Keystone formatting [ 1] for spotlight synthetic aperture radar is an alternative to the popular polar format and reduces the computational burden of resampling the polar data to a rectangular grid before inverse Fourier transformation by a fast Fourier transform—the polar format algorithm. After a brief review of relevant geometry and signals including linear frequency modulation and stepped sines, it is shown how to convert a polar-format radar into a keystone-format radar. Reconstructions from simulated data in both formats are shown and the differing kinds of image aliasing are examined. 1 Introduction SPOTLIGHT synthetic aperture radar is a microwave remote sensing scheme in which a radar on a stand-off moving platform emits structured pulses and collects the reﬂected energy as the antenna dwells on a spot on the ground as shown in Figure 1. Under some broad assumptions dis- cussed below, an image of the reﬂectivity of the illuminated ground patch is formed by processing the reﬂected energy, the main step of the processing being inverse Fourier trans- formation. It is common to assume that the emitted waves are planar over the ground patch and that the the variation in propa- gation attenuation over the ground patch is negligible. Both approximations tend to truth as the distance between the radar and the ground patch is large while the ground patch is small. Methods exist to compensate for nonplanar waves [2] and other methods exist to remove both approximations [3, 4, 5, 6]. Linear frequency modulation (LFM) is commonly used for the transmitted signal and a rough sketch of the signal processing of the received signals is, for each reﬂected pulse, to perform a mixing operation with a modiﬁed version of the transmitted signal [ 7] or a similar operation by matched ﬁl- tering or de-chirping, treating these sampled discrete data as points particularly arrayed in the two-dimensional Fourier transform plane of the ground patch (“formatting”), and sub- sequently performing an inverse two-dimensional Fourier transform. The most popular Fourier formatting is polar formatting. *Originally published on October 18, 2019. Available at engrxiv.org. This paper has not been peer reviewed. Jerry Bauck welcomes comments and can be reached by e-mail at this address after left-shifting each alpha char- acter by one position, e.g., f becomes e: kfssz@cbvdl.ofu. ©2019 Jerald L. Bauck xyg(x,y)Figure 1: Collection geometry for spotlight synthetic aperture radar showing moving platform and ground patch to be imaged. A less common format is keystone or trapezoidal formatting [1] which carries a signiﬁcant advantage over polar format- ting. Figure 2 shows these two formats schematically. The purpose of this paper is to describe these two forms of Fourier formatting of spotlight SAR data, describe how to convert a polar-format radar into a keystone-format radar, and to study the differences in particular with respect to aliasing. In this service, two kinds of waveforms that ap- proximate the LFM will be discussed and one used to create image reconstructions of a single small scatterer at the ori- gin of the ground patch from both polar and keystone data, and then to describe in detail how image aliasing caused by sampling in the Fourier plane differs between the two formats. This knowledge can inform system design aspects that aim to tolerate or reduce the effects of aliasing. 2 Geometry and Waveforms Many kinds of waveforms are used in radar including spot- light SAR. LFM is a popular SAR waveform because it has 1 2 GEOMETRY AND WAVEFORMS 2 kxky (a) kxky (b) Figure 2: Sampling schemes of the Fourier transform of the ground patch g¡ x,y¢ by (a) polar formatting and (b) keystone formatting. many nice properties including, after demodulation, the mapping of range to frequency. A detailed analysis of LFM in spotlight SAR is offered in [ 7] wherein discrete polar samples of the Fourier plane appear. Other waveforms are suitable and stepped frequency or stepped sine signals [ 8,9] will be considered herein for simplicity and clarity of exposition since they, too, provide discrete Fourier plane samples. Sim- ilar principles apply to other waveforms with perhaps some modiﬁcations in post-processing [10]. Deﬁne the ground patch as a complex-valued microwave reﬂectivity function on an x-ycoordinate system, g¡ x,y¢ Æ g(x). Ordinarily the support is assumed to be limited to a circle of a speciﬁed radius by the antenna pattern but that will have no bearing here. Let there be another coordinate system, the x0-y0system, sharing the same origin as the x- ysystem but rotated counterclockwise by an angle µand with the radar at a distant point on the negative x0axis. In the rotated system the ground patch is g0¡ x0,y0¢0Æg0¡ x0¢0 where the0notation relates to rotated systems, axis variables, and functions deﬁned thereon. Let the transmitter emit a harmonic signal of radian frequency !,ej!t. Under the plane wave assumption and with a unit scatterer at xs, this creates a reﬂected, backward-traveling ﬁeld in the vicinity of the radar r¡ x,y,t¢ Æej[!tÅk¢(x¡2xs)](1) after removing the bulk delay between receiver and (0,0) and normalizing the propagation attenuation to one. Here, jkjÆkandkÆ¡ kx,ky¢ Æ(kcosµ,ksinµ)is the propagation vector with wavenumber kÆ!