Rémi Bonnefoi
email: remi.bonnefoi@centralesupelec.fr

Amor Nafkha
email: amor.nafkha@centralesupelec.fr

A New Lower Bound on the Ergodic Capacity of Optical MIMO Channels

In this paper, we present an analytical lower bound on the ergodic capacity of optical multiple-input multiple-output (MIMO) channels. It turns out that the optical MIMO channel matrix which couples the mt inputs (modes/cores) into mr outputs (modes/cores) can be modeled as a sub-matrix of a m × m Haar-distributed unitary matrix where m > mt, mr. Using the fact that the probability density of the eigenvalues of a random matrix from unitary ensemble can be expressed in terms of the Christoffel-Darboux kernel. We provide a new analytical expression of the ergodic capacity as function of signal-to-noise ratio (SNR). Moreover, we derive a closed-form lower-bound expression to the ergodic capacity. In addition, we also derive an approximation to the ergodic capacity in low-SNR regimes. Finally, we present numerical results supporting the expressions derived.

I. INTRODUCTION

With the advent of massive multiple-input multiple-output (MIMO) technologies and the development of Internet of Things (IoT), the fifth generation cellular networks (5G) should be supported by a high quality backhaul. This backhaul can be realized through both wired and wireless technologies. Among all the possible solutions, the deployment of optical fibers is the one that ensures the greatest throughput while maintaining a high level of reliability. Moreover, spacedivision multiplexing (SDM) based on multicore/multimode optical fiber can significantly increase the capacity limit of optical fibers [START_REF] Richardson | Space-division multiplexing in optical fibers[END_REF]- [START_REF] Lin | Achievable information rates calculation for optical OFDM few-mode fiber long-haul transmission systems[END_REF] and it overcomes the capacity crunch. To take full advantage of the potential throughput increase promised by SDM, the in-band crosstalk must be properly managed. This can be done using MIMO signal processing techniques [START_REF] Ryf | Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 x 6 MIMO Processing[END_REF]. Moreover, assuming negligible backscattering and near loss-less propagation, the propagation channel in an optical fiber can be modeled by a complex random unitary matrix [START_REF] Mehta | Random Matrices; 3rd ed. Pure and applied mathematics series[END_REF].

For wireless communications, different models are used to characterize the MIMO propagation channel. Most of the time, the Rayleigh fading model [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF] is used to model the MIMO channel, in this case, the entries of the channel matrix can be modeled by an independent and identically distibuted zero mean Gaussian complex numbers. Moreover, the matrices H † H are referred to as uncorrelated Wishart matrices, where . † is the complex transconjugate. In optical fiber, the channel matrix can be modeled by a Haar distributed matrix [START_REF] Dar | The Jacobi MIMO Channel[END_REF]. Therefore, the matrices follow the Jacobi unitary ensemble [START_REF] Mehta | Random Matrices; 3rd ed. Pure and applied mathematics series[END_REF]. The channel model of the SDM optical fiber was first discussed in [START_REF] Winzer | MIMO capacities and outage probabilities in spatially multiplexed optical transport systems[END_REF], further studied in [START_REF] Dar | The Jacobi MIMO Channel[END_REF], [START_REF] Karadimitrakis | Outage Capacity for the Optical MIMO Channel[END_REF]. In [START_REF] Dar | The Jacobi MIMO Channel[END_REF] the ergodic capacity of the Jacobi MIMO channel was expressed as an integral and sum of Jacobi polynomials.

In this article, the ergodic capacity formula is first rewritten to show that the polynomial part of the integrand consists of the Christoffel-Darboux kernel [START_REF] Richard | Encyclopedia of mathematics and its applications[END_REF]. Then, the Christoffel-Darboux formula is used to reword the expression of the ergodic capacity of loss-less SDM optical fiber channel. The derived new expression reduces the computational complexity of numerical evaluations of the ergodic capacity. Besides, we use this new expression to propose a lower-bound of the ergodic capacity. We finally use the proposed new expression to derive a low SNR first order Taylor expansion of the ergodic capacity. According to [START_REF] Winzer | MIMO capacities and outage probabilities in spatially multiplexed optical transport systems[END_REF] and [START_REF] Dar | The Jacobi MIMO Channel[END_REF] as well as authors knowledge, no existing work addresses the lower-bound and low-SNR approximation of the ergodic capacity of the Jacobi MIMO channel.

The rest of this paper is organized as follows. The system model is introduced in Section II. In Section III, the Christoffel-Darboux formula is used to derive a new expressions for the ergodic capacity and its lower bound. Some numerical results are provided to validate the accuracy of the derived expressions in Section IV. Finally, Section V provides a conclusion.

II. SYSTEM MODEL

In an optical MIMO-SDM system with m t transmit cores/modes and m r receiving cores/modes, neglecting nonlinearities, the expression of the received signal can be expressed as [START_REF] Winzer | MIMO capacities and outage probabilities in spatially multiplexed optical transport systems[END_REF]:

y = Hx + n (1)
Where y ∈ C mr×1 is the vector of received symbols, x ∈ C mt×1 vector of transmitted symbols, n ∈ C mr×1 is the Gaussian with zero mean and unitary variance noise vector and H ∈ C mr×mt is the channel matrix.

A multi-modes/multi-cores optical fiber composed by m modes/cores can be modeled by a m × m matrix denoted G. In the case of m t , m r ≤ m, if m t (m r ) modes/cores are excited for transmission (reception), the channel matrix H is a m r × m t block of G. Without loss of generality, we can take the upper-left corner of G [START_REF] Collins | product of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF]. In optical fiber, H can be modeled as a random Haar distributed matrix [START_REF] Dar | The Jacobi MIMO Channel[END_REF]. This matrix can be achieved by orthonormalizing an independent identically distribution random Gaussian matrix. Moreover, if H is a Haar distributed matrix, H † H follows the Jacobi Unitary Ensemble (JUE).

Let us recall the definition of the Jacobi polynomials as follows. The Jacobi polynomial of degree n is denoted P a,b n (x). Those polynomials are orthogonal with respect to the inner product:

P (x)|Q(x) = 1 -1 (1 -x) a (1 + x) b P (x)Q(x) dx (2)
More generally, with this inner product a Jacobi polynomial of degree n is orthogonal with all polynomials of degree n -1 or lower. Moreover, we recall here the expression of the derivative of a Jacobi polynomial [START_REF]NIST Digital Library of Mathematical Functions[END_REF]:

P a,b′ n (x) = (n + a + b + 1) 2 P a+1,b+1 n-1 (x) (3)
Furthermore, we suppose that the receiver has complete channel state information (CSI) and that the transmitter has no CSI. In this case, the definition of the ergodic capacity of the channel is given by [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF]- [START_REF] Winzer | MIMO capacities and outage probabilities in spatially multiplexed optical transport systems[END_REF]:

C m,ρ mt,mr = E H log 2 det(I mt + ρH † H) (4)
Where E H is the expectation with respect to the density of the eigenvalues of H † H and ρ the signal-to-noise-ratio (SNR) which can be defined by the ratio between the variance of the received power and the variance of the noise.

When the total number of antennas exceeds the number of modes/cores in the fiber, m r + m t > m, the ergodic capacity can be expressed as [START_REF] Dar | The Jacobi MIMO Channel[END_REF]:

C m,ρ mt,mr = (m t + m r -m) log 2 (1 + ρ) + C m,ρ
m-mr,m-mt (5) Thus, without loss of generality, in this paper, our study will be limited to the case m r + m t ≤ m, and the expression of the ergodic capacity is given by [START_REF] Dar | The Jacobi MIMO Channel[END_REF]:

C m,ρ mt,mr = 1 0 λ a (1 -λ) b log 2 (1 + λρ) × r-1 k=0 B -1 k,a,b P a,b k (1 -2λ) 2 dλ (6)
Where P a,b k (x) is a Jacobi polynomial of degree k and:

B k,a,b = ||P a,b k (x)|| 2 2 a+b+1 = 1 2k + a + b + 1 2k + a + b k 2k + a + b k + a -1 (7)
Where

n k = n! k!(n-k)! denotes the binomial coefficient, and a = |m r -m t |, b = m -m r -m t , r = min{m r , m t }.

III. NEW EXPRESSION OF THE ERGODIC CAPACITY

Moreover, the expression of the diagonal Christoffel-Darboux kernel is given by [START_REF] Richard | Encyclopedia of mathematics and its applications[END_REF]:

K n (x, x) = n k=0 1 p k 2 p k (x) 2 (8)
And the Christoffel-Darboux formula:

K n (x, x) = k n k n+1 p n 2 (p ′ n+1 (x)p n (x) -p ′ n (x)p n+1 (x)) (9)
Where k n is the leading coefficient of the orthogonal polynomial p n .

With the variable change x = 1 -2λ, the expression of the ergodic capacity (6) can be rewritten:

C m,ρ mt,mr = 1 -1 (1 -x) a (1 + x) b log 2 1 + ρ(1 -x) 2 × r-1 k=0 P a,b k (x) 2 P a,b k 2 dx (10)
The first theorem of this paper is obtained by applying the Christoffel Darboux formula for Jacobi polynomials on [START_REF] Richard | Encyclopedia of mathematics and its applications[END_REF] and then by computing the derivatives of the polynomials with (3).

Theorem 1. For m r +m t ≤ m with perfect CSI at the receiver and no CSI at the transmitter, the expression of the ergodic capacity of a Jacobi MIMO channel is:

C m,ρ mt,mr = M a,b r 1 -1 (1 -x) a (1 + x) b log 2 1 + ρ(1 -x) 2 × P a,b r-1 (x)P a+1,b+1 r-1 (x) -N a,b r P a,b r (x)P a+1,b+1 r-2 (x) dx (11)
Where,

M a,b r = (r + a + b + 1)!r! 2 a+b+1 (r + a -1)!(r + b -1)!(2r + a + b) , (12)
and,

N a,b r = (r + a + b) (r + a + b + 1) (13)
.

In case where r is large (r ≥ 3), this new expression is more convenient than (6) for numerical evaluations. Indeed, the computation of the capacity with (6) requires the computation of the sum of r -1 products of Jacobi polynomials, whereas, in [START_REF] Collins | product of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF] this sum is replaced by the difference between two products.

The new expression expressed in theorem 1 can be used to derive a lower bound for the capacity: Lemma 1. If m r +m t ≤ m, the ergodic capacity of the Jacobi MIMO channel is lower bounded by:

Lb m,ρ mt,mr = M a,b r 1 -1 (1-x) a (1+x) b log 2 1 + ρ(1 -x) 2 × P a,b r-1 (x)P a+1,b+1 r-1 (x)dx (14)
Proof. To prove that ( 14) is a lower-bound of the capacity, it is sufficient to prove that the quantity:

Q = 1 -1 (1 -x) a (1 + x) b log 2 1 + ρ(1 -x) 2 P a,b r (x)P a+1,b+1 r-2 (x) dx (15) is negative.
First, the use of the inequality of convexity of the logarithm log(1 + x) ≤ x leads to:

Q ≤ ρ 2 ln(2) 1 -1 (1-x) a+1 (1+x) b P a,b r (x)P a+1,b+1 r-2 (x) dx
(16) Moreover, the link between the degree and the coefficients of Jacobi polynomials can be made with [START_REF]NIST Digital Library of Mathematical Functions[END_REF]:

P a-1,b n (x) = (n + a + b) (2n + a + b) P a,b n (x) - (n + b) (2n + a + b) P a,b n-1 (x) (17)
(17) and the orthogonality of Jacobi polynomials allow us to conclude that:

1 -1 (1 -x) a+1 (1 + x) b P a,b r (x)P a+1,b+1 r-2 (x) dx = 0 (18)
This finally proves that Q ≤ 0. This last result proves lemma 1.

Moreover, at low SNR, log(1 + x) ≈ x and thus Q ≈ 0, thus, at low SNR,

Lb m,ρ mt,mr ≈ C m,ρ mt,mr (19)
We can note that, at low SNR, log(1 + x) ≈ x and thus Q ≈ 0 and the proposed lower bound is a good approximation of the capacity.

The novel expression of the ergodic capacity derived in theorem 1 can also be used to propose a low SNR approximation of the ergodic capacity. Indeed, at low SNR, the ergodic capacity can be approximated by a very simple linear expression.

Lemma 2. At low SNR, the expression of the ergodic capacity of the Jacobi MIMO channel is:

C m,ρ mt,mr ≈ ρm r m t m ln(2) ρ ≪ 1 (20)
Thus, to maximize the capacity at low SNR, the number of transmit and receiving cores/modes must be maximized and the number of unused cores/modes must be minimized.

