Generalized Poisson Summation Formula for Tempered Distributions

2015 International Conference on Sampling Theory and Applications (SampTA)

Ha Q. Nguyen and Michael Unser Biomedical Imaging Group, Ecole´ Polytechnique Fed´ erale´ de Lausanne (EPFL) Station 17, CH–1015, Lausanne, Switzerland {ha.nguyen, michael.unser}@epﬂ.ch

Abstract—The Poisson summation formula (PSF), which re- In the mathematics literature, the PSF is often stated in a lates the sampling of an analog signal with the periodization of dual form with the sampling occurring in the Fourier domain, its Fourier transform, plays a key role in the classical sampling and under strict conditions on the function f. Various versions theory. In its current forms, the formula is only applicable to a of (PSF) have been proven when both f and fˆ are in appro- limited class of signals in L1. However, this assumption on the signals is too strict for many applications in signal processing priate subspaces of L1(R) ∩ C(R). In its most general form, ˆ that require sampling of non-decaying signals. In this paper when f, f ∈ L1(R)∩C(R), the RHS of (PSF) is a well-defined we generalize the PSF for functions living in weighted Sobolev periodic function in L1([0, 1]) whose (possibly divergent) spaces that do not impose any decay on the functions. The Fourier series is the LHS. If f and fˆ additionally satisfy only requirement is that the signal to be sampled and its weak ˆ −1−ε derivatives up to order 1/2 + ε for arbitrarily small ε > 0, grow |f(x)|+|f(x)| ≤ C(1+|x|) , ∀x ∈ R, for some C, ε > 0, slower than a polynomial in the L2 sense. The generalized PSF then it can be shown [6], [7] that (PSF) holds pointwise with will be interpreted in the language of distributions. both sides converging absolutely. It is also known [8], [9] ˆ that (PSF) holds pointwise when f, f ∈ L1(R) ∩ C(R) and I.INTRODUCTION fˆ has bounded total variation. Other versions of the PSF on L1(R) can also be found in [9]–[11]. The main limitation of A widely used result in engineering is that sampling in the above mathematical forms of the PSF is that they do not time corresponds to periodization in frequency and vice versa. apply to functions that are non-decreasing, such as our above This beautiful relation is captured in the celebrated Poisson example of a ramp signal or any realization of a stationary summation formula stochastic process, not to mention the non-stationary ones like X X the Brownian motion which may even grow at infinity [12]. f(k)e−2πikξ = fˆ(ξ + k). (PSF) Intuitively, however, there does not seem to be any compelling k∈ k∈ Z Z reason why a continuous signal cannot be sampled even if it In other words, the discrete-time Fourier transform of the sam- is not decaying. pled sequence is equal to the periodization of the continuous- In this paper we attempt to bridge the gap between the time Fourier transform of the analog signal. This identity is engineering and mathematics statements of the PSF. We will the key behind many results in sampling theory including show that (PSF) holds in the sense of distribution theory [13] the classical Shannon’s sampling theorem [1] for bandlimited for a broad class of continuous signals included in a weighted signals. What appears to be missing, however, is a proof of s Sobolev space L2,1/w(R), whose weak derivatives up to order f (PSF) for a general function beyond the classical hypotheses, s are in the weighted-L2 space L2,1/w(R). Here, the decaying which will be detailed later. In engineering texts, see for ex- weight 1/w allows growing signals. In particular, when the ample [2]–[5], the common “proof” of (PSF) for an arbitrary weight 1/w decays polynomially and the order of smoothness function f is based on the following observations: (1) the s is above the threshold 1/2, both sides of (PSF) are well- sampled sequence is obtained by multiplying the analog signal defined periodic distributions and they are equal when acting with a Dirac comb; (2) multiplication maps to convolution on periodic test functions. In light of our new result, it is in Fourier-domain; and (3) the Fourier transform of a Dirac easy to see that (PSF) holds for the ramp function, and more comb is also a Dirac comb. The problem with this argument generally, for any piecewise polynomials that are continuous. is that the multiplication of a Dirac comb with a general For brevity, most of the proofs in this paper are omitted or function is not necessarily a tempered distribution, so that the just sketched. Detailed proofs can be found in our extensive convolution theorem may not be applicable. Take, for example, papers [14], [15] that are currently in preparation.1 x+|x| the popular ramp signal f(x) = 2 . Since f(x) is not The rest of the paper is organized as follows. Section II differentiable at 0, the multiplication between f and the Dirac introduces some notations and definitions that will be used comb P δ(· − k) is prohibited. It means that the above k∈Z throughout the paper. Section III discusses the sampling in argument for (PSF) does not even work for this seemingly weighted Sobolev spaces. The generalized PSF is stated in well-behaved signal. 978-1-4673-7353-1/15/$31.00 c 2015 IEEE 1The manuscripts are available from the authors upon request.

