Interpretation and Identification of Minimum Phase Reflection Coefficients J

Interpretation and identification of minimum phase reflection coefficients J. Gregory McDaniel, and Cory L. Clarke

Citation: The Journal of the Acoustical Society of America 110, 3003 (2001); doi: 10.1121/1.1416903 View online: https://doi.org/10.1121/1.1416903 View Table of Contents: https://asa.scitation.org/toc/jas/110/6 Published by the Acoustical Society of America

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Analytical analysis of slow and fast pressure waves in a two-dimensional cellular solid with fluid-filled cells The Journal of the Acoustical Society of America 139, 3332 (2016); https://doi.org/10.1121/1.4950752 Interpretation and identification of minimum phase reflection coefficients J. Gregory McDaniela) and Cory L. Clarke Department of Aerospace and Mechanical Engineering, Boston University, 110 Cummington Street, Boston, Massachusetts 02215 ͑Received 23 October 2000; revised 7 September 2001; accepted 10 September 2001͒ If the acoustic reflection coefficient is minimum phase, then the causality condition can be used to reconstruct its phase from its magnitude. Such reconstructions are useful when environmental conditions obscure the phase, which is vital to understanding the dynamics of the reflecting object. However, the reconstructions are usually impossible because one cannot be sure that the reflection coefficient is minimum phase. The present work addresses this difficulty by formulating a time-domain description of the minimum phase condition and by identifying classes of submerged objects that always create minimum phase reflection coefficients. Numerical examples are used to confirm the analytical findings and to illustrate their applicability to objects that reflect sound with a wide range of magnitudes. © 2001 Acoustical Society of America. ͓DOI: 10.1121/1.1416903͔ PACS numbers: 43.40.Fz ͓CBB͔

I. INTRODUCTION useful in examples involving the antenna radiation-power patterns and microwave filters, it does not produce correct It is often the case in experiments and sensing activities results for the problems considered here. The fundamental that the magnitude of a reflected or transmitted acoustic reason is that the reflection coefficient phase does not exhibit wave is more easily measured than its phase. While the mag- the same frequency characteristics as the phases in the elec- nitude can be accurately estimated from suitable time aver- tromagnetic examples considered by Sarkar. ages, ambient noise as well uncertainties in target and re- The causality condition has been used to relate portions ceiver locations conspire to contaminate the phase over of transfer functions that arise in acoustic radiation problems. broad frequency bands. These factors typically leave the ex- Such problems are often characterized by the frequency- perimentalist with nothing but an accurately measured reflec- dependent radiation impedance of a wetted surface, defined tion coefficient magnitude over the frequency range of inter- here as the ratio of surface pressure to velocity. Causality est. If the phase of the reflection coefficient was known, the may be generally used to relate the real and imaginary parts complex valued reflection coefficient could be used to deter- of this transfer function, as was first suggested by Mangulis12 mine dynamics properties of the object that enhance its iden- and used by Radlinski and Meyers.13 In these applications, tification. For this reason, the present article is concerned the causality condition requires that an acoustic pressure on with the finding of a reflection coefficient phase from its the surface cannot be produced until it acquires a nonzero magnitude using causality. velocity. Causality requires that the response of a system cannot The one-dimensional acoustic reflection and transmis- precede its cause. Analysis of this condition in the frequency sion problems typically involve two transfer functions, domain yields integral relationships between the real and which make the problems more complex than the radiation imaginary parts of any transfer function. If the function is ones considered by Mangulis and Radlinski and Meyers. The minimum phase, a property that shall be discussed in greater first is the mechanical impedance of the object, defined here detail later, then its magnitude and phase are also related by as the ratio of applied force to velocity. For this transfer causality. Derivations of these relations and their mathemati- function, the causality requirement is best described by con- 1 cal properties are given by Papoulis and in The Transform sidering the object in vacuo and applying a force to produce 2 Handbook. The relations have been useful in physics and a specified velocity. The velocity is the input and the force is engineering, most notably in electromagnetic scattering the output, so causality requires that no force may be mea- 3–6 7–10 theory, and electrical network design. sured until the velocity occurs. The second transfer function Recently, the problem of recovering nonminimum phase is the reflection coefficient, defined here as the ratio of the from the magnitude of a transfer function has received in- reflected wave amplitude to the incident wave amplitude. 11 creased attention. Sarkar presented a method that relies on Here, causality requires that waves may not be reflected from a Fourier series expansion of the real and imaginary parts of an object before an incident wave arrives. Since the object the transfer function. The Fourier coefficients are initially impedance and the reflection coefficient are algebraically re- chosen by recovering the minimum phase and are then itera- lated, causal impedances produce causal reflection coeffi- tively adjusted so that the magnitude of the transfer function cients. These ideas were first explored by McDaniel14 for matches a measured value. While this technique has been one-dimensional acoustic reflections from mass-spring- dashpot systems. a͒Electronic mail: [email protected] The first goal of the present article is to interpret the

