisseron adds : "It is urgent to reintroduce the possibility for everyone to appropriate their own image experiences with their own means. To this end, media education must create spaces for verbal exchanges around images, but also spaces where children are invited to create their own images, to transform and manipulate them, and finally spaces where they can be solicited to stage them in groups in a playful and creative way" (Tisseron, 2002: 140). Whether one makes the most interesting statements possible to testify in a documentary or experiments with filmmaking in a workshop situation, a device and its related method forces each individual to position himself in front of the camera (Cyrulnik, 2016a: 11-12). Whether we start from the more or less displayed ego of the director as an artist, or from that of the filmed person who enters into action in front of the camera, the approach consists in going towards each other, and forging a bond. Bodies reveal their place in relation to each other.

Device and method

Claude Lévi-Strauss describes in "Tristes tropiques" (1955) a relationship of himself living ethnological experiences. In this respect, the book was considered unscientific by some of his colleagues. He had gone on an exploratory trip in 1955 to study virgin populations that would not be contaminated by civilization... Quite quickly, he insisted on the fact that it was necessary to avoid exoticism, but rather to constantly place man at the centre of nature and other men, in order to replace total humanity through ethnology5 . While he did not know much about this discipline when he arrived in Brazil, he gradually invented a method to understand these populations who lived differently in the heart of the forest. If he felt the need to learn this discipline by trying to read all the books on ethnography when he returned home, this new understanding of these unknown territories allowed him to experiment with things and to refine a method of observation and analysis. His intuitive approach at first became analysed and then constructed (Paillé & Muchielli, 2005: 32). It will make it successful. Claude Lévi-Strauss' story introduces in detail the importance of the method for understanding a territory and the way of life of its inhabitants6 .

Guy Gauthier, for his part, pointed out the importance of the method for documentary cinéma : "To film reality is simply to record what the camera is able to record at the various stages of the technique. It is not a question of "truth" -which is likely to be rushed regardless of the technique used -it is a question of method " (Gauthier, 2004: 12). What Claude Lévi-Strauss experienced could correspond to the documentary filmmaker's method. Each director must, each time he enters a new field, invent a way of going there that suits him. An ethical question, essential for the documentary filmmaker, is emerging. The method he will use or invent will correspond to the device he chooses to set up to approach a place. This link between method and device determines a way of representing a territory. The device (Agamben, 2007) assigned to a terrain adapts to it, and a method is associated with it: "The meticulousness of observation, combined with attentive listening, makes life stories all the more interesting and, undoubtedly, the interest of life" (Colleyn, 1993: 28).

To extend this into a more philosophical and political register, Hannah Arendt (1954) explains that it is ultimately the means that are more important than the purpose in trying to understand the world. Moreover, the films often testify, more or less voluntarily, to what happened during the shooting. The system and method determine the means used to open up to the world, as a documentary does (Cyrulnik, 2017b: 139 -140). Methodology and meaning construction go hand in hand.

Mediology and construction of a cinematographic device

Thus a method is affirmed by making all the protagonists of a film interact in order to build a new representation of a territory that is as close as possible to the daily life of its inhabitants (Cyrulnik, 2018b : 4-6) thanks to the presence of the camera. Because this object is there, people will discuss it and enter into a form of mediology [START_REF] Debray | Manifestes médiologiques[END_REF][START_REF] Bougnoux | Introduction aux sciences de la communication[END_REF]. Thanks to him, a territory is discovered through the speech made by the director, the inhabitants and the spectators (Cyrulnik, 2017b: 126) who reveal themselves.

Through mediology [START_REF] Debray | Introduction à la médiologie[END_REF], the link between action and narrative is deepened. This relationship between a language and an object that would tell a story of a territory does not want to oppose technology and culture, but seeks to link ideologies and beliefs to technical and scientific innovations that are ultimately revealed in our behaviour and culture. « In mediology, "medium" does not mean "media" or "medium" but "media", that is the dynamic set of procedures and intermediate bodies that interpose themselves between the production of signs and the production of events » (Debray, 1994 : 29). The construction of a representation of a territory through the camera, which itself already suggests the upcoming film, links all these elements.

Daniel Bougnoux insists on the fact that this mediological approach intervenes "on those environments, inseparably social and technical, which shape and recycle our symbolic representations, and allow us to hold together" (1998: 67). For him, "the digital tool encourages individualism, both in terms of expression, emission and reception of messages ; it also raises the question of how to show the general, the common good or a public space" (Bougnoux, 2007: 46). Dealing with the narrative of a territory must therefore leave a large place to symbolic representations, which themselves contribute to the affirmation of a unifying link for the inhabitants of these territories thus narrated (Cyrulnik & Meyer, 2015b : 8).

