Signal Processing for Digital Beamforming FMCW SAR

According to the limitations of single channel FrequencyModulation ContinuousWave (FMCW) Synthetic Aperture Radar (SAR), Digital Beamforming (DBF) technology is introduced to improve system performance. Combined with multiple receive apertures, DBF FMCWSAR can obtain high resolution in low pulse repetition frequency, which can increase the processing gain and decrease the sampling frequency. The received signal model of DBF FMCW SAR is derived. The continuous antenna motion which is the main characteristic of FMCW SAR received signal is taken into account in the whole signal processing. The detailed imaging diagram of DBF FMCW SAR is given. A reference system is also demonstrated in the paper by comparing with a single channel FMCW SAR. The validity of the presented diagram is demonstrated with a point target simulation results.


Introduction
The combination of Frequency Modulated Continuous Wave (FMCW) technology and Synthetic Aperture Radar (SAR) paves a way to light-weight, cost-effective, high-resolution active microwave remote sensing instrument. This system is suitable for small platforms such as Unmanned Aerial Vehicles (UAV). In the last years, several FMCW SAR systems have been constructed and have been successfully used in ice measurement [1], environment monitoring [2], threedimensional application [3], and so on [4,5].
However, these systems are currently applied in the short range and narrow swath case. According to the single channel FMCW SAR radar equation [6], the most immediate method for improving the system performance is to increase the transmission power. Nevertheless, high power transmission is not a suitable solution because of the isolation problem between transmitter and receiver and the increasing system complexity. Moreover, when operating in high resolution such as 0.1 m mode, the aperture length of antenna in azimuth is very small. This will decrease the antenna gain of single channel FMCW SAR systems, and then it will decrease the Signal Noise Radio (SNR) of SAR images.
In pulsed SAR systems, wide unambiguous swath coverage and high azimuth resolution are also contradicting requirements. Several proposals, such as multiple beam SAR system operating in a squinted imaging geometry [7], displaced phase center antenna technique [8], quad array system [9], and High-Resolution Wide-Swath (HRWS) SAR system [10], have been presented to resolve the aforementioned dilemma. Those systems have their own advantages and disadvantages. However, their basic principle is ambiguity suppression using Digital Beamforming (DBF) technology combined with multiple antennas.
The HRWS SAR concept relies on separated transmit and receive antennas, which is also the characteristic of FMCW SAR systems. So the HRWS concept may be a suitable solution for improving the system performance of FMCW SAR. Now, multichannel FMCW SAR is becoming a hot area of miniature SAR research. And many research institutions have announced their research projects on multichannel FMCW SAR [11][12][13][14].
According to the imaging algorithm of pulsed DBF SAR, the basic steps are firstly the azimuth reconstruction and then single channel imaging. However, the multichannel azimuth reconstruction algorithm [15] is based on "stop and go" hypothesis. Compared with pulsed SAR received signal, the main characteristic of FMCW SAR received signal is that the continuous antenna motion must be taken into account; that is, the "stop and go" hypothesis is not valid for FMCW SAR [16]. So the impact of continuous antenna motion to azimuth reconstruction must be analyzed. Furthermore, there are many imaging algorithms for single channel FMCW SAR. The best algorithm for DBF FMCW SAR when considering some nonideal factors such as sweep frequency nonlinearity should also be chosen carefully.
The remaining sections are organized as follows. Compared with single channel FMCW SAR, the performance improvement of DBF FMCW SAR is given in Section 2. The signal model, the impact of continuous antenna motion, and the complete imaging diagram of DBF FMCW SAR are all presented in Section 3. Next, an example system is designed and the numerical simulation results are given in Section 4. Finally, Section 5 concludes the whole paper.

Performance Improvement
The noise equivalent sigma zero (NESZ) is a quantity directly related to SAR imaging performance, which is defined as the target Radar Cross Section (RCS) when the SNR of the image is equal to one [17]. The NESZ of single channel FMCW SAR is expressed by [6] NESZ = (4 ) 3 4 sin where is the slant range, is the incidence angle, is the Boltzmann constant, 0 is the system noise temperature, is the receiver noise figure, is the noise bandwidth, is the loss factor, is the transmitted signal power, is the antenna gain, and are the azimuth and range resolution, respectively, and and are the azimuth and range processing gain, respectively.

Antenna Gain.
From (1), we know that the transmit and receive antenna gains in single channel FMCW SAR system are the same. However, in DBF FMCW SAR, the receive antenna gain is not equal to that of the transmit antenna. Assum that receive channels are used, is the antenna number in azimuth, and is the antenna number in elevation. The multiple receive apertures are shown in Figure 1.
The DBF in azimuth is applied to suppress the azimuth ambiguity. This does not increase the antenna gain. In elevation, the channels are coherently combined. Performed by the SCan-On-REceive (SCORE) technique, the DBF in elevation can obtain full array gain over a wide area. Thus, under the same resolution and swath width, the maximum receive antenna gain of the DBF system is re = . ( The DBF technique increases the antenna gain by times, compared to the single channel system.