J. Acoust. Soc. Am. 110 (6), December 2001 0001-4966/2001/110(6)/3003/8/$18.00 © 2001 Acoustical Society of America 3003 minimum phase condition for the reflection of a plane wave H(␻) is minimum phase. This concept often arises in the normally incident on a uniform impedance surface. This is context of a causal inverse filter in noise control.16 Zeros in achieved by reviewing existing descriptions of the minimum the upper half of the complex frequency domain create phase phase condition and describing a new time-domain interpre- shifts in the transfer function as frequency is varied along the tation. The second goal is to identify two classes of sub- real axis, which is the hallmark of a nonminimum phase merged objects that produce minimum phase reflection coef- function. In particular, passing beneath a zero in the upper ficients. The first class consists of objects that effectively half of the plane creates a phase shift of (Ϫ␲).17 One can transfer velocity and the second class consists of objects that show that as frequency is varied from infinity to zero, a effectively transfer force. nonminimum phase function experiences a greater positive phase shift than a minimum phase function with the same 18 II. EXISTING DESCRIPTIONS OF THE MINIMUM magnitude. This is the source of the minimum phase adjec- PHASE CONDITION tive. Further insight into the minimum phase condition may In this section, existing descriptions of the minimum be gained by expressing a nonminimum phase transfer func- phase condition have been collected from a variety of disci- tion as a product of a minimum phase function, Hmp , and an plines and are presented first in the frequency domain and 16–19 all-pass function Hap , then in the time domain. The following Fourier transform ͑␻͒ϭ ͑␻͒ ͑␻͒ ͑ ͒ convention is used: Hnmp Hap Hmp . 6 ϱ ␻ Each of the two functions on the right is required to satisfy H͑␻͒ϭF͕h͑t͖͒ϭ ͵ h͑t͒ei tdt, ͑1͒ ͑ ͒ ͑ ͒ Ϫϱ the causality and real-valued conditions in Eqs. 3 – 5 .In addition, the all-pass function must have a unit magnitude at 1 ϱ all frequencies, h͑t͒ϭF Ϫ1͕H͑␻͖͒ϭ ͵ H͑␻͒eϪi␻td␻. ͑2͒ 2␲ Ϫϱ ͉ ␻͉͒ϭ ͑ ͒ Hap͑ 1. 7 Throughout this discussion, h(t) is assumed to be the im- In this way, each nonminimum phase function has an asso- ␻ pulse response of a physical system and H( ) is the transfer ciated minimum phase function that has the same magnitude function for the system. but a different phase. ␻ Before discussing the phase properties of H( ), let us One useful form of the all-pass function is given by recall two conditions on passive linear physical systems. Victor,19 Since h(t) is the response to a Dirac input at tϭ0, causality ϭ Ͻ requires that h(t) 0 for t 0. The implication of this con- ␻͒ϭ i␻D ␻ ͒ ͑ ͒ Hap͑ e ͟ P͑ ,u j . 8 dition for H(␻) is more subtle. When time is negative, the j inverse Fourier integral may be evaluated by closing a con- tour in the upper half of the complex frequency plane. The The exponential term is recognized as a time shift D that causal requirement of no response for negative times leads creates a pure delay in response. For simplicity, we shall ␻ henceforth assumed that time has been referenced to the first directly to the condition that H( ) may not have poles in the ϭ upper half of the complex frequency plane. Further occurrence of an output signal, so that D 0. The factors are analysis1,2 reveals that the real and imaginary parts of H(␻) defined as ␻ ϭ ␻ are related by Hilbert transforms. Writing H( ) Hr( ) ␻Ϫ u j ϩH (␻), causality requires that P͑␻,u ͒ϭ . ͑9͒ i j ␻Ϫ u*j 1 ϱ H ͑y͒ ͑␻͒ϭH͕ ͑␻͖͒ϭ ͵ r ͑ ͒ These factors and their product are sometimes referred to as Hi Hr dy, 3 ␲ Ϫϱ ␻Ϫy 5 the Blaschke factors and product. In order for Hap to satisfy the causality and real-valued conditions, the Blaschke prod- 1 ϱ H ͑y͒ ͑␻͒ϭϪH͕ ͑␻͖͒ϭ ͑ ͒Ϫ ͵ i ͑ ͒ uct must be carefully chosen. To satisfy the causality condi- Hr Hi Hr 0 dy. 4 ␲ Ϫϱ ␻Ϫy tion, the poles should be in the lower plane. This requires that ͕u ͖Ͼ0. To satisfy the real-valued condition, the The second condition is that the impulse response should be I j Blaschke product must have the property indicated in Eq. real-valued, or I͕h(t)͖ϭ0. This condition is met by requir- ͑5͒, ing H͑␻͒ϭH*͑Ϫ␻͒, ͑5͒ ␻ ͒ϭ Ϫ␻ ͒ ͑ ͒ ͟ P͑ ,u j ͟ P*͑ ,u j . 10 j j where the superscripted * denotes the complex conjugate. One description of the minimum phase condition, which This may be achieved by either choosing the Blaschke fac- ␻ is accepted by many as the definition, is that H( ) may not tors in pairs of the form P(w,u j) and P(w,uk), where 1 ϭϪ have zeros in the upper half of the complex frequency plane. R͕u j͖ R͕uk͖, or by requiring a single factor to have a This condition insures that the system’s inverse transfer pure imaginary u j . function is also causal because HϪ1(␻) will have no poles in Time-domain descriptions of the minimum phase condi- the upper plane if H(␻) has no zeros there.9,15 The inverse tion have received far less attention in the literature but are impulse response, F Ϫ1͕HϪ1(␻)͖, is only causal when interesting and useful. The present article shall use a descrip-

