A Synthetic Aperture Radar Imaging Mode Utilizing Frequency Scan for Time-of-Echo Compression

The synthetic aperture radar (SAR) imaging mode described in this article utilizes the available signal bandwidth to form a narrow frequency-scanning transmit antenna beam illuminating the swath of interest from far to near range. The imaging technique is named frequency scan for time-of-echo compression (f-STEC), because, for a proper choice of mode parameters, the radar echo duration is reduced, i.e., compressed. This article provides a detailed analysis of the f-STEC imaging technique and derives the operational and performance parameters as well as—to the authors’ knowledge—for the first time, analytic expressions for the f-STEC timing constraints. Furthermore, the performance and trade space are reported and compared to both conventional and modern, i.e., digital beamforming imaging modes. The f-STEC imaging technique is shown to be specifically advantageous for SAR systems operating at higher carrier frequencies or an attractive add-on for state-of-the-art SAR instruments.


I. INTRODUCTION
T HE development of synthetic aperture radar (SAR) instruments at higher carrier frequency, such as Ka-band [1], [2], [3], [4], [5], is motivated, among others, by the relative wavelength scaling, which provides an increased bandwidth and reduced physical dimensions of the RF front end. The difficulty in the realization of high-performance [here, in terms of swath width, azimuth resolution, and signal-to-noise ratio (SNR)] spaceborne SAR sensors at Ka-band and beyond is traced back to two main reasons. 1) Operating in conventional imaging modes leads to a small antenna area-possibly even violating the minimum antenna area constrain [6]-of extremely large length-to-height aspect ratio. To elaborate on the reason, consider increasing the carrier frequency of the SAR instrument while keeping the imaged swath width fixed. The consequence of the smaller antenna area is a degradation of the SNR at the receiver, even if the peak transmit power, and by this the power density on the ground, is maintained constant. This is because of the reduced receive power due to the reduced effective antenna area [8], [9], [10]. The high antenna aspect ratio is problematic from an electrical design and mechanical stability point of view. Thus, shortening the antenna length may be favorable and comes with the beneficial side effect of allowing for an improved azimuth resolution [11], but it causes a further SNR degradation and a reduced swath width [6], [7].
Frequency-scanning antennas and radar systems have been known for a long time and are utilized, for example, in automotive radar [19], [20]. The imaging mode described hereafter is known in a general form as f-Scan in [21], [22], and [23]; further work focusing on the processing and the data compression has been published in [24] and [25], respectively. A recent publication on f-SCAN can be found in [26]. In line with the new contributions of this article, the imaging mode is (re)named to frequency scan for time-of-echo compression (f-STEC), aiming to emphasize the echo compression characteristic that will be shown to be an essential element of the imaging mode, relative to the frequency-scanning antenna property. 1 The f-STEC mode has the potential to eliminate the abovementioned problems when moving to higher frequency SAR by exploiting the available trade-space parameters. Specifically, it is shown that f-STEC trades (sacrifices) range resolution for (improved) SNR while leading to a moderate antenna area aspect ratio. Furthermore, the beam-scanning parameters of the technique may be chosen to significantly reduce the duration of the radar echo window without sacrificing swath width. This is a major advantage of the f-STEC imaging technique, which can be traded for a longer transmit pulse duration and lower peak-to-average power ratio. At the same time, f-STEC is also advantageous as a possible add-on imaging mode for upcoming lower frequency SAR missions offering sufficient bandwidth [27].
The basic concept behind the f-STEC imaging mode is straightforward and consists of three key aspects.
2) The steering angle of the antenna's main beam is a function of frequency.
3) The main beam scans the imaged swath from far to near range. The angular direction of the radiation pattern's main lobe (referred to as main beam) of a frequency-scanning antenna is a function of the signal's RF frequency. Thus, f-STEC utilizes the available bandwidth to form a narrow frequencyscanning transmit antenna beam illuminating the swath of interest from far to near range. The frequency-scanning receive antenna beam then collects the (compressed) return echo signal reflected from the ground. The scan direction is a prerequisite to yield compression of the echo window length, i.e., the duration of the radar return echo from the ground, which would not occur when scanning from near to far range.
Several options exist for the realization of such an antenna, ranging from passive leaky wave antennas [28] to active phased array antennas where the frequency dependence is imposed by the transmit/receive modules or a reflector antenna combined frequency dispersive feed array. The latter is shown in Fig. 1 for the simplified case of four color-coded frequencies; here, all feed elements illuminate the complete reflector surface and the different feed element positions result in reflector beams pointing to a different direction.
This article is organized as follows. The f-STEC operation principle is introduced in Section II and compared to other imaging techniques. In Section III, the parameters governing f-STEC operation are introduced together with the equations describing these parameters. Signal timing and echo window length, elaborated in Section IV, are important for determining the extent and position of the imaged swath. Section V derives the SNR equations for f-STEC and two other operation modes; the performance of these systems is then compared to each other in Section VI. Finally, Section VII summarizes the f-STEC operation mode and points out further aspects to be investigated.