p Section IV with a sketched proof. Finally, some concluding w(0)/Cw ≥ 1/Cw, for all x. This means that every submul- remarks are made in Section V. tiplicative weighting function is lower-bounded. A prototypical example of submultiplicative weights is w(x) = (1+|x|2)α/2, II.NOTATION AND DEFINITION for some α ≥ 0. For this weight, we can choose Cw to be α/2 Throughout the paper, we deal with complex-valued func- Cα = 2 . tions and sequences. Following the convention in signal pro- Definition 2 (Weighted L and ` spaces). For p ≥ 1 and cessing, we use parentheses for functions and square brackets p p a weighting function w, a function f is in L ( ) if fw is for sequences to specify their values at a particular point. Espe- p,w R in Lp(R); a sequence c is in `p,w(Z) if {c[k]w(k)}k∈Z is in cially, for a function f, f[·] denotes the sequence {f(k)}k∈ . Z `p(Z). The corresponding weighted norms are defined as For p ≥ 1, the spaces Lp(R) and `p(Z) include functions and sequences, respectively, whose p-norms are finite. The scalar kfk := kfwk ; kck := kc w[·]k . Lp,w(R) Lp(R) `p,w(Z) `p(Z) product between two functions in L2(R) is defined as Z Note that if w is submultiplicative then Lp,w(R) ⊂ Lp(R), hf, gi := f(x)g(x)dx, for f, g ∈ L ( ) 2 R and `p,w(Z) ⊂ `p(Z) because w is lower-bounded. R As usual, the notation h·, ·i is also used for the action of a Definition 3 (Weighted hybrid norm spaces). For p, q ≥ 1, the distribution on a test function. Sometimes, it is useful to write hybrid norm space Wp,q(R) includes all functions f : R → C the complex sinusoid e−2πikξ as e−2πih·,ki to imply that it is a whose following norm is finite function in variable ξ. We also adopt standard notations used  !q/p 1/q in Schwartz’s distribution theory [13]. We use the symbols ∧ Z 1 X kfk := |f(x + k)|p dx , and ∨ to denote the forward and inverse Fourier transforms, Wp,q (R)   0 respectively, of a tempered distribution f ∈ S0(R), i.e. k∈Z

D E ∧ with usual adjustments when p or q is infinity. The weighted f,ˆ ϕ = h(f) , ϕi := hf, ϕî , ∀ϕ ∈ S(R), hybrid norm space Wp,q,w(R) w.r.t. a weighting function w is ∨ f,ˇ ϕ = h(f) , ϕi := hf, ϕˇi , ∀ϕ ∈ S(R) defined according to the following weighted norm where kfkW ( ) := kfwkW ( ). Z p,q,w R p,q R ϕˆ(ξ) = (ϕ)∧(ξ) := ϕ(x)e−2πiξxdx; These hybrid (mixed) norm spaces were mentioned in [9]. ZR ∨ 2πiξx They are similar to Wiener amalgam spaces Wfp,q(R) [16]– ϕˇ(x) = (ϕ) (x) := ϕ(ξ)e dξ. [18], where the discrete and continuous norms are mixed in R reverse order, i.e. The set of continuous functions is denoted by C(R), whereas ∞ 1/p Cc (R) = D(R) denotes the space of bump functions that 1 p/q! X Z are infinitely differentiable and compactly supported. For an kfk := |f(x + k)|q dx . Wfp,q ( ) ∞ R 0 interval T of R, C (T) denotes the space of T-periodic test k∈Z functions that are infinitely differentiable. Here, a function These amalgam spaces play a central role in many important ϕ is T-periodic if ϕ = ϕ(· + |T|k), ∀k ∈ Z. The constants throughout the paper are denoted by C with some subscripts results of the non-uniform sampling theory in shift-invariant denoting the dependence of the constants on those parameters. (spline-like) spaces [19]–[23]. Deeper results regarding Wiener amalgam spaces and their generalizations can be found For example, Cx,y,z is a constant that depends only on x, y and z. in [24]–[29]. We note some obvious inclusions of hybrid norm spaces: W ( ) ⊂ W ( ) when 1 ≤ q ≤ q ≤ ∞, The non-decaying signals in this paper will be modeled p,q2,w R p,q1,w R 1 2 and Wp ,q,w( ) ⊂ Wp ,q,w( ) when 1 ≤ p1 ≤ p2 ≤ ∞. as elements of weighted L2 spaces w.r.t. some decaying 1 R 2 R weighting function that controls the growth of the signals. Also, it is easy to see that Wp,p,w(R) = Lp,w(R). We assume implicitly throughout the paper that any weighting Definition 4 (Weighted Sobolev spaces). For s ∈ R, and a function is real-valued, continuous, positive, and symmetric. s weighting function w, the weighted Sobolev space L2,w(R) is Definition 1 (Submultiplicative weights). A weighting func- defined as tion w is called submultiplicative if there exists a constant s ∨ s 0 2 2 ˆ Cw such that L2,w(R) := f ∈ S (R): (1 + | · | ) f ∈ L2,w(R) . w(x + y) ≤ C w(x)w(y), ∀x, y ∈ . (1) w R 2 s In the above definition, we note that (1 + |ξ| ) 2 is an Note that the submultiplicativity of w implies w(x) ≤ infinitely differentiable and slowly increasing function, and so 2 s ˆ Cw w(x)w(0), or w(0) ≥ 1/Cw. On the other hand, w(0) = the multiplication (1 + | · | ) 2 f is a well-defined tempered 2 w(x − x) ≤ Cw w(x)w(−x) = Cw (w(x)) , and thus w(x) ≥ distribution.

