UNCTAINTY PINCIPLS FO ADA AMBIGUITY FUNCTIONS AND MOMNTS PHILIPP JAMING Abstract. In this article, we give an estimate of the zero-free region around the origin of the ambiguity function of a one-dimensional signal u in terms of the moments of u. This is done by proving an uncertainty relation between the first zero of the Fourier transform of a non-negative function and the moments of the function. As a corollary, we also give an estimate of how much a function needs to be translated to obtain a function that is orthogonal to the original function. The paper is completed with a proof of the local uncertainty principe for the ambiguity function.. Introduction Woodward s time-frequency correlation function or radar ambiguity function [Wo, Wi, AT], as defined by + Au)x, y) = u t + x ) u t x ) e iπyt dt plays a central role in evaluating the ability of a transmitted radar waveform e ut)e iω0t) to distinguish targets that are separated by range delay x and Doppler frequency y. Ideally, one would like to have Au)x, y) = δ 0,0) x, y), the Dirac mass at 0, 0), but this desideratum is not achievable because of the so-called ambiguity uncertainty principle, that is the constraint Au)x, y) dx dy = Au)0, 0) = ut) dt). As Au) is continuous when u L ) is a signal of finite energy, it follows that Au) can not vanish in a neighborhood of 0, 0) unless u = 0. Further limitations are given by various versions of the uncertainty principle for the ambiguity function, see e.g. [BDJ, De, Gr, GZ] and the references therein. These state in particular that if one concentrates Au) in the x-variable, then one looses concentration in the y-variable. Since ideal behavior is not achievable, it becomes important to determine how closely to the ideal situation one can come. A first attempt in that direction is due to Price and Hofstetter [PH] who considered the quantity V ) = Au)x, y) dx dy where is a measurable subset of. Under the asumption that V ) V 0 when shrinks to {0, 0)}, they proved that, if A is a convex set symmetric with respect to the origin, then V A) V 0 4 areaa). It should however be mentioned that, for this result to be significant, one has to go outside 99 Mathematics Subject Classification. 4B0. Key words and phrases. zero-free region, ambiguity functions, short time Fourier transform, windowed Fourier transform, local uncertainty principle. esearch partially financed by: uropean Commission Harmonic Analysis and elated Problems 00-006 IHP Network Contract Number: HPN-CT-00-0073 - HAP).
PHILIPP JAMING the class of signals of finite energy. Indeed, when u has finite energy, from the continuity of Au), one gets that V ) area) sup x,y) Au)x, y) {0,0)} area{0, 0)}) Au)0, 0) = 0. So for u L ), V 0 = 0 and the bound in [PH] is trivial. Nevertheless, it is possible to define Au) when u is a Schwartz distribution and the results in [PH, Section II] have then to be read under the implicit assumption that Au) L loc \ {0, 0)}) in which case V 0 may not be 0 and their result is then meaningfull. We will here restrict our attention to signals u of finite energy and have some L -moments. In this context, we strengthen the above remark by giving an upper estimate of V ). This is done by a direct computation when is a large enough ball and by generalizing Faris local uncertainty principle when is the complementary of a small enough ball. In particular we prove the following : Theorem A. Let u L ) such tht tu, ξû L ) and let δ > 0. Then if ρ max tu, tû ) 65 u δ then Au)x, y) dx dy δ ) u 4 ; x +y ρ if tu + ) tû δ u then Au)x, y) dx dy δ ) u 4. x +y ρ The second aim of this paper is to determine more precisely the neighborhood of 0, 0) on which Au) does not vanish. To do so, we prove a new form of uncertainty principle, showing that there is an exclusion relation between the function u having moments and its Fourier transform being 0 near 0. More precisely, we prove the following: Theorem B. For every q > 0 there exists κ q > 0 such that, if u L ) is a non-negative function, u 0, then inf{ξ > 0 : û±ξ) = 0} q inf t t 0 q ut) κ q u. t 0 This result is inspired by a recent result of Luo and Zhang [LZ]. They prove that if a real nonnegative function is supported in [0, + ) then there is an uncertainty principle of the above form relating its moments and the first zero its Fourier transform. It turns out that the ambiguity function, when restricted in a given direction, is always the Fourier transform of a non-negative function, but the support condition of Luo and Zhang is not valid. We thus start by removing that condition in their uncertainty principle, which can be done at little expenses apart from some numerical constants. This then allows to obtain a zero-free region for the ambiguity function Au) when u and its Fourier transforms have L -moments. This region turns out to be a rectangle when one considers dispersions: Theorem C. Let u L ) be such that tut) L ) and ξûξ) L ). Then the ambiguity function Au) of u has no zero in the convex hull of the four points ) ) u u ± π inf a t a ut), 0 and 0, ± π inf ω ξ ω ût). The article is divided in two sections. The first one is devoted to the extension of Luo and Zhang s uncertainty principle. In the second section, after some preliminaries on the fractional Fourier transform and on the ambiguity function, we give a proof of Theorem C. The section is completed with ecall that this is part of the narrow-band asumption of the model in which the radar measures Au), see e.g. [AT].