II. FREQUENCY SCAN FOR TIME-OF-ECHO COMPRESSION
Spaceborne SAR instruments typically transmit pulsed signals of constant envelope and linearly varying instantaneous frequency, known as chirps. Utilizing a frequency-scanning antenna, the instantaneous frequency variation of the chirp over time translates into a changing direction of the antenna's main beam over time. For f-STEC, the (half power) beamwidth is smaller than the angular swath extent. The frequencyvarying (narrow) beam direction thus scans the swath and Fig. 1. Illustration of f-STEC operation principle realized by a reflector antenna and a frequency dispersive feed array. Here, for simplicity, only four discrete color-coded frequencies are assumed each radiated by a distinct feed element resulting in a narrow output antenna beam.
can be thought of as a spatial and temporal spreading of the signal's frequencies over the swath angles. This is shown in Fig. 1 for the simplified case of four discrete frequencies, each radiated by one antenna element of the feed array. When exciting the feed array by a chirp waveform, the radiation center changes with time and frequency along the feed array, which effectively scans the reflector beam.
It is necessary to understand that the abovementioned scanning property is a consequence of the frequency-modulated (chirp) signal in conjunction with the dispersive antenna, but not an intrinsic property of the antenna itself. Each spectral component of the signal at the terminals of the antenna generates its own narrow beam; if all spectral components were imposed simultaneously, then all corresponding antenna beams would also be generated simultaneously. The technique should not be confused with that of frequency diverse antenna arrays, addressed, for example, in [29] and [30], which, although similar, are not suitable for f-STEC as described here.
The transmit (Tx) and receive (Rx) radiation patterns of a passive antenna are identical as a consequence of the reciprocity of Maxwell's equations and electromagnetic theory within an isotropic medium [28], [31]. Assuming that the same antenna is used for transmission and reception implies that, onreceive, the antenna's main beam will inherently point to the direction of the ground echo. Again, the radiation pattern can be thought of as consisting of many (overlapping) main lobes, each pointing to the distinct direction of the corresponding frequency. This can be understood as the antenna collecting the signal frequencies that have previously been spread over the imaged swath.
To summarize, exciting a chirp signal at the terminals of a frequency-scanning antenna will generate a narrow transmit beam scanning the swath over time. The linearity of electromagnetic scattering causes the echo signal to maintain the frequency of the incident wave. Consequently, the receiving radiation pattern's narrow main lobe is instantaneously pointing toward the echo direction independently of its time of arrival.
On a first view, f-STEC may seem similar to the SCORE [12], [13], [14], [32] operation mode. This is, however, misleading as there are substantial differences. 1) Transmit Pattern: SCORE generates a fixed, wide, and low-gain radiation pattern illuminating the complete swath with the full signal bandwidth, while f-STEC utilizes a time-dependent narrow and high-gain pattern scanning over the swath. 2) Receive Pattern: In SCORE, the antenna array's (digital) excitation coefficients are controlled to generate a narrow pattern that follows the pulse echo on the ground, while in f-STEC, the fixed but frequency-dependent radiation patterns are inherently generated by the antenna, where each beam is pointing to the direction of an incoming monochromatic wave. 3) Timing: The duration of the ground echo is primarily proportional to the swath width (geometry) in SCORE, and thus, imaging a wider swath will increase the echo duration. For f-STEC, the echo duration is an operational parameter (cf. Section IV-B), and its minimum value can be a fraction of the transmitted chirp and shorter than the corresponding swath width.

III. IMAGING MODE PARAMETERS
In the following, we proceed by developing equations that describe the mode parameters assuming ideal, i.e., simplified conditions and by this accentuating the trade space. The developed equations may be used to provide a quantitative instrument design and operational parameter values. At this point, it is sufficient to consider the 2-D incidence plane geometry. The antenna beamwidth in elevation, el , is assumed to be equal to the radiation pattern's half-power beamwidth and, for simplicity, independent of the scan angle and is given by where λ is the carrier wavelength, h ant is the antenna height (in cross-track direction), and γ el is a constant of proportionality, which depends mainly on the antenna type, taper, and illumination.

A. Dwell Bandwidth and Dwell Time
The two main performance parameters are the dwell time and the dwell bandwidth that are defined as the duration during which a point target is illuminated by a single radar transmit pulse and the range of frequencies (bandwidth) seen by the point target during that time, respectively.
With reference to Fig. 2, the angular swath extent is denoted by sw = θ far −θ near , while the available signal (system) bandwidth is B sys = | f e − f s | with the instantaneous carrier frequencies f s and f e at the start and end of the chirp, respectively. Since the direction (i.e., scan angle) of the antenna's main beam is determined by the signal frequency, there is a oneto-one correspondence between the instantaneous frequency and the beam scan angle. Specifically, at chirp start time, the instantaneous frequency is f s and the main beam points to θ s , as shown in Fig. 2, whereas at chirp frequency f e , the beam maximum points to θ e . Here, the angles are measured with respect to an arbitrary reference, taken to be the nadir direction. At this point, no explicit assumption has been made as to whether an up chirp f s < f e or a down chirp f s > f e is transmitted, thus not implying the time dependence of the scan direction, which is irrelevant for the operation parameters derived next.
To ensure that each point within the swath is traversed by the complete antenna beamwidth, it is necessary that the angular scan extent sc = |θ e − θ s | is larger than the angular swath extent by an amount equal to the antenna beamwidth, and thus, sc = |θ e − θ s | = sw + el (2) where |·| indicates the absolute value introduced here to ensure a correct expression for all θ s ≷ θ e . A closed expression can be derived by assuming a linear dependence 2 between the scan angle θ and the instantaneous frequency f , cf. straight line in Fig. 2, which is given by where + and − correspond to an up and down chirp, respectively.
To determine the dwell bandwidth, B w , consider a point on the ground at angular position θ 0 . The instantaneous frequencies f a and f b at the edges of the beamwidth, i.e., when the antenna beam moves into and out of the point, are obtained by inserting θ = θ 0 ± el /2 into (3) yielding respectively, from which the dwell bandwidth (cf. Fig. 2) is readily obtained as which, when reformulated by inserting (2) finally gives 3 Although simple, the above expression describes the parameters of the frequency scan technique and is worth understanding. The dwell bandwidth-which determines the range resolution-is proportional to, but smaller than the total (invested) signal bandwidth. The denominator of (6) is larger than 1 by an amount equal to the ratio of angular swath extend to beamwidth; thus, increasing either the imaged swath width or the antenna height h ant will reduce the dwell bandwidth. This shows the main trade space of the f-STEC imaging mode to be range resolution versus swath width and SNR, and the latter is determined by the antenna gain, which is proportional to the antenna height (see Section V). The ratio sw / el can be understood as the number of beamwidths necessary to cover the swath. This is a static quantity, as is the expression in (6) in the sense that it does not involve the beam scanning rate, which thus does not affect the dwell bandwidth and has therefore not been considered up to now.
The trade space described by (6) is shown in Fig. 3. The left ordinate represents the dwell-to-chirp bandwidth, γ w = B w /B sys , where γ w is the dwell factor, in percent versus the abscissa taken to be the f-STEC antenna height normalized to the height of a stripmap SAR antenna imaging the same swath. Note that the height of a stripmap SAR antenna is readily obtained by inserting sw into (1), in which case the two-way pattern will be 6 dB below the maximum at the swath edges. Increasing the antenna height increases its gain without decreasing the swath width (as in stripmap mode), but it also reduces the antenna beamwidth, and by this, the spectral support of a point target positioned within the swath. As such, the underlying trade is SNR versus range resolution.
The right ordinate of Fig. 3 represents the swath-tobeamwidth ratio, sw / el , introduced earlier. The line has a slope of 1 for identical f-STEC and stripmap antenna types, which can be easily verified by inserting (1).
Next, the dwell time duration, defined earlier as the time during which a point target on the ground is illuminated, is derived. Define the pulse start and end times (leading and lagging pulse edge, respectively) by t 1 and t 1 +τ p , respectively, where τ p > 0 is the pulse duration. By following the exact same procedure leading to the dwell bandwidth, we arrive at the expression relating the scan angle and the time: which, following the same reasoning as before, leads to the expression for the dwell time: Note that the dwell factor, γ w , describes the ratio of dwellto-pulse duration as well as dwell-to-system bandwidth 3 An equivalent expression is stated in [21] and [22].   (6) shows an identical form of dependence for the dwell time and dwell bandwidth, which justifies the identical curves for the two quantities. The implication, however, is rather different: a reduced dwell bandwidth is advantageous in terms of SNR since it reduces the noise power, 4 while the opposite is true for the dwell time, since a reduced point target illumination time reduces the average power density and, by this, causes a reduced SNR. In Section V, an expression for the SNR is developed, which quantifies the dependence on the various parameters.
An interesting effect occurs when the beamwidth converges toward the angular swath extent, i.e., el → sw . One would expect that the dwell bandwidth and dwell time converge to the signal bandwidth and pulse duration, respectively as is the case for classical stripmap operation. This is, however, not the case (!) as shown in Fig. 3 where the percentage dwell factor approaches 50% when the f-STEC and stripmap antenna heights become equal. The reason is inherent to the f-STEC operation since the beam is (still) scanned with the start/end steer angle lying outside the swath. Mathematically, this manifests itself through additive 1 in the denominator of (9) such that From (2), it is evident (cf., also Fig. 2) that in this case, sc = 2 · el = 2 · sw . The above scenario indicates the least favorable operation or, stated differently, an efficient f-STEC manifests itself by a large swath-to-beamwidth ratio sw / el 1.
One important aspect worth noting is the shape of the antenna's radiation pattern, which manifests itself through the relation between the assumed beamwidth, el , and the halfpower beamwidth. The temporal and spectral response to a point target is determined by the shape of the radiation pattern, which affects the main performance parameters such as the dwell bandwidth and time and, by this, determines the impulse response function.

