# Radar basics – Part 5: synthetic aperture radar

**Editor’s Note:** *See also* **Part 1**, **Part 2**, **Part3**, *and* **Part 4** o*f this mini-series.*

Synthetic Aperture Radar, or “SAR”, is normally used to map ground features and terrain. It is also known in literature as Synthetic Array Radar. Both names make sense, though “Synthetic Aperture Radar” will be used here. It is used for a wide variety of military and commercial applications. It can be made to map almost arbitrarily fine resolution ground features or used to more coarsely map larger areas in with comparative effort.

**SAR resolution**

The key parameter in ground mapping is the resolution. SAR systems can be designed with abilities to differentiate using dimensions from few centimeters to hundreds of meters, depending on if the purpose is to map a military installation, an urban area, or the contours of a mountain range. The range is basically limited by the transmit power of the radar and can operate at resolutions at much greater than visual, at long ranges, and is unaffected by darkness, haze, or other factors impacting visual detection.

As with video, the quality of images depend upon the pixel density (pixel stands for “picture element”). The equivalent of pixel density in radar is a voxel, or “volume element”. The voxel is defined by the azimuth, elevation, and range. The minimum voxel size is dependent upon the radar resolution capabilities. The voxel spacing is basically the distance that two points on the ground can be distinguished from each other. Radar resolution capabilities, in turn, are dependent upon range resolution and main lobe beam width capabilities.

The voxel spacing or density should in general be at least 10 times the dimensions of the objects being mapped to achieve useful images. A 1 meter resolution is feasible for detecting buildings that are at least 10 meters long and wide.

As precision range detection is a fundamental requirement, high PRF (Pulse Repetition Frequency) operation is unsuitable for SAR due to the range ambiguities. Low PRFs are used instead, to eliminate range ambiguities over the distances from the aircraft to the ground being mapped. Maximum Doppler rates tend to be low, as the only motion is due to the radar-bearing aircraft flight path. Due to the nature of SAR, the relative motion is often substantially less than the aircraft flight speed. Use of a low PRF, while restricting the usable Doppler range, enhances the precision of Doppler frequency detection within that restricted range. This is an advantage in high resolution SAR mapping.

**Pulse compression**

Range resolution is dependent upon the precision of the receive pulse detection arrival delay. This can be achieved by a very short transmit pulse width, which has the disadvantage of low transmit power level due to the short duration. Or very high levels of pulse compression can be used, which allow relatively long transmit pulses and therefore long integration times at the receiver, with the receive operating on higher power returns. This increases the SNR and allows for longer range mapping. A high level of pulse compression can be achieved by using long matched filters (correlation to the complex conjugate of transmit sequence) and transmit sequences with strong autocorrelation properties. The only consequence is a higher level of computations associated with the long matched filter. The speed of light, and therefore radar waves, is about 1 meter per 3 nanoseconds (3x10^{-9}). Since the path is roundtrip, the range appears to become half this. So for about 1 m range resolution requires a 6 nanosecond timing detection precision. To achieve this level of correlation would require a transmit sequence with phase changes with at least 160 MHz rate. This requires radar transmit frequency width of at least the same bandwidth.

The elevation of the antenna main lobe does not need to be narrowly focused. In a SAR radar system, the antenna is directed to the ground at an angle off to the side, as shown in Figure 1. As the elevation angle decreases, the radar beam will be directed at a steeper angle a ground location closer to the flightpath of the aircraft, with a shorter range. The different portions of the beam elevation will therefore map to different ranges, and the return sequence can be directed into different range bins. The precision of the range detection capability translates into the degree of elevation resolution attainable.

**Figure 1. Elevation processing using range binning**

**Azimuth resolution**

The other requirement for precise ground mapping is for a very narrow angular resolution of the main beam in the azimuth. As discussed in a previous installment:

**Radar Basics – Part 3: Beamforming and Radar Digital Processing**, the narrowness of the radar beam depends upon the ratio of the antenna size to the wavelength. To achieve a “pencil” like radar beam requires either a very large antenna or very high frequency (and short wavelength) radar. Both are impractical for airborne radar. The antenna size is necessarily limited by the aircraft size. Extremely high frequency radars tend to be useful only at very short range, due to both atmospheric absorption and scattering.

Instead, a virtual large antenna or a virtual antenna of large “aperture” is used. The forward motion of the aircraft is used to transmit and receive from many different points along the flight path of the aircraft. By focusing the radar main beam the same area of ground during the aircraft motion, the returns from different angles at different times created by the aircraft motion can be synthesized into a very narrow equivalent azimuth main lobe using signal processing techniques. The end result is as if an antenna of great length (up to a kilometer) was used. Because this is done using radar returns over several hundred milliseconds, this technique works for stationary targets, so is ideal for ground mapping.