Proof. For m r + m t ≤ m,
Firstly, with the first order Taylor expansion of the logarithm, ln(1 + x) ∼ 0 x, (11) becomes:

C m,ρ mt,mr = ρM a,b r 2 ln(2) 1 -1 (1 -x) a+1 (1 + x) b P a,b r-1 (x)P a+1,b+1 r-1 (x) -N a,b r P a,b r (x)P a+1,b+1 r-2 (x) dx (21)
It has already been proved that:

1 -1 (1 -x) a+1 (1 + x) b P a,b r (x)P a+1,b+1 r-2 (x)dx = 0 (22)
Then, with (17) and by performing the variable change x = -x, the expression of the capacity at low SNR becomes:

1 -1 (1 -x) b (1 + x) a+1
× AP b,a+1 r-1 (x)P b+1,a+1 r-1

(x) + BP b,a+1 r-2 (x)P b+1,a+1 r-1 (x) dx (23)
Where

A = (r + a + b) (2r + a + b -1) B = (r + b -1) (2r + a + b -1) (24)
Finally, we can compute this integral with [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]:

1 -1 (1 -x) c (1 + x) b P a,b n (x)P c,b m (x) dx = 2 b+c+1 (a + b + m + n)!(b + n)!(c + m)!(a -c + n -m -1)! m!(n -m)!(a + b + n)!(b + c + m + n + 1)!(a -c -1)! (25)
Finally, 25 proves that at low SNR, the expression of the ergodic capacity is:

C m,ρ mt,mr = ρ ln(2) (a + r)r (a + b + 2r) (26)
Which proves lemma 2 in the case where m r + m t ≤ m. This result can be extended to the case m r + m t > m with the first order Taylor expansion of (5).

IV. NUMERICAL RESULTS

In this section, we provide a set of Matlab simulations that illustrate the theoretical results presented in the previous sections. We first suppose that the number of supported modes in the fiber is equal to 32 and that the SNR varies between 0 and 30 dB. In a first step, the ergodic capacity is computed using the expression proposed in theorem 1, then the computation is made using [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF] proved in [START_REF] Dar | The Jacobi MIMO Channel[END_REF]. The obtained results with the two formulas, for different values of m r and m t , are drawn in figure 1.

Figure 1 shows that the two expressions of the ergodic capacity ( Eq.( 6) and Eq.( 11)) produce the same simulation results. Furthermore, the proposed expression of the ergodic capacity reduces the evaluation time without impacting the results. Note that m r and m t have a symmetrical role in the expression of the ergodic capacity. Also, the evaluation time Fig. 3. Simulated ergodic capacity and analytical lower bound against the SNR when m = 32 and mt = mr is drastically reduced when the parameter r = min{m r , m t } is large.

In figure 2, we analyze the effect of unused modes on the channel ergodic capacity. We compare the evolution of the capacity where m = 16 and m = 32 for different values of m r and m t . For a fixed number of transmit and receiving cores/modes, the channel ergodic capacity decreases as m increases. In other words, the capacity decreases as the number of unused cores/modes increases. This observation allows to generalize the observation done at low SNR.

Using the same simulation parameters (m, m r , m t and the SNR variation range) as in figure 1, the channel ergodic capacity and its lower-bound are depicted against the SNR in figure 3. As anticipated, at low SNRs, the lower-bound of the ergodic capacity is very close to the ergodic capacity. However, them increases as the capacity gap between them increases when SNR is large.

In figure 4, we analyze the evolution of the lower-bound of the ergodic capacity for different values of m r and m t when m r + m t = 16 and m = 32. The gap between the ergodic capacity and the proposed lower bound expression decreases as a = |m r -m t | is large. Moreover, this capacity gap reaches a maximum when m r = m t .

V. CONCLUSION

In this paper, we used the Jacobi MIMO channel to analyze the ergodic capacity of the propagation channel in a multi-

  t =2 , formula of theorem 1 m r =m t =4 , formula of theorem 1 m r =5 m t =3 , formula of theorem 1 m r =4 m t =6 , formula of theorem 1 m r =m t =2 , formula of R.Dar and al. m r =m t =4 , formula of R.Dar and al. m r =5 m t =3 , formula of R.Dar and al. m r = 4 m t =6 , formula of R.Dar and al.

Fig. 1 .C

 1 Fig. 1. Numerical evaluation of the ergodic capacity when m = 32

  Fig.[START_REF] Ryf | Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 x 6 MIMO Processing[END_REF]. Simulated ergodic capacity and analytical lower bound against the SNR when m = 32 and mt = mr mode/multi-core optical fiber. We first used the Christoffel-Darboux formula to reformulate the expression of the ergodic capacity. The proposed new expression reduces the computation time to evaluate the ergodic capacity. Moreover, this expression has been used to propose a lower-bound of the capacity. We finally derived a very simple low SNR approximation of the ergodic capacity.

		60
				Capacity, m r =m t =8
				Capacity, m r =10, m t =6
		50		Capacity, m r =12 m t =4
				Lower Bound, m r =m t =8
				Lower Bound, m r =10, m t =6
	Capacity (bit/s/Hz)	20 30 40		Lower Bound, m r =12 m t =4
		10
		0	0	5	10	15	20	25	30
					SNR (dB)

ACKNOWLEDGMENT

Part of this work is funded by Région Bretagne, France.


Philippe Jaming
email: philippe.jaming@math.u-bordeaux.fr

Salvador P Érez-Esteva

THE PHASE RETRIEVAL PROBLEM FOR SOLUTIONS OF THE HELMHOLTZ EQUATION

Keywords: 1991 Mathematics Subject Classification, 42B10, phase retrieval; Helmholtz equation

In this paper we consider the phase retrieval problem for Herglotz functions, that is, solutions of the Helmholtz equation ∆u + λ 2 u = 0 on domains Ω ⊂ R d , d ≥ 2. In dimension d = 2, if u, v are two such solutions then |u| = |v| implies that either u = cv or u = cv for some c ∈ C with |c| = 1. In dimension d ≥ 3, the same conclusion holds under some restriction on u and v: either they are real valued or zonal functions or have non vanishing mean.

Introduction

The phase retrieval problem consists in reconstructing a function from its modulus or the modulus of some transform of it (frame coefficient, Fourier transform,...) and some structural information on the function (e.g. to be compactly supported). This kind of problems occur in many scientific fields such as microscopy, holography, crystallography, neutron radiography, optical coherence tomography, optical design, radar signal processing , quantum mechanics to name a few. We refer to the books [Hu, St], the review articles [KST, Mi, Fi, LBL] and to the introduction of our previous paper [Ja] for descriptions of various instances of this problems, some solutions to it (both theoretical and numerical) and for further references.

The problem can be split into two main questions:

-design algorithms that allow to reconstruct at least one solution.

-obtain uniqueness results (up to obvious invariants of the problem like the multiplication by a constant phase factor).

After having long been ignored by mathematicians, recent progress on the algorithmic aspect of the problem [CSV, WdAM] has triggered a lot of attention to this problem. While the design of numerical algorithms allowing to reconstruct one solution is of course essential, the task can only be complete once one is certain to reconstruct all solutions of interest. This is generally not possible as long as uniqueness is not guarantied and plainly justifies the second part of the problem.

Uniqueness is also usefull in order to have stability results in presence of additive noise. In this paper, we will only deal with the uniqueness aspect of the problem. More precisely, the phase retrieval problem is extremely common in optical sciences, among other reasons, this is due to the lack of sensitivity of optical measurement instruments to phase. It turns out that optical signals are solution of partial differential equations and our aim is to show that this information can be of some use in the phase retrieval problem. In our previous work [Ja], we considered solutions of the free Shrödinger equation. Our aim here is to pursue a similar study for solutions of the Helmholtz equation ∆u + λ 2 u = 0 in a domain Ω ⊂ R d , d ≥ 2 where by domain we mean an open connected subset of R d . Recall that this equation is obtained by reducing the wave equation to monochromatic waves and that |u| 2 when u is a solution of this equation is the intensity of the monochromatic wave and does not vary with time. Note also that a different version of the phase retrieval problem for solutions of the Helmholtz equation has been recently studied in [START_REF] Klibanov | Phaseless inverse scattering and the phase problem in optics[END_REF][START_REF] Klibanov | Phaseless inverse scattering problems in three dimensions[END_REF][START_REF] Klibanov | On the first solution of a long standing problem: Uniqueness of the phaseless quantum inverse scattering problem in 3-D[END_REF][START_REF] Klibanov | Uniqueness of a 3-D coefficient inverse scattering problem without the phase information[END_REF] and for the Shrödinger equation in [START_REF] Klibanov | Romanov Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation[END_REF] and references therein.

Finally, note that, up to a renormalization, we may restrict attention to the case λ = 1. Up to a translation, we may also assume that 0 ∈ Ω. We are thus concerned with the following problem:

Phase Retrieval Problem for the Helmholtz Equation. Let d ≥ 2 and 0 ∈ Ω ⊂ R d be a domain. Let u, v be two solutions of the Helmholtz equation on Ω (H) ∆u + u = 0.

Does |v| = |u| imply that v = cu or v = cū for some c ∈ C with |c| = 1.
Of course, if v = cu or v = cū then v is also a solution of (H) and |v| = |u|. We will say that v is a trivial solution of the phase retrieval solution for u. This problem, as stated still eludes us. Our aim here is to show that, in many instances, the problem has only trivial solutions.

Our main result is then the following:

Main Theorem. Let d ≥ 2 and 0 ∈ Ω ⊂ R d be a domain. Let u, v be two solutions of the Helmholtz equation on Ω ∆u + u = 0 , ∆v + v = 0
on Ω. Assume one of the following holds u and v are real valued; -u has non-zero mean; -The dimension is d = 2; -d ≥ 3 and u, v are zonal functions.

Then there exists a constant c ∈ C such that either v = cu on Ω or v = cū on Ω i.e. v is a trivial solution of the phase retrieval problem.

Recall that a zonal function is a function of the form ϕ( x, x 0 ). Such functions are sometimes also called ridge functions.

The remaining of this paper is organized as follows: in the next section, we gather all information we need about spherical harmonics and Bessel function and then reformulate the problem in terms of spherical harmonic coefficients. The remaining of the paper is then devoted to the proofs of the various statements of the main theorem, Section 4 is devoted to the 2-dimensional cases and Section 3 to the other three cases.

Preliminaries

2.1. Notations. Throughout this paper, d is an integer, d ≥ 2. The Euclidean norm in R d is denoted by |x| = x 2 1 + • • • + x 2 d
1/2 and we set

S d-1 = {x ∈ R d : |x| = 1}.
We denote the standard basis of R d by e 1 , . . . , e d . We denote by N 0 the set of non-negative integers,

N = N 0 ∪ {0}. For α ∈ N d 0 we use the standard multi-index notation, |α| = α 1 + • • • + α d and ∂ α = ∂ α 1 x 1 • • • ∂ α d x d . The Laplace operator is defined by ∆ = d j=1 ∂ 2 x j .
2.2. Spherical harmonics. Let us here gather some information on spherical harmonics as can be found in many books in harmonic analysis. We will here take the notations from [START_REF] Feng | Approximation Theory and Harmonic Analysis on Spheres and Balls[END_REF]Chapter 1]. We denote by P d m the space of homogeneous polynomials of degree m in R d and H d m = {P ∈ P d n : ∆P = 0} the space of harmonic homogeneous polynomials of degree n in R d . Recall that this space has dimension

N(m) = m + d -1 m - m + d -3
m -2 with the standard convention that the second term vanishes for m = 0 and m = 1. In particular N(0) = 1, and for m ≥ 1, N(m) = 2 when d = 2 while N(m) = 2m + 1 when d = 3

We will not distinguish homogeneous polynomials and their restriction to S d-1 . In particular, H d m and H d n are orthogonal subspaces of L 2 (S d-1 ) when n = m.

Recall the following definition:

Definition 2.1. A function f on R d or S d-1 is said to be zonal with respect to some ζ 0 ∈ S d-1 if there exists a function f 0 on R such that f (x) = f 0 ( x, ζ 0 ). Such functions can be described in terms of a so-called zonal basis:

Y j m (θ) = C d/2-1 m ( θ, ζ j m ) where {ζ j m , m ≥ 0, j = 1, . . . , N(m)} ⊂ S d-1 , ζ 1 m = ζ 0 for every m and C λ
m is the Gegenbauer polynomial of degree m and parameter λ are given by

C λ m (z) = ⌊m/2⌋ k=0 (-1) k Γ(m -k + λ) Γ(λ)k!(m -2k)! (2z) m-2k .
For the existence of such a basis, see e.g. [START_REF] Feng | Approximation Theory and Harmonic Analysis on Spheres and Balls[END_REF]Theorem 3.3]. A zonal function has then an expension in L 2 (S d-1 ) in terms of (Y 1 m ) m≥0 only: f is zonal with respect to ζ 0 if and only if f = m≥0 a m (f )Y 1 m . A zonal basis need not be orthogonal however, the orthogonality property of the H d m 's show that there is no convergence issue in L 2 (S d-1 ). Those basis have the very desirable property for our problem to be real valued. We also recall that the zonal basis is extended to R d by homogeneity,

Y j m (x) = |x| m C d/2-1 m x
|x| , ζ j m . We will need the following simple Lemma: ), so that we want to show that ϕ = 0 or ϕ = π.

