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116 J. Opt. Soc. Am. A/Vol. 25, No. 1 /January 2008 Andrew K. Gabriel

Fundamental radar properties. II. Coherentphenomena in space–time

Andrew K. Gabriel*

TechConsulting, 1601 Ramona Avenue, South Pasadena, California 91030, USA*Corresponding author: [email protected]

Received March 17, 2006; accepted June 13, 2007;posted August 13, 2007 (Doc. ID 69118); published December 14, 2007

A previous publication [J. Opt. Soc. Am. A 19, 946–956 (2002)] presented a general formulation of radiativesystems based on special relativity, and properties of imaging radar were derived as examples. Complex anddiverse properties of radar images were shown to have a simple and unified origin when viewed as lower-dimensional (temporal) projections of the space–time structure of a radar observation. A diagram was devel-oped that could be manipulated for a simple, intuitive view of the underlying structure of radar observationsand phenomena. That treatment is here extended to include coherent phenomena as they appear in the lowertime dimensions of the image. Various known coherent properties of imaging radar and interferometry arederived. The formulation is shown to be a generalization of a conventional echo correlation and is extended toa second spatial dimension. From this perspective, coherent properties also have a surprisingly simple andunified structure; their observed complexity is somewhat illusory, also a consequence of projection onto thelower temporal dimension of the receiver. While this formulation and the rules governing it are quite differentfrom the standard treatments, they have the considerable advantage of providing a much simpler, intuitive,and unified description of radiative (radar and optical) systems that is rooted in fundamental physics. © 2007Optical Society of America

OCIS codes: 120.1880, 120.3180, 280.5600, 280.6730, 350.5720.

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. INTRODUCTIONany of the complex and seemingly unrelated properties

f radar returns and other radiative signals are usuallyhought of as separate phenomena. Radar signals in par-icular (Appendix A, Table 1, contains a comparison of ra-ar and optical terminology) were shown [1] to be projec-ions of a remarkably simple higher-dimensional (space–ime) structure onto a lower-dimensional receiver thatas temporal but not spatial properties. Much of the com-lexity of radar echoes was shown to originate not fromhe target (or scene), but from the projection of a signalecho) with multiple dimensions onto a lower dimensionaletector. This effect, “dimensional folding,” can produceomplex, multiple-valued signal functions in recordedata, but these functions originate from simpler, single-alued functions in the higher dimension(s). A veryimple spatial analogy is the shadow (projection) of a ringcircular shape) down one dimension onto a plane from aoint illumination source. In three dimensions, a circle isnly a circle (unified description). Its shadow on a plane,etermined by its orientation, can be a continuum of el-ipses (higher complexity); a line in the degenerate case,or which the points (excluding end points) map into twooints on the original circle (multiple values); or the spe-ial case of a circle (no change except possible rescaling).

In a real observation, as in this analogy, the structuref the observation rather than that of the target [1] deter-ines the dominant characteristics of the recorded echo,hich may itself differ in subtle ways from the physicalcho. The theory presented in [1], derived from the ideasf relativity, showed that the same comment can be madeor many other phenomena in radar (or radiative)

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ignals—that they express primarily the structure of thebservation rather than the characteristics of the sceneand can contain subtle errors that are nearly invisible inore conventional formulations). As an example at opti-

al wavelengths, a distant star seen through a conven-ional telescope produces an image consisting of the famil-ar diffraction pattern from a circular aperture. Such anmage, some scaling of the first Bessel function, primarilyxpresses the structure of the observation rather thanome property of the star: one looks at a star but sees theelescope.

Radar phenomena [2] are quite extensive but can be ex-lained in this manner for both conventional and syn-hetic aperture radar (SAR). A partial list includes rangeesolution, speckle and coherence, slant range obliquity,nite antenna size, and beam- versus pulse-limited con-gurations. Most of these, along with other properties,ere shown [1] to be lower-dimensional projections of a

elatively simple “causal channel” associated with an ob-ervation, most importantly, the curvature of the world-ines that form the channel. That curvature is at the coref a unification of radar phenomena and is relevant to op-ical (and other) wavelengths.

Reference [1] deals primarily with how diverse radarhenomena emerge from this underlying unity andouches only lightly on the coherent, or frequency-ependent aspects of radar, which, inclusive of interfer-metry, constitute a large subject. The purpose of thisork is to generalize the previous paper to phase-

oherent and interferometric phenomena (with occasionalptical analogies) to show how the conventional impulseesponse of a radar, described in [2], can be expressed

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Andrew K. Gabriel Vol. 25, No. 1 /January 2008 /J. Opt. Soc. Am. A 117

ore generally as space–time curvature, to describe howoherent radar phenomena also arise primarily from thebservation rather than the target, and to generalize thepace–time picture to a second scene dimension (azi-uth).The emphasis throughout is on applying the unified

pace–time picture to radar phase measurements, ratherhan on properties that are already extensively docu-ented. The purpose is to show that this visualization is

seful as both a heuristic for understanding coherent ra-ar from a simple and unified point of view and as a toolor doing radar phase calculations; to this end some well-nown results are reproduced. The figures are often exag-erated (especially curvature) to show functional behav-or. This is not necessarily unrealistic; even small effectsan be quite visible in interferometric radar because theavelength is so much smaller than the spatial dimen-

ions of the scene (�10−7 for both a C-band radar satellitend a 5 m optical path with green light).

. SPACE–TIME, LIGHT CONES,ORLDLINES, AND RADIATION

his section summarizes the already brief treatment of1] on some of the basic concepts of relativity; a full treat-

ent may be found elsewhere [3]. In the theory of specialelativity, space becomes a four-dimensional entity wherehe extra dimension is time. Scaling this dimension by c,he constant speed of light, yields the “timelike” fourthength dimension. Any stationary point in Euclideanpace exists for all time, and thus in space–time becomesline, or “worldline,” that runs parallel to the ct axis. An

ccelerating object has a curved worldline (for example, aarabola for a constant acceleration).Isotropic radiation originating at some point �x0 ,y0 ,z0�

n Euclidean space can be thought of as an expandingpherical wavefront. In two spatial dimensions, the inter-ection of any plane [say, �x ,y�] through that point withhe expanding sphere is a circle that is expanding radiallyn the plane. With a time axis established perpendicularo the plane (i.e., replacing the z axis), the locus of pointsn the expanding circle forms a cone with vertex at �x ,y�,xis of symmetry parallel to the ct axis, and central anglequal to 2�� /4�=� /2. This “light cone” is a geometric formepicting the location of radiation in space–time. Likeutgoing radiation, incoming radiation incident onx0 ,y0 ,z0� in space must be on a light cone, now rotated by

so that the open end faces the −ct direction. The sur-aces of the cones are the boundary in space–time be-ween past and future (defined by the speed of light),hich is why they are sometimes called “causal” cones or

urfaces.The simple configuration used in [1] (and continued

ere) is shown in Fig. 1. It consists of an antenna (or anyadiative source) at height z0 emitting a spherical pulse ofadiation of physical duration �0. It produces an illumina-ion pattern over flat (planar) ground that begins as aingle nadir point (x0�0, y0�0) at time t0=z0 /c and ex-ands radially in the �x ,y� plane (a circle) until time t0�0. It then becomes a radially propagating annulus withn asymptotic radial length of c� .
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A function called aT�r , t� in [1] was developed to de-cribe such a pulse of radiation; it is defined to have aalue of unity when there is illumination present at somer , t� and zero otherwise. In the plane �x ,y� where r2=x2

y2, aT� �x ,y , t� and the circles aT� �x ,y , t−�0� form the pe-iphery of the expanding annulus; aT�·� is the two expand-ng circles and the space between them. In two-imensional space–time, for �0=0, aT�x , t� is theyperbolic intersection of the light cone originating at theource (transmitter, T) with the �x , t� plane. An incomingight cone aR� �x , t� (terminating on the receiver, R) also hashyperbolic intersection with the �x , t� plane. The two hy-erbolas of intersection can be identified as the worldlinesf a �0=0 transmitter pulse and �0=0 receiver windowwhich can be visualized as a time-reversed transmitterulse) as they appear in that plane. The hyperbolas willntersect only if � exceeds 2t0=2z0 /c, the nadir round-tripime to the source (antenna). The locus of points wherehe two cones intersect represents the areas of space–timeommon to both aT� �x , t� and aR� �x , t�—that is, where cau-ality allows communication between the transmitter andeceiver. The four light cones of Fig. 1 represent the situ-tion for a nonzero value of �0, now the temporal durationf the outgoing pulse and of the incoming window.