UNCTAINTY PINCIPLS FO ADA AMBIGUITY FUNCTIONS AND MOMNTS 3 the adaptation of the local uncertainty principle to ambiguity function, in particular with the proo of Theorem A. Notation : In this paper d is an integer. measurable functions such that For p < we define L p d ) as the space of f p p := d ft) p dt < +. For u L d ) L d ) we define the Fourier transform as Fuξ) = ûξ) = ut)e iπξt dt, d and then extend it to L d ) in the usual way. ξ. Zero-free regions of the Fourier transform In this section, we restrict our attention to dimension d =. Theorem.. For every q > 0 there exists κ q > 0 such that, if u L ) is a non-negative function, u 0, then inf{ξ > 0 : û±ξ) = 0} q inf t t 0 q ut) κ q u. t 0 emark : One may take for the constant κ = cπ) where c is the smallest constant in quation.) below. q A similar result has been proved in [LZ] but with the extra assumption that u be supported in [0, + ). The constant κ is then explicitely known and better than the one above. ) /q It is enough to prove that if u, t q u L u and u 0, then for ξ < κ q t q u, ûξ) 0. Once this is done, we can apply the result to the translate ut t 0 ) of u. Assume the theorem has been proved for a given q and let q > q. Let u 0 be in L and such that t q u L. From Hölder s inequality, we get that t q u t q u q q u q q and t q u L. It further follows that ) /q ) /q u κ q t q u u κ q/q q t q u and we obtain the theorem for q if we set κ q = κ q /q q. In particular κ q κ q for q. Taking u = on [ a, a] and u = 0 elsewhere gives the bound κ q q + ) q. We will give two proofs of this result. The first one gives an explicit constant and works easily in all dimensions, but is restricted to q. First proof of Theorem.. From elementary calculus, we get that ûξ) û0) ξ sup η [0,ξ] ûη) so that.) ûξ) û0) ξ sup ûη). η [0,ξ]
4 PHILIPP JAMING But ûη) = that sup ûη) π η [0,ξ] ut)e iπtη dt = iπ tut)e iπtη dt as long as t ut) dt <. It follows t ut) dt. On the other hand û0) = ut) dt so that.) gives ûξ) ut) dt π ξ t ut) dt > 0 as long as ξ < u π t u. From the above remarks, we get the desired result with κ q = π). q Second proof of Theorem.. The proof is very similar to that of Luo and Zhang. The idea is that e e ix = cos x x / for all x. Thus e û πtξ) / ) ut) dt. It follows that if ûξ) = 0, then e ûξ) = 0 thus ξ t ut) π ut). The constant π can be improved when u is supported in [0, + ) using a refined version of the inequality cos x x / see [LZ, Proposition.]) and can further be extended by replacing t u by other moments t q u. Our substitute to [LZ, Proposition.] is the following : Fact. For every q > 0, there exists a, c such that, for all x,.) a cos x c x q. It then follows that, if ξ is such that ûξ) = 0, then 0 = ae ûξ) = ut)a cos πtξ dt ut) cπ ξ ) q t q) dt = u cπ ξ ) q t q u. Now, as û is continuous, we may take ξ = ±τ, and get that as claimed. τ q t q u cπ) q u Proof of.). The fact is trivial as long as one does not look for best constants. The simplest way is to take a = and c such that c π q 3 = i.e. c = 3 ) 3 q. π π In this case, for 0 x 3, c x q cos x and for x > π 3, c x q cos x. emark : One can slightly refine this argument, taking a = + η, x η = arccos c x η q = + η) that is c = + η and c such that + η arccos + η) ) q. Then for x < x η, c x q + η) cos x and for x x η, c x q + η) + η) cos x. One may then minimize over η > 0 and some values are given in the following table : q 3 4 5 6 a 3.6 3.94 4.6 5.7 c.34.656.4 0.908