Lemma 2.2. Let m, n, d be integers, m, n ≥ 1, d ≥ 3 λ = d/2 -1, a, b ∈ C \ {0}, ζ 1 , ζ 2 ∈ S d-1 . Assume that aC λ m ( θ, ζ 1 ) 2 = bC λ n ( θ, ζ 2 ) 2 then m = n, ζ 2 = ±ζ 1 and a = b. Proof. Let ζ 3 , ζ 4 ∈ S d-1 be such that (i) ζ 1 , ζ 3 , ζ 4 are form an orthonormal basis of its span; (ii) ζ 2 ∈ span{ζ 1 , ζ 3 }. Write ζ 2 = cos ϕζ 1 + sin ϕζ 3 with ϕ ∈ [0, 2π
Let θ = cos s cos tζ

1 +sin tζ 3 +sin sζ 4 ∈ S d-1 then θ, ζ 1 = cos t cos s while θ, ζ 2 = cos(t-ϕ) cos s and aC λ m ( θ, ζ 1 ) 2 = bC λ n ( θ, ζ 2 ) 2 reduces to (2.1) aC λ m cos t cos s 2 = bC λ n cos(t -ϕ) cos s 2 .
As powers of the function cos x are linearly independent, it remains to look at the highest order term in cos s to see that (2.1) implies

a 2 m Γ(m + λ) m! 2 cos 2m t = b 2 n Γ(n + λ) n! 2 cos 2n (t -ϕ).
Using again the linear independence of powers of cos, this implies that m = n and thus reduces to

a cos 2m t = b cos 2m (t -ϕ). As 2 2m cos 2m t = (e it + e -it ) 2m = 2m j=0 2m j e i(2m-2j)t
we get a = be -i(2m-2j)ϕ for j = 0, . . . , 2m. Therefore a = b and ϕ = 0

or ϕ = π that is ζ 2 = ±ζ 1 . Remark 2.3. Note that if m = 0 or if a = 0, we get a = b but, of course, we can not conclude that ζ 2 = ±ζ 1 .
From e.g. [START_REF] Feng | Approximation Theory and Harmonic Analysis on Spheres and Balls[END_REF]Theorem 1.9], we can construct another real valued basis for

d ≥ 3. If α ∈ N d 0 is obtained as follows: α ∈ N d 0 , |α| = m (2.2) p α (x) = (-1) m 2 m d-2 2 m |x| d-2+2m ∂ α |x| -d+2 = x α + |x| 2 q α (x)
with q α an homogeneous polynomial of degree m -2 (when m = 0 or 1, q α = 0). Then

{p α : α ∈ N d 0 , |α| = m, α d = 0 or 1} is a basis of H d m , that is not orthonormal. For j = 0 or j = 1, let us write N j m = {α ∈ N d 0 , |α| = m, α d = j}, and write N m = N 0 m ∪ N 1 m . Lemma 2.4. For every integer m ≥ 0, the set of polynomials {p 2 α : α ∈ N d 0 , |α| = m, α d = 0 or 1} is linearly independent. Proof. Note that p 2 α = x 2α + 2x α |x| 2 q α (x) + |x| 4 q α (x) 2 . Assume that (2.3) α∈Nm λ α p 2 α = α∈Nm λ α x 2α + 2|x| 2 α∈Nm λ α x α q α + |x| 4 α∈Nm λ α q 2 α = 0.
In this sum, the terms α∈N 0 m λ α x 2α are clearly linearly independent from from all other terms (since x 2e d factors in them), thus

α∈N 0 m λ α x 2α = 0 and finally λ α = 0 if α d ∈ N 0 m . But then (2.3) reduces to x 2e d α∈N 1 m λ α x 2α-2e d + 2|x| 2 x e d α∈N 1 m λ α x α-e d q α + |x| 4 α∈N 1 m λ α q 2 α = 0.
Again, the first term is linearly independent from the two others, thus α∈N 1 m λ α x 2α-2e d = 0 and finally λ α = 0 for α ∈ N 1 m as well.

2.3. Bessel functions. Here we gather some basic facts about Bessel functions (of integral order) that can be found e.g. in [AS] and [Wa].

Let n ≥ 0 be an integer and α > 0. Recall that the Bessel function J n+a can be defined via the power series

J n+α (r) = r 2 n+α ∞ k=0 (-1) k k!Γ(n + α + k) r 2 2k . Alternatively J 0 (r) = 1 π π 0 cos(r sin θ) dθ and, for n ≥ 1, J n (r) = 2r n √ π2 n Γ(n + 1/2) 1 0 (1 -t 2 ) n-1/2 cos(rt) dt.
Finally, recall that for a negative integer n < 0, we may define

J n = (-1) n J -n .
From these expressions, one immediately deduces that |J 0 (r)| ≤ 1 and, for n

∈ Z \ {0} |J n+α (r)| ≤ 2 √ πΓ(n + α + 1/2) r 2 n+α .
In particular, if (a n ) ∈ ℓ 2 (Z), the series

n∈Z a n J n (r)
is uniformly convergent over every bounded interval. Moreover, all formal computations that we will use are directly justified with these estimaes.

Next, if n, m ∈ N 0 , then (see [AS,Formula 9.1.14], [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF]p 147])

(2.4) J n+α (r)J m+α (r) = r 2 n+m+2α ∞ k=0 (-1) k Γ(n + m + 2a + 2k + 1) Γ(n + a + k + 1)Γ(m + a + k + 1)Γ(n + m + 2a + k + 1) r 2 2k .
As a consequence, J n (r)J m (r) ∼ r n+m+2α when r → 0. Moreover, using the pointwise bound of J n , we see that if (c n,m ) n,m≥0 is of moderate

growth then r -2α n,m≥0 c n,m J n+α (r)J m+α (r)
is holomorphic in a neighborhood of 0. Moreover if, for some η > 0 and for 0 ≤ r < η, n,m≥0

c n,m J n+α (r)J m+α (r) = 0 then, for every j ≥ 0, n+m=j c n,m J n+α (r)J m+α (r) = 0. But then, using [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF]p150]:

J n+α (r)J m+α (r) = 2 π π/2 0 J n+m+2α (2r cos θ) cos(n -m)θ dθ we have 0 = π/2 0 J j+2α (2r cos θ) j n=0 c n,j-n cos(2n -j)θ dθ = r j+2α ∞ k=0 (-1) k k!Γ(j + 2α + k)! r 2k × π/2 0 j n=0 c n,j-n cos(2n -j)θ cos j+2k θ dθ.
It follows that, for every j, k ≥ 0,

j n=0 c n,j-n π/2 0 cos(2n -j)θ cos j+2k θ dθ = 0 that we rewrite (2.5) π/2 0   cos j θ 0≤n≤j/2 c n,j-n + c j-n,n 2 cos(2n -j)θ   (cos 2 θ) k dθ = 0.
We will now use the following simple lemma:

Lemma 2.5. Let f : [a, b] → [0, 1] be C 1 , strictly monotonic and onto, then {f k } k∈N is dense in L 2 [a, b]. Proof. According to Hahn-Banach's, it is enough to show that, if ϕ ∈ L 2 [a, b] is such that (2.6) b a ϕ(s)f k (s) ds = 0 for every k ≥ 0 then ϕ = 0. But b a ϕ(s)f k (s) ds = 1 0 ϕ f -1 (t) f ′ f -1 (t) t k dt thus, (2.6) implies that 1 0 ψ(t)t k dt = 0 for every k ≥ 0 where ψ(t) = ϕ f -1 (t) f ′ f -1 (t) . As {t k } k∈N in L 2 [0, 1], we get ψ = 0 thus ϕ = 0.
Applying Lemma 2.5 to (2.5) implies that, for every j ≥ 0 and every θ ∈ (0, π/2)

cos j θ 0≤n≤j/2 c n,j-n + c j-n,n 2 cos(2n -j)θ = 0.
This implies that for every j ≥ 0 and every 0 ≤ n ≤ j, c n,j-n + c j-n,n = 0. We have thus proved the following lemma:

Lemma 2.6. Let α, η > 0. Let (c n,m ) n,m≥0 be a sequence with at most polynomial growth. The following are equivalent:

(i) for every 0 ≤ r < η, n,m≥0
c n,m J n+α (r)J m+α (r) = 0.

(ii) for every j ≥ 0 and every 0 ≤ k ≤ j, c k,j-k + c j-k,k = 0.

We will apply this lemma in the form

n,m≥0 c n,m J n+α (r)J m+α (r) = n,m≥0
cn,m J n+α (r)J m+α (r)

if and only if, for every j ≥ 0 and every 0

≤ k ≤ j, c k,j-k + c j-k,k = ck,j-k + cj-k,k .
2.4. Reduction of the problem. Let us first make the following observation: solutions of the Helmholtz equation are real analytic in Ω, a connected set. Therefore, if u = cv (resp. u = cv) in a ball B(0, ε) ⊂ Ω, then u = cv (resp. u = cv) in the whole of Ω. We may thus assume that Ω is a ball centered at 0. We will now need to describe the solutions of (H) ∆u + u = 0 in polar coordinates. As is well known and easily shown, in a neighborhood B(0, ε) ⊂ Ω of 0, a solution u of (H) can be expanded as a series (2.7)

u(rθ) ∼ (2π) 1/2 r -(d-2)/2 ∞ m=0 N (m) j=1 a m,j (u)J ν(m) (r)Y j m (θ)
where ν(m) = m + (d -2)/2 and {Y j m } j=1,...,N (m) is a basis for the spherical harmonics of degree m in R d .

Throughout the remaining of this section, u, v will be two solutions of (H) in Ω such that |u| = |v|. Now (2.7) implies

(2.8) |u(rθ)| 2 = 2π r d-2 ∞ m,n=0 c m,n (u)J m+ d-2 2 (r)J n+ d-2 2 (r)
where (2.9)

c m,n (u) = N (m) j=1 N (n) k=1 a m,j (u)a n,k (u)Y j m (θ)Y k n (θ). Note that c m,n (u) = c n,m (u). Therefore, Lemma 2.6 implies that |u| = |v| is equivalent to ℜ c m,n-m (u) = ℜ c m,n-m (v) for all 0 ≤ m ≤ n.
ℜ c m,n (u) = ℜ c m,n (v) . Note that c m,m (u) = N (m) k=1 a m,k (u)Y k m (θ) 2 . In particular, (2.10) ℜ c m,m (v) = ℜ c m,m (v)
is the classical phase retrieval problem for trigonometric polynomials.

We are thus facing a family of classical phase retrieval problems (2.10)

with compatibility relations ℜ c m,n (v) = ℜ c m,n (v) , n = m.
The main difference between the 2 dimensional problem and the higher dimensional one is that the 2 dimensional one has only trivial solutions (for fixed m). We will show that the compatibility relations imply that the same trivial solution has to be chosen independently of m.

Further, if {Y k m } k=1,...,N (m) is an orthonormal basis of H d m then, integrating over S d-1 and using orthogonality, we obtain (2.11)

N (m) k=1 |a m,k (u)| 2 = N (m) k=1 |a m,k (v)| 2 .
As a consequence if u is a trigonometric polynomial in the sense that {a m,j (u)} has finite support, so has {a m,j (v)} and v is also a trigonometric polynomial. We have thus proved the following lemma:

Lemma 2.8. Let d ≥ 2 and 0 ∈ Ω ⊂ R d . Let u, v be two solutions of the Helmholtz equation (H) on Ω.
Assume that u is a trigonometric polynomial (K-finite) in the sense that its expansion in spherical harmonics (2.7) has only finitely many terms. If |u| = |v|, then v is also a trigonometric polynomial.

The case of dimension d ≥ 3

3.1. The real case. Let us now start by showing that the problem is very simple if one restricts it to real valued solutions. In this case, |u| = |v| is equivalent to u 2 = v 2 thus (uv)(u + v) = 0. Thus, one of u = v or u = -v occurs on a set of positive measure and, as u, v are analytic, either u = v or u = -v.

According to [Le, Sj], (see also [JK]) in dimension d = 2, and to [GJ] for general dimension, solutions of the Helmholtz equation in R d are uniquely determined by their restriction to two generic hyperplanes. It follows from [FBGJ] that this result also holds for solutions on domains. Those results extend to the phase retrieval problem as follows:

Proposition 3.1. Let d ≥ 2 and 0 ∈ Ω ⊂ R d . Let u, v be two solutions of the Helmholtz equation (H) on Ω. Let θ 1 , θ 2 , θ 3 ∈ S d-1 and assume that 1 π arccos θ j , θ k / ∈ Q when j = k ∈ {1, 2, 3}. Assume that -u, v are real valued; -|u| = |v| on the hyperplanes Ω ∩ θ ⊥ j , j = 1, 2, 3 (a fortiori if |u| = |v| on Ω).
Then either u = v or u = -v.