B. Echo Time Reversal
To derive the condition for the echo time reversal, the geometry and timing are considered in detail. At this point, the earlier assumption of arbitrary scan direction is dropped and the antenna's main beam radiation pattern is specified to scan from far to near range, i.e., θ e < θ s .
Take as reference a sphere centered at the antenna and just large enough to contain the complete antenna structure. Let t 1 and t 1 + τ p mark the time instances where the leading and lagging "edge" of an electromagnetic wave generated by a transmitted pulse of duration τ p passes the sphere.
The aim is to determine the time of arrival of various "portions" of the transmit pulse ground echo. This can best be calculated by considering a Dirac delta impulse transmitted at time t = t 1 +t, where 0 ≤ t ≤ τ p . The steering direction of the beam center, i.e., the maximum of the antenna's radiation pattern, as a function of time is determined by solving (7) for θ 0 (t)-the subscript 0 is added to indicate the beam center-yielding where, as expected, the scan limits are θ s and θ e at times t = 0 and τ p , respectively. To determine the time delay, τ h , after which the beam-center points to the far-range angle of the swath, set θ 0 (τ h ) = θ far and solve (11) for t = τ h , and thus, where the last two terms in (12) have been recast in terms of the antenna beamwidth el introduced earlier and the dwell time from (9). The above expression is intuitively comprehensible since the dwell time is the duration any target remains within the antenna main beam; hence, after t = τ w /2 seconds, the beam center has moved by half a beamwidth and is pointing to θ far corresponding to slant range R far . The ground echo return will arrive at the reference sphere at time instance The same reasoning is used to determine the time t = τ p − τ h at which the main beam is pointing toward the near range of the swath θ near ; the corresponding echo from slant range R near will then arrive at time The f-STEC technique allows for an echo time reversal, which occurs if the far-range signal echo arrives earlier than the near-range echo. At the reversal point, both near and farrange echos arrive at the same time instance. Setting t far = t near from (13) and (14) and substituting for τ w from (12) yield the following condition: (15) where τ 0 is named the intrinsic duration and depends on the swath geometry and antenna pattern beamwidth. When the duration of the transmitted pulse is equal to the intrinsic duration, both near-and far-range echoes arrive at the same time instance. The above conveys that echo time reversal, t far < t near , occurs for long transmit pulse durations and small swath widths.
The ratio of transmit-to-intrinsic pulse duration is a relevant imaging mode parameter, as it allows specifying the f-STEC operation point, defined by When t far = t near , the system operates at the reversal point for which O p = 1 and the echo window length is minimized as will be shown later. The system operates in echo reversal for O p ≥ 1, while the near-range echo arrives before the far-range echo when O p < 1. Note that, as mentioned earlier, the echo window length compression is inherent to the f-STEC imaging and occurs, independently of the value of the operation point, as a consequence of the far-to-near beam scanning.
It should be pointed out that for a classical SAR, t far is always larger than t near such that an echo time reversal will not occur. This is because the transmitted pulse propagates as a spherical wave illuminating the swath from near to far range; a time-angle correspondence as given by (11) is not applicable. It is worth mentioning, however, that a similar echo compression can be achieved by using subpulse techniques [33], [34], [35], [36] as suggested in [37].