The camera is a means of connecting people. The audiovisual tool is a stimulator and generator of particular communication situations: "The expression of juice requires the press and the grape; in the same way, an environment of objects that resist as well as an emotion and an internal impulse are needed to constitute an "expression" of emotion" (Dewey, 1915: 125). The object promotes expression by experimenting with art. Mediology is in line with the artistic and experiential dimension that documentaries enhance. Working on images and sounds requires communication; these techniques linked to the camera encourage reflection in relation to mediology. It is through the prism of the camera that we see a reality, and that we can give it to see. Capturing and transmitting make it possible to represent the world. When making a documentary film, one of the priorities is the human relationship: it is the human relationship that will make something happen and the subject become interesting (it will be a pretext to exchange later). Jean-Paul Colleyn even cites the method used with Jean Rouch's or Pierre Perrault's cameras because they have "as a prerequisite the love of humanity", this object provoking "performances in spontaneous interactions" (Colleyn, 1993: 88). The audiovisual tool offers a technique that must serve this idea above all : it is part of a mediation process, more than just a medium. Because the notion of transmission7 , which has just been mentioned and which is underlying within the very word "documentum", is essential in a media approach. It must be taken even more into account with regard to the media, which themselves participate more or less happily in changing worldviews, particularly social representations [START_REF] Moscovici | sous la dir.) La psychologie sociale[END_REF]. The technical associated with the media also reveal a culture based on their use, in relation to the "mediacultures" (Maigret & Macé, 2005) already mentioned. The documentary, a media object since it uses the camera and microphone to testify to the world while trying not to lock itself in too precise a format, offers a critical and alternative vision of the use of the media (television, and increasingly on the Internet) to get out of stereotypes. The documentary then proposes a mediological approach that goes hand in hand with the affirmation of a critical spirit about the media and the culture represented as well. It promotes the analysis of a method for the construction of a representation. Technical and audiovisual material thus contribute to the understanding of a culture, by constantly associating themselves with a reflection to build meaning through the different levels of representation. To study the social changes often present in suburbs undergoing urban change, the audiovisual tool seems to be an appropriate support8 . This documentary method therefore fully fits into this mediological approach, through the interactions it offers between technology and culture. All this is also based on the intrinsic beliefs of documentary : the relationships between the protagonists of a documentary are based on the acceptance that these images come from reality [START_REF] Lioult | A l'enseigne du réel, penser le documentaire[END_REF]. Technical and beliefs go hand in hand, thus favouring a collective : that of the people linked to the camera during the shooting, in the movie theater during the debate, or in front of the film quite simply to receive social representations that are nuanced. Starting from technical considerations, we come to the positioning of people in their bodies and their status, which influences an interpretation of places, and the political dimension of documentary is affirmed when it helps man to find his place in the world (Cyrulnik, 2017b : 145 -146).

Experience of a territory through art

The definition of a device, and of a related method, also requires that the documentary proposes another experience, through art [START_REF] Dewey | L'art comme expérience[END_REF], of a territory.

Moving from the intimate to the universal Philippe [START_REF] Meirieu | Libre parole, réalisé par Jean-Pierre Daniel, DVD -Carnet de route de l'Alhambra Cinémarseille[END_REF] in educational science, Denis [START_REF] Guénoun | Le théâtre est-il nécessaire ?[END_REF] in comparative literature, or Pierre-Michel Menger in sociology, define being an artist in much the same way when they explain that it consists in making something intimate universally perceptible. This articulation between the intimate and the universal therefore seems to be part of any artistic process, whatever the discipline that analyses it. Something intimate resonates for others... This immediately questions the place of the body in relation to others : the place of the words spoken, of course, but also a particular management of space (E.T. [START_REF] Hall | Au-delà de la culture[END_REF] and behaviours. Claudio Pazienza, for example, clearly explains that when he chooses to film himself next to his father who has just died to start shooting " Wild boar hunting scenes "9 (46 ', 2007), it is an extremely intimate procès ; but that he does so by telling himself that it will resonate in others. The lying body and his body sitting next to him, almost curled up on his microphone to testify to what he was feeling at that moment, led the spectator into a very personal, even disturbing, register of exchange. Does the resonance of these images bring something to the viewer? In any case, he questions this relationship between the intimate and the universal in an extreme way.