The hyperbolic intersections of aT� �x , t� and aR� �x , t� withhe (x , ct) plane are shown in Fig. 2. Most of the analysisn this study, like that of [1], is based on examination and

anipulation of Fig. 2 as it represents different radar (orimilar optical) situations. In Fig. 2, the time has beencaled to �0, and the distance is shown only as function ofhe bandwidth, ��0�−1. The scaling of these physical quan-ities plays a large role in radiative phase sensing, espe-ially in the relativistic picture herein. The image axesx , t� as presented scale together because both are scaled

ig. 1. (Color online) Light cones representing the space–timeegions of an outgoing radar pulse and an incoming receiver win-ow. As observed along the antenna worldline, the pulse starts athe leftmost vertex and stops at the vertex �0 later; similar com-ents apply to the incoming window aR�x , t�. The pulse length �0

s then the distance along the time axis between the like-orientedone vertices. The cone intersections with the �x , t� plane are theyperbolas aT�x , t� and aR�x , t�. The small diamond-shaped win-ow is the region where causality allows the transmitter and re-eiver to communicate.

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o �0. However, the distance z0 is independent of x, so theyo not in general scale together. Similar comments applyf there are different time scales, for example, a band-idth that is not derived from the carrier (center) fre-uency but from some independent time scale, such as thentegration time of a filter.

In [1], the diamond-shaped area in Fig. 2, the productf aT�x , t� and aR�x , t−�� is referred to as the causal chan-el ��x , t�. It was shown that the output of the receiver�·� is the correlation

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� aT�x,t�� aR�x,t�. �2.1�

�x , t� is nominally the space–time area of ��x , t�, whichetermines its scattering power. Changes to the offsetariable � is a translation of aT�x , t� with respect toR�x , t� and performing the correlation ��x , t� over someew integration region where ��x , t� is nonzero. In the fareld (��0 and ��z0 /c), the limit where most radars op-rate, if the pulse is band limited, then the resolution isefined by the above integration along the target world-ine within the channel ��x , t�. Directly below the trans-

itter at the “nadir” or “subradar” point (the axial centern optics), when �=�0, ��x , t� is a maximum, resulting in aarge, bright resolution element (rezel); the usual radarerm or this effect is “layover.”

. VISUALIZATION OF PHASE INPACE–TIME

n Figs. 1 and 2, the unscaled time axis would have di-ensions of time �t� and the unscaled space axes would

ig. 2. Space–time components of radar light cones in the �x , t�lane. The transmitter pulse worldlines aT�x , t� are the two linesising from left to right, representing the apparently superlumi-al motion on the ground of a transmitted pulse of length �0. Thehase velocity c� is infinite at nadir and is asymptotic to c forarge x. Similarly, the receiver worldlines aR�x , t� fall from left toight. The offset variable � is the interval along the time axis be-ween the outermost two traces, equal to 2z0 /c when the emittedulse first returns to the receiver. The diamond-shaped area ishe spacetime channel, and the horizontal line is the worldline ofstationery target at some distance xt.

ave dimensions of length �L�. The phase velocity of light� [the slope of the a*��x , t� functions] has dimensions of/ t, as does the usual physical velocity c (slope at x→�).he units are a matter of choice but determine the nu-erical values of c (relative scaling of L and t) and of c�.hase is dimensionless (some count of rotations usuallycaled by 2�) but is proportional to time when one periods identified along x=0 with some fixed interval �. Simi-arly, when a cycle of phase is tied to some fixed length �,t is proportional to distance. There are subtleties of thehysical meaning of transmitter or receiver phase thatill become apparent (Sections 6 and 7), which stem from

he fact that the receiver and transmitter can be phase se-ial rather than time serial. In that case transmitter andeceiver time is inferred from what is actually measured,hich is a numerical count of rotations of an oscillator,ot physical time. Time can be calculated from the phasebserved along the time axis as �= t /�. However, a localhase ��x , t� depends on the curved isophases that are de-ermined by the a*�x , t� functions, requiring a correctionlong with any other sources of phase error in order to ar-ive at the correct time. More subtle and complex errorsan also arise (below) when an uncorrected echo phase issed as feedback to infer other properties such as inter-erometric alignment.

. Phase along the t Axisor visualization purposes, Fig. 3 treats phase as a quan-

ity separate from time or space by assigning it to an axisrthogonal to the t and x axes of Fig. 2 (note that t is innits of �0, x in units of c�0). Adding the transmitter andeceiver phases (sloped lines in the plane through x=0 inhe phase plane) together in the above diagram is equiva-ent to frequency conversions such as carrier removalheterodyning); it is only meaningful when aT� �x=0, t�0 /2� and aR� �x=0, t−�0 /2� intersect (equivalently when

�x , t� is nonzero). It is possible to visualize the phase re-ationships in the scene �x0� by viewing the temporalhase as a time shift of the vertex of a light cone. For ex-

ig. 3. Visualization of phase as an extra dimension of space–ime (phase can be measured only along x=0). The transmitterhase is chosen to increase in time; the receiver phase is chose toecrease. The pitched line segments beneath the �x , t� plane arehe phases of the outgoing and incoming pulses (temporal length0) referenced to zero at the phase axis origin, which was chosens the halfway time between transmission and reception (theonvention for the phase origin is usually �0=0 at the antennaenter with unspecified temporal origin).

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mple, in Fig. 2, the time at the center of the outgoingulse can be represented by an identical curved line mid-ay between the two lines defining the edges of aT�x , t�nd similarly for the center of the receiver window andR�x , t�. The midlines are then the locus of points inpace–time �x , t� corresponding to the times at the centersf the temporal windows a*�0, t�. Expanding, the phase inpace–time can now be visualized for the time period �0ithin the windows aT�0, t� and aR�0, t�. The transmitterr receiver oscillator phase (i.e., t axis scaled by the car-ier period 2� /c) becomes a contour of constant phase inpace–time (isophase) that runs parallel to the worldlines*�x , t�; this is shown in Fig. 8 below.

ig. 4. (Color online) Visualization of phase as an extra dimen-ion of space–time (rotated for better perspective). The transmit-er phase is represented as an increasing linear ramp imposed onhe outgoing aT�x , t�; the receiver phase is represented as a con-ugate (decreasing) ramp imposed on the incoming aR�x , t�. Thentersection area shown in the far field is the causal channel. Theombined (heterodyne) phase inside the channel increases withistance but does not change in time, as is expected for a station-ry target.

ig. 5. (Color online) Space time rezel in the far field with trans-itter and receiver phases visualized as an extra dimension;ithin the channel the phases are added to represent the re-oval of the carrier frequency. The phase of a target (measured

t the receiver) is the result of the temporal integration over thehannel [1].

The space–time phase surface ��x , t� is shown in Figs. 4nd 5 (both rotated for perspective), where the transmit-er or receiver oscillator phase is represented as a linearincreasing or decreasing) ramp within unit rect(·) func-ions of width �0 that represent the temporal windows athe antenna location, aT�0, t� and aR�0, t�. This is slightlyifferent from the phase shown in Fig. 3, where the trans-itter and receiver phases (pitched line segments) are

ffset from the �=0 plane; this amounts to ignoring somerbitrary initial phase.The intersection of the worldline functions aT�x , t� and

R�x , t� in Fig. 4 will be recognized as the causal channel�·�, which itself defines a range rezel [1], again withhase represented as a dimension as in Fig. 3. Figure 5 isplot of the causal channel and combined transmitter

nd receiver phases therein. In Fig. 5, the phase increasesinearly with distance x and for �0=0, �=2��x� /�, where �

s the range; �=�x2+y2. Also, the phase is constant inime (constant shade across the rezel in the time direc-ion) because the rezel is stationary; that is, a point targetorldline has only one shade in the rezel, expressing thexed phase of a nonmoving target.As noted above, the changes in the rezel with range

equivalently, with look angle or scene slope) determineecorrelation, resolution, layover, and various other phe-omena. Figure 6 shows the phase in the same manner asig. 5 but at the nadir (closest point), which is the limit-

ng case associated with layover [1].

. Wrapped Phasehe above visualizations can be extended to include�-ambiguous or “wrapped” phase. In Figure 3, this isone by changing the monotonic phase inside the channelo a sawtooth, shown in Fig. 7. This phase wrapping cane included in Figs. 5 and 6 as shading, which results inhe contour plots in Figs. 8 and 9.

Inside the space–time channel ��·� in Fig. 8, carrier re-oval [implicit in the choice of opposite phase directions

or aT�x , t� and aR�x , t�] appears as a rescaling of the x di-ension; that is, each small element in the causal chan-el is a miniature of the one in Fig. 5 (when visualized inhe far field, where there are insignificant curvature ef-ects).

Similarly, Fig. 9 shows wrapped phases, now for thelosest rezel (at nadir). Figure 9 once again reveals tem-

ig. 6. (Color online) Transmitter and receiver phases visual-zed as an extra dimension in space–time for the nadir (closest)ezel.