SAR radar directs the radar beam at ninety degrees to the plane’s flight path. The width of this radar beam does not have to be exceptionally narrow – in fact, the wider beam covers more ground and allows more processing gain in the SAR algorithm. When a large angle main lobe is used, the maximum length of the synthetic antenna is increased. Therefore, small antennas can work well with the SAR technique, as long as the antenna gain is sufficient to meet the SNR requirements for the range involved. The antenna will illuminate a large swath on the ground, typically an oval shape due to the aspect ratio of the beam being aimed outward from the aircraft flight path at downward angle.

To start, let’s assume that we can build an antenna as large as necessary to meet our azimuth resolution requirement. The rule of thumb governing antenna size is:

d_{azimuth}≈ λ R / L, where

d_{azimuth}= resolvable distance in the azimuth direction

λ = wavelength of radar

R = range

L = length of the antenna

(Note, for reasons not explained here, this expression is valid for conventional antennas. A SAR antenna actually has half the resolvable azimuth limit as a real antenna. In other words, as SAR antenna needs to be twice as large as a real antenna for the same resolvable distance.)

If a 1 meter aperture at a 10 km range is needed, with a 3 cm (X band) radar, this requires an antenna length of 300 m.

Imagine there were such an antenna mounted along the fuselage of a 300 meter long plane. Each pulse could be focused with an azimuth width of 1 m at the 10 km range, with a wide elevation, allowing the radar to scan a narrow (1 meter) strip of land each PRF, as the plane travels forward.

This 300 meter long antenna could be composed of many radiating elements along its length (for example 301 separate elements, spaced every meter). The antenna steering is accomplished by setting the phase of each element to ensure that the radar wave transmitted from each element is at a common phase when arriving at the 1 meter strip at 10 km distance. The phases would have to be adjusted, or focused, at the distance to the 1 meter strip of land will be slightly different for each element, due to the offset relative from the center element.

In order to aim the antenna beam at a very narrow region, the phase relationship of the different antenna elements must be carefully controlled. In the example, the wavelength is 3 cm. If a radar round trip path is 1.5 cm longer or shorter than the middle element path, it will be 180 degrees out of phase and add destructively, or cancel. Therefore the roundtrip path length must be controlled within a few millimeters for each element. The phase error that occurs due to the plane’s straight flight path as compared to an arc must be compensated for. This is shown below in Figure 2. The phase correction relative to the middle element of the antenna works is approximately:

Θ

_{n}= (2? π ? d

_{n}

^{2}) / (λ ? R) where

Θn = phase error of nth antenna element (in radians)

d

_{n}= distance between middle element and n

^{th}antenna element

(In SAR literature, this term Θn is sometimes called the “point target phase history”).

The return echo would be reflected from the ground at all locations and travel back to each element. Due to the reciprocal path, it would arrive at the same phase offset that was transmitted and if the same phase compensation is performed on the receive element signal prior to being summed together, the result will be that only the reflections from the 1 meter width azimuth portion will arrive in phase with all other ground returns being canceled out or at least severely attenuated.

Now suppose the same process is done in sequence rather than all at once (since a 300 meter antenna is clearly impractical). Starting with an element at the one end of the antenna, transmit a pulse, and receive the return, using only this element. All the other elements are inactive. Both transmit and receive signals are modified by the phase compensation as before. The return sequence is stored in memory. Then repeat this process with each separate antenna element in turn, until all the 301 return sequences are saved. Remember, these return sequences are complex numbers with a magnitude and phase. Now if all the complex results at the end are summed, then the same result is attained as if everything was in parallel at once. Nothing else has changed – the situation on the ground is assumed static. This is a simplified version of the process the SAR radar performs. Imagine as the plane flies forward, the PRF is such that 301 pulses are transmitted and received along 301 meters of flight path as shown in Figure 2.

**Figure 2. Azimuth processing forward motion of aircraft**

The radar could then effectively map the 1 meter wide strip at right angles to the flight path. However, while this solves the azimuth resolution problem, this is still not workable because only 1 meter wide of strip ground is mapped perpendicular to the plane’s flight path, every 301 meters.