UNCTAINTY PINCIPLS FO ADA AMBIGUITY FUNCTIONS AND MOMNTS 5 The constant c may be slightly improved by a further refinement. It is of course enough to establish the inequality for 0 x π. Now that we know that for each η > 0, there exists a constant c such that c x q + η) cos x for all x, it is enough to take the biggest such constant c η. The curves y = c η x q and y = + η) cos x then have a contact point at which they are tangent. Thus there exists x η such that + η) cos x η = c η x q η and + η) sin x η = c η qx q η. This implies that cos x η + q x η sin x η = + η. It is then enough to show that this equation has a unique solution x η in [0, π] which then gives c η = + η) sin x η qx q. η Indeed, set ϕx) = cos x + x sin x q then ϕ0) = and ϕπ) = so that ϕx) = + η +η has at least one solution in [0, π]. Moreover ϕ x) = sin x + q x cosx) + sinx) ). lementary calculus and the study of the comparative behaviour of sin x and q x cosx) + sinx) ) shows that, if q then ϕ is strictly decreasing over [0, π] and if q <, there exists x such that ϕ is increasing over [0, x ] and strictly decreasing over [x, π]. In both cases, this implies that ϕx) = / + η) has a unique solution x η in [0, π]. To obtain the optimal constant is then difficult as one has to minimize c η over η but this is not explicitely known. It should be noted that η 0 may not be optimal. For instance, a computer plot will convince the reader that.0 cos x 0.5x 3/,. cos x 0.4x. Notation For q > 0, we write q u) = inf t t0 q f. t 0 d To conclude this section, let us give a first application of this result. We ask whether a translate f a t) = ft a) of f can be orthogonal to f. But 0 = ft)ft a) dt = fξ) e iaξ dξ = F[ f ] a). Similarily, if the modulation f ω) t) = e iπωt ft) of f is orthogonal to f then F[ f ]ω) = 0 From Theorem., we get the following: Corollary.. Let q > 0 and κ q be the constant of Theorem.. Let f L ). Assume that + ξ ) q/ f L. Then for f and its translate f a to be orthogonal, it is necessary that a q q/ f) κ q f. Assume that + t ) q/ f L. Then for f and its modulation f ω) to be orthogonal, it is necessary that ω q q/ f) κ q f. 3. Ambiguity functions and moments 3.. Preliminaries on the fractional Fourier transform. Here again, we restrict our attention to dimension d =. For α \ πz, let c α = exp ) i α π sin α be a square root of i cot α. For f L ) and α / πz, define F α fξ) = c α e iπξ cot α ft)e iπt cot α e iπtξ/ sin α dt = c α e iπξ cot α F[ft)e iπt cot α ]ξ/ sin α)
6 PHILIPP JAMING while for k Z, F kπ f = f and F k+)π fξ) = f ξ). This transformation has the following properties :. F α fξ)f α gξ)dξ = ft)gt)dt which allows to extend F α from L ) L ) to L ) as a unitary operator on L );. F α F β = F α+β ; 3. if f a,ω t) = e iπωt ft a) then and 4. if f L is such that tf L then F α f a,0 ξ) = F α fξ + a cos α)e iπa cos α sin α iπaξ sin α ; F α f 0,ω ξ) = F α fξ + ω sin α)e iπω cos α sin α+iπωξ cos α ; F α [tf]ξ) = ξf α fξ) cos α + i[f α f] ξ) sin α. 3.. Preliminaries on the ambiguity function. Let d be an integer. Let us recall that the ambiguity function of u, v L d ) is defined by Au, v)x, y) = t + x ) v t x ) e iπ t,y dt. d u Closely related transforms are the short-time Fourier transform also known as the windowed Fourier transform, defined by S v ux, y) = e iπ x,y Au, v)x, y) and the Wigner transform W u, v)η, ξ) = ξ + t ) v ξ t ) e iπ t,η dt d u which is the inverse Fourier transform of Au, v) in d. Further W u, v)η, ξ) = d Au, ˇv)ξ, η) where ˇvt) = v t). We will now list the properties that we need. they are all well-known e.g. [Al, AT, Wi]:. Au, v), W u, v) L ) with Au, v) L ) = W u, v) L ) = u v and are continuous;. if we denote Au) = Au, u) then Au)0, 0) = u where it is maximal; 3. for a, ω d let u a,ω t) = e iπ ω,t ut a), then ) Au a,ω, v b,η )x, y) = e iπ ω+η)x+a+b)y ω+η) Au, v)x a + b, y ω + η); 4. in dimension d =, AF α u, F α v)x, y) = Au, v)x cos α + y sin α, x sin α y cos α); 5. W û, ˆv)η, ξ) = W u, v)ξ, η). Property 4 was proved in [Wi] when the fractional Fourier transform is defined in terms of Hermite polynomials and in [Al] with the above definition of the fractional Fourier transform. 3.3. Zero free region around the origin of Au) in dimension d =. In this section, we restrict our attention to the one-dimensional situation. Noticing that Au) is a Fourier transform and in particular that Au)0, y) = F[ u ]y), we get from Property 4 of the ambiguity function that Au)y sin α, y cos α) = F[ F α u ]y). Definition. Let us define, for θ ]0, π[ τ θ = inf{t > 0 : Au)t cos θ, t sin θ) = 0 or Au) t cos θ, t sin θ) = 0}.