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orally constant phase for nonmoving targets; however, inontrast to Fig. 8, the phase as a function of x is nonlin-ar, expressing high curvature in the near field associatedith the phase velocity c�c, the local slope of the world-

ines a*��x , t�.It has been emphasized in the foregoing that the phase

an be thought of as the scaling of the time or space axiso some interval � or fixed length �, which turns out (be-ow) to be an important notion. It is possible for phaseimply to scale with t and x; this occurs if there are no in-ependent times, time constants, or lengths, which is thease for a light cone representing free propagation. Ifhere is curvature (not scalable) in the worldlines a*�x , t�,hase does not scale with x or t. Such a phase shift ishat is measured in interferometry, where the interfer-meter baseline is an independent length [for example anbservation from a different altitude z1�z0+h, where h isn independent physical length, implies different curva-ures in a*�x , t�]. Finally, even if curvature is present, it isossible that a combination of curvatures will cancel, re-ulting in some scalable set of phases in some part of (or

ig. 7. Sawtooth phase representing 2� (here [0,1]) phase wrap-ing with four cycles of phase.

ig. 8. (Color online) Shaded visualization of phase wrappingn aT�x , t� and aR�x , t� worldlines (phase dimension perpendicu-ar to the page, shown as shading). As in the previous plots, theutgoing phase is presented as having the opposite slope of thencoming phase in order to visualize carrier removal.

ll of) the scene. This useful property, which results fromancellation of worldline curvatures [4], occurs in somebservation geometries and is the mathematical core ofeformation interferometry; [4] contains a discussion andome experimental results.

. TEMPORAL IMPULSE RESPONSE ANDORLDLINES

f the one-dimensional reflectivity of a scene is the func-ion �x�, where x is the distance from the nadir, the sceneimension x�t� is given by the hyperbola

x�t� = ��ct�2 − z02. �4.1�

urther, if the radar emits some temporal waveform w�t�tarting at t=0, the reflected signal later received back athe antenna [2] is R��, the correlation of w�t� and �t�:

R�� = �x�t�� w�t�,

here �� indicates temporal correlation with offset vari-ble �. When R�� is then correlated against the conjugateaveform w*�t�, there results a compressed (in radar,

ange compressed) response, also called the image ��,hich is easy to confuse (below) with �x�.The waveform w�t� may or may not be scalable in time;

or example, if w�·� is a fixed number of temporal phaseycles, it does scale. If w�·� is a chirped waveform with anite bandwidth 1/�0, it does not scale if �0 is an indepen-ent time constant (not originating in a master oscillatorhat controls the transmitter and receiver). Similarly, thenitial phase of w�t� is derived from a master oscillator orefined independently, which may introduce phase curva-ure and errors. In either case, a measured phase can beelf-referential (tautological), which in turn can mask er-ors (below). The Hubble Telescope mirror had a spatialrror of this type; the mirror was misshapen because aeference interferogram (spatial) of the mirror’s surfaceas compared with the flawed apparatus that generated

t, causing global phase cancellation (i.e., tautologicalhase), which masked the independent shape of the mir-or.

By the above definition, the image ��·� is given by

�� = �x�t�� w�t�� w*�t�� = �x�t�� I�t�, �4.2�

here the temporal impulse response associated with theaveform w�·� is

I�� = w�t�� w*�t�. �4.3�

he space–time picture [1] is expressed in the precedingime-domain derivation through the relationship betweenand t, as some global offset �, as a spatially dependent

ffset ��x� (conventional view), or as a property of theorldlines a*�x , t� (space–time view).

. Approximation 1: Conventional Far-Field Responsehe far field of x�t� is where ct�z0. In this case, Eq. (4.1)an be approximated as

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nd the image becomes

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hich is a standard result. It can be visualized in space–ime as the waveforms w�t� and w*�t� translating alonghe far-field worldlines a*��x , t�= ±ct, temporally offset by �Eq. (4.3)]. In the far field, both x and t are linearly re-ated to �, meaning there is no implicit curvature in x�t�.his situation can be seen in Fig. 8, where the phase lines

isophases) are straight and at angles ±� /2 to the timexis. The four ramp functions shown are identified withhe waveforms w�x−ct� and w*�x−ct�. Changes in the cor-elation offset � of Eq. (4.4) can be visualized as the out-oing isophases shifting horizontally against the incom-ng isophases. The possible independence of some initialscillator phase �0 is hidden in this visualization becausehe four ramps were chosen to start at zero phase regard-ess of time; in a real waveform, the phase can changeith � either independently or in synchrony with the os-

illator. The latter case can also have a periodic initialhase, as from a nonzero heterodyne frequency.A point target �xp� at some xp has a horizontal world-

ine (Fig. 2); the integrand of Eq. (4.4) is nonzero only foralues of � where xp, w�·�, and w*�·� overlap. The length ofhe segment of the target worldline inside the rezel is ariangle function [1] with a half-width (resolution) of �0.he phase of the echo at the outside edges of the ramp ishe phase of the receiver ramp at t±� /2 less the phase ofhe transmitter ramp at �t /2.

This conventional picture only describes a far-field re-el [no curvature in a*�x ; t�] but can be extended to ap-roximate the general case. In either, the signal from theeceiver is a projection of the causal channel onto a lower-

ig. 9. (Color online) Space–time visualization of wrapped phasel (and, so, x rezel size) and creates a spatial phase chirp, visivenly spaced in the x direction (the tick marks drawn to the righn x).

imensional time-serial (or phase-serial) receiver along=0. The central point of the space–time formulation ishat the curvature of the a*�x , t� worldlines determineshe characteristics of the projection and presents a unifiedescription of coherent radar (or radiative) phenomena.

. Approximation 2: Limited Worldline Curvature at theubradar (Closest) Pointhe distance x�t� in Eq. (4.1) is often approximated as

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he image �� from Eqs. (4.3) and (4.4) correspondinglyecomes

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his implicit expression means that some curvature [non-inear x�t�] in the image �� will appear on the time scale� t0=z0 /c (near nadir) where the quadratic correction inq. (4.5) is significant. A better way to understand this isy solving Eq. (4.1) for t�x�:

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hich in the near field x�z is approximately

adir, where the worldline a*�x , t� both increases the causal chan-m the isophase contours (shade) in the channel, which are notlaced at each full cycle of phase in x; they are not evenly spaced

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ow the image � can be written as

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here t�x� is given by Eq. (4.6), yielding the implicit de-endence ��x�.The correlation offsets (t and �) are deliberately left in

q. (4.7) to indicate that the correlation is in time, not x;hat is, the curves w�x� are displaced in time [a generali-ation of the standard correlation of Eq. (4.4) to include aonlinear relationship between x and t]. Equation (4.7) isritten this way because the receiver has only time-serial

or phase-serial) properties (in interferometric configura-ions, there is a spatial baseline in x or z but which mighte inferred from temporal data).In Eq. (4.7), the spatial properties of the image ��x� are

ow expressed in the reference functions w�t�x�� and*�t�x�� in Eqs. (4.6) and (4.7), rather than in the scene�x�t�� as in Eq. (4.5). That is, the temporal impulse re-ponse becomes space variant in x when the worldlinesssociated with the illumination phases are curved. Theunctions w�t� have now been generalized to w�t�x��,hich can be seen to be equivalent to the functions*�x , t�. This formalism keeps the basic conceptual struc-ure of temporal signal processing [Eqs. (4.2) and (4.3)]hile including the space–time curvature that produces

uch phenomena as large near-field rezels and speckle.he change to a higher dimension appears as the seman-ics of the algebra; viz., a*�x , t� replaces the implicit�t�x��. The change assigns the effects of curvature, origi-ating in independent physical times and lengths, to theeference functions a*�x , t�, which are defined on a domainf space–time, whereas w�t� was defined only in time.hus the spatial variance ��x� of Eq. (4.4), which is both

mplicit and awkward, is simplified when set in more gen-ral terms. This distinction is mostly formal, since the re-ult ��t� is unaffected, but is the connection between theonventional and the relativistic pictures.

. Exact Image Solution: Curved Worldlinesmore formal description of the one-dimensional space-

ariant impulse response may now be constructed by not-ng that the function ��·� is itself [Eq. (2.1)] a space-ariant generalization of the impulse response I�� in Eq.4.3):

�� = �x�� aT�x�,t��2aR�x�,t�� = �x�� x�,t�,

�4.8�

here the quantity �2 in the second correlation indicatestemporal offset (i.e., making the spatial structure in Eq.

4.8) a dependent quantity). All Eq. (4.8) really says ishat the impulse response, usually thought of as a tempo-al, invariant property of the transmitter and receiver,an be generalized as ��x , t� by incorporating the tempo-al reference functions w�t� in the space–time structure of*�x , t�. The previous implicit nonlinear dependence ��x�

n Eq. (4.3) becomes explicit as curvature of w�t�x��; this

an be seen in Fig. 2, where the distance x to the causalhannel x has a nonlinear dependence on the offset � be-ween aT�x , t� and aR�x , t�.