**SAR processing**

Going further, some conventions are needed. Assign an index to each 1 meter strip of land oriented at right angles to the flight path, designated “n”. At the PRF when the aircraft is physically aligned with strip

_{n}, the real antenna will receive a complex range sequence

_{n}. This same index applies to the virtual or synthetic antenna element that is directly perpendicular to the 1 meter strip. The next synthetic antenna element forward would be index n+1, continuing on up to n+150. The synthetic antenna element behind would be n–1, extending to n–150. There are complex weighting factors of proper phase and amplitude for each index, W

_{–150}through W

_{150}. W

_{0}is always equal to 1.

To calculate the image for strip

_{n}, receiving range sequences at index n–150 must be started and sorted into range bins according to arrival time. This is using the single real antenna with wide beam angle (6 to 12º is typical). This will continue for 300 more PRF intervals and result in 301 stored receive sequences in memory, indexed from –150 to +150. After PRF

_{150}, processing can start. The 301 stored, binned range sequences will be multiplied or scaled by W

_{–150}through W

_{150}, respectively. Note that both the values in the binned range sequences and the weighting factors are complex. Each range bin is multiplied separately by the weighting factor. Then the 301 complex range sequence results can then all be summed across each range bin and will result in a range binned sequence that has an azimuth of 1 m. The range bins correspond to the individual values across the elevation of the 1 meter strip

_{n}.

For strip

_{n+1}, wait until PRF

_{151}and the plane has advanced 1 meter on its flight path. Start processing again. The binned range sequence

_{151}has been saved and can discard binned range sequence

_{–150}. This updated set of 301 binned range sequences is again multiplied by the weighting factors W

_{–150}through W

_{150}. In this case, there is an offset as follows:

In this manner, each of the strips can be computed, one after the other, using a single broad beam antenna, but using a long synthetic array to achieve a narrow azimuth. The synthetic antenna can be made arbitrarily long, by using more PRF cycles, more memory and higher processing rates.

The signal processing achieves the same cancellation of signals coming from azimuths outside out desired 1 meter ground strip as an actual 300 meter antenna would do. This processing technique is known as line-by-line processing.

**SAR Doppler processing**

Another alternative method to perform SAR processing is to incorporate Doppler processing. Due to the use of the efficient FFT algorithm, this will lead to a much lower level of computations than line-by line processing.

Consider the patch of ground being illuminated by the radar pulse directed at right angles to the aircraft flight path. This patch may a 2000 m or more wide (azimuth direction), depending upon the range and antenna beam width azimuth. At the mid-point, the Doppler frequency is exactly zero, as the radar is moving parallel to this point and has no relative motion. At the two azimuth end points the scan area will have:

positive Doppler frequency = sin (azimuth angle) · (aircraft velocity) / (wavelength)

negative Doppler frequency = –sin (azimuth angle) · (aircraft velocity) / (wavelength)

As an example, with a range of 10 km, and a ground scan area of +/– 1000 m, this equates to an angle of +/– 5.71º. If the aircraft is flying at 250 m/s, this works out to +/– 829 Hz in for the 10 GHz radar band. This is shown in Figure 3.

**Figure 3. Doppler effect due to azimuth angle**

The Doppler frequency variation is not linear across the azimuth, due to the “sin (azimuth angle)” in the equation. At small angles, sin (θ) ≈ θ or approximately linear. As the angle increases, the effect becomes more non-linear, until at 90 degrees, the Doppler frequency asomtopically approaches the familiar (aircraft velocity / wavelength), or 8333 Hz. However, the Doppler frequency response wants to be completely linear across the azimuth range. This can be compensated for by using phase correction multiplier, known as “focusing”. The purpose is to make the Doppler frequency variation linear across the azimuth angle, rather than proportional to the sine of the angle. Once the frequency spacing per unit length on the ground is made linear, it allows the use of Doppler filters with equally spaced main lobes along the frequency axis. This filtering is the familiar DFT, which can be implemented using the FFT algorithm. This is known as SAR Doppler processing. The advantage of this is that the computational load is made much more manageable than the line by line processing technique, by virtue of the FFT algorithm efficiency.

As a side note, this Doppler linearity is issue only for SAR radar. For conventional radar, the radar is aimed towards the horizon, and there is less variation due to the aspect angles (in this case, θ

_{elavation}is close to 0º, although θ

_{azimuth}can have significant variation) and the sensitivity requirements are much less than for SAR.

**Figure 4. Doppler digital signal processing**

At each PRF, the return sequence is multiplied by a phase correction (focusing). Each range bin stores a complex value, representing the phase and magnitude of the return at that range. For each range, the values are loaded into all the Doppler frequency filters matching the azimuth angles for each ground element, as shown in Figure 4.