IV. ECHO WINDOW TIMING
The imaging mode parameters presented in Section III are used in the following to compute quantities related to the timing of an SAR system operating in the f-STEC mode. A graphical representation of the timing diagram (also known as diamond diagram) is commonly used in order to determine the possible extent (width) and position of the imaged swath. The expressions for the blockage caused by the transmit pulse and the nadir return are derived after introducing the time-range equation.

A. Range-Time Characteristics
For a spherical Earth of radius r E , there is a one-to-one correspondence between the off-nadir look angle, θ , and the slant range, which is given by (cf. e.g., [38]) where r O = r E + h sat is the radius of the satellite orbit and h sat is the orbit height. Consider a Dirac delta transmitted-passing the reference sphere-at time instance t = t 1 + t when the beam center is scanned to θ 0 (t) given by (11). Inserting into (17) yields the beam-center range as a function of time The return echo from range r 0 (t) will be received-pass the reference sphere-after a time delay 2r 0 (t)/c 0 at time The last two expressions give an profound insight into the f-STEC mode and are therefore considered in detail in the following. First, note that the expressions are with respect to the beam center. The slant range of the leading (or lagging) beam edge is easily obtained by subtracting (or adding) el /2 to θ 0 (t) in (18). The temporal quantity corresponding to el /2 is τ w /2, as such the leading beam edge moves into the far swath at t = 0, while the lagging beam edge moves out of the near swath at t = τ p . As a consequence of the above (cf. the argumentation leading to (12) in Section III-B), the transmit times of the Dirac delta pulse such that the beamcenter points at the far and near edges of the swath are given by t = τ w /2 and τ p − τ w /2, respectively. Fig. 4 shows the slant range r 0 (t) versus the echo arrival time as determined from (19). The shaded region marks the time duration during which a point at arbitrary range r 0 (t) is within the antenna patterns main beamwidth and is equal to the dwell time τ w . The total echo duration T echo is the difference between the time of the last and first return, as detailed in Section IV-B.
When operating beyond pulse reversal, the far-range echo arrives first, i.e., t far < t near as shown in Fig. 4(a) for O p = 1.14. At the reversal point, O p = 1, both near-and far-range echoes arrive at the same time instance, as shown in Fig. 4(b). It is interesting that the minimum echo arrival time is smaller than t far and t near by an amount denoted as the residual time, τ res ≥ 0. Thus, it should be considered that although each range is illuminated by a constant dwell time and t near = t far , the resulting echo duration is larger than τ w (as detailed in Section IV-B). In the third case, shown in Fig. 4(c), the f-STEC system operates slightly below the reversal point at O p = 0.95. As mentioned before, the echo compression still occurs, although t near < t far . Furthermore, note the earliest arrival time is slightly smaller than t near by the residual time τ res .
To compare, Fig. 5 shows the range-time diagram for a conventional stripmap SAR exhibiting a linear range-time dependence and the near-range echo arriving well before the far-range echo.

B. Echo Window Length
The duration of the echo (typically referred to as echo window length or echo window time) is the time during which the scattered echo signal from the imaged swath arrives at the radar. It is given by where t far and t near , introduced earlier, mark the echo return times at which the beam center moves into and out of the swath, respectively, with the modulus | · | ensuring validity for t far ≶ t near ; the dwell time τ w , defined in (8), is added to account for the illumination time of a point scatterer, while the last term is the residual time, which is introduced to consider that the earliest echo return may arrive before, i.e., earlier than, any of the far or near-range echo returns. Inserting (13) and (14) into the above expressions and rearranging terms give (21) which expresses the echo duration in terms of known operational parameters and the residual time τ res . The latter is detailed next as it may affect the echo window length.

1) Residual Time:
To determine τ res , the echo return time in (19) is differentiated with respect to t, and thus, 5 ∂t ∂t Inserting the beam-center range and steering angle from (18) and (11), respectively, and after some simplification yields To obtain the saddle point, the above expression has to be set to zero and solved for angle θ eta , which gives the direction of earliest time of arrival (eta), i.e., the direction from which the earliest return echo arrives. A numerical solution for (23) is easily implemented yielding θ eta . In the Appendix, a closedform solution is derived through a rotation of the coordinate system and approximating the Earth curvature near the swath center by a local flat geometry; the results are shown to be in very good agreement to the numerical solution.
Once the angle θ eta is known, inserting into (18) and (11) yields the corresponding slant range and time delay t eta , respectively. Inserting into (19) yields the earliest (minimum) echo time of arrival t eta . Finally, the residual time τ res is defined as the difference between the smaller of t near and t far , and t eta yielding The residual time is shown in Fig. 6 plotted versus the operation point for two different configurations imaging a swath of 50 and 126 km. The following is concluded.
1) The maximum residual time is at the reversal point. 6 O p = 1.
2) The value of the maximum residual time (at O p = 1) increases proportionally to the swath width. 3) When O p ≶ 1, the residual time falls off (decreases) at a rate inversely proportional to the swath width R far − R near . 4) The residual time curve is slightly asymmetrical with respect to O p = 1. With the known residual time, the echo window length is fully determined by (21). Fig. 7 shows T echo versus operation point for a fixed intrinsic duration. 7 At the reversal point, the residual-free echo time (dotted curve in Fig. 7) is at its minimum; this is a general case for f-STEC operation since 0 < γ w < 1 as can easily be verified from (21). Furthermore, the echo time, i.e., including the residual time, is shown in Fig. 7 to reach its minimum value at the reversal point, where

C. Pulse Repetition Frequency
A crucial parameter of SAR is the pulse repetition frequency (PRF), which is the reciprocal of the pulse repetition interval (PRI), i.e., f prf = 1/T pri . It must be ensured (see also later discussion in Section IV-D) that the duration of the latter is sufficient to accommodate the transmit pulse duration τ p and the echo window length T echo , in addition to a margin (guard time) 2τ g after the fall time and before the rise time of the transmit pulse Inserting (21), the minimum PRI may be expressed in terms of the intrinsic echo duration τ 0 and the f-STEC operational parameters as