Without necessarily going that far, the relationship with the public is in any case present as soon as you choose to put your camera somewhere. The place of the bodies tells a story and the spectator is more or less disturbed by it. It is also the experience in his own body that he will do that must be analyzed. Experiencing (common) through art [START_REF] Dewey | L'art comme expérience[END_REF], both during filming and screening, is a strength of cinema that will help us to better understand a place or situation.

Experimenting through art Experience and expertise are linked, making each inhabitant a specialist in the analysis of their own city. But the fact of proposing to make it a new experience with the help of a camera will precisely provoke a situation that forces him to enter, in terms of method, into a phenomenological reformulation (Paillé & Mucchielli, 2005: 72-75). This in turn will transform his experience into knowledge, for him and for the spectators afterwards. The social and intimate dimension of this experience through documentaries is affirmed in the field. A form of pedagogy is developing to better understand the territory and daily life in the related cities.

John Dewey claims "art as an experience" (1915) and helps us to better understand the issues involved in acquiring knowledge about a territory through documentary, in a common experience.

To define what it means to live an experience through art, he explains: "The primary emotion on the part of the postulant can be at the beginning of hope or despair, and, in the end, joy or disappointment. These emotions give unity to the experience. But, as the interview progresses, secondary emotions develop, such as variations of the underlying primary emotion. It is even possible for every attitude, every gesture, every sentence and almost every word to express an emotion that is more than just a fluctuation of the main emotion; that is, to express a change in nuance and hue in its quality" (Dewey, 1915: 93). In the context of a documentary, this would mean using all these steps, as an emerging method, to formulate states felt in the body, to state intentions, in order to compose a narrative for the film (Cyrulnik, 2016b : 31). Each brushstroke gives the painter (as well as the filmmaker or the filmed) the opportunity to put meaning into it and to apprehend the reception of his work (or himself). The creator exercises his thought, becomes demanding, and builds a representation, himself and his environment at the same time. He connects (Dewey, 1915: 97). The artistic dimension refers to the act of production, that of aesthetics to the act of perception and pleasure (Dewey, 1915: 98). Here again, we are in an intuitive approach that will be analyzed and understood later (Paillé & Muchielli, 2005: 32). In the case of the artistic experience, it is the action that matters, but the fact of having to build a representation in the end favours the consideration of the other's point of view and the notion of pleasure. The two dimensions come into play together based on the co-presence of the film's protagonists. What is experienced, with all the emotions that go with it, will be transformed in one way or another into a narrative for the film. From an intimate emotion at the beginning, a work of art is created.

The territory and the senses For the Greeks, the arts were directly associated with the experience we were having together ! That is why they gathered at the theatre of Epidauros:" (...) Existence takes place in an environment; not only in that environment but also because of it, through its interactions with it" (Dewey, 1915: 45). We are at the heart of our problem. What could be called "set" for a film, even a documentary, contributes to the construction of the film. The context is largely responsible for this. In a french suburbs, for example, this "set" is necessarily a driving force that helps creation because it is so strong and present. It is the subject of the documentaries films in the series "Living into the territory" since they question these special places. "Art as an experience" connects man to his environment, to his world. The aesthetic experience10 that is made in his body at first offers a form of knowledge that helps to better understand the world (Cyrulnik, 2016b: 33).

" The term "significance" covers a wide range of content : sensory, sensational, sensitive and sentimental, not to mention sensual. It includes almost everything from the pure emotional and physical shock to the meaning itself, that is, the meaning of the things present in the immediate experience " (Dewey, 1915: 59) : all these terms from "significance" must be considered as the premises of a formulation on how to live in one's french suburbs for a resident who comes into action and testifies. But, because he is in front of the camera, with all the viewers it suggests in the future, he will rather spontaneously use these first "sensations" to extract a meaning, then a story, for the film. The position of his body itself will be transformed by the presence of the camera : he stands in front of it, he stands a little straighter, articulates a little more, makes humorous strokes to try to make himself understood as best as possible... in any case this is what happens most of the time during the first exchanges in front of the camera. The documentary is inspired by this first state. It promotes exchange between all, analyses its environment (or at least refers to the world in general), in order to affirm an individual and collective dimension : "Experience is the result, sign and reward of this interaction between the organism and the environment which, when completed, is a transformation of interaction into participation and communication" (Dewey, 1915: 60). The experience through art places the body within its environment, and transforms it into a mode of communication with the world outside the city in our case. Art thus prefigures the very "process" of existence (Dewey, 1915: 63), and documentaries offer a sensitive new experience of french suburbs. This is achieved through the body and its sensations: "Art is the best proof of the existence of a union, realized and therefore achievable, of the material and the ideal" (Dewey, 1915: 68). This is in line with the media mentioned above. From this link between nature and spirit can arise from meaning and spirituality (Cyrulnik, 2016b: 33).