Each pulse has its return processed by azimuth and range, which allows separation over all locations in the radar beam, with resolution determined by the number of range bin and Doppler filter frequency banks. This is repeated at each PRF with the next phase correction value and the results are accumulated or integrated. Over the set of N PRFs equal to the number of N elements in the synthetic antenna, this is repeated.

After each set of N PRFs, the process repeats. The same point is measured N times and complex values representing both magnitude and phase are integrated over the measurements for each point. In this architecture, the number of virtual elements in the synthetic array is equal to the number of Doppler filters, which can also be set equal to the number of range bins. This is also the number of times each point is measured, and the results integrated. However, each of the different measurements for a given point is done at a different azimuth angle.

In reality, these two methods provide equivalent results, although the processing steps are different. The first method is conceptually easier for most people to understand. The second method has the advantage of lower computational rate. An intuitive diagram of the two different approaches is shown in Figure 5.

**Figure 5. Line verses Doppler SAR processing**

Another way to look at this is that in line-by-line, a new narrow beam at right angles to the flight line is synthetically created each PRF. With Doppler processing many different azimuth beams are generated by each Doppler frequency bank during each PRF and the returns from each beam are summed over multiple PRFs.

**SAR impairments**

Several factors can degrade SAR performance. One of the most significant is non-linear flight path of an aircraft. We have seen how sensitive the phase alignments are to proper focusing, in fractions of the radar wavelength. Therefore, any deviations in flight path away from the parallel line of the radar scan path must be determined and accounted for. This motion compensation can be done using inertial navigation equipment and by using GPS location and elevation measurements. Another consideration is sidelobe return. When the sidelobe return from the ground beneath the plane is integrated over a wide azimuth and elevation angles, this can become significant despite the low antenna gain at the sidelobes. The design of the synthetic antenna, just like a real antenna, must take this factor into account. There are methods, similar to windowing in FIR filters, which can reduce sidelobes, but at the expense of widening the main lobe and degrading resolution.

Another issue is that the central assumption in SAR is that the scanned area is not in motion. If vehicles or other targets on the ground are in motion, they will not be resolved correctly and be distorted in the images. Shadowing is impairment. This occurs when a tall object shields another object from the radar’s illumination, causing a black or blank spot in the range return. This becomes more prevalent when very shallow angles are used, which occurs when the aircraft is at low altitude and scanning at long ranges. At high altitudes, such as satellite mounted SAR, this is much less of an issue.

**SAR radar implementation**

SAR radar processing tends to be a bit less demanding than most other military detection and tracking radars. Many SAR radars are used in commercial applications, such as ground mapping. In some cases, SAR radars can be implemented in software, using floating point processors or DSPs. For higher performance SAR radars, FPGAs once again can provide a much higher level of signal processing and throughput.

It should be noted that FPGAs come in a variety of performance and density levels. Traditionally, the FPGA vendors offered highend and low cost FPGA families. Any high performance DSP application tended to use the high end FPGAs. Several years ago, FPGA vendor Altera added a mid-range FPGA family, named Arria. Now in its third generation, the 28nm Arria V FPGAs offer compelling DSP performance. Following the trend, FPGA vendor Xilinx recently introduced their mid-range FPGA family, called Kintex.

Mid-range FPGAs share common architectures with highend FPGAs. The differences are primary in logic density and I/O speeds. For example, Altera Stratix V FPGAs have transceivers operating at 13 and 28 Gbps, whereas Arria V FPGAs have transceivers operating at 6 and 10 Gbps.

Of course, in radar processing, DSP resources and architecture are of paramount importance. The Arria V FPGAs, like Stratix V FPGAs, also use Altera’s new Variable Precision architecture. Fully one half of the ten Arria V FPGA device family members contain over 2000 multiplier resources. None of the competing Kintex devices have this multiplier density. Further, Arria V FPGAs are capable of several hundred GFLOPs of floating point single precision processing, leveraging the same Fused Datapath floating point toolflow and Variable Precision DSP architecture as the larger Stratix devices.

**Conclusion**

This concludes the 5-part article series on radar basics. I sincerely hope this information has been and will continue to be useful, while encouraging the reader to continue studying this fascinating and rapidly evolving field. Please feel free to comment on this article series and I will do my best to respond.

**About the author**

As senior DSP technical marketing manager, Michael Parker is responsible for Altera’s DSP-related IP, and is also involved in optimizing FPGA architecture planning for DSP applications.

Mr. Parker joined Altera in January 2007, and has over 20 years of DSP wireless engineering design experience with Alvarion, Soma Networks, TCSI, Stanford Telecom and several startup companies.

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