2 2015 International Conference on Sampling Theory and Applications (SampTA)

III.SAMPLINGIN WEIGHTED SOBOLEV SPACES Young inequalities we obtain kfk = sup hf ∗ ϕ , gi L2,1/w(R) s s kgk =1 For a slowly increasing weight w, the sampling of a slowly L2,w(R) increasing signal in L2,1/w(R) might be troublesome because fs the sampled sequence might be rapidly increasing even when = sup , w · (g ∗ ϕs) kgk =1 w the signal is continuous, and so its discrete-time Fourier L2,w(R) ≤ sup kf k C kgk kϕ k transform as in the LHS of (PSF) might not be well-defined. It s L2,1/w(R) α L2,w(R) s L1,w(R) kgk =1 is therefore desirable to impose some order of smoothness on L2,w(R) s = C kf k kϕ k < ∞, the signal to be sampled by switching L2,1/w(R) to L2,1/w(R). α s L2,1/w(R) s L1,w(R) In what follows, we show that if the signal is continuous and which implies that f ∈ L2,1/w(R), completing the proof. the order s of the Sobolev space is above the threshold 1/2 then the sampling is stable in the sense that the discrete `2,1/w- Before stating the central result of this section we need the norm is bounded by the continuous L2,1/w-norm. following key lemma whose proof can be found in [14]. We first want to make an important observation: any Lemma 1. Let w be a submultiplicative weighting function. s distribution f ∈ L2,w(R) can be written as a convolution If f ∈ L2,1/w(R) and ϕ ∈ W1,2,w(R) then f ∗ ϕ is a well- s ∨ 2 ˆ defined continuous function and its sampled sequence {c[k] := f = fs ∗ ϕs, where fs := (1 + | · | ) 2 f ∈ L2,w(R), (f ∗ϕ)(k)} belongs to ` ( ), and the following bound 2 − s ∨ k∈Z 2,1/w Z and ϕs := (1 + | · | ) 2 . In the literature, the function holds ϕs when s > 0 is often referred to as Bessel potential kernel. kck ≤ Cw kfk kϕk , The following properties of this kernel will be useful later. `2,1/w(R) L2,1/w(R) W1,2,w(R) First, it is easy to see that ϕs(x) is a real and symmetric where Cw is the constant given in (1). function. Second, it is well-known from Sobolev space theory 2 α/2 (see, for example, [30, Prop. 6.1.5]) that ϕs(x) is a positive Theorem 1. Let s > 1/2, and w(x) = (1 + |x| ) for some function that decays exponentially outside a neighborhood of α ≥ 0. The sampling operator f 7→ f|Z is then bounded from s the origin. More precisely, there exists a constant Cs such that L2,1/w(R) ∩ C(R) to `2,1/w(Z), i.e., there exists a constant Cα,s such that s −π|x| 1 kf[·]k` ( ) ≤ Cα,s kfkLs ( ), ∀f ∈ L ( ). ϕ (x) ≤ C · e , ∀ |x| > . (2) 2,1/w Z 2,1/w R 2,1/w R s s π Proof. From Proposition 1, we know that f = fs ∗ ϕs almost everywhere, where f ∈ L and ϕ ∈ W ( ). It then Third, it is also known that ϕ ∈ L ( ) whenever s > 1/2. s 2,1/w s 1,2,w R s 2 R follows from Lemma 1 that f ∗ ϕ is continuous. Combining Based on these properties we can show the following. s s this with the fact that f is continuous, we deduce that f = Proposition 1. Suppose s > 1/2, and w(x) = (1 + |x|2)α/2 fs ∗ ϕs everywhere. for some α ≥ 0. The Bessel potential kernel ϕs is then an Now we can safely write f(k) = (fs ∗ϕs)(k), for all k ∈ Z, and again invoke Lemma 1 to get element of the weighted hybrid norm space W1,2,w(R).