Art thus proposes a form of method insofar as it is the very process that is questioned. The action offered by the experiment makes it possible to analyse the production procedures of a work. A more or less spontaneous perception of what is happening is revealed in each shot, for both the filmmaker and the filmed, in co-presence. The experience becomes aesthetic (Dewey, 1915: 104). Art as an experience offers a device and a method that is reflected in the work in the end, whatever the more or less visible presence of the aesthetic dimension put in place (even more in documentaries !). The founding perception of this artistic experience is for the artist, the one being filmed, and the spectator. Human relations become witnesses to the artistic act (Dewey, 1915: 123).

Experimenting and symbolizing a territory Through the strength of the experience (of a territory) that art fosters, it is both action, appropriation and symbolization that are called upon. The place of man on the ground, the situation in which he finds himself and which makes him move, the position he chooses to take, will lead him to build his personal projections and what he tells himself about it. When a young girl in " A meadow in the city " (44', 2002) says that she dreams for later of a man who "is kind and doesn't drink too much" she reveals a lot of things about her daily life, but who can be said because it's in a film... The symbolic dimension of art, also possible through documentary cinema, offers a new openness, it being understood that the symbol keeps a natural link with the thing (Berque, 2010: 289). This precision links art to nature. Moreover, Augustin Berque, in his approach, which is primarily geographical but also philosophical, insists on "trajectories": " It is this double process of technical projection and symbolic introjection. It is the coming and going, the existential pulsation which, animating the mediation, makes the world matter to us. It is important to us carnally, because it comes from our pulpit in the form of technical and it comes back to us in the form of symbols. This is what makes us human, this is what the ecumene is, and this is why the world makes sense " (2010: 208). Everything he has theorized about human environments leads us to an approach to symbols for which the artistic dimension is a plus. Art is a possible support to talk about man and his environment. The documentary is a help to talk about territory more precisely. John Dewey returns to the links between art and the environment in the broad sense of the term : "The interaction between the environment and the organism is the direct or indirect source of all experience; these controls, resistances, additions, balances which, meeting in the appropriate way the energies of the organism, constitute the form" (1915 : 250).

In this quest for meaning, and concerning this apprehension of a reality, Nishida Kitarô (1911) starts from the theme of pure experience, to come to the " logic of place " or " logic (of identity) of the predicate " that would promote the identity of the subject. Because things (places) are named, qualified by a relationship or built by our hands, they unfold our world (Berque, 2010: 234). Augustin Berque develops further : " If I, domiciled in Sendaï, do not like the idea of building an aerodrome in Adelie Land, it is not because I experience its decibels or kerosene vapour; it is because the symbolic systems of modernity make them present in my animal body; and if they make them present to me, it is indissolubly because the technical systems of the same modernity, very materially, produce and can reproduce such artifacts, as well as those that physically connect me to them (communication networks etc). The Adelie Earth is not only through symbols that it is part of my existence; it is really part of it as long as I exist within modern civilization " (2010 : 254 -255). The symbolic dimension then evokes the place of the documentary film spectator who thinks of the world in which he is, at the same time as the place he finds there. Experience leads us to symbolism. Documentary as an art affirms the place of the viewer, and as a genre it contextualizes and validates an intrinsic rhetorical approach [START_REF] Soulez | Quand le film nous parle, Rhétorique, Cinéma, Télévision[END_REF].

In summary and to conclude this section on the relationship between community and identity through film, documentary has emerged as a socio-technical device (Cyrulnik, 2015a) that helps to improve "being in the world" [START_REF] Schaeffer | Pourquoi la fiction ?[END_REF]. Jean-Marie [START_REF] Schaeffer | Pourquoi la fiction ?[END_REF] said this to justify our need for fiction; to the extent that the documentary tells us (it romances) the world, and a particular territory, it participates in a certain way in this anchoring in society. It is part of an approach that often oscillates between the social and the territorial, even the political (Cyrulnik, 2017b : 148 -153). From an intuitive approach to the body, we come to value the symbolic dimension of documentary work and to consider its political significance.