The core idea is that the image �� is derived from thecene �x� by passing through a time-domain receiver, sohere must be an implicit time dependence, i.e., ��x�t��.he above formalism rewrites this as ��a*�x , t��, estab-

ishing the worldlines as generalized reference functionsith curvature.

. Nonscalable Bandwidth and Temporal Curvaturehe bandwidth 1/�0 expresses properties of the transmit-

er and receiver that may or may not be independent ofrequency. A fixed hardware resister–capacitor (RC) timeonstant can be independent, in which case Eq. (4.8)cales to �0:

��/�0� = �x��/�0aT�x�,t�/�0��2/�0

aR�x�,t�/�0�

= �x��/�0��x�,t/�0�. �4.9�

ut a bandwidth defined as a certain number of oscillatoreriods will depend on physical properties of the oscilla-or, including any errors. Phase errors can thus becomenvisible (self-obscuring) if a dependent property is inommon mode with the master oscillator. Further, if im-ge phases are used for feedback to derive or tune systemarameters (a universal practice in interferometry withadar or light), a local phase error can be undetectabletautological) and can propagate to other parts of the sys-em. Even the simplest ranging experiment can lose accu-acy because of errors masked by a tautological phase.andwidth time scaling, a related topic, is treated in de-

ail in Section 7, where a more general formulation of thempulse response is also constructed.

A transmitted signal may incorporate purely temporalhase curvature such as a temporal frequency changechirp) or Doppler shift imposed on the radiation. If suchemporal phase modulation is present, the compositehase in the causal channel (Fig. 2) will result from twondependent curvatures, the temporal chirp and theorldline curvature, which originates in fundamental

spatial) lengths. The composite curvature is projectednto the lower dimension of the receiver, where it appearss conventional radar (or optical) properties. In someases, one curvature may be used to cancel the other, re-ulting in an image (or portion) where “phase-affine” ef-ects such as deformation interferometry will emerge; [4]overs this is more detail and has experimental results.

. VISUALIZATION OF A SECOND SPATIALIMENSIONhe preceding discussion and most of what is in [1] dealith one spatial dimension, but in real radar or radiative

ystems there are two or three. Conventional treatmentsf SAR use three spatial dimensions; however, sceneeight z�x ,y� is nominally treated as a perturbation

�z�x ,y��z0�. In conventional interferometric SAR,eight is derived from the receiver (temporal) phase butsually described as spatial phase. The space–time pic-ure of phase makes a clear distinction between the scene�x ,y� as a two-dimensional spatial quantity and the im-

aht

AIs(ttacs1wt

sclevwda1oiai

iy(riptaFcml�t

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iTottp

cagifiy

sa

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Ftod


ge �� as a one-dimensional (temporal) quantity. Thisas subtle consequences, like the nuanced difference be-ween Eqs. (4.8) and (4.9) (covered in Section 6).

. Space–Time Plane „x ,y , t…n a plane �x ,y� orthogonal to z (e.g., a radar scene), apherical pulse of length �0 emitted from some distanceheight) z0 will illuminate an annulus that expands inime. The illuminated area is the generalization of aT�x , t�o aT�x ,y , t�; similarly the incoming receiver windowR�x , t� becomes aR�x ,y , t�, a corresponding annulus thatontracts in time. This situation is shown for the mono-tatic case of colocated transmitter and receiver in Fig.0. Figure 11 in [1] shows the comparable bistatic case,here the transmitter and receiver are in different loca-

ions (occasional in radar but common in optics).Since the functions a*�x ,y , t� represent the regions of

pace–time accessible to both the transmitter and re-eiver, in the situation of Fig. 10, where there is no over-ap of aT�x ,y , t� and aR�x ,y , t� (null causal channel), nocho can appear from those regions at that time withoutiolating causality. In Fig. 2, the equivalent situationould be for the target worldline to be at some x where itoes not intersect the causal channel, though it may do sot some other time (different offset �). Similarly, in Fig.0, at some later time the outgoing aT�x ,y , t� annulusverlaps the contracting aR�x ,y , t� annulus; then causal-ty allows the transmitter and receiver to communicate,nd there will be an echo if there are suitable scatterersn the causal channel.

It is reasonably straightforward to incorporate timento Fig. 10 by using Fig. 2 with a second scene dimension. Noting from Fig. 10 that x and y are interchangeablesymmetric), the causal zones must become the figures ofotation of the a*�x , t� functions around the time axis des-gnated ax��x ,y , t�. Figure 10 then is of the plane �x ,y� per-endicular to the time axis. The usual offset variable � de-ermines the relative temporal offset of aT� �x ,y , t−� /2� and

R� �x ,y , t+� /2�; a change in � appears in the �x ,y� plane ofig. 10 as expansion or contraction of the annuli. Theausal channel as it appears in the �x ,y� plane is then theutual annulus of overlap, which is nonzero only for a

imited range of � ; in �x ,y , t� space–time the channel�x ,y , t� is the figure of rotation of ��x , t� of Fig. 2 aroundhe t axis.

ig. 10. (Color online) Perspective view of the annulus-shaped whe scene plane �x ,y�. The aT�x ,y , t� ring expands in time, while thf such a time-delimited annulus. When the rings overlap, theyimensions, i.e., ��x , t� and ��x ,y , t�.

Figure 10 was made with some arbitrary choice of z0,hich may now be regarded, as in Figure 2, as a funda-ental length that determines the curvature of the

*�x ,y , t� functions. However, z0 can be a time scaling t0z0 /c; in a receiver that has only temporal properties,hase can then be measured against (scaled by) the quan-ity fct0, where fc is the primary (carrier) frequency. Fig-re 11 shows cutaway visualizations of the functions*�x ,y , t� for three different values of z0 (all with pulseemporal width �0�0). The x and y dimensions scale iden-ically (and have identical curvatures for different valuesf z0) within each plot. However each of Figs. 11(a)–11(c)as a different z0 (different curvature), and so does notiffer from the others by just a scale factor.In Figs. 11, the curvature of the a*�x ,y , t� functions,

dentical in x and y, results in a light cone with curvature.o complete the visualization, Fig. 12 is a generalizationf Fig. 11 to include nonzero pulse length. As noted, if theransmitter and receiver bandwidths are not derived fromhe frequency of the master oscillator, �0 creates an inde-endent time scaling, just as z0 does in space.In Fig. 12, it is possible to visualize the changes in the

ausal channel as a function of correlation offset �. For ex-mple when ��0, the rezel becomes a large circular re-ion on the plane directly below the source. In SAR, thiss the rezel size at the subradar point; in optics it is therst two Fresnel zones (full 2� phase shift) around (x=0,=0) in the �x ,y� plane for a source at �0,0,z0�.In Fig. 6 the rezel is a slice �x ,y ,�0� of the flying saucer

hape that is the figure of rotation of the causal channelround the t axis.

. SPATIAL AND TEMPORAL PHASES INWO DIMENSIONS

n Section 3 it was demonstrated how temporal phaseay be visualized in space–time. This is distinct from

urely spatial phase, wherein some distance, usually theange �, is viewed as a static spatial quantity such as

� = �x2 + y2 + z02, �6.1�

hich is then scaled by 2� /�. The phasor �kx−t� implieshe far-field form of temporal phase as a linear scaling ofhe spatial phase, k�x− � /k�t�. This is not so much inac-

ne functions aT�x ,y , t� and aR�x ,y , t� for some t as they appear in,y , t� ring contracts; Fig. 5 of [1] shows the changing radial widtha planar �x ,y� slice of the causal channel which has two space

orldlie aR�xform

cdfAcasr

ATcwd

th

wfdr

BAi

Fd(m


urate as incomplete, since many basic properties can beerived this way. For example, the range phase can be dif-erentiated spatially [5] to include a simple misalignment.

spatial picture of phase [6] was used to derive what hasome to be called deformation interferometry. Other ex-mples of basic ‘spatial’ radar quantities are propertiesuch as altitudes derived from interferometric phase andesolution.