3004 J. Acoust. Soc. Am., Vol. 110, No. 6, December 2001 J. G. McDaniel and C. L. Clarke: Minimum phase reflection coefficients tion that was presented by Eisner15 and involves the output then the minimum phase and log magnitude are related by energy of the system. Let the energy contained in an arbitrary ␪ ͑␻͒ϭH͕␣ ͑␻͖͒, ͑20͒ response g(t) up to time T be defined as mp mp ␣ ͑␻͒ϭ␣͑ ͒ϩH͕␪ ͑␻͖͒ ͑ ͒ T mp 0 mp . 21 ϭ ͵ ͉ ͑ ͉͒2 ͑ ͒ Eg g t dt. 11 ␻ Ϫϱ This description is of course related to the others. If Hmp( ) has no zeros in the upper plane, then one can create a new Let us further define h(t)tobeg(t) convolved with an all- transfer function by taking the logarithm of Eq. ͑19͒. The pass and causal function p(t), so that g(t)ϭh(t)*p(t). real and imaginary parts of this new transfer function are Then one can show that related by the Hilbert transform relations in Eqs. ͑3͒ and ͑4͒, у ͑ ͒ leading to the above equations. Eh Eg . 12 This implies that any multiplication by a causal all-pass function has the effect of decreasing the energy that has ar- III. A RESPONSE-BASED DESCRIPTION OF THE rived up to a time T. The key implication of this result for the MINIMUM PHASE CONDITION present work is that the response energy in a minimum phase In this section, a time-domain description of the mini- system integrated up to time T is greater than that of a non- mum phase condition is presented that complements the de- minimum phase system with the same frequency-domain scriptions in the previous section by giving physical insight magnitude. into the time-domain response of nonminimum phase func- For convenience of referral, we include the proof of Eq. tions. For clarity, the discussion begins with a simple form of ͑12͒ that was presented by Eisner. A truncated function g (t) T the all-pass function in Eq. ͑8͒. Following discussion of this is defined according to example, a generalization is presented. g͑t͒, tрT, Consider a single Blaschke factor, ͑ ͒ϭͭ ͑ ͒ gT t 13 0, tϾT. ␻Ϫu ␻͒ϭ ͑ ͒ Hap͑ . 22 This allows Eg to be expressed as an infinite integral instead ␻Ϫu* of the semi-infinite integral given in Eq. ͑11͒, The real-valued condition in Eq. ͑5͒ leads to the condition ϱ that u be pure imaginary. Writing uϭiu , the function is ϭ ͵ ͉ ͑ ͉͒2 ͑ ͒ i Eg gT t dt. 14 Ϫϱ expressed as Ϫ ␻ Since we now have an infinite integral, Parseval’s formula is 2ui i2 H ͑␻͒ϭ1Ϫu . ͑23͒ applied to get ap i 2ϩ␻2 ui ϱ ͑ ͒ ϭ ͵ ͉ ͑ ͉͒2 ͑ ͒ When this function is taken into the time domain by Eq. 2 , Eg GT f df. 15 20 Ϫϱ the following simple form results: ͉ ͉ϭ 0, tϽ0, If p(t) is an all-pass function, then P( f ) 1. Inserting this ͑ ͒ϭͭ ͑ ͒ h t Ϫ 24 ͑ ͒ ap ␦ ͒Ϫ uit у unity factor in the integrand of Eq. 15 and again applying ͑t 2uie , t 0. Paresval’s formula results in The frequency-domain multiplication of the all-pass and ϱ minimum phase functions in Eq. ͑6͒ amounts to a convolu- ϭ ͵ ͉ ͑ ͒ ͑ ͉͒2 ͑ ͒ Eg gT t *p t dt. 16 Ϫϱ tion in the time domain, ͑ ͒ϭ ͑ ͒ ͑ ͒ ͑ ͒ As the integrand is always positive, hnmp t hap t *hmp t . 25 ͑ ͒ T Using Eq. 24 , the nonminimum phase impulse response is у ͵ ͉ ͑ ͒ ͑ ͉͒2 ͑ ͒ Ϫ Eg gT t *p t dt. 17 ͑ ͒ϭ ͑ ͒Ϫ ͓ uit ͑ ͔͒ ͑ ͒ Ϫϱ hnmp t hmp t 2ui e *hmp t . 26 Therefore, the nonminimum phase response is a sum of the Recalling that p(t) is causal, one can show that gT*p minimum phase response and the response of the minimum ϭg*pϭh for tрT. Inserting this relation into the above equation gives phase system to an additional exponential forcing function. Generalization of this result to the general case is con- T ceptually straightforward. The all-pass function in Eq. ͑8͒ is у ͵ ͉ ͉2 ͑ ͒ Eg h dt, 18 Ϫϱ cast in the form ͑␻͒ϭ Ϫ ͑␻͒ ͑ ͒ which is Eq. ͑12͒. Hap 1 Hrem . 27 Finally, the minimum phase is often described as the Substituting Eq. ͑27͒ into Eq. ͑6͒ gives phase obtained by taking the Hilbert transform of the loga- ͑␻͒ϭ ͑␻͒Ϫ ͑␻͒ ͑␻͒ ͑ ͒ rithm of the magnitude.13 If the minimum phase function is Hnmp Hmp Hmp Hrem , 28 written as or, in the time domain, ͑␻͒ϭ ͑␣ ϩ ␪ ͒ ͑ ͒ ͑ ͒ϭ ͑ ͒Ϫ ͑ ͒ ͑ ͒ ͑ ͒ Hmp exp mp i mp , 19 hnmp t hmp t hrem t *hmp t . 29