1) Stripmap SAR PRF and Echo Window Length:
It is worth comparing the maximum f-STEC PRF, i.e., minimum PRI, to that of a classical stripmap SAR imaging the same swath width. The stripmap echo window length is given by where the superscript sm is added to indicate the stripmap operation. Note that the first term in the above expression is proportional to the intrinsic duration τ 0 given in (15). The stripmap PRI is then readily calculated to be where a constant pulse duty cycle ρ sm has been assumed such that τ sm p = ρ sm T sm pri . The left ordinate in Fig. 8 shows the maximum PRF normalized to the stripmap SAR PRF, f prf / f sm prf , versus the operation point, O p = τ p /τ 0 . The reversal point is for τ p = τ 0 , and thus, the system operates in echo time reversal for O p > 1, i.e., abscissa values larger than 1. For the chosen parameter values, the PRF is higher compared to the stripmap case for O p 1.12; this allows a more square-like antenna shape and a better azimuth resolution.  Table I. 2) Pulse Duty Cycle: A further quantity of interest is the pulse duty cycle, which is the percentage of time the system is transmitting. The expression for the maximum possible duty cycle is obtained from (27) through which is plotted in Fig. 8 (right ordinate). The peak of the duty cycle is at the time reversal point and exhibits exceptionally high values in the order of 90%, much higher (about 5-10 times) than what is usually possible for spaceborne SAR. 8 The high duty cycle values allow for low peakto-average transmit power and are believed to be a major advantage of f-STEC operation as it significantly reduces the complexity of the transmitter unit and allows for large average transmit power thus improving the SNR, one of the main challenges of spaceborne Ka-band SAR. The large duty cycle of f-STEC may be utilized in combination with more advanced imaging techniques such as subpulse operation [33], [34], [35], [36]. Transmitting multiple subpulses to illuminate different subswathes within the period of one PRI becomes feasible. This multibeam f-STEC imaging technique, advantageous also for lower frequency SAR, can be used to increase the imaged swath width. Note that introducing subpulses in a conventional operation mode will reduce the available echo window length and, by this, limit the imaged swath width.
It turns out by varying the instrument parameters, e.g., the dwell time has a minor influence on the performance values shown in Fig. 8. It is mainly the operation points O p and sw / el that determine the SAR performance. In terms of trade space, this offers the knowledgeable instrument engineer the flexibility to choose the optimum parameters.

D. Swath Position Limits
It was shown that SAR utilizing the f-STEC technique may operate at an extremely high pulse duty cycle while imaging a wider swath than what would be possible with a conventional stripmap operation mode of the same duty cycle. This, however, does not answer the question as to where the swath is positioned within the possible access range. Conventionally, the imaged swath width and position are determined from the timing (diamond) diagram so as to avoid an overlap of the signal echo and the transmit pulse (cf. [39] for details on the imaging gap caused by the transmit pulses). It turns out that the identical approach is not applicable to f-STEC SAR because of the inherent interrelation between the swath width and various operational parameters. This manifests itself in (21), which shows that the echo duration T echo , the swath width ∝ (R far − R near ), and the transmit pulse duration τ p are not independent parameters.
Nevertheless, a graphical representation similar to the (well known) timing diagram can be obtained, if the f-STEC operation point defined in (16) by O p = τ p /τ 0 is known. This allows the system engineer to start with the required swath width and later determine the (start/stop) position of the swath from the timing diagram.