. Spatial Phase: Conventional Formulationhe usual derivation of azimuth footprint and resolutiononsists of finding the spatial diffraction limit associatedith the antenna aperture and, in the case of SAR, theiffraction limit associated with the azimuth �y� length of

ig. 11. (Color online) Cutaway views of aT� �x ,y , t� and aR� �x ,y , t�,imension y (time is now vertical) for �0=0 (�-function pulse). ThR). Each panel represents a different value of length z0; the resuade identical by scaling.

he synthetic aperture. For an aperture of length L, thealf-width of the main lobe footprint is nominally

yf = ��/L, �6.2�

hich becomes z0� /L at nadir. For SAR, identifying theull footprint size 2yf as the synthetic aperture, theiffraction-limited rezel size (half-width) is the standardesult,

��/2yf = L/2. �6.3�

. Spatial Phase: Alternative Derivationn different derivation of SAR azimuth �y� properties [7]

s more useful in attempting to connect with the relativ-

neralizations of worldlines aT�x , t� and aR�x , t� to a second spatialof symmetry is the worldline of the transmitter (T) and receiverurvature changes in the a*�x ,y , t� mean that the cones cannot be

the gee axislting c

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stic picture. An azimuth rezel of size L /2 at nadir (theull rezel spans y=−L /2 to y= +L /2), y=0, has an associ-ted distance �=�z0

2+y2, which varies a small amountrom y=0 to y=L /2. The spatial phase shift as y increasesrom y=0 to y=L /2 is

d��y = 0� =2�

�d� =

2�

��y = L/2� − ��y = 0��

�2�y2

�z0=

2��L/2�2

�z0. �6.4�

imilarly, for the near rezel edge at some y0, by com-aring the phase at the near and far sides of the rezel, thehase shift for the round trip is

��d��y�� 2�

��d��y� − d��y = 0��

4�

�

�L/2�y

z0. �6.5�

sing the phase definition that d� changes by � in L /2Nyquist limit for L /�) yields the nadir azimuth diffrac-ion limited footprint:

yf =z0�

2�L/2�. �6.6�

he same argument of double-difference phase changed� now for a � phase shift across yf (half of the syntheticperture 2yf) yields the azimuth diffraction-limited reso-ution L /2. To anticipate what is developed below, this isn example of a definition of a coherent property derivedrom phase (here instead of lobe half-width amplitude);his property emerged when a physical length and a spa-ial phase scaling were in resonance, i.e., 2� in L, which islso noted to be a nominal statement of the Nyquist cri-erion.

. Temporal Phase from Spatial Phasepatial phase is conventionally defined in radar as twice

in optics, once) the distance � from the source to a target,easured in units of �. The curvature of an isophase in

pace–time is hidden in the nonlinear expression ��x0

2+z02, and the rezel size does not appear. This has to

e reconciled with temporal phase, which, as in [1] is

ig. 12. (Color online) Cutaway visualization of curved lightones aT�x ,y , t� and aR�x ,y , t� including finite pulse length �0 ashe distance between like-facing cones. The causal channel isow the region of rotation of the small diamond shape (rezel).

reated as a property of a master oscillator (reflecting theractice of radar). Such reconciliation can be representedn Fig. 2 by taking either of the symmetric pairs of edges

T� �x , t� and aR� �x , t�, measuring the time delay betweenhem along x=0 (time is measured only there), aT� �0, t�nd aR� �0, t�, dividing by the radar wavelength and per-aps scaling to 2�. Implicitly, this definition is for the fareld, where the curvature of aT�x�z0 , t� and aR�x�z0 , t�

s small. However, Fig. 2 is more general; the near-fieldhase has space–time curvature, which is a geometricroperty of a cone intersecting a plane. It expresses thendependent length z0, formally assigned to the a*�x , t�unctions [Eq. (4.7) et seq.].

A temporally generated phase then has curvedsophases ��x , t�=constant on the domain �x , t� identicalo the worldlines a*��x , t� but offset by some amount along=0. In the near field, the temporal phase observed inpace (i.e., as a function of x) is thus not linear; it has ahirp (a term usually used in the time domain). A fixedemporal phase change (e.g., 2� at the frequency c of theocal oscillator) does not have an associated fixed length in; rather, it is space variant. Similarly, a phase that is de-ned to be linear in space �x� (as might occur, for example,or emitters spaced along the arm of an antenna) does notap linearly into the time measured along x=0. A 2� spa-

ial phase change (defined by some fixed length �) doesot correspond to a fixed interval on the t axis, but is timeariant. In the far field, it is approximately constant andlso maps to a constant length that partly determines theezel size, or resolution, of the radar. The full picture alsonvolves temporal integration of the echo that is impliedy a finite bandwidth (discussed in Section 4).Spatial resolution in optics or radar is usually specified

s the far-field half-width of the amplitude or power of aentral diffracted lobe of radiation emitted from an aper-ure. From the foregoing, that half-width can be seen toe equivalent to a phase shift of � across a half-aperture,hich is nominally a spatial statement of the Nyquist cri-

erion. That is, the spatial resolution of an aperture islso the Nyquist criterion, two observations per 2� ofhase at the highest frequency, applied spatially in theransverse direction to the main lobe. A more precisetatement would be to invoke the band-limited Nyquistriterion for the spatial bandwidth 1/L, a topic that wille treated elsewhere. The a*�x ,y , t� functions are a map-ing of space into time, linear in the far field x�z0. Therehe temporal Nyquist criterion, familiar in the signal-rocessing description of radar, can be identified as a lin-ar mapping of a spatial diffraction limit. If a linear map-ing is applied (inaccurately) to an observation wherehere is a phase shift from worldline curvature, the re-ulting distortions of temporal phase can subtly violatehat criterion, creating a phase error that will then propa-ate (e.g., the altitude z0 is inferred from interferometrichase or from phase-based alignment of an optical sys-em). This can happen (above) if the detection hardwarencorporates an independent (of the local oscillator) physi-al time constant, making an integration time longer than� phase shift (undersampling). This is in addition to theossible tautological phase occurring from the use of feed-ack in parameter estimation and similar errors intro-uced into the phase scaling operation of heterodyning.

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. Differencing the Phase in Space–Timen Fig. 13, an extension of Fig. 2 can be used in combina-ion with the alternative derivation of spatial phaseabove) to construct a unified space–time picture of spa-ial phase differences as they appear in the time domainhere they are measured in a radar (or in another time-ependent system such as an optical heterodyne detec-or).

In Figure 13, the basic temporal structure of the spatialhase-differencing alternative method of Eqs. (6.2) and6.3) is shown (compare Fig. 2). A black (A) and a gray (B)bservation are made of the same rezel, where the trans-itter and receiver are separated in x by the amount x0.

f the black observation is described as

a*�x,y,t�A = a*�x,y,t�, �6.7a�

hen the gray observation is described by

a*�x,y,t�B = a*�x − x0,y,t − �2�xt,x0��, �6.7b�

here �2�xt ,x0� is a dependent temporal offset that alignsn some way the A causal channel and the B causal chan-el over a target. Per [1], there is more than one way ofefining alignment; the rezel size varies with x, so the twoannot be exactly the same unless x→�.

Equation (6.7) includes a target at some arbitrary xt inhe B observation, which is at a different location on theame worldline (i.e., same z0 and same curvature) as theobservation. Because the local (translated) curvature is

ifferent, the delay (time � between aT�x=0,y , t−�� andR�0,y , t�) of the gray echo differs from that of the blackcho nonlinearly in xt and in x0. The B echo delay is not a

ig. 13. Space–time phases of different observations made fromocations x=0 (black, A) and x=x0 (gray, B). In this visualization,he causal channels of the two observations have been overlaidy shifting the B lines by x0 relative to the A lines. The same lo-ation xt in the scene is observed at two different locations on theame isophases (and accordingly different causal channels fromhe same isophases). The different curvatures result in differenthases across the rezel when observed along the t axis.

caling of the A echo delay by xt /x0, but has a more com-lex nonlinear dependence designated in Eq. (6.7) as2�xt ,x0�.

Figure 14, a corollary to Fig. 13, shows symbolically theorldlines a*�x ,y , t�, for two different observation alti-

udes z0 (black) and z1=z0+Bz (gray) (this geometry islso equivalent to rotating x0 in Fig. 13 by � /2 around theime axis, then substituting Bz for z0). In this unifiediew, as in Fig. 13, the A curvature is different from the Burvature, expressing the new altitude z1, which is itself aecond independent physical length. The A versus B cur-ature change can be seen in Fig. 14 to mean that there is

nonzero double-difference time [Eq. (6.5)] associatedith the two observations, which is the physical basis of

he interferometric phase associated with a z translationbaseline) Bz.

A hypothetical pair of observations can be described us-ng Fig. 13 by asserting the geometry as known indepen-ently (of a master oscillator) and fixed, then consideringhe measurement of echo delays that correspond to differ-nt locations of the causal channel in the domain �x , t�.he B echo from the near edge of the rezel arrives at theeasured delay t4, and from the far edge at delay t3. Theecho has the measured near and far delays labeled t2

nd t1. The phase differences across the rezel are

d�A = ��t1� − ��t2�, d�B = ��t3� − ��t4�, �6.8�

hich correspond to Eq. (6.4) for the length differencecross a rezel.For two observations, the double-difference phase cor-

esponding to Eq. (6.5) is then

��d�� = d�A − d�B = ��t1� − ��t2� − ��t3� + ��t4�.