kf[·]k` ( ) ≤ Cα kϕsk kfsk Proof. See [14]. 2,1/w Z W1,2,w(R) L2,1/w(R)

= Cα kϕsk kfk s , W1,2,w(R) L2,1/w(R) | {z } s Cα,s Note that L2,1/w(R) is identical to L2,1/w(R) when s = s which is the desired bound. 0. Is it true that every element of L2,1/w(R) is an L2,1/w function? The following result gives an affirmative answer to IV. GENERALIZED POISSON SUMMATION FORMULA that question when s ≥ 0 and the weighting function 1/w is Now we are ready to generalize the Poisson summation decaying polynomially. formula for functions in weighted Sobolev spaces and show Proposition 2. If s > 0, and w(x) = (1 + |x|2)α/2 for some that both the discrete-domain Fourier transform of the sampled s sequence and the periodization of the continuous-domain α ≥ 0, then L2,1/w(R) ⊂ L2,1/w(R). Fourier transform are well-defined periodic distributions and Proof. As a special case of Proposition 1, we have ϕs ∈ they are equal in S0(T), for T := [0, 1]. First we need to W1,1,w( ) = L1,w( ), for all s > 0. Let f be an element P ˆ R R define the periodization k∈ f(· + k) by means of a unitary of Ls ( ). By duality we can write ∞ Z P 2,1/w R function ψ ∈ Cc (R) such that k∈ ψ(· + k) = 1. In P ˆ Z particular, the action of k∈ f(· + k) on a periodic test ∞ Z kfk = sup hf, gi . function ϕperio ∈ C (T) is given by L2,1/w(R) kgk =1 L2,w(R) * + X ˆ X D ˆ E f(· + k), ϕperio := f(· + k), ϕperioψ . (3) k∈ k∈ Using the submultiplicativity of w, Cauchy-Schwarz and Z T Z

3 2015 International Conference on Sampling Theory and Applications (SampTA)

The next result shows that the RHS of (3) is finite for V. CONCLUSION appropriate weighted Sobolev spaces, whereas Theorem 2 We have provided a distributional proof of the Poisson guarantees that this definition is independent of the choice summation formula for non-decaying signals whose weak of ψ. derivatives up to order 1/2 + ε are slowly growing. These Proposition 3. For s > 1/2, and w(x) = (1+|x|2)n, for some signals are modeled as tempered distributions in weighted + s P D E Sobolev spaces. Although our results are stated only for the n ∈ Z , if f ∈ L (R) then the series fˆ(· + k), ϕ 2,1/w k∈Z sampling on , they can be easily extended to T for an is absolutely convergent, for every bump function ϕ ∈ C∞( ). Z Z c R arbitrary sampling period T by the scaling property of the Proof. See [15]. Fourier transform. This generalized PSF may be used as a foundational result for the sampling theory of non-decaying Theorem 2 (Generalized PSF). Suppose s > 1/2, and w(x) = signals. It also confirms the general validity of the statement: (1 + |x|2)n for some n ∈ +. For all f ∈ Ls ( ) ∩ C( ), Z 2,1/w R R “A periodization in the frequency domain maps into a sampling the following equality holds in the time domain and vice versa.” X X f(k)e−2πih·,ki = fˆ(· + k), ACKNOWLEDGMENTS k∈Z k∈Z ∞ The research leading to these results has received funding in the distributional sense, i.e., for all ϕperio ∈ C (T) and every unitary function ψ, we have from the European Research Council under the European D E D E Union’s Seventh Framework Programme (FP7/2007-2013) / X −2πih·,ki X ˆ ◦ f(k)e , ϕperio = f(· + k), ϕperioψ , ERC grant agreement n 267439 and the Swiss National T k∈Z k∈Z Science Foundation under Grant 200020-144355. (4) with both sides converging absolutely. REFERENCES [1] C. Shannon, “Communication in the presence of noise,” Proc. IRE, Sketch of Proof. 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