Proof. As we have already noticed, we have (uv)(u + v) = 0 on Ω ∩ θ ⊥ j . Therefore, for each j = 1, 2, 3 at least one of the following two cases holds -either u = v on a subset of Ω ∩ θ ⊥ j of positive d -1-dimensional Lebesgue measure on the hyperplane, thus by analicity, on Ω ∩ θ ⊥ j ; -or u = -v on Ω ∩ θ ⊥ j . We thus have 2 cases to consider -either u = v on at least 2 of θ ⊥ j , j = 1, 2, 3. -or u = -v on at least 2 of θ ⊥ j , j = 1, 2, 3. Up to replacing v by -v the second case reduces to the first one so that we can assume that u = v on θ ⊥ 1 , θ ⊥ 2 . According to [FBGJ], this implies u = v everywhere.

3.2. Solutions of the Helmholtz equation with non vanishing mean in dimension d ≥ 3. In this section, the basis of spherical harmonics will be the real basis given by (2.2).

Theorem 3.2. Let d ≥ 3 and 0 ∈ Ω ⊂ R d be a domain. Let u, v be two solutions of the Helmholtz equation (H) on Ω. Assume that

S d-1 u(ηθ) dσ(θ) = 0 for some η > 0 such that B(0, η) ⊂ Ω and J d/2-1 (η) = 0.
If |v| = |u| then there exists c ∈ C with |c| = 1 such that, either u = cv or u = cv.

Proof. As Y 1 0 = 1 and all other spherical harmonics have vanishing spherical means,

a 0,1 (u) = η d/2-1 √ 2πσ(S d-1 )J d/2-1 (η) S d-1 u(ηθ) dσ(θ) = 0
is, up to a multiplicative constant, the mean of u over ηS d-1 . But, (2.9) for m = n = 0 reduces to |a 0,1 (v)| 2 = |a 0,1 (u)| 2 . In particular, v as non-zero mean as well. Further, up to changing u and v by unimodular multiples, we may then assume that a 0,1 (u) = a 0,1 (v) and that this quantity is real.

Next, note that c 0,n (u) =

N (n) k=1 a 0,1 (u)a n,k (f )Y k n (θ) since Y 1 0 (θ) = 1. Thus ℜ c 0,n (u) = a 0,1 (u) N (n) k=1 ℜ a n,k (u) Y k n (θ)
As the Y k n 's are linearly independent, (2.9) for m = 0 implies that

ℜ a n,k (v) = ℜ a n,k (u) thus ℜ(v) = ℜ(u). As |v| 2 = |u| 2 we deduce that |Im (v)| 2 = |Im (u)| 2 . But Im u, Im v are real solutions of the Helmholtz equation. Proposition 3.1 then im- plies that either Im v = Im u or Im v = -Im u, that is v = u or v = ū.
3.3. The sparse case and the zonal cases in dimension d ≥ 3. In this section, d ≥ 3 and we fix an orthonormal basis of spherical harmonics {Y j m } m≥0,j=1,...,N (m) that is either zonal or is of the form (2.2). In particular, it is real and has the property

that if a(Y j m ) 2 = b(Y k n ) 2 on S d-1
, then m = n, j = k and b = a. For the zonal basis this is due to Lemma 2.2 and is a consequence of Lemma 2.4 for the other basis.

Definition 3.3. We will say that u ∈ L 2 (S d-1 ) is sparse (in the basis {Y m j }) if, for every m there exists at most one j = j(m) such that a m,j (u) = 0.

Example 3.4. A zonal function is sparse in a zonal basis.

Proposition 3.5. Let d ≥ 3 and 0 ∈ Ω ⊂ R d be a domain. Let u, v be two solutions of the Helmholtz equation (H) on Ω.

Assume that both u and v are sparse in a common real orthonormal basis of spherical harmonics

{Y j m , m ≥ 0, 0 ≤ j ≤ N(m)}. If |v| = |u| then there exists a c ∈ C with |c| = 1 such that either v = cu or v = cu.
Proof. Let u, v be two sparse functions:

u(rθ) = √ 2π r d/2-1 m≥0 a m,j(m) (u)J ν(m) (r)Y j(m) m (θ) and v(rθ) = √ 2π r d/2-1 m≥0 a m,k(m) (v)J ν(m) (r)Y k(m) m (θ)
and assume that |u| = |v|. Note that some of the a m,j(m) (u)'s may still be zero. For simplicity, we set j(m) = 1 when a m,j (u) = 0 for all j and k(m) = 1 when a m,k (v) = 0 for all k.

First, (2.9) for n = m implies that

a m,j(m) (u) 2 Y j(m) m 2 = a m,k(m) (v) 2 Y k(m) m 2 .
This implies that k(m) = j(m), i.e that a m,k (u) and a m,k (v) have same support, and that a m,j(m) (u) = a m,j(m) (v) . Let M = {m ≥ 0 : a m,j(m) (u) = 0} and m 0 = min M. Then, up to replacing u, v by unimodular multiples, with may assume that a m 0 ,j(m 0 ) (u) = a m 0 ,j(m 0 ) (v) is real (and non zero).

Next, note that, for m ∈ M,

c m,n (u) = a m,j(m) (u)a n,j(n) (u)Y j(m) m (θ)Y j(n) n (θ)
and similarily

c m,n (v) = a m,j(m) (v)a n,j(n) (v)Y j(m) m (θ)Y j(n) n (θ) Further, Y j(m) m (θ)Y j(n) n
(θ) is a non-zero real polynomial. Thus (2.9) reduces to

ℜ a m,j(m) (u)a n-m,j(n-m) (u) = ℜ a m,j(m) (v)a n-m,j(n-m) (v) for every 0 ≤ m ≤ n. Taking m = m 0 and n = 2m 0 + k we get ℜ a m 0 +k,j(m 0 +k) (u) = ℜ a m 0 +k,j(m 0 +k) (v) .
It follows that ℜ(u) = ℜ(v) and, as in the previous proof, this implies that either u = v or u = v.

The 2 dimensional case

In this section, we will specifically treat the 2 dimensional case for which uniqueness is

Theorem 4.1. Let 0 ∈ Ω ⊂ R 2 be a domain. Let u, v be two solutions of the Helmholtz equation (H) on Ω be such that |v| = |u|. Then there exists c ∈ C with |c| = 1 such that either v = cu on Ω or v = cū on Ω.
Proof. A basis of spherical harmonics is given by the usual Fourier basis {e ikθ , k ∈ Z} so that we may write u in polar coordinates as a Fourier series

u(rθ) = √ 2π k∈Z u(k)J |k| (r)e ikθ . Note that c m,n (u) =      |û(0)| 2 if m = n = 0 û(0) û(-n)e -inθ + û(n)e inθ if m = 0, n > 0 û(-m)e -imθ + û(m)e imθ û(-n)e -inθ + û(n)e inθ if m, n > 0 .
We will now exploit Lemma 2.7, that is ℜc m,n (u) = ℜc m,n (v) for every 0 ≤ m ≤ n.

We have already treated the case u(0) = 0 so that we will now assume that u(0) = 0. This implies that c 0,0 (v) = c 0,0 (u) = 0, thus v(0) = 0.

Next c m,m (u) = û(-m)e -imθ + û(m)e imθ 2 is real so that (4.12) v(-m)e -imθ + v(m)e imθ 2 = û(-m)e -imθ + û(m)e imθ 2

for every m ≥ 1. We thus need the following lemma:

Lemma 4.2. Let m, n ∈ N. Let a, b, c, d ∈ C be such that, for every θ ∈ R, (4.13) ae imθ + be -imθ 2 = ce imθ + de -imθ 2 then there exists κ ∈ C such that either ae imθ + be -imθ = κ ce imθ + de -imθ for every θ or ae imθ + be -imθ = κ ce imθ + de -imθ for every θ.

Proof of Lemma 4.2. This lemma is folklore in the subject, for sake of completeness, let us here give the proof. Expanding the square in (4.13) we see that |a| -resp. ae imθ + be -imθ = e iϕ de imθ = e iϕ (ce imθ + de -imθ ). (iii) If d = 0 then b = 0 -resp. a = 0-and ae imθ + be -imθ = e iϕ (ce imθ +de -imθ ) -resp. ae imθ +be -imθ = e iψ (ce imθ + de -imθ ). The proof of the lemma is thus complete.

Applying this lemma, we may now distinguish two cases: Type I. We will say that m is of type I if û(m), û(-m) = ( 0 Also, ĝ(m), ĝ(-m) = (0, 0) if m has a type. We will now need the following lemma:

Lemma 4.3. Let m = n ∈ N. Let a, b, c, d ∈ C be such that (a, b), (c, d) = (0, 0). Let κ, κ ′ ∈ C with |κ| = |κ ′ | = 1. Assume that ℜ κ ′ κ(ae imθ + be -imθ )(ce inθ + de -inθ ) = ℜ (ae imθ + be -imθ )(ce inθ + de -inθ )
for every θ ∈ R. (4.14) κ ′ κ(ae imθ + be -imθ )(ce inθ + de -inθ ) = (ae imθ + be -imθ )(ce inθ + de -inθ ) for a set of positive measure of θ's thus, by analycity, for all θ.

Expanding and comparing the coefficients of e ±imθ±inθ (recall that m = n) this is equivalent to (i) κ ′ κac = bd, (ii) κ ′ κad = bc, (iii) κ ′ κbc = ad and (iv) κ ′ κbd = ac. -i(ϕ+ψ) ac(e i(mθ+ϕ) + e -i(mθ+ϕ) )(e i(nθ+ψ) + e -i(nθ+ψ) ) = ace -i(ϕ+ψ) (e i(mθ+ϕ) + e -i(mθ+ϕ) )(e i(nθ+ψ) + e -i(nθ+ψ) ) which reduces to κ ′ = αβ αβ κ.

As a consequence of the lemma we get that, if m, n are of type I (resp. of type C) and one of them is not of type R, then κ m = κ n . We thus obtain the following:

-either all m are of type R and then (4.15)

           u(rθ) = √ 2π ∞ m=1 β m J m (r) cos(mθ + θ m ) v(rθ) = √ 2π ∞ m=1 κ m β m J m (r) cos(mθ + θ m )
where β m = u(m)e -iθm and for every m, n for which u(m) u(n) = 0,

either κ m = κ n or κ m = β m β m β n β n κ n .
-or there exists exactly at least one m that is not of type R and at least one m of type R (thus of type R and C) and then κ m = κ n := κ for every m, n and We will now show that in the two last cases one of the type I or type C sums in (4.16)-(4.17) is empty. This follows from the following lemma and the fact that ℜc m,n (v) = ℜc m,n (u) with m of type I but not R and n of type C but not R: Proof of Lemma 4.4. We will use the fact that two complex numbers of same modulus and same real part are either equal of conjugate of one an other. Therefore -either (4.18) κ ′ κ(ae imθ + be -imθ )(ce inθ + de -inθ ) = (ae imθ + be -imθ )(ce inθ + de -inθ ) -or (4.19) κ ′ κ(ae imθ + be -imθ )(ce inθ + de -inθ ) = (ae imθ + be -imθ )(ce inθ + de -inθ )

Note that, by analycity, if one of the alternatives holds for a set of positive measure of θ's, then it holds for every θ. Further, up to exchanging the roles of the two factors, (4.19) reduces to (4.18). But, expanding the factors, we see that this equation is equivalent to (i) κ ′ κa d = ac (ii) κ ′ κac = ad (iii) κ ′ κbc = bd (iv) κ ′ κb d = bc. Now, if c = 0 then d = 0 and (i) implies that a = 0 while (iv) implies b = 0 which is excluded. Using (ii) and (iii) we can also exclude d = 0.

Further (i),(ii) imply that |a||c| = |a||d| while (iii),(iv) imply |b||c| = |b||d|. As one of a, b = 0 we get |c| = |d| which was excluded.