�6.9�

or the far-field case of free propagation, conversion be-ween time or distance and phase involves simply scalingto a time (inverse frequency 1/ fc) or scaling x to a lengthc. Either way, the units of the speed of light c (slope of

he far-field worldline; note this is c and not c�) then yieldhe other quantity. In Eq. (6.9), ��d�� (A versus B differ-

ig. 14. (Complementary diagram to Fig. 13) A scene is ob-erved at some xt from two different altitudes z0 (gray, B) and z1black, A), resulting in two different curvatures for the worldlines

�x ,y , t�.
*

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nce in phase), measured as temporal phase, appear asonlinear (curved or chirped) phase changes ��x�; equiva-

ently, �d�� has isophase contours that are curved on thearger domain �x , t�. The double difference ��d�� can bedentified [6] as the interferometric phase from two obser-ations spatially separated by x0 (or by Bz), convention-lly called the baseline. A baseline corresponding to a �ouble-difference phase at the receiver is [5] the criticalmaximum) baseline allowed by phase decorrelationithin a rezel and is also the phase at the Nyquist limit

or ��d�� across a rezel.

. Azimuth Phenomena as Space-Variant Rangezimuth �y� properties can be derived by noting that thease of two observations separated by x0 is structurallyimilar to interferometry, to a bistatic configuration, ando a single observation of single rezels, one at x=0 and onet x=x0. The y-dependent properties of those rezels [i.e.,*�0,y , t� versus a*�x0 ,y , t�] then describe azimuth �y�roperties, including phase, at different distances x.In the preceding paradigm, which asserts an indepen-

ent (known) geometry, for the case of nonzero x0, let

r = �x2 + y2, � = �r2 + z02.

ifferentiating,

�d� = rdr = xdx + ydy.

t any fixed x=x0, dx=0, so

d� =y

�dy. �6.10�

oting that for y�x0, ��x02+z0

2 and r�x0, a require-ent of d�=� (which is �d� across two rezels) is equiva-

ent to requiring that d�=�; designating �dy�� as the cor-esponding (small) change in y, Eq. (6.9) becomes

d�

�= 1 ⇒ y =

��

�dy��

.

earranging,

�dy�� = ��

y. �6.11�

omparing with Eq. (6.6), this can be recognized as theiffraction-limited half-width (length) in the far field. Forn antenna of size L=2�dy�� (now spanning a phase of 2�)his the standard result for radar (SAR) azimuth resolu-ion in the far field. It was derived from phase, and theiffraction limit emerged where a spatial phase of �caled to a physical length (spatial Nyquist criterion).

. PHASE CURVATURE, SCALING,YQUIST RATES. Spatial versus Temporal Phasehe spatial description of phase as ��x ,y ,z0� scaled inavelengths (start of Section 6) can yield basic coherentzimuth and range properties (e.g., the diffraction limit)erived from phase. In Section 3 the transmitter temporalhase was introduced, only to be canceled by the conju-

ate receiver phase (resulting in the temporally constantut spatially varying range phase of Fig. 5). This raiseshe question of what the interpretation of temporal phaseeally means.

Phase, a numerical (dimensionless) count of rotationss usually what is transmitted and detected rather thanime. The transmitter and receiver may be phase-serialevices (possibly time-serial if there is an independentime constant) with no spatial properties except perhapshe antenna size, which on the scale of z0 is very smallbelow). Basic relativity says that in the frame of refer-nce of the radiation, moving at the speed of light, thelock (phase) is stopped. The phase of a radar echo or ofn optical interferometer is almost always thought of inpatial terms (i.e., 4�� /�). In radar, phase is actuallyome rotation of the local oscillator at whatever time theulse echo arrives. More subtly, the relativistic view ishat the physical speed of light is axiomatic (the relativ-stic definition is c�1) [3]; measurements of phase or timere then derivative rather than physical. Distance in ra-ar (range), extracted from spatial phase that is impliedy the temporal phase, is even further removed. This isery much at odds with what is intuitive, that an oscilla-or is a physical clock and that c is constant, so x and t areinearly related. It matters only when there are curvedhases (which do not scale). The situation is only slightlyifferent in optical systems, where spatial phase curva-ure can introduce nonuniform delays, then distorting orven degrading interferometric fringes because of finiteoherence times.

Phase information in a recorded radar echo is thusolely a property of the local oscillator (as opposed to apatial property of the target). In a very simple examplef nonozero spacetime curvature, Eq. (4.5), a distortion ofhe image �� occurs. Even though that image �� ex-resses only error-free temporal data, it contains nonlin-ar phase changes from the curvature in the a*�·� world-ines. That curvature expresses the fundamentalphysical) length z0; accordingly, a physical length mea-ured by radar (ranging) has in general a nonlinear map-ing through phase or time into physical length. As coun-erintuitive as it seems, physical lengths and times do notecessarily have a one-to-one correspondence with inde-endent length or time constants as they appear in con-entional formulations of radar. In radar, which is physi-ally temporal, observed phase shifts originating inigher-dimensional isophase curvature cannot be inter-reted correctly solely in terms of the lower-dimensionallocal) metric of receiver time or phase. While the struc-ure of such derivative errors is straightforward in space–ime, their effects on the image may be complex andubtle.

Similar comments can be made about other errors,uch as an inaccurate processing parameter or a temporalrift in an oscillator frequency, and they may or may note scalable. The curvatures of the a* functions alone canake simple errors exhibit complex image effects in the

ower dimension of the receiver, as shown in [1]. Accord-ngly, in attempting to make phase-based measurementsrom radar echo data, the distinction between spatial er-ors and their temporal expression, and how it is inferred,s central.

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Even in a simple ranging measurement there can beisinterpretation of phase and resulting error. A first

cho returned to an antenna at altitude z0 arrives at timez0 /c. The measurement of that interval implicitly comesrom some clock with an internal time constant. However,f the phase of a master oscillator is used as the clock,oth what is taken as physical time and the signal phaseriginate in the same place. This means that in measur-ng an echo, a phase might be used to measure itself,hich would mask any deviation of the oscillator. Thus, inny scenario with isophase curvature, time (or an inde-endent clock) can have a phase shift compared with alock derived from oscillator phase, especially if oscillatorhase was used (feedback) to infer system properties. Theistinction between such phase time and an independentime is important (especially for the small phase shiftshat occur in interferometry) because of the possibility ofnonunique origin, implying possible hidden ambiguities

r errors in phase measurements. This in turn implieshat inferring worldline curvature from echoes can be im-ossible. Additionally, phase errors are very unlikely to beonfined or obvious; as noted in [1], most radar phenom-na can be explained in terms of a*�·� curvatures, so er-ors therein can potentially affect almost any radar prop-rty. Further complication can come from the complexardware and software systems that make up real radars,hich are likely some hybrid of phase-based and indepen-ent notions of time.An obvious way phase errors can propagate globally is

hen there are parameters derived from echoes (feed-ack) that are then used to characterize the larger sys-em. A particularly important radar example is interfero-etric baseline estimation by using (possibly multiply)

eterodyned phases; an optical example is active hetero-yne stabilization of an optical cavity. Pixel averaging,ommonly used to increase the fringe signal-to-noise ration interferometry, has intrinsic limits when there are (im-licitly) errors in the curvatures of the a*�·�. An autofocuslgorithm in a radar processor estimates a temporalandwidth by examining spectra of echoes; image or in-erferogram focusing is then performed with the result.dding misfocused images will reduce speckling, but isery unlikely to improve image focusing, the more soince such errors are systematic (non-noiselike). A fewycles of phase in an echo or interferogram can in generalave global effects; however, knowledge of the details ofny processing is necessary to know the full effect. Theomplexity of errors that can thus appear in an imageakes post facto correction seem an almost hopeless task;

owever, the simplicity of the unified view above indicatestherwise, since multiple effects can have a single origin.ccordingly, the best general strategy for maximizinghase accuracy is to characterize the system by using for-ard methods rather than inverse methods, the latteraving extensive uniqueness issues that originate in theeceiver’s lower-dimensional view of the causal channel.

. Scalingn [1] and in the current work, the notion of scaling be-ween space and time was mentioned in several examplesnd plays a fundamental role. The relative scaling of the xnd t axes (the dimensions of c) is what determines phase

s a simple scale transformation that removes the dimen-ion time; equivalently, it dedimensionalizes distance �x�nto a count of lengths �. In the more abstract relativisticiew, c is taken as fundamental, and space and time arequivalent component dimensions. As long as there is freeropagation (no independent time or length quantities af-ecting the light cone), the choice of units for space andime is arbitrary and basically insignificant. The nonlin-ar case is very different, with universal consequences.hen an independent length z0 was introduced above, the

sophase worldlines became curved, creating a nonlinearelationship between x and t; then the mapping of x into ts through some local isophase slope c�, the phase velocityn the �x ,y� plane. In the treatment of azimuth as a space-ariant translated range (Section 6), a second indepen-ent spatial quantity x0 was the source of a second non-inear phase mapping, which was shown to be the root ofhe coherent azimuth properties of a radar.