J. Acoust. Soc. Am., Vol. 110, No. 6, December 2001 J. G. McDaniel and C. L. Clarke: Minimum phase reflection coefficients 3005 by noting that the fluid to the left of the object requires the ϭϪ ϭ␳ ratio F1 /V1 Zfl , where Zfl cA. Algebraic combination of this result with Eq. ͑32͒ gives the effective impedance seen by the incident wave, Z2 ϭ Ϫ 12 ͑ ͒ Zeff Z22 ϩ . 33 Z11 Zfl The effective impedance is the drive-point impedance presented by the object and the fluid to the left of the object if the fluid to the right of the object were removed. It has FIG. 1. A schematic of the one-dimensional reflection and transmission of been shown that the drive-point impedance of any linear sys- acoustic waves by a passive linear object. tem, electrical or mechanical, belongs to a class of ‘‘positive real’’ functions.1,21 These functions possess many interesting These results indicate that the nonminimum phase impulse properties, but the most important to the present discussion is response is the sum of the equivalent minimum phase im- that the real part of a positive real function is positive in the pulse response and the response of the minimum phase sys- upper half of the complex frequency plane and on the real tem to a forcing function of hrem(t). frequency axis. This property, which is a direct consequence It is interesting to note that this forcing function, of causality and passivity, will be used in the following sec- hrem(t), is causal and real-valued. One may observe the real- tions to identify minimum phase reflection coefficients for valued property by evaluating the real and imaginary parts of two special cases. ͑ ͒ Eq. 27 at negative and positive frequencies. These evalua- When an incident wave of unit amplitude strikes the tions lead to object, a reflected wave is created that propagates away from ͑␻͒ϭ ͑Ϫ␻͒ ͑ ͒ the object. The acoustic pressure in the fluid due to this re- Hrem Hrem* . 30 flected wave is The remainder function is recognized as causal by taking the ͑␻ ͒ϭ ͑␻͒ ik(␻)x ͑ ͒ inverse Fourier transform of Eq. ͑27͒, Pr ,x R e , 34 ͒ϭ␦ ͒Ϫ ͒ ͑ ͒ where R is the reflection coefficient and kϭ␻/c is the acous- hrem͑t ͑t hap͑t . 31 tic wavenumber. Continuity of force and displacement at the ␦ Because (t) and hap(t) are both causal, hrem(t) must be right-hand piston face yields an algebraic relationship be- causal. tween the reflection coefficient and the effective impedance,22 IV. A MINIMUM PHASE CONDITION FOR THE ␻͒Ϫ REFLECTION COEFFICIENT zeff͑ 1 R͑␻͒ϭ , ͑35͒ ␻͒ϩ In this and the following sections, the minimum phase zeff͑ 1 condition is explored for the reflection coefficient of a pas- where the normalized effective impedance is sive object immersed in an infinite waveguide. For clarity, ␻͒ϭ ␻͒ ␳ ͒ ͑ ͒ the one-dimensional problem shown in Fig. 1 will serve as zeff͑ Zeff͑ /͑ cA . 36 the basis for discussion. In this figure, the object is posi- The reflection coefficient is minimum phase when it has tioned in an infinite waveguide of cross-sectional area A that ␳ no zeros in the upper half of the complex frequency plane. contains a fluid with sound speed c and mass density . One means of satisfying this condition is to require the real Forces are transmitted between the object and the fluid via part of the numerator to be strictly positive or negative in this two rigid massless pistons. The object is ensonified by an region, so that it cannot pass through a zero. If either incident wave, creating reflected and transmitted waves that ␻͒ Ͼ ␻ у ͑ ͒ propagate away from the object. R͕zeff͑ ͖ 1 for I͕ ͖ 0 37 In the frequency domain, the dynamic properties of the or object are completely described by an impedance matrix that ͕ ͑␻͖͒Ͻ ͕␻͖у ͑ ͒ relates forces and velocities at the ports labeled 1 and 2 in the R zeff 1 for I 0, 38 figure according to then R is minimum phase. Evaluation of these criteria re- ␻͒ ␻͒ quires a detailed knowledge of the generally complex-valued F ͑␻͒ Z11͑ Z12͑ V ͑␻͒ 1 ϭ 1 ͑ ͒ ͭ ␻͒ͮ ͫ ͬͭ ␻͒ͮ , 32 impedance matrix. In the envisioned applications, where F2͑ Z ͑␻͒ Z ͑␻͒ V2͑ 21 22 only the magnitude of the reflection coefficient is measured, where F1 and F2 are the forces applied to the object by the one usually knows very little about the impedance matrix fluid in the directions of the velocities V1 and V2 , all of and the above criterion are not helpful. This difficulty is which are defined as positive in the x direction. Reciprocity overcome in the special cases described in the following two requires the impedance matrix to be symmetric, so that sections. ␻ ϭ ␻ Z12( ) Z21( ). The effective impedance seen by the inci- Before considering these special cases, let us develop ϭ dent wave is simply Zeff F2 /V2 . some additional physical intuition of the minimum phase This effective impedance is found by noting equal and condition on the reflection coefficient by specializing the opposite forces between the object and the fluid at port 1 and time-domain descriptions presented in the previous two sec-