1) Transmit Pulse Blockage:
The approach is to formulate the constraint for the allowable echo time limits within the mth PRI period, where m is an integer that depends on the number of traveling pulses. The time constraint is then used to obtain the slant range limits, which, assuming spherical Earth, has a one-to-one correspondence to the look (off-nadir) and incidence angle.
With reference to Fig. 9, the limits are set such that the receive echo does not coincide with the transmit instances. This yields the mth minimum and maximum allowable echo times where τ g , introduced earlier, is the applicable guard time before and after any transmit pulse, whereas τ p = τ p + τ g incorporates both the pulse duration as well as the guard time.
Since the two quantities of interest are the (discrete) swath limits, a closed expression may be derived by considering, again, the leading and lagging transmit pulse edges intersecting the reference sphere at times t 1 and t 1 +τ p while steered toward the far and near range, respectively. Referring to Fig. 4, the conditions for t near and t far to avoid an overlap of the echo signal with the transmit pulse are readily expressed as where the shift by ±τ w /2 accounts for t near and t far being relative to the beam center, thus transforming the conditions to be with respect to the earliest/latest echo arrival time.
Consider first the case of echo pulse reversal where O p > 1 and the far-range echo arrives first, and thus, t far < t near . Inserting (31) into (33) with min{t near , t far } = t far and substituting for t far as in (13) yield 2R far c 0 ≥ mT pri + τ p + τ res (35) which is formulated with respect to near range and dwell factor using (9) and (15) to yield the minimum allowable near range in (36). Furthermore, to avoid an overlap of the echo with the (m + 1)th transmit pulse, the condition in (34) is applied; thus, inserting (31) and (14) and rearranging as before yield the maximum allowable far range Following the identical procedure yields the expressions for the limits in terms of the slant range when O p < 1 as: The above conditions (alternatively expressed in terms of the look or incidence angle [40], [41]) are valuable for indicating the flexibility, i.e., margin, for placing the imaged swath within the access range. The look angle swath limits are shown in Fig. 10 plotted versus the operation point. Here, O p = 0.95 (indicated by the vertical line) and the PRI is slightly higher than the minimum allowable value given by (27). The blue shaded areas between the minimum near range and maximum far-range swath limits violate the timing conditions; the yellow shaded area marks a 50-km swath. Clearly, from the swathposition point of view, operating the instrument beyond echo reversal provides less flexibility; conversely, a PRI value higher than the minimum T min pri allows for variable swath positions.
2) Nadir Echo: The nadir echo signal is caused by nearspecular reflection of high amplitude that may saturate the receiver if not sufficiently attenuated by the (two-way) antenna pattern. The nadir return may be mitigated by avoiding the swath ranges blocked by the nadir return through the timing. In the following, closed expressions for the slant ranges masked by the nadir return are derived.
The ground area contributing to the nadir echo is defined by a cone of height h sat and half-angle ϑ nd yielding the minimum and maximum ranges h sat and l nd = h sat / cos ϑ nd , respectively (valid for ϑ nd ≤ 10 • ).
Consider the nadir echo of a pulse lasting τ p seconds transmitted at t = t 1 . The earliest and latest nadir echo  Table I, specifically here T pri = 1.06T min pri at O p = 0.95. The blue area marks angle ranges blocked by the Tx pulse while the yellow area corresponds to the look angle limits for a 50-km swath. return will arrive, i.e., pass the reference sphere defined earlier, at times t 1 + 2 h sat /c 0 and t 1 + τ p + 2 l nd /c 0 , respectively. Clearly, the nadir echo duration is much larger than the SAR echo, cf. (21), as a consequence of τ p > τ w . To determine the time interval and corresponding slant ranges during which the wanted SAR echo is masked by an unwanted nadir echo, the same approach as for the transmit pulse blockage is followed. The SAR instrument transmits successive pulses at T pri intervals, and the nadir echo return may intervene with the SAR echo of a pulse transmitted n ·T pri seconds earlier. The minimum and maximum allowable SAR echo times are limited by the nadir returns t n1 = t 1 + τ p + nT pri + 2l nd c 0 (41) t n2 = t 1 + (n + 1)T pri + 2h sat c 0 (42) and the conditions for t near and t far to avoid an overlap of the echo signal with the nadir return are readily expressed as Proceeding as before yields the swath limits (46) and  Table I, specifically here T pri = 1.06T min pri at O p = 0.95. The green area marks the ranges blocked by the nadir return echo, while the yellow area corresponds to the look angle limits for a 50-km swath.
The expressions above are used to determine the portion of the imaged swath masked, i.e., blocked, by the nadir echo, which, for a given terrain topography, is determined through the respective look angle or slant range interval. The swath limits are plotted in Fig. 11 and are seen to exhibit a nearly identical dependence with respect to the operation point as the transmit pulse blockage (cf. Fig. 10). As such, the same reasoning as before applies to the trade space in terms of swath position and echo reversal operation. Note that placing the swath limits to avoid transmit pulse blockage does not imply a nadir-free echo and vice versa. When positioning the swath, the system engineer needs to take care to independently avoid both nadir return and transmit pulse blockage.
3) Nadir Mitigation Techniques: The constraint imposed by the nadir echo may be relaxed in some cases. For reflectorbased SAR, the nadir return may be sufficiently attenuated by the Rx/Tx antenna pattern to avoid contaminating the (wanted) SAR echo. If the nadir return collected by the antenna does not saturate the receiver, then it is sufficient to consider the (shorter) compressed nadir return duration, which relaxes the timing constraint. Furthermore, techniques for the removal of the nadir echo within postprocessing exist, which are based on the alternation of the transmitted waveform [42], [43].

V. SNR PERFORMANCE
The increased antenna height of an f-STEC SAR system is expected to result in a higher SNR when compared to a conventional stripmap SAR imaging the same swath. Furthermore, it is worth comparing the SNR to that of a digital beamforming system utilizing SCORE. In the following, the SNR expression is derived for these cases giving an insight into the particularities of f-STEC operation mode.
The starting point is the well-known radar equation for extended targets [8], [10] given by where P av is the average transmit power, which is related to the peak power through P av = P t τ p f prf = P t ρ; A e = λ 2 G/4π is the effective antenna area written in terms of the antenna gain G [28] where the superscript T and R are added to refer to the transmit and receive antenna, respectively; k is Boltzmann's constant; T N is the system noise temperature; B sys is the bandwidth; L f is the system loss; χ ρ and χ ϑ are the slant range and angular pulse extent, respectively [38]; σ 0 is the backscatter coefficient; B D is the processed Doppler bandwidth; and az and C az (φ) are the azimuth beamwidth and normalized radiation pattern, respectively. The last two terms in the above expression represent the azimuth power reduction factor and the two-way pulse extension loss (PEL). The former considers that only the power within the azimuth beamwidth az , which corresponds to the processed Doppler bandwidth, contributes to the SNR as detailed in [8]. The latter considers the loss of the two-way elevation antenna pattern 9 within the angular pulse extent as derived in [38].

A. Stripmap Operation Mode
In the case of the stripmap SAR, the slant range pulse extent is χ ρ = c 0 τ p /2. Furthermore, the PEL is negligible, i.e., the elevation pattern can be assumed constant within the angular pulse extent, which allows approximating the last term in (49) such that the SNR expression becomes where A e = A T e = A R e , i.e., identical antennas for transmission and reception are assumed.

B. Score Operation Mode
For a system operating in SCORE mode, the SNR improves due to the increased receive antenna height A R > A T . Furthermore, the SNR loss at the swath edges is reduced with respect to the stripmap mode. This gives SNR score = P av A T e A R e 8πλ 2 σ 0 c 0 kT N B sys L f az B D R(ϑ) 3 sin η i 9 In [38], only the effect of the receive, i.e., one-way antenna pattern within the angular pulse extent, is considered, as applicable for a DBF system utilizing SCORE. For f-STEC, the expression is modified to consider both the receive and transmit antenna patterns resulting in a |C el (ϑ)| 4 dependence.
where the last term represents the one-way PEL of SCORE accounted for by the square of the elevation pattern [38].

C. f-STEC Operation Mode
To arrive at the SNR expression for the f-STEC case, the particularities of this operation mode have to be considered. Remembering that any target "sees" the elevation beamwidth during the dwell time suggests an equivalent slant range pulse extent of χ ρ = c 0 τ w /2. This effect is not insignificant and reduces (worsens) the SNR by a factor τ p /τ w in comparison to the stripmap case. On the other hand, the f-STEC technique allows increasing the transmit and receive antenna height by a factor τ p /τ w − 1, see Section III-A and (8), which improves the SNR by a factor ≈ (τ p /τ w ) 2 and (over) compensates the earlier effect. However, the two-way PEL also has to be considered for f-STEC, whereas SCORE is only affected by a one-way PEL. Assuming that the pulse power within the elevation beamwidth el is processed, i.e., each target "sees" the elevation beamwidth, allows the following approximation for the integral limits: Then, the expression for the f-STEC SNR becomes where the noise contribution within the processed resolution bandwidth B w is considered, which works in favor of improving the SNR with respect to stripmap since B w < B sys .