/cand cthe speed of light. The receiving antenna acts to sample the ﬁeld at a single point, generating the spatial phasor (remove ej!t) receiver signal r0 x¡ x0¢ Æej(kx0¡2k¢xs). (2) (Reference [ 10] contains a full development from ﬁrst prin- ciples of these basic results and others mentioned herein.) An important concept is that a harmonic illumination of the ground patch with transmitter signal ej!tsamples its Fourier transform G(k)atG¡ 2kˆx0¢ where ˆx0Æ(cosµ,sinµ) is a unit vector in the direction of the x0and kxaxes. The plane can be sampled at arbitrary points by adjusting the angular position µof the radar and the sinusoid frequency !. Spotlight SAR functions to place many such samples of G(k)over a limited region of the kplane and then to ﬁnd an esti- mate of the bandlimited ground patch by Fourier inversion, either by the so-called direct Fourier inversion also known as the polar format algorithm (PFA) [ 2] or by backprojection [11,10]. For example, backprojecting (2) reconstructs the phasor version of ﬁeld (1); performing such backprojections from all µand all !reconstructs a Dirac impulse at xs. How- ever, the usually preferred Fourier inversion method, the polar format algorithm, proceeds by interpolating the po- lar formatted data to a rectangular sampling grid and then inverting by using a fast Fourier transform (FFT). It is the ability to use an FFT that offers a computational advantage over backprojection that makes this preferable, although some architectures [ 11,12] and algorithms [ 13] can reduce that advantage. Even so, the two-dimensional interpola- tion required by the polar format algorithm represents an additional computational stage, usually proceeding by two one-dimensional steps where the ﬁrst step resamples the polar data along lines of constant µ, resulting in a format similar to Figure 2(b). The second one-dimensional inter- polation is along lines parallel to the kyaxis in Figure 2(b) yielding the desired rectangular sampling grid. A family of discrete-frequency, or stepped, sines, can pro- duce the same constellation of polar samples in the Fourier plane as samples of demodulated LFM signals from discrete values of µ, even though some details differ which are not important here; inverting the constellation in either case is the same, however. The process of generating keystone formatted data di- rectly will be described shortly. It potentially somewhat increases the transmitter burden of synthesizing the trans- mitted signals which is the tradeoff for reducing the interpo- lation burden later. An additional advantage of generating keystone-formatted data directly is that the simpler interpo- lation stage should generate less error in the image. There are two kinds of stepped sine waveforms. The ﬁrst is sequentially stepped sines whereby a single pulse is divided into sub-pulses each of which is a sine segment with a frequency different than all the others. Let fÆ!/2¼ be the cyclical temporal frequency and lÆk/2¼Æ¡ lx,ly¢ the cyclical spatial propagation vector. Cyclic and radian quantities will be used as convenient, with similar nota- tional accoutrements, without prejudice. With uniform amplitude, a waveform in which each sub-pulse has a higher frequency than the preceding sub-pulse, where the beginning frequency is fbÆf0, the ending frequency is feÈfb, the number of frequencies and sub-pulses is N, ¢fÆ¡ fe¡fb¢ /(N¡1), and the sub-pulse length is Tp, is, along with its Fourier transform, s(t)ÆN¡1X nÆ0rectµt¡nTp Tp¶ ej2¼(fbÅn¢f)(t¡nTp) S¡ f¢ ÆTpN¡1X nÆ0sinc£¡ f¡fb¡n¢f¢ Tp¤ e¡j2¼f Tp where rect (x)is unity for jxj·1/2 and zero otherwise and 3 POLAR-TO-KEYSTONE FORMAT MAPPING 3 sinc (x)Æsin(¼x)/¼x. (A variation allows gaps between the sub-pulses.) Another kind of stepped-sine waveform is the stacked sine version whereby sines of all the discrete fre- quencies are transmitted simultaneously. With the pulse length Tp, the time- and frequency-domain signals are be s(t)Ærectµt Tp¶N¡1X nÆ0ej2¼(fbÅn¢f)t Ærectµt Tp¶sin¡ N¼¢f t¢ sin¡ ¼¢f t¢ej2¼(fbÅ¢f(N¡1)/2)t(3) S¡ f¢ ÆTpN¡1X nÆ0sinc£¡ f¡fb¡n¢f¢ Tp¤ . The