3006 J. Acoust. Soc. Am., Vol. 110, No. 6, December 2001 J. G. McDaniel and C. L. Clarke: Minimum phase reflection coefficients tions. The reflection coefficient may be interpreted as the Fourier transform of the reflected pressure due to an incident wave that applies a Dirac pressure distribution to the object at time tϭ0. Since the magnitude of the acoustic intensity vector for a plane wave is Iϭp2/(␳c), the energy quantity defined in Eq. ͑11͒ is proportional to the integrated acoustic energy that has left the object up to time T. Therefore, an object with a minimum phase reflection coefficient has the FIG. 2. A schematic of the system in Fig. 1 under the assumption that the property of reflecting the acoustic energy faster than an ob- object transmits velocity. ject with the same magnitude of reflection coefficient but with a nonminimum phase. ϭ ϩ The response-based description of the minimum phase where the net force acting on the object is Fnet F1 F2 and property presented in the previous section also provides a V is either V1 or V2 . The velocity transfer impedance is ϭ physical basis for interpreting the reflection coefficient. In simply Zvt 1/Y vt . This impedance acts in parallel to the particular, Eq. ͑29͒ can be applied to the reflection problem acoustic impedance of the fluid to the left of the piston because the velocities of the fluid and the object are identical. by interpreting hnmp as the pressure reflected by a nonminimum phase object. This pressure is a sum of two terms. The Alternatively, the structural admittance acts in series with the first is identified as the pressure reflected when a Dirac inci- fluid admittance. dent wave ensonifies a minimum phase object with the same A schematic of the reduced system is shown in Fig. 2. reflection coefficient magnitude. The second term is the pres- The normalized effective impedance defined by Eqs. ͑33͒ sure radiated by the same minimum phase object when and ͑36͒ is simply Ϫ forced by a pressure distribution of hrem(t) applied to the z ϭ1ϩz . ͑43͒ object. eff vt Now, we use the previously quoted result that the drive-point impedance, zvt , of a passive linear system is a positive real function. Its real part is always positive in the upper half of V. SYSTEMS WITH PERFECT VELOCITY TRANSFER the complex frequency plane and the real part of zeff is al- In this section, the reflection coefficient is shown to be ways greater than unity. Therefore, the condition in Eq. ͑37͒ minimum phase when the velocities at points 1 and 2 in Fig. is met and the reflection coefficient is always minimum 1 are equal. This situation is sometimes referred to as ‘‘per- phase. fect velocity transfer.’’ While it is tempting to interpret this To illustrate this result, let us consider objects that sat- as a rigid body condition, one must bear in mind that the isfy the velocity transfer assumption. For these examples, the object may have internal degrees-of-freedom due to internal impedance of the object is generated by the following Fou- inertia and flexibility. The condition of perfect velocity trans- rier series with real-valued Fourier coefficients: fer only requires that the boundary velocities at points 1 and N in␲␻ 2 are equal. One example of such an object would be a very ͑␻͒ϭ ϩ ͩ ͪ ͑ ͒ zvt a0 ͚ anexp ⍀ . 44 stiff container with an assemblage of masses, springs, and nϭ1 dashpots inside. This function satisfies the causality condition in Eqs. ͑3͒ and In order to understand the conditions that lead to perfect ͑4͒, since the Hilbert transform of sin(n␲␻/⍀)is velocity transfer, Eq. ͑32͒ is inverted to the form cos(n␲␻/⍀). For nϾ0, the coefficients are chosen by ␻͒ ␻͒ V ͑␻͒ Y 11͑ Y 12͑ F ͑␻͒ ͭ 1 ͮ ϭͫ ͬͭ 1 ͮ , ͑39͒ a ͑1Ϫ2r ͒ V ͑␻͒ ͑␻͒ ͑␻͒ F ͑␻͒ ϭ 0 n ͑ ͒ 2 Y 21 Y 22 2 an , 45 N where the admittance matrix is the inverse of the impedance Ϫ1 matrix, ͓Y ͔ϭ͓Z͔ . Equating the velocities leads to the where rn is a random number that is uniformly distributed р р condition over the interval 0 rn 1. This selection insures that R͕z ͖Ͼ0 and the system is therefore passive. In all of our ͑ Ϫ ͒ ϭ͑ Ϫ ͒ ͑ ͒ vt Y 11 Y 12 F1 Y 22 Y 12 F2 . 40 ϭ simulations, we set N 50. The leading term, a0 , was used For this equation to hold for arbitrary frequency-dependent to control the magnitude of the impedance. ϭ forces, every element in the admittance matrix must be iden- In the first example, we set a0 0.1 to simulate an ap- tical so that proximately pressure-release surface. The real and imaginary parts of the object impedance obtained by using this value in Y vt Y vt ͑ ͒ ͑ ͒ ͓Y ͔ϭͫ ͬ, ͑41͒ Eqs. 44 and 45 are shown in Fig. 3. The reflection coef- Y vt Y vt ficient was evaluated using Eq. ͑35͒. The magnitude and phase of the reflection coefficient are shown in Fig. 4. In the where the subscripted vt denotes the assumed ‘‘velocity envisioned applications, the phase would not be known and transfer’’ property of the object. Substitution of this result causality would be used to predict it from the magnitude. To into Eq. ͑39͒ yields the simple relation this end, a causally predicted phase is computed by inserting ϭ ͑ ͒ ͑ ͒ Fnet ZvtV, 42 the magnitude of the reflection coefficient in Eq. 21 . This