  Finally, replacing n by n+m and using the symmetry c n,m (u) = c m,n (u), we obtain the following: Lemma 2.7. Let d ≥ 2 and 0 ∈ Ω ⊂ R d . Let u, v be two solutions of the Helmholtz equation (H) on Ω. Then |u| = |u| if and only if for every 0 ≤ m ≤ n

  2 + |b| 2 = |c| 2 + |d| 2 and a b = c d. In particular |a||b| = |c||d|, so that |a| 2 , |b| 2 have same sum and product as |c| 2 , |d| 2 . There are thus two possibilities, either |a| = |c| and |b| = |d| or |a| = |d| and |b| = |c|. We may thus write a = ce iϕ and b = de iψ (resp. a = de iϕ and b = ce iψ ) so that c de i(ϕ-ψ) = c d (resp. c de i(ϕ-ψ) = c d). We now distinguish 3 cases: (i) If c d = 0 then e iϕ = e iψ and the conclusion is straightforward. (ii) If c = 0 then a = 0 -resp. b = 0-and ae imθ + be -imθ = e iψ de -imθ = e iψ (ce imθ + de -imθ )

  , 0) and if there exists κ m ∈ C with |κ m | = 1 such that v(-m)e -imθ + v(m)e imθ = κ m û(-m)e -imθ + û(m)e imθ .Type C. We will say that m is of type C if û(m), û(-m) = (0, 0) and if there existsκ m ∈ C with |κ m | = 1 such that v(-m)e -imθ + v(m)e imθ = κ m û(-m)e -imθ + û(m)e imθ .It should be noted that m can be of both types simultaneously. This happens precisely when |û(-m)| = |û(m)|, in this case, we may write û(-m) = e -2iθm û(m) so that û(-m)e -imθ + û(m)e imθ = e -iθm û(m) cos(mθ + θ m ). Type R. We will say that m is of type R if |û(m)| = |û(-m)| = 0 and if there exists κ m ∈ C with |κ m | = 1, θ m ∈ R such that û(-m)e -imθ + û(m)e imθ = e -iθm û(m) cos(mθ + θ m ) v(-m)e -imθ + v(m)e imθ = κ m e -iθm û(m) cos(mθ + θ m )

  (i) If |a| = |b| or |c| = |d|, then κ ′ = κ. (ii) If |a| = |b| and |c| = |d|, write b = e -2iϕ a, d = e -2iψ c and α = e -iϕ a, β = e -iψ c. Then either κ ′ = κ or κ ′ = αβ αβ κ. Proof of Lemma 4.3. Let us first observe that, if ℜ(uz) = ℜ(z) with u, z ∈ C, |u| = 1 and z = 0 then either uz = z and u = 1 or uz = z. We thus have to prove that the second case only occurs when |a| = |b| and |c| = |d| and that κ, κ ′ are then related by κ ′ = αβ αβ κ.

First

  , if a = 0 (resp. b = 0) then, as b = 0 (resp a = 0), the two first equations imply d = c = 0, a contradiction. Thus a = 0 and b = 0 and any of the equations then shows that c = 0 and d = 0 since otherwise they would both be 0. Next, (i)/(ii) reads c d = d c thus |c| 2 = |d| 2 and comparing modulus in (i) then shows that |a| = |b|. We then write b = e -2iϕ a, d = e -2iψ c. But then, (4.14) reads κ ′ κe

m

  of type I not R J m (r) û(-m)e -imθ + û(m)e imθ + m of type R û(-m)e -iθm J m (r) cos(mθ + θ m ) + m of type C not R J m (r) û(-m)e -imθ + û(m)e imθ   ;-or no m is of type R and then there exists κ, κ such that (4.17) v(rθ) = κ √ 2π m of type I J m (r) û(-m)e -imθ + û(m)e imθ + κ m of type C J m (r) û(-m)e -imθ + û(m)e imθ .

  Lemma 4.4. Let m = n ∈ N. Let a, b, c, d ∈ C be such that (a, b) = (0, 0) and (c, d)= (0, 0). Let κ, κ ′ ∈ C with |κ| = |κ ′ | = 1. If ℜ κ ′ κ(ae imθ + be -imθ )(ce inθ + de -inθ ) = ℜ (ae imθ + be -imθ )(ce inθ + de -inθ ) for every θ ∈ R then |c| = |d|.We postpone the proof of the lemma to the end of the proof of the theorem. Let us first conclude with the first case (4.15). Let m 0 = inf{m : β m = 0} and let κm 0 = βm 0 βm 0 κ m 0 . We may then write M 0 := {m :β m = 0, κ m = κ m 0 }, M 1 := {m : β m = 0, κ m = κ m 0 }. If M 1 = ∅ then g = κ m 0 f otherwise let m 1 = min M 1 . Then for every m ∈ M 1 , κ m β m = κm 0 β m . Further, if n ∈ M 0 and m ∈ M 1 then κ n = κ m thus κ n β n = β n β m κ m β m = κm 0 β n . But then g(θ) = κ m 0 m∈M 0 β m cos(mθ + θ m ) + κm 0 m∈M 1 β m cos(mθ + θ m ) = κm 0 m≥1β m cos(mθ + θ m ) = κm 0 f .

Acknowledgments

This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the Investments for the future Programme IdEx Bordeaux -CPU (ANR-10-IDEX-03-02). P.J. acknowledges financial support from the French ANR program ANR-12-BS01-0001 (Aventures) from the Austrian-French AMADEUS project 35598VB -ChargeDisq and from the Tunisian-French CMCU/Utique project 15G1504. S.P.E. acknowledges financial support from the Mexican Grant PAPIIT-UNAM IN102915.


Guillaume Jacques

Khal- Ing Tonogenesis

Enrique L In

Jean Palancar

Tonogenesis and tonal alternations in Khaling

Keywords: Tonogenesis, Syllable reduction, Obstruents, Kiranti, Khaling

In this paper, we present a historical account for the tonal alternations observed in the Khaling verbal system. We show that most of these alternations can be explained by two series of sound change. First, level and falling tones appeared in closed syllables with sonorant and obstruents codas respectively. Second, the loss of vowels in specific contexts lead to the development of additional falling tones. A residue of falling tones remain unaccounted for by the present reconstruction, but several hypotheses are proposed to explain their origin.

Introduction

Khaling,1 like other Kiranti languages, has a very complex templatic verbal morphology (see for instance [START_REF] Bickel | Free prefix ordering in Chintang[END_REF][START_REF] Jacques | Agreement morphology: the case of Rgyalrongic and Kiranti[END_REF]. It is unique in the Kiranti subgroup however in having relatively recently innovated a tonal system. 2Tone in Khaling is not only phonemic in that it is used to distinguish between lexical minimal pairs, it also plays a critical role the structure of the verbal inflectional system of the language. Complex patterns of stem alternations involving vowel, consonant and tonal changes are observed in this language. These alternations are not correlated in isomorphic fashion with any morphosyntactic feature. The distribution of stems3 (and of tonal alternations) in the Khaling verbal system is influenced by transitivity, person and number of one or two arguments, tense (past vs non-past) in an intricate way. The complete list of conjugation classes, as well as a computational model to automatically generate them is provided in [START_REF] Jacques | An overview of Khaling verbal morphology[END_REF]. The conjugation classes with the greatest number of stems has up to ten of them, while those with the fewest number of stems only have two (but such classes are very limited).

Table 1 illustrates some typical examples of stem alternations. All the forms in the table comprise a stem followed by one unstressed short vowel suffix (which has no tonal contrast). These data show that while in some verb conjugation classes the tone is constant in all stems (for instance, in the case of the verb 'jump' we have level tone, indicated by a macron, in all forms), in most conjugation classes we observe alternations between falling tone (marked with a retroflex accent) and level tone (with a macron, see the conjugation of 'touch') or between falling tone and short vowel syllables where the tonal contrast is neutralized ('go' and 'hit').

Table 1 also illustrates the fact that tonal alternations in Khaling are not completely independent from vowel and final consonant alternations: in the case of the verb 'touch' for instance, we see that the level tone is correlated with the stem final consonants -j and -ʦ, while the falling tone appears in stems with final -n. In addition, we observe that stem distribution and tonal alternation cannot be accounted for by a single factor. [START_REF] Walther | The Opacity-Compactness Tradeoff: Morphomic Features for an Economical Account of Khaling Verbal Inflection[END_REF] has shown that stem alternations is Khaling are better accounted for by a morphomic account, rather than in terms of transparent morphosyntactic features.

The apparent chaos of verb stem alternations in general, and the opacity of tonal alternations in particular, can however be partially accounted for by a historical account based on a combined application of the comparative method and of internal reconstruction. Most of the tonal alternations in Khaling can be straightforwardly explained as the mechanical result of two major sources of tonal contrasts: the transphonologization of stem final consonant contrasts into tonal contrasts, and the creation of falling tones following the loss of some final syllables.

Given the fact that tonal alternations are not completely independent from segmental changes in Khaling, it is necessary to opt for an approach that does not treat tones in isolation from vowel and consonants, and therefore part of the paper will provide the minimal quantity of information on segmental alternations and their origins that are necessary to make sense of the tonal alternations.

First, we describe Khaling synchronic phonology, including segmental inventories, phonotactic rules and tonal contrasts.

Second, we present a simplified picture of stem alternations in Khaling,4 and propose to reconstruct a stress pattern in pre-Khaling to account for vowel lengthening in some stem types. Although this section does not directly discuss tonal alternations in Khaling, it is a pre-requisite for the complete tonogenesis model elaborated in the following section.

Third, we provide a detailed account of tonogenesis in Khaling. Although a partial discussion of the origin of tones can be found in [START_REF] Michailovsky | Notes on the Kiranti verb (East Nepal). Linguistics of the Tibeto Burman Area[END_REF], the present paper, based on more reliable and complete data, shows that tones in Khaling have more than one origin and describes for the first time the development of falling tones from simplification of disyllables. Some sound laws are demonstrated on the basis of data from nouns rather than verbs, to avoid circularity in the internal reconstruction of the verbal system.

Fourth, we show how the model of tonogenesis proposed in the previous section can account for most tonal alternations in the Khaling verbal system. Pre-Khaling as reconstructed in this model was more similar to other Kiranti languages in having a less elaborate system of stem alternation, and suprasegmentals were restricted to stress shifts.

Fifth, we point out the presence of a residue of forms which cannot be explained as the result of regular sound changes using the laws elaborated in the tonogenesis section. Several potential lines of explanation involving analogical leveling are explored.

Synchronic phonology

As shown in [START_REF] Jacques | An overview of Khaling verbal morphology[END_REF]Jacques et al. ( : 1098)), Khaling has eighteen vowel phonemes (Table 2) and 27 consonant phonemes (Table 3). The only wordinitial clusters allowed are labial or velar stop + l and r.

The consonant ç only appears as the first element of a word-internal cluster (as in seçki 'we kill it'), never word-initially or word-finally. Only unvoiced obstruents and sonorants -p, -t -k, -m, -n, -ŋ, -r, -l, -s and -j can occur as codas. Clusters involving two unvoiced stops or affricates (such as pt, kt etc), as well as geminates (tt, pp, kk and ʦʦ), are realized with preaspiration when the preceding vowel is short. The diphthong oɔ is treated as a unique phoneme. Khaling has a two-way tonal contrast on open syllables with long vowels or closed syllables with a sonorant coda. There is a high level tone (in-dicated by a macron)5 and a falling tone (circumflex accent). A marginal contrast between surface low level and high level tone is also attested in a very restricted morphosyntactic environment (see below), but it is irrelevant for the description of verbal flexion and its phonological interpretation is deferred to future research.

The tonal contrast is neutralized on short and / or unstressed vowels. Note that in sonorant-final syllables, vowels are always long, and vowel length is not indicated as it is redundant. Table 4 illustrates possible tonal contrasts on monosyllables.

In polysyllables, only stressed syllables receive tone, and verbs and nouns alike only have one stressed syllable, except in the case of compound verbs, which are not discussed in this paper. In addition, a contrast between high level tone and a phonetically low tone is attested in the purposive construction with the locative suffix -bi followed by a motion verb. Verbs whose root ends in a sonorant consonant m, n, ŋ, l or r use the infinitive stem with level tone before the suffix -bi as in example 1. Root-final -n changes to -j in this context.

(1) bʌĵ cow ʔu-gʰas 3sg.poss-grass kɛm-bi chew-loc kʰɵs-tɛ go-pst:3sg 'The cow went to chew the grass.' Verbs with roots ending in obstruents use the nasalized infinitive stem (with final -n or -ŋ respectively) with falling tone in this construction.

Noun and verb forms with falling tone remain unchanged as in 2: the free noun 'elk' kɛm has the same form as when occurring with the locative suffix.

(2) kɛm-bi elk-loc kʰɵs-tɛ go-pst:3sg

'He went (to hunt) for the elk.'

Monosyllabic nouns with level tone, however, develop a low tone in the purposive construction when suffixed by -bi, as kɛm 'work'6 in example 3.

(3) kɛm-bi work-loc kʰɵs-tɛ go-pst:3sg 'He went for his work.' This highly restricted, syntactically determined low alternant of the level tone will not be considered further in the present paper, as its synchronic analysis in still unclear (in particular, it remains to be determined whether a similar contrast is found in other grammatical constructions), and it is not necessary to the analysis of the verbal morphology.

In the following, we show the origin of the tonal and length contrasts in Khaling. We point two origins for the falling tone: loss of obstruent codas and syllable reduction.

Vowels and stress in pre-Khaling

In this section, we provide a justification for the reconstructions used in the following sections, in particular for the stress patterns reconstructed for pre-Khaling, which have a direct effect on tonogenesis. Thus, this section mainly discusses vowel alternations from pre-Khaling (an earlier stage of Khaling closer to the common ancestor of Khaling and Dumi) to attested Khaling.