structure of the sequential and stacked sines differs in that the sequential sines, in both time and frequency do- mains, contain an extra phase term. In the frequency do- main this term can cause vigorous interference between sinc lobes if ¢fis too small or if Tpis too small; otherwise, the oscillations are partially limited by the sinc sidelobes and are further reduced in effect due to partial cancellation from phase terms of neighboring sinc sidelobes. For the stacked time-domain signal, if Tpis long enough, the various sines build nicely into the Dirichlet function sin(N¼x)/sin(¼x) [14]. Either version will convey the salient points of this pa- per; since the stacked version is less complicated and easier to interpret it will be preferred here. The sines will be consid- ered to be long enough to avoid signiﬁcant lobing effects and ground patch edge effects. It is not the purpose of this paper to discuss differences in transmitter and receiver design or signal-to-noise ratio for the two types of stepped sines. 3 Polar-to-Keystone Format Mapping Keystone formatting can be obtained by conceptually start- ing with a polar format. As seen in Figure 2, polar-format points in the plane are reduced in the lx-direction by cosµ relative to a point representing a wave propagating in the µÆ0 direction; thus, if a new frequency f0Æf/cosµand consequently l0Æl/cosµare used, then the new sampling point is (l,ltanµ)so that the ﬁrst coordinate is constant for allµ. (Assume jµjÇ¼/2.) When this adjustment is made to all relevant points in the 2D frequency domain, the keystone format is established. It is not necessary to retain the same set of angles when converting a polar plan to a keystone plan. LFM waveforms for keystone format are readily obtained by adjusting the beginning frequency, ending frequency, and chirp rate [ 1]. Consider a prototype LFM signal, also suitable forµÆ0, s(t)Æej¼®t2 between tband te,tbÇte, and zero elsewhere. The instan- taneous frequency fi(t)ÆdÁ/2¼dtÆ®t,fi(tb)ÆfbÆ®tb, fi(te)ÆfeÆ®te, and ®Æ¡ fe¡fb¢ /(te¡tb). Adapting this signal to the keystone format, the parameters become func- tions of µ:f0 b(µ)Æfb/cosµ,f0 e(µ)Æfe/cosµ, and ®0(µ)Æ¡ f0 e(µ)¡f0 b(µ)¢ /(te¡tb), the latter assuming that tband te are the same as the prototype. The frequency domain signal is shifted to the right and stretched1. The intention is that the prototype corresponds to µÆ0 so that fbÆf0 b(0)etc. The same rule as derived in the previous paragraph can be applied to any signal s(t)with frequency support be- tween fbandfe—move the beginning point to f0 band stretch the spectrum by a factor of ¯Æ¡ fe¡fb¢ /¡ f0 e¡f0 b¢ , that is, S0¡ f¢ ÆS¡ ¯¡ f¡f0 bÅfb¢¢ . By now the adaptation to keystone format for a stepped sine signal should be obvious. For a collection of fre- quencies used for polar format at a particular angle µ,FÆ© fnÆf0Ån¢f,n20,1,... N¡1,f0Æfb,fN¡1Æfeª , deﬁne a new collection for keystone format as F0Æ© f0 nÆfn/cosµ,n20,1,... N¡1,f0 0Æf0 b,f0 N¡1Æf0 eª . 4 Polar and Keystone Format Aliasing To demonstrate the keystone waveform principle, several plane waves composed from stacked sines were summed over the simulated ground patch in the manner of Section §2 in the discussion of (1) and (2), mimicking the coherent sum- mation that is normally done in the receiver. This method avoids all interpolation artifacts which is desirable since the present interest is not in interpolation studies but in alias- ing artifacts speciﬁc to the format style. This was done for both polar and keystone formats using FandF0respectively. The ground patch is a square region ranging over ±50 spatial units on each side with the origin in the middle. There was no 1D interpolation in the backprojection because the back- projected functions are known exactly at all points. (This luxury is not available in an actual radar.) The summed ﬁeld was computed over a 256 £256 grid overlaid on the ground patch. There was no rect (¢)applied to the signals so the only windowing in effect is due to the ground patch limits. The scatterer position is in the middle, xsÆ(0,0). Angle µranged from¡30±toÅ30±from 120 angles. Temporal frequencies ranged from fbÆ2 to feÆ6; with cÆ10, spatial frequen- cies for polar and prototype keystone signals ranged from lxbÆ0.2 tolxeÆ0.6. The number of stepped sine frequen- cies NÆ26 so that ¢lxÆ0.016. These are not typical radar parameters but they were chosen for expository purposes. The results are shown in Figure 3 where parts (b) and (d) are the frequency domain versions of parts (a) and (c) re- spectively, computed by a 2D DFT merely to conﬁrm that the constituent plane waves are in fact from the expected formats. (Each plane wave is represented as a near-impulse in the frequency plane. However, since they are not distributed uniformly, each impulse must be weighted according to how much area it represents. The Jacobian for polar format ! rectangular format is of course frequency radius ½; the Ja- cobian for keystone format !polar format is cosµ, and so 1Alternatively, the sample rate of the received baseband signal can be adjusted pulse-to-pulse [1, Section 3.4]. 