J. Acoust. Soc. Am., Vol. 110, No. 6, December 2001 J. G. McDaniel and C. L. Clarke: Minimum phase reflection coefficients 3007 FIG. 3. Real and imaginary parts of normalized object impedance, z , for vt FIG. 5. Real and imaginary parts of normalized object impedance, z ,for an object that transfers velocity and has a low magnitude of impedance. vt an object that transfers velocity and has a high magnitude of impedance. phase coincides with the actual phase in Fig. 4, indicating that the reflection coefficient is indeed minimum phase. ton’s second law then requires the net force on the object to To further illustrate this result, an object with a large be zero. One example of such a system is a layer of air in reflection coefficient magnitude was simulated by choosing water with a normally incident wave. ϭ Applying this condition to Eq. ͑32͒ results in a0 50. The real and imaginary parts of the resulting impedance are shown in Fig. 5, where we see a magnitude that ͑Z ϩZ ͒V ϭϪ͑Z ϩZ ͒V . ͑46͒ exceeds 0.95 everywhere. The magnitude and phase of the 11 12 1 12 22 2 reflection coefficient are shown in Fig. 6. Again, the causally For this equation to hold for arbitrary frequency-dependent predicted phase coincides with the actual phase because the velocities, the impedance matrix must have the form reflection coefficient is minimum phase. Ϫ Zft Zft ͓Z͔ϭͫ ͬ, ͑47͒ ϪZ Z VI. SYSTEMS WITH PERFECT FORCE TRANSFER ft ft where the subscripted ft denotes the assumed ‘‘force trans- This section describes the minimum phase reflection co- fer’’ property of the object. efficient that results when an object transmits force perfectly. Substitution of this result into Eq. ͑32͒ yields the simple To achieve this limit, the forces applied to the object by the ϭϪ relation fluid must be equal but opposite, F1 F2 . Physically, this condition results when the mass of the object is small. New- ϭϪ ϭ Ϫ ͒ ͑ ͒ F1 F2 Zft͑V2 V1 . 48

FIG. 4. Magnitude and phase of the reflection coefficient for an object that FIG. 6. Magnitude and phase of the reflection coefficient for an object that transfers velocity and has the impedance shown in Fig. 3. Causality is used transfers velocity and has the impedance shown in Fig. 5. Causality is used to recover the phase from the magnitude. to recover the phase from the magnitude.