Previous scholarship on comparative proto-Kiranti (van Driem 1990, [START_REF] Michailovsky | Manner vs place of articulation in the Kiranti initial stops[END_REF], Starostin 1994-2000[START_REF] Opgenort | A grammar of Jero, with a historical comparative study of the Kiranti languages[END_REF]) has focused on the reconstruction of proto-Kiranti initial consonants, but no work has been published on the vowel correspondences between Khaling and other Kiranti languages. In Khaling, as we will see, the evolution of vowels, codas and the development of tonal contrasts are closely related, so that an account of the basic vocalic sound laws is necessary for any further work on the topic.

Internal reconstruction suggests that the complex vowel system of Khaling was innovated from a simpler system, as can be ascertained from the vowel alternations in the verbal system and the complementary distributions between vowels and codas [START_REF] Michailovsky | Notes on the Kiranti verb (East Nepal). Linguistics of the Tibeto Burman Area[END_REF], Jacques et al. 2012).

As shown in [START_REF] Michailovsky | Notes on the Kiranti verb (East Nepal). Linguistics of the Tibeto Burman Area[END_REF], [START_REF] Jacques | An overview of Khaling verbal morphology[END_REF] and Michailovsky (2012) using internal reconstruction, in both Khaling and Dumi, no more than five vowels have to be postulated in verbal roots. Complex morphophonological alternations yield all 18 vowels in different contexts. Table 5 illustrates the most important alternations: closed syllable verb stems can be classified into two major categories, weak and strong. Weak stems are found mainly in forms with vowel initial suffixes (the exceptions are shaded in Tables 7, 8 and 9 and discussed in section 5), and strong stems are found with consonant-initial suffixes except in the case of verb stems with final clusters, which always have t as second element (see the conjugation of CVCt roots in Table 9 and the discussion in section 3.4). Some stem forms (3sg:n.pst→3sg and 3sg:n.pst→3sg) have a lengthened vowel (see Table 6), which we interpret as reflecting radical stress (vs suffixal stress) in pre-Khaling. Reconstructing stress for pre-Khaling has other important consequences, in particular for the Syllable Reduction Rule (section 4.2). Tables 7, 8 and 9 illustrate the distribution of the stems in the Khaling verbal paradigms. Irregular cases, where a weak stem is found with a consonant-initial suffix, are shaded in grey. In addition to weak and strong stems, we also find lengthened weak stems in the transitive CVC paradigm.

The vowel alternations in Tables 5 and6 suggest the existence of three (historical) vowel shifts in Khaling: fronting, lowering and backing. We provide an account of the Khaling vowel shifts on the basis of CVC intransitive and transitive paradigms in addition to some comparative data. Then, we discuss the case of verbs with roots in final clusters (which can only be CVCt) where additional minor sound laws have to be proposed. Finally, we tackle some additional issues concerning the Khaling vowel system in historical perspective.

Fronting

The non-front vowels *a, *o and *u in pre-Khaling are fronted to ɛ, ɵ and ʉ respectively in open syllables. This shift occurred in open syllable roots, 1di weak-i but also in weak stems, where (in pre-Khaling) suffixes are vowel-initial and the coda is resyllabified as the onset of the next syllable. Thus for instance the proto-form *loḿ-u (search-3sg:n.pst→3sg) is resyllabified as *lo.mu and undergoes the shifts *o → ɵ and *u → ʉ to *lɵ.mʉ and the stress on the first syllable causes vowel lengthening to the attested form lɵːmʉ 'he searches for it'.

ŋɵk-i weak-iti ŋɵk-iti 1de weak-u ŋɵk-u weak-utu ŋɵk-utu 1pi strong-ki ŋok-ki strong-tiki ŋok-tiki 1pe strong-kʌ ŋok-kʌ strong-tʌkʌ ŋok-tʌkʌ 2s ʔi-strong ʔi-ŋoː ʔi-weak-tɛ ʔi-ŋɵk-tɛ weak-je ŋɵk-je 2d ʔi-weak-i ʔi-ŋɵk-i ʔi-weak-iti ʔi-ŋɵk-iti weak-ije ŋɵk-ije 2p ʔi-strong-ni ʔi-ŋoː-ni ʔi-weak-tɛnu ʔi-ŋɵk-tɛnu weak-nuje ŋɵk-nuje 3s strong ŋoː weak-tɛ ŋɵk-tɛ 3d weak-i ŋɵk-i weak-iti ŋɵk-iti 3p strong-nu ŋoː-nu weak-tɛnu ŋɵk-tɛnu
1s→3 weak-u ʔob-u weak-utʌ ʔob-utʌ 1di→3 weak-i ʔɵp-i weak-iti ʔɵp-iti 1de→3 weak-u ʔɵp-u weak-utu ʔɵp-utu 1pi→3 strong-ki ʔoɔp-ki strong-tiki ʔoɔp-tiki 1pe→3 strong-kʌ ʔoɔp-kʌ strong-tʌkʌ ʔoɔp-tʌkʌ 2s→3 ʔi-weak.length-ʉ ʔi-ʔɵːb-ʉ ʔi-weak.length-tɛ ʔi-ʔɵːp-tɛ weak.length-e ʔɵːb-e 2d→3 ʔi-weak-i ʔi-ʔɵp-i ʔi-weak-iti ʔi-ʔɵp-iti weak-ije ʔɵp-ije 2p→3 ʔi-strong-ni ʔi-ʔoɔm-ni ʔi-weak-tɛnu ʔi-ʔɵp-tɛnu weak-nuje ʔɵp-nuje 3s→3
This shift does not occur in regular velar final verbs with |a| vocalism such as the transitive verb |kak| 'peel', which has a weak stem in ʌ (kʌg-u 'I peel it') or aː (kaːg-ʉ 'He peels it') instead of expected ɛ and ɛː. Thus, the fronting shift was limited to rounded vowels when occurring before velars. 7Only one verb does not fit the pattern in Table 5 : |jal| 'strike', which displays a / ʌ alternation like roots with velar codas (jʌl-u n.pst:1sg→3sg, ja l-ki n.pst:1pi→3sg). This exception is due to the fact that this verb was borrowed from Thulung (a neighbouring Kiranti language) after the vowel shift took place.

The fronting shift as defined here is confirmed by the fact that some borrowings from Nepali do present the sound shifts that have been postulated on the exclusive basis of internal reconstruction.

Lowering

The high vowels *i, *u in pre-Khaling are lowered to ʌ, and *o changes to oɔ in closed syllables with a non-velar coda. Closed syllables occur in forms with consonant-initial suffixes, where resyllabification is not possible.8 For instance, in the pre-Khaling form *lom-ki (search-npst:1pi→3s), the coda *m cannot be resyllabified, and since the verb stem remains a closed syllable, its vowel cannot undergo the fronting shift. This form fulfils the conditions for lowering to occur and *o regularly changes to the diphthong oɔ in this context, yielding the attested form loɔm-ki.

This sound change predicts that there should not be in Khaling any words with the vowels i, u, o or their fronted equivalents ʉ and ɵ in closed syllables. Yet, such words are indeed attested, but come from different origins (reduction of polysyllables and borrowings).

Backing

The only vowel affected by the backing change is i. This vowel changes to ʌ in closed syllables with a velar coda followed by an obstruent. The pre-Khaling rhymes *-iŋ and *-ik change to -uː and -uː respectively when followed by a sonorant. When occurring at the end of a word, *-iŋ undergoes vowel backing to -ʌŋ while *-ik changes to -uː.

For instance, the pre-Khaling forms *siŋ-na 'ask, infinitive' and *siŋ-ki 'ask.n.pst. 1pi→3sg' respectively become suː-nɛ (before a sonorant) and sʌŋ-ki (before an obstruent).

In the southern Khaling dialects, *-iŋ and *-ik in the verbal paradigm verbs appear as -ʌː and -ʌː respectively before sonorants. This is however most probably an analogical development unifying the vocalism of all strong stem forms.

The case of CVCt roots

As all Kiranti languages, Khaling has transitive roots with complex codas, the second element of which is always t. Most of these roots are historically analysable as containing the -t causative/applicative suffix (see Jacques 2013 for a description of its morphosyntactic functions).

These complex codas affect the application of the sound laws defined above: we find strong stems almost everywhere in the paradigm, even before most vowel-initial clusters, where the final -t of the verb root is resyllabified with the suffix. For instance, |kript| 'cut' + -ʉ 3sg→3 gives krʌp.t-ʉ 'He cuts it' with a strong stem (with backing rule).

Weak stems are restricted to inverse forms, as well as first and second dual, and second plural in the past and imperative paradigms. The reason for this cluster simplification is unclear and does not warrant positing a historical sound law of cluster simplification to account for its distribution. The most likely explanation is that this allomorphy is analogical with corresponding simple coda paradigms, especially in the case of inverse forms.

Tonogenesis in Khaling

The contrast between level and falling tones has at least two distinct diachronic origins. First, some contrasts originate from the evolution of codas. Second, other tonal contrasts were created by the reduction of disyllables into monosyllables through loss of the final vowel.

In this section, sound laws are demonstrated whenever possible on the basis of data from nouns rather than verbs, and using comparative data from other Kiranti languages. This way of establishing sound laws limits the risk of circular reasoning, which is particularly serious when doing internal reconstruction in a single-language basis.

Tonal contrasts from codas

The first origin of tonal contrast in Khaling is the evolution (preservation or change) of codas, as discovered by [START_REF] Michailovsky | Notes on the Kiranti verb (East Nepal). Linguistics of the Tibeto Burman Area[END_REF]. In pre-Khaling, the eight codas can be reconstructed on the basis of internal reconstruction: three stops *-p, *-t and *-k, three nasals *-m *-n and *-ŋ and two nonnasal sonorant *-r and *-l. Of these eight codas, four (*-t, *-k, *-n and *-ŋ) undergo changes in some contexts.

Obstruent vs sonorant codas develop respectively falling tone and level tone in monosyllables and stress syllables within polysyllabic words.

Obstruents and falling tones

The final stops *-k and *-t induce falling tone and change respectively to compensatory lengthening and -j in word-final position and before sonorants, as can be assessed by internal reconstruction [START_REF] Jacques | An overview of Khaling verbal morphology[END_REF], confirmed with comparative data from Limbu and Dumi (see Table 11, Dumi and Limbu data respectively from [START_REF] Van Driem | A Grammar of Dumi[END_REF][START_REF] Michailovsky | Limbu-English Dictionary[END_REF]. The coda -k also induces backing and rounding in the case of the vowels *a and *i: pre-Khaling *-ak and *-ik become -oː and -uː respectively. It is interesting to note that this sound change should lead us to expect a / ʌ / -oː in the Khaling system but only a / ʌ / -aː is observed as was shown in the previous section. This suggests that the paradigms of verbs with roots in -ak have undergone analogy, and remade the falling tone forms.9 The coda *-p remains a stop in Khaling, but can be nasalized to -m with falling tone if followed by a nasal-initial suffix as in the infinitive and various verbal forms. Pre-Khaling *-t followed by a nasal is likewise nasalized to -n with falling tone.

Sonorants and level tones

The final sonorants -m, -r, -l are preserved without change and develop level tone in word-final position, as shown in Table 13. The pre-Khaling coda *-n changes to -j word-finally and before almost all consonants; 11 *-ŋ is preserved word-finally, between vowels and before obstruents, but it manifests as compensatory lengthening of the previous vowel before sonorants (inducing the same vowel shift *-aŋ, *-iŋ → -oː, -uː as final *-k). All closed syllables from syllables with the pre-Khaling coda *-n and *-ŋ word-finally have level tone.

If the loss of obstruents were the only origin of falling tones, and the sonorant codas the only origin of level tone, we would expect a nearcomplementary distribution between the two tonal categories: only rhymes |kʰot| 'go'. For instance, the first person plural forms (with the velar-initial suffixes -ki and -kʌ) of these verbs always have |-t| → -ç alternation (kʰoɔç-ki 'we pe go').

11 It is possible that it is preserved before dental stops, but the evidence requires some analysis; this question is discussed in section 5. Yet, there appear to be no gaps in the distribution of tones in Khaling: the contrast is present in all long vowel open syllables and in all syllables with sonorant codas. Thus, at least one other origin should be postulated to account for the types of falling and level tone syllables not predicted by the sound laws proposed in this section.

Falling tones originating from the reduction of disyllables

The reduction of disyllables into monosyllables is another common origin for falling tones. In inherited disyllabic nouns, the vowel and coda of the second syllable are lost in certain conditions (see below), and either the coda of the first syllable or the onset of the second syllable becomes the coda.