Such nonlinearity (curvature) can be a well-hiddenource of phase shifts in radar echoes. A thought experi-ent involving phase sensing illustrates this; a point tar-

et at distance D, probed by radiation of wavelength �, isndistinguishable from a target at 2D probed by radiationf wavelength 2� without independent knowledge of time:he phase count is the same. A more realistic exampleight involve a phase shift of only a few interferometric

ringes between two observations, implying very smallhanges in c� and a small spread of frequencies. From the�1 point of view of relativity, distance is only the scalingf the x axis (similar comments for time); curvature in*�·� does not imply a different c (i.e., c�) because c�1 isxiomatic. It does imply different local scalings of x and t,ut erroneously: there is a nonunique c� (phase velocity)eing asserted mistakenly as c�=1. Curvature then im-lies a range of scalings for the axes, which in a temporaleceiver is equivalent to shifting phase; for a phase-ensing receiver, there can be a spread of units of length,ime, or both.

Wavenumber shift methods in radar [8] exploit differ-nt scalings of time. Specifically the upper and lowerarts of a range bandwidth are used to form two images,nd phase shifts on sloped areas between the two result-ng images are detected. In general, the two will haveome difference in a*�·� curvature that cannot be ex-lained by scaling and that can then used for focusing oregistration.

In implementation, temporal and spatial phase scalingan easily embody the erroneous identification of phaseith time or length. Temporal phase scaling is usually

alled heterodyning and can be accomplished by compu-ational signal processing or (in hardware) by a mixer.patial phase scaling, or spatial heterodyning, usuallyppears in radar as the (apparent) removal of a constantpatial frequency (visualized as a plane sloped relative tohe scene plane). In the temporal case, there is the ques-ion of whether hardware-based heterodyning scales theime or the phase; in software, the sources of system pa-ameters are suspect. Spatial scaling, where ad hoc planeemoval is thought of as spatial when it is in fact based onime or phase, implies possible feedback-based phase er-or. Imaging radars have only antenna size, altitude, andossibly interferometric baseline as spatial properties (a

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otated coordinate system that implies a different z0 issed to account for scene slope as well in [1]). As noted,hose three lengths can actually express hidden temporalr phase dependency if the local oscillator was used inheir determination (feedback)—hence the carefullyarsed notions of known geometry in Section 6.Beyond the scope of this paper, but worth mentioning,

s imposed temporal phase modulation such as a rangehirp, which will change the curvatures of the a*�·� func-ions. Also, the initiation and termination of such a signalan occur at a time determined by an independent clocknonscalable) or by dependent synchronization to a phase;imilar comments apply to the scalability of the temporalandwidth.While a*�·� curvature generally makes global scaling

mpossible, space–time curvatures can cancel, resultingn phases between two coherent observations that cancel.

simple example is a radar interferogram formed by aadar moving twice on the exact same flight path; differ-ncing the interferometric phases (i.e., homodyne) in-okes common-mode phase cancellation, including twice-ade errors, with a global result of constant phase in the

mage ��t�. Deformation interferometry [6] can behought of as scaling the linear far-field phases of two in-erferograms to make them cancel; phases representingegions with deformations (independent physical lengths)ill not so cancel; they do not scale because the two inter-

erograms have different isophase curvatures in the de-ormed region. A more general case is the presence of twoifferent a*�·� with curvatures that cancel in a local regionf an image. In such a region, the two phases are alwaysynchronized, or tautological, meaning that the oscillatorrequency has no effect on the combined phase. Reference4] has an example of this occurring in a single interfero-ram because of nonparallel orbits of two radar observa-ions.

The antenna length L is a fundamental y length (pre-umably known from a tape measure, not phase feed-ack). In general L is very small compared with any otherundamental length and does not significantly affect theorldline curvature of a*�x , t� versus that of a*�x−L , t�.or L�x0 manipulation of Eqs. (6.10) et seq. yieldsy= �L /x0�dx or dy= �L /�x0

2+z02�dx, meaning that azimuth

istance in the limit of small L scales with L. Accordingly,2� shift across L defines azimuth resolution from theyquist criterion as L /2, which is independent of any

ther fundamental length (such as ��z0� or system param-ter, the classic SAR property.

. Nyquist Rates and Interferometryhe Nyquist sampling requirement in essence says thathe phase must be sampled at least every time it changesy �; for a conventional temporal signal, this translates towice the highest frequency in the signal, whether it ap-ears as spatial or temporal phase in space–time. Forand-limited signals, the requirement is for sampling atwice the rate of the highest frequency of the information-earing (modulated) part of the signal. It is straightfor-ard to connect spatial and temporal Nyquist rates for

calable properties by synchronizing the sampling with s

he phase. For nonscalable phases, the connection is moreomplex, but the basic requirement is the same.

Azimuth �y� resolution L /2 is fundamentally spatial,eriving from the antenna size L. An antenna’s lengthay or may not be known without phase dependence.ven such an obvious quantity may be tainted with tau-

ological phase if, for example, the antenna was cali-rated using phase methodology. As an analogy, to mea-ure L with an unscaled tape measure by counting theotations (phase) of the reel seems possible, but thehanging diameter, uncorrected, causes a phase chirp.

Range resolution is fundamentally temporal, connectedo the fundamental time of the inverse signal bandwidth,hich may itself express multiple independent times, de-ending on the details of the hardware. However, a tau-ological phase could creep in if, for example, the band-idth is determined as a certain number of cycles of thescillator.

The corresponding spatial Nyquist rate for the rangex� is twice per diffraction-limited rezel, since a range re-el is the length where d��z0��2�. Whatever frequencyorrectly scales the time axis into the 2� phase for given*�·� functions at that rezel is half the temporal Nyquistrequency for that rezel. Azimuth �y� Nyquist criteria arelmost identical to those for range in the space–time for-ulation. If a target in the scene plane at �x ,0� is dis-

laced by some amount in the y direction, the length�x�=�x2+z0

2 becomes ��x ,y�=�x2+y2+z02. The same no-

ion of two samples per 2� of phase also then implies aemporal scaling for some �x ,y� (usually called the pulseepetition frequency).

In the far field, the spatial Nyquist frequency is ap-roximately constant because of the constant rezel size.ut at closer ranges, rezel size changes (Fig. 9) because of

he curvature of the a*�·� functions that express the fun-amental length z0. An arbitrary rezel at some distance x1as a (spatial) phase shift of � occurring over some length�x1�, which maps into a � temporal phase shift. The

icks on the right-hand side of Fig. 9, which each indicatene cycle of isophase in the observation, have nonconstantpatial separations. This shows that the dependent timenterval corresponding to � is nonconstant, depending onhe local slope (value of c�). But this interval is usuallydentified with the integration time (inverse bandwidth)f the receiver, which is constant. A value of c�c will re-uce the interval over which a � phase shift occurs, so ahift greater than � can occur in that interval. A hiddenampling error may thus occur if the receiver bandwidths derived from an independent physical time intervaluch as a resistor–capacitor (RC) time constant as op-osed to the phase of a master oscillator. A more completeiew of resolution involves discarding the incorrect notionf independent spatial phase � /� and considering insteadhe temporal origin of the phase across a rezel. In thatase, the preceding argument implies subsampling whenwas used to calculate near-field phases instead of c�.ore succinctly, from the relativistic point of view �c�1�

ny error in the a*�·� worldline functions implies an incor-ect mapping between space and time that is scene (slope)ependent and incurs the possibility of subsampling. The
ize of such errors depends on how the a*�·� are con-

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tructed, which can be quite subtle; there may be multipleependencies on scalable and nonscalable quantities thatay be buried as some implicit construction in a com-

uter processing program.Interferometric phase, the phase difference between

wo observations (A versus B in Figs. 13 and 14) of theame rezel from different locations is, in the space–timeiew, a consequence of the different curvatures of theorldlines. The worldlines aT� �x ,y , t�A and aT� �x ,y , t�B can-ot be overlaid globally by linear rescaling. Accordingly,he double difference phase �d� corresponding to a rezelill vary in some nonlinear way on the domain �x ,y , t�.hat irreducible (nonscalable) phase difference is the in-erferometric phase and defines whether the rezel exhib-ts correlation between observations �d��x ,y�� or deco-relation past the spatial Nyquist limit, �d��x ,y��.his construction also defines the Nyquist limits for alti-ude �z� measurements deduced from interferometricringes, as noted in [1]. Therein a slope (equivalent to aotation of the scene) implies a different z0, different a*�·�urvatures, and different Nyquist rates; however, the un-erlying structure is the same as that for interferometrichase (i.e., no varying altitudes in the scene). Phase un-rapping seems a natural candidate for the notion of

lope as a change in the curvatures of a*�·�; finding theorrect a*�x ,y , t� for the two observations would allow op-imal focusing for a given slope as well as correct hetero-ying (implemented temporally as opposed to the sloped-lane approximation) in the image.