Sharing the sensitive to engage

Intimacy is the process of forging a bond with another participant or with the director to reveal something about oneself... which will then resonate in a more universal way when the spectator becomes aware of it. The fact of sharing, with several people (whether during the shooting, through the articulation of the story in the editing, during the screening or during the debate that follows), these experiences and emotions offer a basis for building a representation of them, which are transformed into reflection, before becoming a film. This process is gradually maturing. The participant's commitment is confirmed as the film is being made. It is by seeking to analyse, decipher, interpret images, his environment or representations, that an operation is carried out in the person and that he affirms a position that evokes a form of commitment. By affirming his place in the world, the person who participates in films also anchors his posture as a citizen (Cyrulnik, 2016a: 108).

Sharing the creation

Already the way of sharing the creation of the film extends the chosen method and encourages human interaction between the protagonists that gives substance to the film.

Workshop situations more often correspond to a form of co-creation insofar as all participants are on equal terms (director -accompanying person included). The film is invented between all of them. The director accompanies this act of creation by constantly stimulating them, while withdrawing as an artist, so that the participants become full-fledged authors. The film is written during the shooting, all together, and by positioning oneself individually and collectively at the same time. From intuitive practice to understood analysis (Paillé & Mucchielli, 2005: 32), the film takes shape from the moment it is shot, anticipating the editing insofar as we are already thinking about the construction of the film. Thus, after an afternoon strolling through the streets of the « Cité Berthe » to find the inspiration for a new film, the older sisters seized the audiovisual material as soon as they saw a little boy of Senegalese origin offer a bouquet of wild flowers that he had just picked during this walk to a pretty Algerian girl. From this scene, it was therefore decided collectively that the film to be made this year would take place exclusively in this large abandoned field, investing it as a separate space where we should "only talk well, not say swear words, and only say intimate and sincere things to each other". "A meadow in the city" (2002, 44') was being born. Through this scene that became fictionalized (but whose main interest was documentary since the invented characters made it possible to make them say many things about everyday life 11 ), an 11 It is not a question of opposing fiction and documentary but on the contrary of signifying to what extent one is constructed in relation to the other [START_REF] Guynn | Un cinéma de Non-fiction, Le documentaire classique à l'épreuve de la théorie[END_REF]. That said, over the years, participants in these workshops (Cyrulnik, 2016a) have come to make real documentaries because they wanted to show themselves more. As Jacques Rancière says: "[The film] brings to its highest power the double resource of silent printing that speaks and editing that calculates the powers of significance and the values of truth. And documentary cinema, cinema dedicated to "reality" is, urban and human reflection emerged in connection with the suburb in full mutation. The principle is that each participant contributes his own little stone to the creation of the film because he is forced to position himself in front of or behind the camera, thus affirming himself in one way or another both individually and collectively. These young people invented a new understanding of their territory by mobilizing their friends and enacting common rules of conduct to better live together. If they did not formulate them like that, these criteria became the first steps in a scenario that was built on the logic of intimacy to be revealed in all confidence, to be shared... From a filmed secret, new representations are set up as people affirm themselves. By choosing the subject to be addressed in the strong setting of the territory, the participant reflects what is important to him to tell about this space. The participatory process in a workshop is the basis for the construction of the film.

These emerging interactions are also reflected in the collaboration in the making of a documentary insofar as it is finally the relationship that has been established between the filmmaker and the filmed person that gives rise to the testimony that the spectator will ultimately see on the screen. These situations of exchange make it possible to co-construct an urban representation (the territory also appears through what the participants tell about it), filmic (the documentary), social (the comprehensive approach makes it possible to get out of the stigma on the suburb), and self (the filmed person catalyzes his behaviour) at the same time [START_REF] Cyrulnik | Rénovations urbaines mises en récit pour de nouvelles représentations des cités[END_REF]. All these levels of representation attest to the different contributions of documentary. Through this set of representations, the filmed person, by collaborating through his testimony, finally also participates in the creation of the film: he feeds it, he also thinks it by testifying to his daily life, he tells his city by making all his part of subjectivity and the part of romance that goes with it show through, he puts himself on stage in front of the camera, etc. When a young person walks towards the camera by hiding behind a hood and rolling the mechanics, he reproduces the clichés in which he feels locked, reproducing the television codes of the " scum that should be cleaned with a karcher 12 ". But, after a few minutes of discussion, he finally pulled down his hood and began to tell his life and feelings 13 in a more intimate register. A part of creation emanates from him through his way of being in front of the camera. It is the enhancement 14 of this one that will make the film all the more interesting and will touch the spectator, thus encouraging him to feel concerned in his turn. The emergence of the comments accompanied by this self-direction of the filmed, the narrative that is composed live in front of the camera, the resonance of the comments collected with each other, and the one that the spectator will have in the end, propose stages of collaboration that arise from the possible interactions between the different protagonists of the film.