3008 J. Acoust. Soc. Am., Vol. 110, No. 6, December 2001 J. G. McDaniel and C. L. Clarke: Minimum phase reﬂection coefﬁcients FIG. 7. A schematic of the system in Fig. 1 under the assumption that the object transmits force.

ϭϪ Recalling that F1 ZflV1 , one finds that this impedance acts in series with the acoustic impedance of the fluid to the left of the piston. Therefore, the normalized effective impedance is

zft z ϭ . ͑49͒ eff ϩ 1 zft FIG. 8. Real and imaginary parts of normalized object admittance, y ft , for A schematic of the reduced system is shown in Fig. 7. This an object that transfers force and has a high magnitude of admittance. reduced system can also be represented by parallel admittances, so that the effective normalized admittance is y eff coefficient. Again, the agreement between the actual phase ϭ ϩ ͑ ͒ 1 y ft . The reflection coefficient is found from Eq. 35 , and the causally predicted phase confirms the analytical result that the reflection coefficient is minimum phase. Ϫ 1 y eff Rϭ . ͑50͒ 1ϩy eff VII. CONCLUSIONS It has been shown21 that the drive-point admittance is also a A response-based description of the minimum phase positive real function. Therefore, its real part is always posi- condition in the time domain has been developed and applied tive in the upper half of the complex frequency plane and the to the one-dimensional interactions of acoustic waves with a real part of y is always greater than unity. The condition in eff submerged object. This description decomposes the nonmini- Eq. ͑38͒ is met and the reflection coefficient is always minimum phase reflection into the reflection from an equivalent mum phase. minimum phase structure and the radiation created by a forc- This finding will be illustrated by generating random ing function applied to the object. An energy-based descrip- admittances in the same way that random impedances were tion of the minimum phase condition presented by Eisner has generated in the previous section. The admittance of the ob- also been valuable. In particular, this description reveals that ject is generated by the following Fourier series with real- an object with a minimum phase reflection coefficient returns valued Fourier coefficients: energy to the fluid sooner than an equivalent nonminimum N in␲␻ phase object. ͑␻͒ϭ ϩ ͩ ͪ ͑ ͒ y ft a0 ͚ anexp , 51 nϭ1 ⍀ which is causal and real-valued in the time domain. The passivity condition is satisfied by the choice of an given in Eq. ͑45͒. Again, we set Nϭ50 in all of the examples. ϭ Choosing a0 50 creates a large object admittance that results in a small reflection. The real and imaginary parts of the object admittance obtained by using this value in Eqs. ͑51͒ and ͑45͒ are shown in Fig. 8. In order to show that the reflection coefficient for this object is minimum phase, it was evaluated using Eq. ͑35͒. The magnitude and phase of the reflection coefficient computed from Eq. ͑50͒ are shown in Fig. 9. As in the previous section, a causally predicted phase is computed by inserting the reflection coefficient magnitude in Eq. ͑21͒. This phase coincides with the actual phase in Fig. 9, indicating that the reflection coefficient is indeed minimum phase. This finding persists even when the object admittance is very low. Choosing a ϭ0.1 creates a low admittance whose 0 FIG. 9. Magnitude and phase of the reflection coefficient for an object that real and imaginary parts are shown in Fig. 10. Figure 11 transfers force and has the admittance shown in Fig. 8. Causality is used to shows the magnitude and phase of the associated reflection recover the phase from the magnitude.

J. Acoust. Soc. Am., Vol. 110, No. 6, December 2001 J. G. McDaniel and C. L. Clarke: Minimum phase reflection coefficients 3009 suring the magnitude of reflected sound. Second, the results will provide a basis for analyzing more complex three- dimensional scattering problems. For example, monostatic scattering measurements for three-dimensional structures can be modeled by replacing the structure with an effective impedance screen that is perpendicular to the incident wave front. This will be the subject of a future paper.

ACKNOWLEDGMENTS This work was supported by the Ofﬁce of Naval Re- search under Grant No. N00014–99–1–1017. The authors are grateful for the encouragement and technical advice from Dr. Geoffrey Main and Dr. Luise Couchman during the course of the work. Portions of the work were inspired by conversations with Dr. Richard Weaver. Insights from Dr. Robert Burridge were useful in clarifying the mathematical properties of positive real functions.