UNCTAINTY PINCIPLS FO ADA AMBIGUITY FUNCTIONS AND MOMNTS 7 Let q > 0 and κ q be given by Theorem.. Applying this theorem, we obtain 3.3) inf t t0 q F θ π/ u κ Fθ π/ q u = κ q u. τ q θ t 0 Let us now show that in the specific case q = a more precise result can be obtained: if u L is such that tu L and tû L then from Property 4 of the fractional Fourier transform tf α u = tut) cos α) iu sin α tut) cos α + u sin α = tut) cos α + ξûξ) sin α. In particular, the ambiguity function Au) of u has no zero in the region { } u t sin α, t cos α) : 0 < α < π, t π tut) cos α + ξûξ) sin α ). This region is a rombus with endpoints ) u ± π tut), 0 and ) u 0, ± π ξût). Further, changing ut) into ut a)e iωt leaves the modulus of Au) unchanged, and so are the zero-free regions of Au). We have thus proved Theorem 3.. Let u L ) be such that tut) L ) and ξûξ) L ). Then the ambiguity function Au) of u has no zero in the convex hull of the four points ) ) u ± π u), 0 u and 0, ±. / π û) / The area of that rombus is π u u) / û) / 4 according to Heisenberg s uncertainty principle. Note also that the numerical constant π can be improved to 0.48 using the inequality. cos x 0.4x instead of cos x x /. 3.4. The local uncertainty principle. In this section, we switch back to the higher dimensional situation d. The first result we will prove may be well-known, but we do not know an reference for it. Let µu) = t ut) dt. Writing h x t) = u t + x ) v t x ) we get as for [BDJ, Lemma 5.]: u d x Au)x, y) dx dy = x ĥxy)) dy dx = x ĥxt)) dt dx d d d = r µu) + µv) s ur) vs) dr ds d = r µu) u v + u r µv) v = u) v + u v).
8 PHILIPP JAMING Using Property 4 of the ambiguity function, we get that x Au)x, y) dx dy = û) v + u ˆv). It follows that x +y Au)x, y) dx dy As Au) = u v we get the following : x + y ) Au)x, y) dx dy u) + û) ) v + u v) + ˆv) )). Proposition 3.. Let u, v L d ) and let 0 < δ <. Then for u) + û) ) / max, v) + ˆv) ) ) / u v δ, then x +y Au, v)x, y) dx dy δ ) u v. We will now prove a similar result about the L -mass of the ambiguity function outside small sets thanks to the local uncertainty principle. Let us recall that Faris [Fa] and Price [Pr] proved that for 0 < α < d/, d of finite measure and f L d ), 3.4) fξ) dξ K α,d α/d x α f, where K α,d = d + α) α) 4α/d d α)α/d. On the other hand Price [Pr] proved that for α > d/, d of finite measure and f L d ), 3.5) fξ) dξ C α,d x α f d/α f d/α, where C α,d = πd/ α d ) d/α αγd/) d α and this constant is optimal. ) Γ ) d Γ d ) α α Corollary 3.3. Let u, v L d ) and d be a set of finite measure. Then for α < d, Au, v)x, y) dx dy α K α,d α/d α u) + α û) ) v + u 3.6) α v) + α ˆv) )) while for α > d, α Au, v)x, y) dx dy d C α,d u) + α û) ) v + u α v) + α ˆv) )) d/α u ) d/α 3.7) v