The director remains the conductor in all circumstances, but it is the relationships he knows how to establish that will best reflect the daily life in the cities. The importance of the I of "I am another" by D. W [START_REF] Winnicot | accompagné par Natacha Cyrulnik -« Ceux qui pensent le projet urbain, ceux qui le vivent[END_REF] suggests that one is never alone, that one must think of the other in oneself ; documentary cinema and these workshops help to do this. This is a question that is put under tension especially for adolescents, at a time when their bodies are changing. To touch the principle of creation with your finger at that moment emanates. These situations are examples that show how living and experiencing these feelings related to the moments of creation, going through in this sense, capable of a stronger fictional invention than "fiction" cinema, easily dedicated to a certain stereotypy of actions and characters" (Rancière, 2010: 60 -61). 12 Expression comes from Nicolas Sarkozy's words that the young people finally appropriated and diverted in their own way and now call themselves the "Kaïras" (verlan de "scum"). 13 This is a visible sequence in particular, since it is a relatively common practice that testifies to the importance of "mediacultures" (Maigret & Macé, 2005), in "La revendication d 'un regard" (2008, 39') and in "Ceux qui pensent le projet urbain, ceux qui le vivent" (2011, 80'). 14 The filmmaker, behind the camera, directly means by his gestures and facial expressions during the shooting, that what the filmed person tells is interesting, in order to encourage him to develop his idea. A relationship is forged ; it enhances the value of the filmed. In any case, this is what happens most of the time for the takes that will be retained during editing.... all these stages, taking action and taking shape, open up a view of the wider world. D. W Winnicot (1975) speaks of a transitional object as a reference point for opening up to the outside world. The creation of the objects or images just mentioned is part of this logic (Cyrulnik, 2016b: 63).

Without positioning themselves as authors, this set of interactions encourages the construction of several levels of representation and engagement in a collaboration of which the documentary film is intended to be both the goal and the relay for a better understanding of these territories. Codevelopment, combined with a form of co-responsibility due to speaking in public, makes it possible to make these words better heard about everyday life in the suburbs. This is combined with a form of recognition for the inhabitants that corresponds precisely to what they claim above all, so that a less caricatural view is taken of them (Cyrulnik, 2018a). Through these situations exposed, the logics of co-creation, collaboration, and participation in all these forms show that the positioning of bodies plays a role according to their status, their way of assuming it, and especially their commitment to it.

Aesthetics, relationship and politics

Jacques Rancière approaches « the sharing of the sensitive » (2000) as an exchange within a community 15 . From the title of the book, the relationship takes root as the foundation of our relationship to the world. Aesthetics would be in the logic of emancipation, like a battle to be fought. Schiller, in his « Letters on the Aesthetic Education of Man » (1943), presented " a specific mode of living in the sensitive world that must be developed by "aesthetic education" to train men likely to live in a free political community " (quoted by Rancière, 2010: 39-40). Art helps to find its place in the world by making an act of creation; it is in any case what many young people from the suburbs of the South of France have been able to experience during these years. As if to deepen these issues of the workshops of artistic practices of cinema, Jacques Rancière adds : " To make a way of technique -whether it is a use of words or the camera -qualified as belonging to art, its subject must first be qualified as belonging to art " (2010 : 49). He extends it with : " The ordinary becomes beautiful as a trace of truth " (2010: 52). Cinema thus becomes a new art of storytelling and self-engagement, in all sincerity. And this spontaneity is confirmed after the first ten minutes of exchanges necessary to forget the camera 16 . It is on this sincerity and complicity that the making of a film is based. It becomes useful to the person in the sense that it brings him something, thus a process of transformation takes place. Jacques Rancière comes to describe" [...] art as the transformation of thought into a sensitive experience of community " (2010: 71) (Cyrulnik, 2016b: 107-108). This process of transformation of the film's protagonists takes on a social and political dimension insofar as it is what is at stake between them, together, that matters.