. UNIFICATIONhe description of radiative systems using light cones iselatively simple and explains a broad range of radarroperties, as shown in [1] and in this work. The resultingiew of the properties of radiation-based observations asbservable lower-dimensional projections of the cone in-ersections is correspondingly simplified, or unified. Nev-rtheless, the composite picture of the various phenomenand associated errors as they are observed in the spatialr temporal domains can be complex and sometimes quiteubtle. This is especially true with regard to accountingor dependent versus independent variables, scaling andonscaling signals, and implied limits to feedback-based

nference and other inverse methods. Nevertheless, theimple underlying structure of the causal channel [9] ap-lies to errors as well as to signals, implying the possibil-ty of provable or rigorous methods for global optimiza-ion.

The basic concepts associated with the relativistic for-ulation of radiative observation are that isophase cur-

ature and the causal channels are almost universally aeterminant of lower-dimensional coherent properties, es-ecially in the near field. The temporal or spatial diffrac-ive properties correspond to a phase scaling of � across aausal channel that incorporates a fundamental time orength. The inequality �d��x0 ,z0 , . . . ,�0 ,1 / fc . . . �� (thellipsis after the space and time constants indicate otherndependent physical constants) is a basic requirement ofhe Nyquist criterion as it applies to space–time.

In this view, interferometric measurements at the re-eiver are the projection of two different causal channels

nto the lower time dimension, where they are in someanner compared, creating the phases of an interfero-

ram. The apparent complexity of phenomena in the timeomain is now due to folding two (or more) dimensionsnto just one, and it occurs twice. Interferometric phase inhe mathematical sense is the irreducible curvature mis-atch of the causal channels associated with two obser-

ations. The interferometric phase, though conceptuallyimple in the higher dimensions (i.e., Fig. 14), is in prac-ice constructed from complex one-dimensional temporalfolded) signals. Construction of an interferogram is inhis sense attempting to reconstruct the causal channelsrom ambiguous lower-dimensional information to obtainhe irreducible (nonscalable) phase curvatures. Further,urved isophases can cancel locally, creating tautologicalnterferometric phases, which are scalable and havenique properties, including deformation (double differ-nce) interferometry.

This unified formulation produces many radar proper-ies from these few ideas, as a partial list, temporal range,emporal subsampling errors, temporal azimuth Nyquistate (pulse repetition rate), range resolution, azimuthesolution, antenna footprint, azimuth reference chirps,oppler centroid and unfocused processor resolution, in-

erferometric baselines, decorrelation and height resolu-ion, deformation baseline scaling, decorrelation wave-umber shift, and split-spectrum phase shift [8].One possible way to apply the unified formulation isanipulation of the isophase curvature for focusing free

f phase errors. This is a generalization of the more lim-ted conventional focusing that is done in lower dimen-ions. Different values for isophase curvatures focus dif-erent slopes in a scene (conventional focusing assumes aat scene). Possibly such worldline-based focusing, prop-rly implemented, would make phase unwrapping muchasier, perhaps even unnecessary if local slopes are opti-ally focused.

. SUMMARY AND CONCLUSIONdescription of radar and radiative phenomena as the

rojection overlapping regions of light cones onto theower dimensional (time-serial) receiver was given in [1].t was shown that many of the apparent properties of ra-ar images are due to the structure of the observationather than of the scene. The radar image propertieslower-dimensional projections) are often quite complex,uch as the shadow of the edges of a simple three-

imensional frame cube can cast complex shadows on awo-dimensional plane. It was noted that the curvaturesf the worldlines associated with the phases of a radiativeulse were primarily responsible for the complexity of therojections. Then, implicitly, this is true of radar phenom-na, much as the orientation of the frame cube is respon-ible for the complexity of the shadow.

In the present paper, that description [1] was extendedo include coherent radiative phenomena, especially ra-ar. A visualization of radiated phase treated as a scalingf the time was presented and applied to some commonadar scenarios. The range (distance from emitter) imag-ng properties of a SAR were summarily described usinghe conventional approach of signal processing. Using the

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ross-correlation of a waveform with an arbitrary reflec-ivity function �x ,y� as a model for the radar echo, tem-oral correlation against a conjugate waveform was in-oked to calculate a temporal image ��. The temporalmpulse response of the radar was identified as the con-entional cross correlation of the waveform and its conju-ate, and the image �� was shown to be the limiting far-eld case x=ct, where there is no isophase curvature. Aimple approximation showed how such curvature causedistortions in the image, and it was shown that the cur-ature could be accounted for by generalizing the impulseesponse to be space variant. An approximation-free im-ulse response was constructed and shown to be equiva-ent to the causal channel construction of [1].

An examination of coherent properties in two- or three-imensional space–time yielded a basic distinction be-ween scalable phenomena, which are independent of thepatial or temporal scaling of the phase (units of c), andonscalable phenomena, which have nonscalable curva-ure in the isophases of the observations. A simple andeneral graphical representation from [1] was extended toescribe spatial or temporal phase measurements as ex-ressions of such curvature, the interpretation of whichas shown to express rather subtle properties of hard-are and possibly the associated processing. A third

pace–time dimension orthogonal to the z direction washown to produce the basic azimuth �y� phenomena as aonsequence of a second independent physical length x0.he very small length of a radar antenna was shown as a

imiting case that scales the azimuth resolution. Severalorms of an impulse response with different levels of pre-ision were formulated. Numerous coherent properties ofadar were shown to be consequences of worldline curva-ure and the Nyquist requirement that phase not changey more than � across the scaling implied by fundamentalength(s) or time(s). A surprising property of an observa-ion’s phase, that it can be self-referential, or tautological,as developed along with a discussion of applications andossible hidden consequences. The description of radar,ptical, and radiative phenomena based on the projectionf curved isophases in space–time was shown to be a fun-amental physical description that unifies the varied ob-ervable properties of radiative systems.

PPENDIX A: RADAR AND OPTICALERMINOLOGY

Table 1. Terms

Radar Optics Radar Optics

ntenna Aperture Modulation Waveformltitude,eight �z�

Axial distance(L or d)

Nadir, subradarpoint

Center(0,0)

ange,zimuth

x ,y Rezel Fresnelzones (2)

andwidth Bandwith,linewidth

Slant range �r� Paraxialdistance �r�

arrierrequency

Centerfrequency

Scene Subject

hirp Nonlinearphase

TransmitterReceiver

SourceDetector

PPENDIX B: NOTATION ANDERMINOLOGY

c � speed of lightc� � phase velocity of light; c��c; c�=c in

the far fieldWorldline � displacement �x� versus time �unspeci-

fied time origin�Isophase � line of constant phase; herein, equiva-

lent to the worldline of a phase�0 � integration time of a transmitter or

receivera*��·� ,a*�·� � generic expression for light cone or re-

gion of space-time bounded by lightcones

a*��x , t� � worldline in two-dimensional space–time: the line �isophase� where a planeintersects either a light cone or the lightcone boundary of a region ofspace–time

a*�x , t� � region of two-dimensional space–timebetween two bounding light cones usu-ally separated by �0

aT��·� ,aR��·� � light cones associated with a transmitteror receiver

*��x ,y , t� ,a*�x ,y , t� � ax��·�, ax�·� in three-dimensional

space–time��x , t� � intersection of aT�·� and aR�·�, the

causal channel�� ,��x�t� ,�� correlation, space-variant correlation

over the causal channel Eq. �2.1�� x� , �x ,y� � reflectivity of observed scene

�� radar imagew�t� � gating �rect� function or reference

function

CKNOWLEDGMENTShe author acknowledges gratefully the ongoing encour-gement and interest of Richard Goldstein, Giorgioranceschetti, Thomas Prince, and Fred Shair.

EFERENCES1. A. K. Gabriel, “Fundamental radar properties: hidden

variables in space-time,” J. Opt. Soc. Am. A 19, 946–956(2002).

2. G. Franceschetti and R. Lanari, Synthetic Aperture RadarProcessing (CRC Press, 1999).

3. E. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed.(Freeman, 1999).

4. A. K. Gabriel, “Unification of radar phenomena asspacetime curvature: prediction and observation of affine-phase effect,” Opt. Lett. 29, 1533–1555 (2004).

5. A. K. Gabriel and R. M. Goldstein, “Crossed orbitinterferometry: theory and experimental results from SIR-B,” Int. J. Remote Sens. 9, 857–872 (1988).


6. A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, “Mappingsmall elevation changes over large areas: Differential radarinterferometry,” J. Geophys. Res. 94, 9183–9191 (1989).

7. A. K. Gabriel, “A simple model for SAR azimuth speckle,focusing and interferometric correlation,” IEEE Trans.
Geosci. Remote Sens. 40, 1885–1888 (2002).
8. F. Gatelli, A. Monti-Guarnieri, F. Parizzi, P. Pasquali, C.Prati, and F. Rocca, “The wavenumber shift in SARinterferometry,” IEEE Trans. Geosci. Remote Sens. 32,855–864 (1994).

9. T. Gilliam, Time Bandits (Handmade Films, 1981).

fundamental radar properties. ii. coherent phenomena in space-time

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