Evidence for this syllable reduction mainly comes from comparison with Dumi, where the lost vowel is still present. The very name of the Khaling people, kʰɛ l, illustrates this sound change: the pronunciation *khaliŋ was borrowed into Nepali before syllable reduction took place, a fact which suggests that it occurred very recently. This sound change is best illustrated by the minimal pair pʰɛr 'trap, snare' vs pʰɛr 'near', which corresponds to phari 'trap ' vs phar 'near' in Dumi (data from Rai 2011).

All good examples (see Table 15) involve words with a high vowel in the second syllable. Mid-high and high vowels in the second syllable are immune to this sound change. These examples develop falling tone when the consonant preceding the lost vowel is a sonorant. 12In some cases, the second syllable could be of the type *-Ci (as in the time ordinal nɛm 'in two days') or *-iC (as in kʰɛ l Khaling') but in most cases only an open syllable needs to be reconstructed.

The reduction of disyllables does not always result in a syllable with a final sonorant. When the internal cluster in pre-Khaling only contained obstruents, we obtain closed syllables with obstruent codas without tonal contrasts. Thus, in mos 'bear' from *moksu, the vowel remains o (does not change to oɔ) due to the backing effect of the velar coda; the stop is then lost, leaving only an indirect trace, and only the second element -s of the cluster remains after loss of the second vowel. Words ending in obstruent codas -t, -k, -s in Khaling all originate from such reduced disyllables (or from loanwords), never from pre-Khaling monosyllables. The geminate stop codas have also been generated by this rule: they originate from aspirated stops or geminated stops that became word-final after the Syllable Reduction Rule.

Reconstructing the pre-Khaling form is impossible on Khaling-internal evidence: comparison with Dumi (and possibly other closely related languages) is necessary to recover the lost vowels and consonants. The dual inclusive with a suffix -i in Dumi and Khaling instead of -si or -su (from *si+u) in other Kiranti languages suggests that a cluster reduction rule took place as early as the common ancestor of Khaling and Dumi, as summarized in Table 17:15 word-internal Proto-Kiranti *stop+s clusters became intervocalic unvoiced stops in pre-Khaling (and Khaling), while intervocalic unvoiced stops in proto-Kiranti were voiced in Pre-Khaling (but after the split from Dumi and Khaling).

Other word-internal clusters (such as *sonorant+s) were simplified by dropping the *s without a trace. This rule entails that all word-internal clusters Cs in pre-Khaling (and even more in Khaling) are secondary, and cannot be inherited from proto-Kiranti. It also explains why the dual forms of verbs with -Ct roots lost the -t-in dual forms and why the weak stems are found there: it results from an early reduction of the proto-Kiranti *-Ctscluster into simple *C in pre-Khaling.

For instance the first dual inclusive of the root |kript| 'cut' is kripi in Khaling. The following stages can be proposed: Proto-Kiranti *kript-si → *krippi → pre-Khaling *kripi ́→ Khaling kripi.

In verbal paradigms, the Syllable Reduction Rule only applied to the second syllable of trisyllabic forms, when this second syllable contained a high vowel and was not stressed.

It applies for instance with the 3→3du and 3→3pl of the transitive paradigm, as illustrated by the data in Table 18.16 On the other hand, it does not apply to first singular and dual forms, where the ending was stressed, as shown in Table 19 (except for -t and -n roots, such as the verb |set| 'kill', past dual inclusive ses-ti).

Tonogenesis and verbal morphology

Most of the tonal alternations observed in the Khaling verbal system (see [START_REF] Jacques | An overview of Khaling verbal morphology[END_REF]) can be accounted for by the two sound changes described above, namely the development of falling vs level tone from obstruents vs sonorants and the Syllable Reduction Rule. The falling tones originating from obstruents (in verbs with roots in stop codas) occur in two situations (see Table 20).

First, they are found in forms that had nasal-initial suffixes in pre-Khaling, i.e. the infinitive and second plural non-past of all verbs, the first singular third plural non-past of intransitive verbs and the 1sg→2sg and 3→1sg forms of transitive verbs. In these forms, the final obstruents -p(t) and -t(t) are nasalized to -m and -n respectively with falling tone. Final -k(t) is dropped with compensatory lengthening and falling tone. The first singular suffix -ŋʌ works differently from other nasal suffixes: it nasalizes -p(t) and -k(t) to -m and -ŋ with falling tone (for -k(t) it can also optionally change to compensatory lengthening) , while -t(t) changes to -j with falling tone before it.

Second, falling tone forms occur in all stems that appear word-finally (first and second person singular or intransitive verbs and 3→2sg of transitive verbs) for verbs with roots in -t(t) and -k(t). The coda of verbs in -p(t) does not disappear word-finally. Most of the tonal alternations not explained by the loss of obstruents can be accounted for by the Syllable Reduction Rule. In the verbal system, this does not apply to disyllabic forms, but does occur in tri-syllabic forms (section 4.2): the second syllable of a trisyllabic verb form in pre-Khaling drops if it had a high vowel *i or *u.

Table 22 illustrates the application of the Syllable Reduction rule in 3→3pl forms (in comparison with 3sg→3sg forms, where it does not occur). The third person plural of intransitive paradigm is -nu, and appears to be superficially similar to that of the transitive paradigm. Yet, no falling tone occurs in this form for verbs with roots in sonorant coda. For instance |bʰer| 'fly', the intransitive counterpart of |bʰert| 'cause to fly' has a third person plural non-past bʰer-nu. This difference is due to the fact that the endings of the transitive and intransitive paradigms were different in pre-Khaling: the former had a compound suffix *-u-nV17 while the latter had simple *-nV. In the third plural of the intransitive paradigm, there was no high vowel between the verb stem and the plural suffix, and hence the Syllable Reduction Rule could not apply to generate a falling tone.

The Syllable Reduction Rule also applies in past tense forms (third and second person direct), as illustrated by Table 23. In the paradigm of verbs with roots in -t coda, the final obstruent drops before the past tense suffix -tɛ, resulting in an open syllable long vowel with falling tone, avoiding a long vowel before geminated tt. For instance, the following intermediary steps can be posited for the third person past form of the verb 'kill': seː-tɛ ← *seːttɛ ← *seːdʉtɛ ← *seduta).

All tonal alternations in the direct transitive paradigm and the inverse non-past paradigm can be accounted for using the rules explained above: SF (stop coda to falling tone), N (nasalization of stop coda), SR (Syllable Reduction Rule), L (lengthening of accented vowel in open syllable). The forms where each of these rules is applied in the transitive paradigm of verb with stop final roots and sonorant final roots are presented in Tables 24 and25 respectively.

Thus, nearly all tonal alternations in the Khaling verbal system are the result of a mechanical application of sound changes whose existence can be independently proven by applying the comparative method to Kiranti languages.

Unexplainable alternations

There remain a residue of forms with falling tones that cannot be accounted for by either scenario, in the past tense of the intransitive paradigm, the As presented in Table 26, singular and plural second and third person past and imperative forms have a falling tone in paradigms of verbs with roots in sonorant codas.

These forms appear to be superficially identical the corresponding ones of the transitive paradigm. For instance, the third singular of the past tense ʦʰɵm-tɛ 'he danced' has the same alternation as that of the transitive verb |lom| 'search', lɵm-tɛ 'he looked for him'. However, this resemblance is misleading. If we observe the intransitive paradigm of verb whose root has a stop coda, as in Table 27, we see that the verb stem has a short vowel in the corresponding forms.

The difference between the transitive and intransitive paradigms of verbs with roots in stop coda can be illustrated by the third person past tense of |pʰrok| 'untie' and that of |ʣʰuk| 'flee'. The former pʰrɵːk-tɛ 'he untied it' has a long vowel and non-contrastive falling tone, while the latter has a short vowel ʣʰʉk-tɛ he fled'. Hence, it is not possible to reconstruct, for the intransitive past tense, a trisyllabic form with word-initial stress (like pʰrɵːk-tɛ ← *pʰroǵuta). It could be possible to account for the intransitive paradigms in Table 26 and 27 by positing a stress-final pattern in the past tense of intransitive verbs in pre-Khaling, such as ʦʰɵm-tɛ ← *ʦʰom-itáand ʣʰʉk-tɛ ← *ʣʰuk-ita. The stress not being placed on the verb stem, vowel lengthening does not occur, and not being placed on the second syllable, the Syllable Reduction Rule can apply and generate the falling tone in verbs with roots in sonorant codas.

However, there is no compelling comparative evidence from any Kiranti ʔi-loɔm-ki ʔi-loɔm-tiki loɔm-kʌje language for a high vowel in any of these forms between the verb stem and the suffix. One possible reason for this is that some languages underwent vowel reduction in trisyllables like Khaling. In Dumi, as can be seen in Table 28, all vowels located between the verb stem and the Non-Past18 -t suffix have been lost, even in forms where Khaling has preserved them. 19Until clear evidence is found in another Kiranti language for reconstructing *-ita rather than -*ta in the second and third singular and plural past forms, the hypothesis laid out in this section must be regarded as tentative.20

Conclusion

In this paper, we have shown that most tonal alternations in Khaling can be accounted for by two main groups of sound changes: the development of tonal contrast following the loss of codas and the Syllable Reduction Rule. Some tonal alternations in the intransitive paradigm cannot be explained yet, and future investigation on this topic will have to take into account

Figure 1 :

 1 Figure 1: Map of Kiranti languages (from Opgenort 2011)

  i-strong-ŋʌ ʔi-ʔoɔm-ŋʌ i-weak-ʌtʌ ʔi-ʔɵp-ʌtʌ weak-ʌje ʔɵp-ʌje 3s→1pi i-strong-ki ʔi-ʔoɔp-ki i-strong-tiki ʔi-ʔoɔp-tiki strong-kʌje ʔoɔp-kʌje

  -ni ʔi-soɔm-ni ʔi-weak-tɛnu ʔi-sɵp-tɛnu i-strong-ŋʌ ʔi-soɔm-ŋʌ i-weak-ʌtʌ ʔi-sɵp-ʌtʌ weak-ʌje sɵp-ʌje 3s→1pi i-strong-ki ʔi-soɔp-ki i-strong-tiki ʔi-soɔp-tiki strong-kʌje soɔp-kʌje

  ʔi-seĵ-ŋʌ SF ʔi-ses-tʌ seʦ-ʌje 3s→1pi ʔi-seç-ki ʔi-seç-tiki seç-kʌje inverse transitive paradigm and the direct past tense 2pl→3. We illustrate the problem with data from the intransitive paradigm exclusively.

  Distribution of the stems in the intransitive paradigm of the verb |ʣʰuk| 'flee', stop final reflexive paradigms and doubly conjugated verbs. Yet, additional data from the languages that are the most closely related to Khaling, in particular Dumi and Koyi, will be necessary to undertake this research.NotesI would like to thank Aimée Lahaussois, Jean-Léo Léonard, Boyd Michailovsky and Enrique Palancar for useful comments on previous version of this work, and Dhana Khaling Rai and his family for teaching me Khaling. I am responsible for any remaining error. Abbreviations follow the Leipzig glossing rules, except for the following: C consonant V vowel, di / pi dual / plural inclusive de / pe dual / plural exclusive. This research was funded by the HimalCo project (ANR-12-CORP-0006) and is related to the research strand LR-4.11 ''Automatic Paradigm Generation and Language Description'' of the Labex EFL (funded by the ANR/CGI).Abbreviations follow the Leipzig glossing rules, except for the following: C Table 28: Comparison between Khaling and Dumi (intransitive |ʦʰom| 'dance' vs transitive |lom| 'look for'consonant V vowel, di / pi dual / plural inclusive de / pe dual / plural exclusive.