Freedom, education and politics

Still in this emancipatory and democratic logic, Hannah Arendt, in « between past and future » (1954), makes comments that resonate with our situations. She speaks of processes (1954: 80-84), truth being challenged according to a capacity for action (action being obviously the driving force behind the making of these films in these streets). She then wondered what freedom was and made free will a priority (1954: 188), thus recalling the critical spirit of the beginning as a necessity, while linking it to artistic considerations. It distinguishes the performing arts from the creative arts of manufacturing : " [...] It is not the free creative process that ultimately appears and imports into the world, but the work of art itself, the final product of the process. Performing arts, on the other hand, have a strong affinity with politics. Both need a publicly organized space for their "work", and both depend on others for the performance itself " (1954: 199-200). Documentary films, like workshop films, bear witness to a reality and debate it ; they are therefore considered as performing arts. They display both their freedom and their place in the world. They fully participate in an affirmation of identity in the political sensé : " If, therefore, we understand politics in the sense of politeness, its purpose or purpose would be to establish and preserve in existence a space where freedom as virtuosity can appear. This is the area where freedom is a reality of the world, tangible in words that can be heard, in actions that can be seen, in events that are spoken of, remembered and transformed into stories before being incorporated into the great book of human history. Everything that happens on this stage is, by definition, political, even when it is not a direct product of action " (Arendt, 1954 : 201).

Education and culture are at the crossroads of these implemented films. Yet what is put in this word " culture " is a little far from the artistic dimension that makes all the difference. The risk is that it becomes a value of exchange by losing " the power to stop our attention and move us " (Arendt, 1954 : 261). " Culture is destroyed to generate leisure " (Arendt, 1954: 266) : in this respect, film workshop owners must be careful not only to "occupy" children who are bored in the suburb, but to ensure that something really happens for them, something that transforms them all a little, something artistic..." Culture and politics then belong to each other, because it is not knowledge or truth that is at stake, but rather judgment and decision, the judicious exchange of opinions about the sphere of public life and the common world, and the decision on the kind of action to be taken there, as well as the way of seeing the world in the future, and the things that must appear there " (Arendt, 1954: 285) : workshops in artistic film practice and documentary experiences, by developing a form of critical thinking that leads to the creation of a filmic representation of a particular territory, contribute to this form of affirmation, which is also of a political nature.

" Political thought is representative. I form an opinion by considering a given issue from different points of view, by going to present to the mind the positions of those who are absent ; that is, by representing them. This process of representation does not adopt a different perspective ; it is not about sympathy as if I am trying to be or feel like someone else, or to count and join the voices of a majority, but to be and think in my own identity where I am not really. The more positions of the people I have in mind while I reflect on a given issue, and the better I can imagine how I would feel and think if I were in their shoes, the stronger my ability to think representatively and the more valid my final conclusions, my opinion will be " (Arendt, 1954: 307). Finally, Hannah Arendt helps to demonstrate that building a representation, as is the case in workshops or for documentaries, is part of a form of political engagement (Cyrulnik, 2016b: 109-112), and "aesthetic empathy" [START_REF] Lemarquis | L'empathie esthétique, entre Mozart et Michel-Ange[END_REF] would be a major factor (Cyrulnik, 2017a : 37-38).

Conclusion

From the need to foster a critical spirit in the face of images that are so easily consumed, from the experience that makes it possible to move from something intimate to a more collective vision, the artistic and political dimensions of these cinematographic situations are becoming more apparent. It is a question of fully assuming the fact that these steps involve all persons, participants and project leaders alike. This social and artistic commitment has a real political dimension. The process of artistic and political transformation takes place during the shooting, and continues thereafter. The lived experience matures over time. This is also the strength of these artistic experiences in film. Citizen engagement began with emotions that are transformed into different forms of expression: through speech, framing, the situation to be filmed, or self-direction in front of the camera, among others.

In addition, there is the question of the " cult of the banal " : is it better for these young people from suburbs to blend into a mass or to emerge in a particular situation such as that of making a film ? Reality TV has tended to value the banal [START_REF] Jost | Le culte du banal. De Duchamp à la télé-réalité[END_REF], the process of creating a film offers an original situation that encourages interaction. This aspect, which may seem playful [START_REF] Winnicot | accompagné par Natacha Cyrulnik -« Ceux qui pensent le projet urbain, ceux qui le vivent[END_REF], seals these interactions in a form of identity affirmation, where bodies assume their place, whether they are in front of or behind the camera, or sitting in the cinema....