VI. INSTRUMENT PERFORMANCE COMPARISON
In this section, we compare the performance of a Ka-band SAR instrument operating in the f-STEC mode to a conventional SAR operating in the stripmap mode and to a, more advanced, digital beamforming SAR utilizing SCORE. The three instruments are designed to image a 50-km swath, the average transmit power is assumed to be 100 W, and the spatial 2-D image resolution is to be 7.4 m 2 .
The resulting instruments and operation parameters are listed in Table I. The values of some of the common parameters, e.g., system noise temperature and losses, do not take the instrument hardware differences into account and may be considered optimistic. However, these have no impact on the relative performance comparison between the systems, which is the purpose here.

A. Swath Timing and PRF
The timing diagram of the f-STEC system shown in Fig. 12(a) is computed based on the equations derived in Section IV-D; it appears similar to the well-known diamond diagram but for an instrument operating at a low pulse duty cycle, whereas f-STEC benefits from the return echo compression and operates at a high duty cycle of 82%. To compare, the timing diagram common for the stripmap and SCORE instrument modes is shown in Fig. 12(b); the transmit blockage appears much wider although the pulse duty cycle is only 15%. The imaged swath off-nadir angular position is chosen to be the same as for the f-STEC systems; the swath is shown to be intersected by nadir returns, which is allowed (assuming that it can be suppressed [42]) in favor of a better comparison between the different instruments.

B. Spatial Resolution
The instruments and mode parameters are designed to yield the same 2-D resolution of 7.4 m 2 for all systems. Since (for nonsquinted SAR) the 2-D resolution is the product of range and azimuth resolution [44], it affects the processed range (chirp) and Doppler bandwidths. The latter is directly determined by the PRF assuming an 120% azimuth oversampling. For the f-STEC system, the range resolution results from the dwell factor and the system bandwidth, while the chirp bandwidth of the stripmap and SCORE systems is chosen to yield the required 2-D resolution.

C. Antenna Sizes
The Tx/Rx antenna size of the f-STEC system results from following the design procedures elaborated in the previous sections. Thus, the dwell factor together with the available chirp bandwidth determines the beamwidth and, by this, the antenna height. The antenna length mainly results from the timing diagram, i.e., the PRF suitable for imaging the required swath width.
The stripmap instrument antenna height is determined from (1) so as to illuminate the complete swath width. The antenna length is determined by the PRF, which is lower compared to the f-STEC case resulting in a longer antenna for the stripmap system.
The transmit antenna of the SCORE system is identical to that of the stripmap instrument and the same is true for the Rx antenna length. Theoretically, the height of the Rx antenna may be of an arbitrarily high value; practically, it is limited by the complexity since increasing the antenna height requires more digital channels to implement SCORE and at some point also requires dispersive onboard beamforming to mitigate the PEL [38], [45]. Here, the antenna height was chosen so as to limit the PEL to about 2.5 dB.

D. Data Rate
The data rate is a relevant system parameter as it determines the required onboard memory and downlink capacity of the satellite. The data rate in sample per second, i.e., independent analog-to-digital (ADC) converter resolution, is computed according to: where γ s B sys is the ADC sampling frequency and γ s is the oversampling factor. The reported values in Table I indicate a 43% higher data rate for the f-STEC instrument compared to the other systems, although all systems image an identical swath width with equal 2-D resolution. This is attributed to two main effects.
1) The finer azimuth resolution requires an increased azimuth sampling for the f-STEC system of T sm pri /T pri , where the superscript sm indicates the stripmap parameter. With the values of Table I, this increases the data rate by a factor ≈ 1.2.
2) With O p = 0.95, the instrument is operating slightly below the reversal point, the increased echo duration causes a proportionally higher data rate, which is not "compensated" by the lower system bandwidth. The relative contribution of this effect is B sys T echo /(B sm sys T sm echo ) amounting to an increased data rate by a similar factor ≈ 1.2. The effect of the latter point can be compensated since the instantaneous bandwidth of the received echo is smaller than B sys ; this reduction becomes significant when O p ≶ 1 and offers the opportunity for onboard data rate reduction as discussed in [25], [46], and [47]. However, we maintain that this type of data reduction merely compensates for the aforementioned increased echo window length.

E. Noise-Equivalent Sigma Zero
The three systems are compared with respect to their noise-equivalent sigma zero (NESZ), calculated according to NESZ = σ 0 | SNR=1 [8], [9]. Fig. 13 shows the NESZ versus the swath-centered look angle. The f-STEC system is characterized by a nearly constant NESZ over the swath, which is due to the Tx/Rx beam scanning property. Note that the loss associated with the short point target pulse duration, cf. factor τ w /τ p in (54), degrades the NESZ by a factor of 11 dB with respect to stripmap and SCORE for the parameters given in Table I.
Nevertheless, f-STEC clearly outperforms the stripmap system, the main reason being the small antenna area of the latter, which further suffers from an extremely high length-to-height aspect ratio of 35 (cf. Table I).
The SCORE system shows a reasonably good performance nearly reaching that of f-STEC at the swath center. Despite its good performance, the SCORE system can be considered an inferior choice at higher carrier frequencies (Ka-band and above) because of its high complexity and technology demand (multichannel RF hardware and onboard digital processing unit) [3], [15], [16], [17], [18].
The roll-off at the swath borders of the stripmap and SCORE SAR systems is due to the fixed low-gain transmit pattern. This well-known effect occurs mainly for planar phased direct radiating antennas and can be mitigated by increasing the antenna height and introducing dedicated phase tapering techniques [48], [49] known as phase spoiling. This will flatten the radiation pattern shape, thus improving the performance near the swath edges at the expense of degradation near the swath center.
Last but not least, it is noted that assuming the same average transmit power P av = 100 W for all three systems penalizes f-STEC since it does not benefit from the high pulse duty cycle of 82%. Stated differently, the peak-to-average power ratio of the stripmap/SCORE instruments is 5.5 times higher than that of the f-STEC instrument.

