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A Necessary and Sufficient Condition for Minimum Phase And IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 13, NO. 9, SEPTEMBER 2014 1 A necessary and sufficient condition for minimum phase and implications for phase retrieval Antonio Mecozzi, Fellow, IEEE Abstract—We give a necessary and sufficient condition for a an optical system using square-law detection at the receiver band-limited function E(t) being of minimum phase, and hence are finally discussed. 2 for its phase being univocally determined by its intensity jE(t)j . While emphasis will be given to examples belonging to the This condition is based on the knowledge of E(t) alone and not of its analytic continuation in the complex plane, thus greatly field of telecommunications only, the new condition discussed simplifying its practical applicability. We apply these results here can be of help in all fields of physics and applied sciences to find the class of all band-limited signals that correspond to where the problem of phase reconstruction from the intensity distinct receiver states when the detector is sensitive to the field only is an important one [4]. intensity only and insensitive to the field phase, and discuss the performance of a recently proposed transmission scheme able to linearly detect all distinguishable states. II. A NECESSARY AND SUFFICIENT CONDITION FOR Index Terms—Mathematical methods in physics, Phase re- MINIMUM PHASE trieval, Coherent communications, Modulation. In this paper, we define the Fourier transform of functions F (t) 2 L1 T L2 as I. INTRODUCTION Z 1 HASE retrieval is a longstanding problem in many fields F~(!) = dt exp(i!t)F (t): (1) P of physics and applied sciences [1], [2], [3], [4]. Sufficient −∞ conditions ensuring that the phase of the signal and hence The fact that F (t) 2 L1 insures that F (!) is uniformly the full E(t) can be reconstructed from the knowledge of continuous for ! 2 R [9]. the intensity profile only jE(t)j2 are well known [2], [3], 1 T 2 [5], [6]. A necessary and sufficient condition is also well Lemma 1. Assume a function Es(t) 2 L L , such that its ~ ~ known, and it is based on the position in the complex plane Fourier transform E(!) is Es(!) = 0, 8! < 0. Then we have of the zeros of E(z), the analytic continuation of E(t). Such Z 1 0 i Es(t ) 0 a condition however is of limited practical use, because the Es(t) = p:v: dt ; (2) π t0 − t analytic continuation of a function is an ill posed problem and −∞ hence far from being amenable to simple numerical solutions where with p.v. we refer to the Cauchy’s principal value of the [7]. Even when analytic continuation is possible, like for integral. instance for bandwidth limited E(t), finding the zeros in the Proof: Condition E~ (!) = 0, 8! < 0 implies that complex plane is a difficult numerical task. Being instead the s E~ (!) = u(!)E~ (!), where u(!) is a Heaviside unit step numerical evaluation of E(t) for t real relatively trivial, a s s function. If we write necessary and sufficient condition based on the properties of E(t) for t real only would be a very useful tool. To the best E~s(!) = lim u(!) exp(−ω)E~s(!); (3) of the author’s knowledge, however, such a condition has not !0+ been reported yet. The purpose of this paper is to derive such arXiv:1606.04861v5 [cs.IT] 24 Oct 2016 inverse Fourier transformation gives a condition, which we will see is very similar to the well- Z 1 established Nyqvist stability criterion [8] of control theory. i 1 0 0 Es(t) = lim 0 Es(t )dt ; (4) We will consider for simplicity to band-limited signals only, 2π −∞ !0+ t − t + i and leave generalizations to wider classes of signals to future that is studies. Z 1 0 The outline of this paper is the following. After presenting i t − t − i 0 0 Es(t) = lim 0 2 2 Es(t )dt : (5) the derivation of a necessary and sufficient condition ensuring 2π −∞ !0+ (t − t) + that the phase of a band-limited field can be retrieved from 0 2 2 0 its intensity profile, we apply this condition to find the class Being lim!0+ /[(t − t) + ] = πδ(t − t) where δ(·) is of band-limited signals corresponding to distinct states when the Dirac delta distribution, we obtain the receiver is insensitive to the signal phase and sensitive to i Z 1 t0 − t E (t) = lim E (t0)dt0; the field intensity only. The implications for the capacity of s 0 2 2 s (6) π −∞ !0+ (t − t) + Antonio Mecozzi is with the Department of Department of Physical and that is relation (2). Chemical Sciences, University of L’Aquila, 67100 L’Aquila, Italy e-mail: [email protected]. Consequence of Eq. (2) is that the real and imaginary parts Manuscript received April 19, 2005; revised September 17, 2014. of Es(t) = Es;r(t) + iEs;i(t) are the Hilbert transform of one IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 13, NO. 9, SEPTEMBER 2014 2 another The Fourier transform of (15) is Z 1 0 1 1 0 Es;r(t ) Es;i(t) = p:v: dt ; (7) X 0 E~s(!) = an exp (inT !) H~0 (! − π=T ) ; (17) π −∞ t − t 1 Z 1 E (t0) n=−∞ E (t) = − p:v: dt0 s;i : s;r 0 (8) where π −∞ t − t p !T H~ (!) = TC ; (18) These relations are known in spectroscopy as Kramers Kronig 0 0 2π relations [10], [11]. and C (x) = lim C (x) is a function equal to 1 in the Let β be a strictly positive constant with 0 < β ≤ 1 and 0 β!0 β p open interval (−1=2; 1=2), equal to 1= 2 for x = ±1=2 and let us define B = (1 + β)=T . Let C~ (0;B) be the class of β zero elsewhere. Being for !; !0 2 (0; 2πB) functions Es(t) of the form X 1 H~ (!)H~ (!0) exp [−inT (! − !0)] = 2πδ (! − !0) ; X π(t − nT ) 0 0 E (t) = a exp −i(1 + β) H (t − nT ); n s n T β n=−∞ (19) 2 (9) any function Es(t) 2 L band-limited to the interval (0; 2πB) where we assume that the sequence of an 2 C has a compact can be expressed in the form (15), where the an are given by ~ support, namely for any Es(t) 2 Cβ(0;B), exists an N 2 Z Z 0 ~ 0 2π ~ 0 d! such that a = 0 for jnj > N. The orthogonal set of functions an = H0(! ) exp −in Es(! ) : (20) n T 2π fHβ(t − nT ); n 2 Zg are defined as ~ t t t Although the functions of the form (15) belonging to C(0;B) 1 sin π T (1 − β) + 4β T cos π T (1 + β) 2 1 Hβ(t) = p h i : are in L , they are in general not in L . For this reason, in the T t t 2 π T 1 − 4β T following where we refer to the class of band-limited functions (10) C~(0;B), we will assume that its components are members 1 T 2 ~ The functions in C~β(0;B) belong to L L . Their Fourier of the class Cβ(0;B) for β > 0, hence always in the class transform is L1 T L2, approaching arbitrarily close the limit β = 0. N ~ X Theorem 1. Let us assume Es(t) 2 Cβ(0;B), and a constant E~s(!) = an exp (inT !) H~β [! − (1 + β)π=T ] ; (11) E¯ 6= 0, and define E(t) = Es(t) + E¯ with E¯ 6= 0, such that n=−N E(t) 6= 0, 8t 2 R and the trajectory of E(t) never encircles where the origin for t 2 (−∞; 1). Then, the number of zeros of p !T H~β(!) = TCβ ; (12) E(t + iτ) with τ < 0 is equal to the winding number (i.e, the 2π number of windings) around the origin of the curve described with by E(t) when t runs over the entire real axis from t ! −∞ to t ! 1. 8 1; jxj ≤ 1−β ; <> h i 2 π 1−β 1−β 1+β Proof: The function E(z) is an entire function. Let Γ be Cβ(x) = cos 2β jxj − 2 ; 2 < jxj ≤ 2 ; > a contour encompassing the lower complex plane z = t + iτ, : 0; jxj > 1+β : 2 t = −∞ 1 (13) which incorporates the real axis from to and −∞ The functions fH (t − nT ); n 2 g have a square-root returns to from the lower complex half-plane by a β Z C ρ ! 1 raised cosine spectrum, and their orthogonality can be directly semicircle with radius . By the Cauchy’s argument verified in the Fourier domain principle, we have Z 1 I _ 2 0 d! 1 E(z) jHβ(!)j exp [i(n − n)T!] = δn;n0 ; (14) IΓ = dz = Nzeros − Npoles; (21) −∞ 2π 2πi Γ E(z) P _ where the integral is facilitated by noting that n jHβ(! − where E(t) = dE(t)=dt, Nzeros is the number of zeros and 2 2πT n)j = T and using the periodicity of exp(inT !). Being Npoles is the number of poles of E(z) encircled by Γ. The 1 T 2 Es(t) 2 L L , its spectrum is a continuous function of !. function E(t + iτ) does not have poles for τ < 0, Npoles = 0.
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