FIG. 10. Real and imaginary parts of normalized object admittance, y ft , for 1 A. Papoulis, The Fourier Integral and Its Applications, Electronic Science an object that transfers force and has a low magnitude of admittance. Series ͑McGraw-Hill, New York, 1962͒. 2 S. L. Hahn, ‘‘Hilbert transforms,’’ in The Transforms and Applications A condition on the impedance matrix was derived that Handbook, edited by A. D. Poularikas, Studies in Applied Mechanics ͑CRC and IEEE, New York, 2000͒. guarantees a minimum phase reflection coefficient. While the 3 H. A. Kramers, ‘‘La difusion de la lumiere par les atomes,’’ Estratto dagli condition is impossible to evaluate in cases where only the Atti del Congresso Internazionale de Fisici Como 2, 545–557 ͑1927͒. reflection coefficient magnitude is measured, it did lead to 4 R. de L. Kronig, ‘‘On the theory of the dispersion of x-rays,’’ J. Opt. Soc. Am. 12, 547–557 ͑1926͒. two special cases. In particular, it was shown that submerged 5 H. M. Nussenzveig, Causality and Dispersion Relations ͑Academic, New objects that either transmit force or velocity from one body York, 1972͒. of fluid to another always create minimum phase reflection 6 J. Hilgevoord, Dispersion Relations and Causal Description ͑North- coefficients. The proofs were based on the positive real prop- Holland, Amsterdam, 1960͒. 7 Y. W. Lee, ‘‘Synthesis of electric networks by means of the Fourier trans- erties of drive-point impedance and admittance. The numeri- forms of Laguerre’s functions,’’ J. Math. Phys. ͑Cambridge, Mass.͒ 11, cal examples confirmed these analytical results for structures 83–113 ͑1932͒. with small and large reflection coefficients. 8 H. W. Bode, Network Analysis and Feedback Amplifier Design ͑Van Nos- The results of this one-dimensional study are expected trand, Princeton, New Jersey, 1945͒. 9 E. A. Guillemin, Synthesis of Passive Networks ͑Wiley, New York, 1957͒. to be useful in two contexts. First, the results will be appli- 10 T. T. Wu, ‘‘Some properties of impedance as a causal operator,’’ J. Math. cable to problems involving acoustic transmission through Phys. 3͑2͒, 262–271 ͑1962͒. nonhomogeneous layers that satisfy either the force or veloc- 11 T. K. Sarkar, ‘‘Generation of nonminimum phase from amplitude-only ͑ ͒ ͑ ͒ ity transfer condition. Here, the results allow one to charac- data,’’ IEEE Trans. Microwave Theory Tech. 46 8 , 1079–1084 1998 . 12 V. Mangulis, ‘‘Kramers-Kronig or dispersion relations in acoustics,’’ J. terize the frequency-dependent layer properties by only mea- Acoust. Soc. Am. 36, 211–212 ͑1964͒. 13 R. P. Radlinski and T. J. Meyers, ‘‘Radiation patterns and radiation impedances of a pulsating cylinder surrounded by a circular cage of parallel cylindrical rods,’’ J. Acoust. Soc. Am. 56, 842–848 ͑1974͒. 14 J. G. McDaniel, ‘‘Applications of the causality condition to one- dimensional acoustic reflection problems,’’ J. Acoust. Soc. Am. 105, 2710–2716 ͑1999͒. 15 E. Eisner, ‘‘Minimum phase for continuous time and discrete time functions,’’ Geophys. Prospect. 32, 533–541 ͑1984͒. 16 M. Tohyama and T. Koike, Fundamentals of Acoustic Signal Processing ͑Academic, New York, 1998͒. 17 R. H. Lyon, ‘‘Statistics of phase and magnitude of structural transfer functions,’’ in Random Vibration—Status and Recent Developments, Studies in Applied Mechanics, edited by I. Elishakoff and R. H. Lyon ͑Elsevier, Amsterdam, 1986͒, pp. 201–208. 18 B. C. Kuo, Automated Control Systems, 6th ed. ͑Prentice Hall, Englewood Cliffs, NJ, 1991͒. 19 J. D. Victor, ‘‘Temporal impulse response from flicker sensitivities: causality, linearity, and amplitude data do not determine phase,’’ J. Opt. Soc. Am. 6, 1302–1303 ͑1989͒. 20 K. B. Howell, ‘‘Fourier transforms,’’ in The Transforms and Applications Handbook, Studies in Applied Mechanics, edited by A. D. Poularikas ͑CRC Press and IEEE Press, 2000͒. 21 E. A. Guillemin, The Mathematics of Circuit Analysis ͑Wiley, New York, FIG. 11. Magnitude and phase of the reflection coefficient for an object that 1949͒. transfers force and has the admittance shown in Fig. 10. Causality is used to 22 A. D. Pierce, Acoustics, An Introduction to Its Physical Principal and recover the phase from the magnitude. Applications ͑Acoustical Society of America, Woodbury, NY, 1989͒.

3010 J. Acoust. Soc. Am., Vol. 110, No. 6, December 2001 J. G. McDaniel and C. L. Clarke: Minimum phase reﬂection coefﬁcients