4 POLAR AND KEYSTONE FORMAT ALIASING 4 (a) (b) (c) (d) Figure 3: Impulse reconstructions and discrete Fourier transforms collected over a restricted range of look angles and signal band- width. (a) Impulse response from polar format data showing alias- ing; (b) Discrete Fourier transform of (a); (c) Impulse response from keystone format data showing aliasing; (d) Discrete Fourier transform of (c). the composite Jacobian needed here for correct weighting is ½cosµ. The reason the mostly non-black parts of Figure 3(b) and (d) are relatively uniform is because each pixel of the image contains several impulses, thus computing a kind of graphical total which is nearly constant across pixels.) Salient features of Figure 3 are sharp, well-deﬁned central spikes for both polar and keystone impulse responses along with lower-level circular-arc ridges appearing in the polar spatial plot and aliased spikes in the keystone spatial plot— these artifacts were left in purposefully and will be discussed shortly. 4.1 Keystone Discussion The keystone reconstruction shows several interesting fea- tures. If the same set of angles is used for both polar and keystone Fourier data, the area covered in the latter case is somewhat greater which likely would cause somewhat in- creased resolution. But for the same reason, the keystone format might result in under-sampling in the region of larger lxandlydue to sample points being farther apart. Taking this idea into the realm of DFT inversion to yield the im- age, the same phenomenon might cause wrap-around in thely-direction of the DFT that wasn’t present in the polar version. The keystone display of Figure 3(d) shows a periodic dis-crete structure in the lxdirection that is not present in the corresponding polar frequency plot of Figure 3(b), imply- ing a periodic discrete structure in the image along the x axis which indeed appears in Figure 3(c). The lx-period is ¢lxÆ0.016 and thus the x-period is 1/2 ¢lxÆ31.25 which is the spacing between spikes in Figure 3(c). (The 2 derives from two-way propagation and carries over from e.g. (1) or (2).) This implies a sample-rate maximum in ¢lxto avoid image aliasing in keystone-formatted data. Interpret G(l)as a series of impulsive ±-ridges parallel to the lyaxis and apply the projection-slice theorem. A ly-projection of G(l), ¯G0(lx)ÆZ1 ¡1G¡ lx,ly¢ dlyÆN¡1X nÆ0±(lx¡lxb¡n¢lx) the inverse Fourier transform of which is ¯g0(x)ÆN¡1X nÆ0Z1 ¡1±(lx¡lxb¡n¢lx)ej4¼lxxdlx ÆN¡1X nÆ0ej4¼x(lxbÅn¢lx) Æsin(2¼N¢lxx) sin(2¼¢lxx)ej4¼(lxbÅ¢lx(N¡1)/2)x(4) where the factor 2 has been attached to spatial variable xfor round-trip travel. (The projection-slice theorem employs the direct Fourier transform; thus, ¯g0(x)expressed above is reversed in xcompared to a central slice of g(x).) When a spatial impulse response like Figure 3(c) is made but with plane waves extending in µover 0±to 360±and a central slice is made at zero degrees and compared to (4), they are essentially identical. The periodicity is not a result of DFT processing but is the periodicity of the transmitted signal. The aliased spikes of Figure 3(c) will disappear if a rect¡ t/Tp¢ window is placed on the transmitted signal (3) such that cTpÇ50, the distance to the edge of the ground patch, or if ¢lxis decreased enough, to 0.01 in this case. 4.2 Polar Discussion The circular-arc ridges in the polar image Figure 3(a) are the same distance from the origin as the spurious spikes in Figure 3(c). The real and imaginary parts of this image exhibit classic caustic formation but the ﬁne details get lost in the magnitude plot of Figure 3(a). If the µcoverage is ex- tended over 0±to 360±, the partial-angle ridges of Figure 3(a) develop into a full circle of ridges at the same radius. The ridges of a full-circle simulation can be predicted with a bit of analysis. 2D Fourier transforms of circularly- symmetric functions such as a full-circle summation of plane waves of various frequencies mentioned immediately above reduce to a 1D transform of the radial variable in both domains, the Hankel transform [ 15]. Let the radial spa- tial variable be rand the radial frequency variable be ½. A continuous-in- µimpulsive ring of radius ain the frequency domain is ±¡ ½¡a¢ . The frequencies Fare used so that there