Table 1 :

 1 Examples of tonal alternations in the Khaling verbal system Infinitive Non-past 1di Non-Past 3pl Past 3sg Meaning

	bʰoɔj-nɛ ʦoɔj-nɛ kʰoɔn-nɛ roɔn-nɛ	bʰɵːʦ-i ʦɵːʦ-i kʰɵʦ-i rɵʦ-i	bʰoɔn-nu ʦoɔj-nu kʰoɔn-nu rɵːt-nu	bʰoɔn-tɛ ʦɵːs-tɛ kʰɵs-tɛ rɵː-tɛ	touch (tr) jump (intr) go (intr) hit (intr)

Table 2 :

 2 List of Khaling vowel phonemes

	i iː e eː ɵ ɵː ʉ ʉː ɛ ɛː	a aː	u uː o oː ʌ oɔ

Table 3 :

 3 List of Khaling consonantal phonemes

	p pʰ tʰ t b d bʰ dʰ m n s w l	ʦ ʦʰ kʰ k ʣ g ʣʰ gʰ ŋ ç r j	ʔ ɦ

Table 4 :

 4 Minimal pairs between level tone and falling tone in monosyllables

	Form Meaning
	mɛm mɛm	he mother
	mɛː mɛː mɛ	there (ideophone) rolling quickly that

Table 5 :

 5 Basic vowel alternations in the Khaling verb

	root vowel open	velar	velar	non-velar non-velar
		syllable (strong) (weak) (strong)	(weak)
	a e i o u	ɛ e e ɵ ʉ	a e ʌ o u	ʌ e i ɵ ʉ	ɛ e ʌ oɔ ʌ	ɛ e i ɵ ʉ

Table 6 :

 6 Examples of basic vowel alternations in the Khaling verb

	Root	Meaning weak stem	weak stem (lengthened) strong stem
			(1di:n.pst→3sg) (3sg:n.pst→3sg)	1pi:n.pst→3sg
	|pʰrok| untie |lom| search |lop| catch	pʰrɵk-i lɵm-i lɵp-i	pʰrɵːg-ʉ lɵːm-ʉ lɵːb-ʉ	pʰrok-ki loɔm-ki loɔp-ki

Table 7 :

 7 Distribution of stems in the intransitive paradigm

	S	non-past		past	imperative
	1s	strong-ŋʌ	ŋoŋ-ŋʌ	weak-ʌtʌ	ŋɵk-ʌtʌ

Table 8 :

 8 Distribution of stems in the CVC transitive paradigm

	A→P	non-past	past	imperative

Table 9 :

 9 Distribution of stems in the CVCt transitive paradigm

	A→P	non-past	past	imperative

Table 10 :

 10 Borrowings from Nepali exhibiting the fronting vowel shift

	Nepali	Meaning Khaling
	kam kʰorsani chilli work buɖʰi wife dudʰ milk	kɛm kʰɵrsɛĵ bʉri dʉːt

Table 11 :

 11 Examples of falling tones originating from final stops Pre-Khaling Limbu Dumi Khaling Meaning

	*bit *met *rak *pak *ʔik	pit met yak phak ik	bhiʔi meːʔe meĵ bʌĵ roː poʔo poː ʔuː	cow wife cliff pig field
	Before stops, *-k remains -k, while *-t changes to ç before labials and velars. 10

Table 12 :

 12 Examples of falling tones originating from the nasalization of obstruents (infinitive verb forms)

	Pre-Khaling Limbu Dumi Khaling Meaning
	*lop-na *rep-na *set-na	yɛps-sɛt-	lop-nɨ loɔm-nɛ repnɨ rem-nɛ setnɨ sen-nɛ	to catch to stand to kill

Table 13 :

 13 Examples of level tones in syllables with sonorant codas Pre-Khaling Limbu Dumi Khaling Meaning

	*lam *rɒm *lem	lam yam	lam ram leːm	lɛm roɔm lem	road body tongue
	*ser *kʰur *kɒr		seːr kʰɨr kar	ser kʰʌr koɔr	louse hand wound
	*del *tulna	tɛn	deːl tɨlnɨ	del tʌlnɛ	village raise (cattle in an enclosed space)

Table 14 :

 14 Examples of level tones in syllables with *-n and *-ŋ

	Pre-Khaling

sonorant should be level tone, and all word-final open syllable long vowels should be falling tone, and only include -oː, -eː and -uː.

Table 15 :

 15 Examples of falling tone or obstruent final monosyllables originating from the simplification of disyllabic nouns Several verbal suffixes had high vowels in pre-Khaling: the first singular direct -u 13 , the inclusive and exclusive dual -i and -u and the third/second transitive -ʉ.14

	Pre-Khaling	Dumi	Khaling Meaning
	*tsili *nini *meri *rali *sali	tsili nini miri raːli sali 'nail, talon' sɛl ʦi ̂l ni ̂n mer rɛl	anger aunt (FZ) tail centipede foot
	*noru *nolu	nurɨ nulɨ	nɵr nɵl	tiger daytime
	*kʰaliŋ *nam-ni	naːmnɨ	kʰɛ l nɛm	Khaling in two days
	*pipi *moksu *pokʰu	pipi moksɨ pɨkʰɨ	pip mos pɵk	grandmother bear dirt
	Exceptions to the Syllable Reduction Rules in nouns are extremely rare,
	and may reflect borrowings, or special contexts where the rule does not
	apply; this question is left for future investigations.
	Table 16: Exceptions to the Syllable Reduction Rule
	Pre-Khaling Limbu Dumi Khaling	Meaning
	*sendi ???	sendi	syendi sendʉ, sendi nail mupu mupu belly
	The application of the Syllable Reduction Rule appears to operate dif-

Table 17 :

 17 Exceptions to the Syllable Reduction Rule

	Proto-Kiranti Limbu Dumi (past) Pre-Khaling Khaling Meaning
	*lop-si *set-si *kok-si *lom-si	lupʰi a-sɛt-su sitsi kukʰi lumi	*lopi ́lɵpi *seʦi ́seʦi *koki ́kɵki *lomi ́lɵmi	we di catch for him we di kill him we di are able to do it we di look for him
	*lop-u *set-u *kok-u *lom-u	sɛr-u	lupʰɨ sidɨ kukhɨ lumɨ	*lobu *sedu *koǵu *loḿu	lɵːbʉ seːdʉ kɵːgʉ lɵːmʉ	he catches him he kills him he is able to do it he looks for him

Table 18 :

 18 The application of the Syllable Reduction Rule in the verbal

	system
	Pre-Khaling Khaling Meaning
	*lobu-nV *sedu-nV *loḿu-nV	lɵːp-nu seːt-nu lɵm-nu	they catch him they kill him they look for him

Table 19 :

 19 The Syllable Reduction Rule in forms with stress-bearing endings

	in pre-Khaling
	Pre-Khaling Khaling Meaning
	*lop-i ́ti *lom-i ́ti	lɵp-iti lɵm-iti	we di caught him we di looked for him
	*seʦ-i ́ti	ses-ti	we di killed him

Table 20 :

 20 Falling tones from lost stops in the Khaling verbal system Verbs with sonorant final roots present level tone in the corresponding forms, as illustrated by Table21.

	Verb root Meaning Infinitive Non-Past 1sg	3sg 1pi
	|ʔɛt| |kʰot| |pʰuk| |rep|	say go get up stand	ʔɛn-nɛ kʰoɔn-nɛ pʰuː-nɛ rem-nɛ	ʔɛĵ-ŋʌ kʰoɔĵ-ŋʌ pʰuŋ-ŋʌ / pʰuː-ŋʌ pʰuː ʔɛĵ kʰoɔĵ kʰoɔç-ki ʔɛç-ki pʰuk-ki rem-ŋʌ reːp rep-ki

Table 21 :

 21 Level tones in sonorant stems in the Khaling verbal system

	Verb root Meaning	Infinitive Non-Past 1sg 3sg	1pi
	|ʦʰom| |ʦon| |kʰoŋ|	dance jump come (from below) kʰoː-nɛ ʦʰoɔm-nɛ ʦʰoɔm-ŋʌ ʦoɔj-nɛ ʦoɔj-ŋʌ kʰoː-ŋʌ	ʦʰoɔm ʦʰoɔm-ki ʦoɔj ʦoɔj-ki kʰoŋ kʰoŋ-ki

Table 22 :

 22 Falling tones from lost syllables in the Khaling verbal system

	Verb root Meaning	Non-Past 3sg	Non-Past 3pl
	|set| |pʰrok| |lom|	kill untie look for	seːd-ʉ ← *sedu pʰrɵːg-ʉ ← *pʰroǵu pʰrɵːk-nu ← *pʰroǵu-nV seːt-nu ← *sedu-nV lɵːm-ʉ ← *loḿu lɵm-nu ← *loḿu-nV
	|bʰert| |sent|	cause to fly bʰerd-ʉ ← *bheŕdu bher-nu ← *bheŕdu-nV see send-ʉ ← *seńdu sen-nu ← *seńdu-nV

Table 23 :

 23 Falling tones from lost syllables in the Khaling verbal system,

	Past forms
	Verb root Meaning	Past 3sg
	|set| |pʰrok| |lom|	kill untie look for	seː-tɛ ← *seduta pʰrɵːk-tɛ ← *pʰroǵuta lɵm-tɛ ← *loḿuta
	|bʰert| |sent|	cause to fly bʰer-tɛ ← *bheŕduta see sen-tɛ ← *seńduta

Table 24 :

 24 Tonal alternations in the transitive paradigm of the verb |set| 'kill'

	A→P	Non-past Origin Past	Origin Imperative Origin

Table 25 :

 25 Tonal alternations in the transitive paradigm of the verb |lom|

	'search'
	A→P	Non-past Origin Past	Origin Imperative Origin
	1s→3 1di→3 1pi→3 2s→3 2d→3 2p→3 3s→3 3d→3 3p→3	lom-u lɵm-i loɔm-ki ʔi-lɵːm-ʉ ʔi-lɵm-i ʔi-loɔm-ni lɵːm-ʉ lɵm-su lɵm-nu	lomu-tʌ lɵm-iti loɔm-tiki ʔi-lɵm-tɛ ʔi-lɵm-ti ʔi-lɵm-tɛnu lɵm-tɛ L+SR lɵm-tɛsu L L L+SR lɵm-tɛnu	L+SR lɵːm-e lɵm-ije lɵm-nuje L+SR L+SR L+SR	L
	1s→2 2/3s→1s ʔi-loɔm-ŋʌ SF loɔm-nɛ	loɔm-tɛni ʔi-lɵm-ʌtʌ	lɵm-ʌje
	3s→1pi

Table 26 :

 26 Distribution of the stems in the intransitive paradigm of the verb |ʦʰom| 'dance', sonorant final

	S	Non-past	Origin Past	Origin Imperative Origin

Khaling is spoken by around 15000 speakers in Solukhumbu district, Nepal, see Figure1.

Tonogenesis and most vowel changes postdate the start of Nepali influence in the eighteenth century, as will be shown in section 4.1.1.

In the present paper, we use the term 'root' to designate abstract forms from which the entire paradigms can be mechanically generated, and the term 'stem' to refer to the part of the inflected verbs remaining once prefixes and suffixes have been removed, in other words, roots to which morphophonemic changes have been applied.

For reasons of space, since the present paper is focused on tone, this section is limited to the strict minimum necessary to follow the tonogenesis model. An exhaustive account of alternations can be found in[START_REF] Jacques | An overview of Khaling verbal morphology[END_REF].

We opt for the macron rather than the acute accent for transcribing the level tone, as the acute accent is used in this paper for indicating stress in (non-tonal) pre-Khaling reconstruction, and we want to avoid confusion.

Borrowed from Nepali kam.

The intransitive verb |bʰak| 'go (honorific)'[START_REF] Jacques | An overview of Khaling verbal morphology[END_REF](Jacques et al. : 1115) ) presents the stem /bʰɛ/ in dual forms (bʰɛ-ji go-n.pst:1pi) with irregular loss of the final k showing the a → ɛ shift (*bʰʌk-i would be expected for the n.pst:1pi). While the reason for the coda loss in this form is unclear, it this verbs confirms the conditioning of the *a → ɛ proposed here.

See section 3.4 for the case of CVCt verb roots.

Probably on the basis of forms preserving the -k, such as first plural.

The effect of this sound law is still visible in the paradigm of |-t| coda verbs such as

A third origin for falling tones results from vowel fusion after the loss of intervocalic *w. It is found in words like gʰrɛː 'crab' ← *ŋgrawa and ʦɵː ← *tsowo, compare Dumi[START_REF] Rai | Dumi-Nepali-English dictionary[END_REF]) grawa 'crab' and tsuwo 'heart'.

This ending corresponds to -uŋ in other Kiranti languages; pre-Khaling *-uŋ normally remains -uŋ word finally in Japhug (and -uː before sonorants). This irregularity is left for further research.

This suffix regularly reflects pre-Khaling *-u.

This rule is unrelated to the Syllable Reduction Rule seen above, whose application is entirely different as in mos ← *moksú'bear'. The Syllable Reduction Rule only applied to secondary clusters in compound nouns, not to the more ancient verbal morphology. Khaling still has -Cs-clusters, but these result from the Syllable Reduction Rule, as we explain below.

The reconstruction of the dual and plural -su and -nu in pre-Khaling is uncertain; the expected vowel in these suffixes should have been either i or ʉ.

The reconstruction of the vowel in pre-Khaling for the plural suffix -nu is still unclear.

In Dumi, the Past and Non-Past paradigms are reversed in comparison with other Kiranti languages. Thus, the Khaling Past paradigm corresponds to the Dumi Non-Past.

The syllable Reduction Rule that took place in Dumi is of course very different, and independent from, that found in Khaling, since it does not apply to disyllabic nouns.

The Past tense suffix probably originates from an independent verb (perhaps the verb 'put', in Khaling |ta|). Evidence for this comes from Bantawa(Doornenbal 2009: 165- 72), where the cognate past tense suffix -dacan still occur as an independent word and receive the second person prefix tɨ-. Thus, this question cannot be satisfactorily answered without an account of the history of doubly conjugated verbs in Khaling and other Kiranti languages. This topic is not discussed in this paper, due to lack of data on most relevant languages.