Everything was built on the basis of a complicity between the filmmaker and the filmed person at the beginning. This process, which values otherness, promotes a form of recognition as well. If the audiences affected are sometimes in difficulty, the commitment is still there. The principle of recognition combines both societal and emotional values. Axel Honneth concludes his book on recognition by defining it in relation to an experience of love : " the experience of love thus gives access to self-confidence, the experience of legal recognition to self-respect and the experience of solidarity, and finally to self-esteem " (Honneth, 2010: 208). Is it a " journey " [START_REF] Ricoeur | Parcours de la reconnaissance, Gallimard, Folio Essais Schiller[END_REF] or a " struggle for recognition " [START_REF] Honneth | La lutte pour la reconnaissance[END_REF] ? In any case, the documentary genre is part of a collaborative process that promotes this (Cyrulnik, 2018a: 2-5), and that goes beyond the film alone.

Current digital reconfigurations accentuate this understanding of shared intimacy. If identity is constructed in the other's eyes, film is an artistic tool to talk about oneself, and to allow something of the body to be tied by speech (or any form of expression). Modesty, decency and indecency, ethics (already mentioned and essential to make a documentary film), or empathy, are all feelings that contribute to this body policy.

The content of the films, which has not been discussed much here, also often reflects precarious situations, especially in suburbs. The last film in the series " Living in the territory " shot in the Cité Berthe, "Adieu Berthe ! "(2013, 39'), testifies to the commitment of the participants after all these years of films as they decide to implement " citizen barbecues " as they say, to re-mobilize and federate all the people who have lived in these districts in full mutation. This film is a form of cry from the heart, from the inhabitants and the director whose subsidies have been withdrawn, putting an end to this work, and ends up mixing the destruction of the towers filmed over fifteen years with the images shot there with such flame, joy and enthusiasm in these places. All these forms of destruction are mixed in this film, still combining creation with the political dimension it implies.

To conclude and try to overcome these claims, from emancipatory potential to new forms of critical and political thought, new forms of sociability now offer new perspectives to these cinematographic situations. " Participation", " co-productions ", " co-creations ", and other collaborative and cooperative situations, become the breeding ground for new communities that federate and take charge of things, thus building other meetings and situations of sharing. They are the ones who ultimately remain the priority of all these artistic adventures that cinema, and art in general, offer... (Cyrulnik, 2017a : 39) A little girl with whom I worked in 2005 to make " La parole de l'Art " with her, (23', 2005) told me almost the same thing. While at first she found my question incongruous, which consisted in knowing what art is and what it is for, she concluded the film by letting go of her own reflection that had developed as the film was being built : " You are obliged to love art, because otherwise you only love yourself, and you can't stay alone. You have to love art because you have to love others..." (Cyrulnik, 2016a : 117).

Bibliographie Agamben, G. (2007). Qu'est-ce qu'un dispositif ? Rivages Poches, Petite bibliothèque Arendt H. (1954) (pour la version française), La crise de la culture, Paris, Folio essais, 1989. Avenel, C. (2005). Sociologie des « quartiers sensibles », A. Colin.

Éric Maigret and Éric Macé (2005 : 10) propose to "link media and culture in order to find the intersection point of meaningful construction practices, to decompartmentalize media, culture and representation studies".

Since reality TV, it is therefore mainly young people who are moving towards the camera, but these films are all interested in an intergenerational and multicultural way.

More informations on http://www.lacompagniedesembruns.com

Suburban youth often characterize themselves primarily as "animals living in the jungle" when they see a camera, as they clearly state in " The word of Art " (2005, 23') among others. In this respect, the documentary resonates with visual anthropology.

The analysis of this method was the subject of my thesis defended in 2008 "Representing the territory and acting with it, the creative documentary method". It is therefore not a question of rewriting it but of extending it.

Even if it can be understood as making a donation, sending, and teaching.

It should also be remembered that documentaries often work over a much longer period of time than reports, thus offering a trace of life in social housing that is more appropriate to bear witness to a memory of places, their practices and their evolution.

When Claudio Pazienza came to meet the students of the art school of Villa Arson in Nice (master class on November

10, 2011 organized by ECLAT), he testified about his work, his reflection, his experience and his journey.

" Because experience is the accomplishment of an organism in its struggles and achievements in a world of objects, it is the embryonic form of art. Even in its rudimentary forms, it contains the promise of that exquisite perception that is the aesthetic expérience "(Dewey, 1915: 55).

" The surface of the "painted" signs, the duplication of the theatre, the rhythm of the dancing choir : these are three forms of sharing the sensitive structuring the way in which the arts can be perceived and thought of as arts and as forms of inscription of the meaning of community "(Rancière, 2010:

16). The experience favours the sensitive division of the community and then raises the question of the relationship between aesthetics and politics(Rancière, 2010 : 25) 16 As for example at the beginning of " La revendication d'un regard " (2008, 39').