UNCTAINTY PINCIPLS FO ADA AMBIGUITY FUNCTIONS AND MOMNTS 9 Proof. Using Property 3 of the ambiguity function, it is enough to prove 3.6)-3.7) with x α replacing α ). Indeed, once this is done, it is enough to apply it to u a,ω, v b,η and a,b,ω,η = + a b, ω η). Now recall that a + b t t a t + b t ). We will use the following computation which further generalizes [BDJ, Lemma 5.]: Let g x t) = u x + ) t v x t ). Then x α W u, v)x, y) dx dy = x α ĝ x y) dy dx = x α g x t) dt dx d d d d d = x α u x + t ) v x ) t dt dx d = α r + s α ur) vs) dr ds d d r α u v + u 3.8) s α v Simarily, using the symetry W û, ˆv)x, y) = W u, v) y, x) and Parseval s identity, we get y α W u, v)x, y) dx dy r αû v + u 3.9) s αˆv d Now if α < d, from Faris Inequality 3.4), Au, v) dx dy = K α,d α/d x, y) α W u, v). L d ) But x, y) α = x + y ) α α x α + y α ). Thus x, y) α W u, v) = x, y) α wu, v)x, y) dx dy L d ) d α x α W u, v)x, y) dx dy d ) + y α W u, v)x, y) dx dy d and 3.8)-3.9) allows to conclude. On the other hand, for α > d, Price s Inequality 3.4) gives Au, v) dx dy = C α,d x, y) α W u, v) d/α W u, L d v) d/α ) L d ). Now W u, v) d/α L d ) = u v ) d/α and the other term has been estimated above. xample. For d =, α = / thus K α,d = 9) and u = v, we first get from Cauchy-Scwarz that / u) u) / u so that Au)x, y) dx dy 8 / u) + û) ) / u v + v) + ˆv) ) / u v u) + û) ) / v) + ˆv) ) ) / δ so that if ρ max, u v 36 π then Au, v)x, y) dx dy δ ) u v. x +y ρ )
0 PHILIPP JAMING eferences [Al] L. B. Almeida The fractional Fourier transform and time-frequency representations. I Trans. Sign. Proc. 4 994), 3084 309. [AT] L. Auslander &. Tolimieri adar ambiguity functions and group theory. SIAM J. Math Anal, 6 985), 577 60. [BDJ] A. Bonami, B. Demange & Ph. Jaming Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. ev. Mat. Iberoamericana, 9 003), 3 55. [De] B. Demange Uncertainty principles for the ambiguity function. J. London Math. Soc. ), 7 005), 77 730. [Fa] W. G. Faris Inequalities and uncertainty principles. J. Mathematical Phys. 9 978) 46 466. [Gr] K. Gröchenig Uncertainty principles for time-frequency representations. Advances in Gabor analysis, 30, Appl. Numer. Harmon. Anal., Birkhuser Boston, Boston, MA, 003. [GZ] K. Gröchenig & G. Zimmermann Hardy s theorem and the short-time Fourier transform of Schwartz functions. J. London Math. Soc. ) 63 00), 05 4. [LZ] S. Luo & Z. Zhang, stimating the first zero of a characteristic function. C.. Acad. Sci. Paris, Ser. I 338 004) 03 06. [Pr] J. F. Price Inequalities and local uncertainty principles. J. Math. Phys. 4 983) 7 74. [Pr] J. F. Price Sharp local uncertainty inequalities. Studia Math. 85 986) 37 45. [PH]. Price &. M. Hofstetter Bounds on the volume and height distribution of the ambiguity function. I Trans. Info. Theory 965), 07 4. [Wi] C. H. Wilcox The synthesis problem for radar ambiguity functions. MC Tech. Summary eport 57 960), republished in adar and Sonar part I eds.. Blahut, W. Miller and C. Wilcox), I.M.A. vol in Math. and its Appl. 3, 9 60, Springer, New York, 99. [Wo] P. M. Woodward Probability and Information Theory with Applications to ADA Pergamon, 953. Université d Orléans, Faculté des Sciences, MAPMO - Fédération Denis Poisson, BP 6759, F 45067 Orléans Cedex, France -mail address: Philippe.Jaming@univ-orleans.fr