F. Azimuth and Range Ambiguities
The azimuth-ambiguity-to-signal ratio is approximately the same for all three systems, given that the processed Doppler oversampling was fixed to 120%.
The range-ambiguity-to-signal ratio (RASR) estimation [50] is shown in Fig. 14. The best performance is for the f-STEC system followed by the SCORE instrument and with the stripmap system having the poorest performance. This is attributed to the respective two-way elevation pattern beamwidths. However, in general, the ambiguity level is low for all three systems since the performance is timing-limited. Nevertheless, both SCORE and f-STEC systems could be operated in a multiswath imaging mode [39], without sacrificing RASR.

VII. CONCLUSION
This underlying article provides an in-depth insight into the f-STEC imaging technique and its utilization for SAR. This article derived the essential mode and instrument parameters necessary for a thorough understanding of the f-STEC technique. With this, two main purposes were served: 1) simple analytic formulas were provided and analyzed in detail to allow for the design of an f-STEC instrument and 2) the equations governing the performance of an f-STEC system were derived and used to provide a comparison between three SAR systems utilizing in different categories of imaging modes. The reported performances suggest that f-STEC may be an especially advantageous choice for SAR systems operating at higher carrier frequencies such as X-band, Ka-band, and above. In fact, given the current technology readiness level of space-qualified Ka-band components, f-STEC might be considered the only feasible choice.
This article revisits an imaging technique, which in its general form has been referred to as f-Scan [21], [22], [23], [24]. The new contributions of this article, which introduces a particular realization of f-Scan, can be summarized as follows.
1) The entire set of analytic equations describing the f-STEC imaging mode parameters is derived from the basic principles. 2) The echo window timing and blockage types are treated in detail deriving the governing mathematical expressions. This enables the system engineer to design the timing for a specific swath width and position instead of relying on simulations. 3) Closed-form expressions for the SNR performance and data rate are provided allowing a straightforward comparison to other imaging modes. An aspect worth further investigation is the utilization of nonlinear time-to-angle steering or time-frequency chirps. For example, it is straightforward to change the steering law to obtain a constant ground range resolution over the swath (corresponding to a variable slant range resolution). The theoretical groundwork has been established in this article but used (only) to derive the expressions for the residual time in Section IV-B. We recognize a high potential of utilizing (22) to fine-tune the system and further improve the performance or, even more, for an adaptive f-STEC SAR adjusting the scan rate to image interesting swath portions at finer resolution and higher SNR. This adaptive approach may conveniently be combined with multibeam f-STEC technique, c.f. Section IV-C2, to image multiple subswathes at different resolutions.
Two topics have not been covered since they are addressed in previous publications and would exceed the scope of this article: the realization and technology for frequency-scanning antennas [21], [22], and processing of f-STEC data to yield the SAR image [24]. Both are considered important and are subject to further research intended for future publications.

APPENDIX LOCAL FLAT EARTH GEOMETRY APPROXIMATION
For a moderate swath width, a good approximation of the slant range can be made assuming local flat Earth geometry at the swath center. As shown in Fig. 15, the satellite coordinates are rotated by the angle by β c with respect to nadir to form the right angle triangle from the slant range at swath center R c , the tangential to the Earth surface, and the modified satellite heightĥ sat . The unknown parameters are determined from the geometry shown in the figure to be β c = η ic − θ 0c (56) h sat = R c cos(θ 0c + β c ) where η ic and θ 0c = (θ near + θ far )/2 are the local incidence and look (off nadir) angles at swath center, respectively. The equation for the approximate slant ranger 0 (t) as a function of beam-center look angle θ 0 (t) then becomes r 0 (t) =ĥ sat cosθ 0 (t) withθ 0 (t) = β c + θ 0 (t).
Inserting into (19), differentiating with respect to t, and setting the result equal to zero give ∂t ∂t = 1 + 2 c 0 ∂r 0 (t) ∂t ! = 0 ⇒ 1 − 2ĥ sat sc c 0 τ p tanθ 0 (t) secθ 0 (t) Rearranging and formulating the result in terms of anglê θ eta = β c + θ eta = β c + θ(t eta ), i.e., the angle at which the main beam is pointing to the direction of minimum echo For a not-too-wide swath, the angular extent of θ 0 (t) is limited and the left-hand side of (60) can be approximated by a straight line, finally yielding where the slope B is = sec(β c + θ 0c ) tan 2 (β c + θ 0c ) + sec 2 (β c + θ 0c ) . From 1992 to 1999, he was with the Ludwig Maximilians University, Munich, where he conducted multidisciplinary research on neuronal modeling and nonlinear information processing in biological and technical vision systems. Since 1999, he has been with the Microwaves and Radar Institute of the German Aerospace Center (DLR), Oberpfaffenhofen, Germany, where he started as a Research Associate developing signal processing algorithms for a novel forward-looking radar system employing digital beamforming on receive. From 2001 to 2007, he led the New SAR Missions Group which pioneered the development of advanced bistatic and multistatic radar systems, such as TanDEM-X, as well as innovative multichannel SAR techniques and algorithms for high-resolution wide-swath SAR imaging. Since 2008, he has been the Head of the Radar Concepts Department which hosts about 40 scientists focusing on new SAR techniques, missions and applications. He has moreover been serving as Mission Engineer for TanDEM-X and he made also major contributions to the development of the Tandem-L mission concept, where he led the Phase-0 and Phase-A studies. Since 2019, he is moreover a professor at the Friedrich-Alexander-University Erlangen, Germany. Gerhard Krieger is author or co-author of more than 100 peer-reviewed journal papers, 9 invited book chapters, more than 400 conference papers, and more than 20 patents. Prof. Krieger