Range Processing Gain.
Range processing of FMCW SAR is analog with pulsed SAR. They are all based on pulse compressing using matched filter. Theoretically, range processing gain is about the following: where is the pulse width and is the signal bandwidth. For FMCW SAR, the value of is equal to 1/ , where is the pulse repetition frequency (PRF). If we decrease the system PRF using receive channels in azimuth, we can get times signal processing gain compared with single channel system. The signal processing gain is increased at the cost of long processing time.

Sampling Frequency.
In order to maintain the low-power and lightweight characteristics of FMCW SAR, the sampling frequency should be chosen as lowly as possible. According to the Nyquist theorem, the lowest sampling frequency for FMCW SAR is where is the chirp modulation rate and max and min are the maximum and minimum slant range, respectively. The sampling frequency is also directly proportional to the PRF. That is to say, the sampling frequency of multichannel FMCW SAR is 1/ times of that of single channel FMCW SAR. The DBF FMCW SAR operates with a low PRF to increase the swath width. To decrease the PRF, the significant and challenging processing is the azimuth ambiguity suppression. In the next section, we focus on the multichannel signal reconstruction in azimuth.

Received Signal Model.
Let the transmitted signal of FMCW SAR be written in complex form as follows: where is the center frequency, is the sweep number, and is the time variable. Because signal amplitude has little effect in SAR signal processing, so for simplicity, antenna gain, amplitude effects of propagation on the signal and any additional time delays due to atmospheric effects are ignored in the following discussions.
The received signal is a delayed version of the transmitted one. The channels of received signals in azimuth are processed. The th received signal is where 0 is the velocity of light and , is the roundtrip instantaneous slant range which can be written as where ( ; 0 ) and , ( ; 0 ) are transmit range and receive range, respectively. In the single channel case, ( ; 0 ) is equal to , ( ; 0 ). However, it cannot be valid in the multichannel case. This is the main difference between the multichannel and the single channel FMCW SAR received signal.
Dechirp on receive is often used in FMCW SAR. In order to further decrease the sampling frequency, and then to decrease the data in range dimension, the radar demodulates the received signal by mixing it with a replica of the transmitted waveform delayed by a time ref = 2 ⋅ / 0 , where is the slant range distance of the swath center. The reference signal is The intermediate frequency signal that results from mixing (8) with (6) is Using Taylor series, the round-trip range , ( ; 0 ) can be expanded as where = and = − represent azimuth time and range time, respectively. V is the velocity of the platform, 0 is the closest distance between antenna and the scatter. Δ is the distance between the transmit antenna and the th receive antenna, = 0 / is the wavelength, and is the approximate Doppler frequency shift during the transmission of one sweep as follows: ] . (11) Different from pulsed SAR, introduced by the continuous antenna motion should be taken into consideration, for the "stop and go" hypothesis is not valid for FMCW SAR. This can be seen from the simulated results in Figure 5.
After substituting (10) into (9), the received signal of channel can be rewritten as where ( ; 0 ) is the azimuth dependent distance to the scatter which can be expressed by In (12), the first to third exponential terms are the azimuth phase history of multichannel FMCW SAR. The fourth term represents the range cell migration due to continuous antenna motion, that is, the Doppler impact in one sweep. The fifth term represents the range signal, which is a sinusoidal signal with a constant frequency value corresponding to the azimuth dependent distance to the point target. The sixth exponential term represents the residual video phase (RVP) term.

The Impact of Continuous Antenna Motion.
As mentioned above, the main characteristic of FMCW SAR is the continuous antenna motion. The range cell migration (RCM) introduced by the continuous antenna motion may result in reconstruction mismatch. Even worse, the extraphase term introduced by the continuous antenna motion may degrade the quality of the focus image.
Multichannel azimuth reconstruction algorithm [15] is a suited approach to process pulsed DBF SAR data in azimuth.

Mathematical Problems in Engineering
According to the algorithm [15] and the received signal model of DBF FMCW SAR described as (12), the reconstruction filters P( ) for DBF FMCW SAR can be calculated by the following equation: where H( ) is ] .
Using basic matrix theory, the system matrix H can be decomposed as a matrix Q multiplied by a diagonal matrix D, where only the elements of D are dependent on the slant range 0 . And the diagonal elements of D can be written as In the SAR case, the range history is not a line but spans several range cells, which results in mismatch between the signal and the reconstruction. In FMCW SAR, the additional mismatch introduced by the continuous antenna motion is in which Δ dfm represents the maximum RCM introduced by the continuous antenna motion of channel . The mismatch can be derived from (14) to (16). Figure 2 shows a simulated RCM introduced by the continuous antenna motion using the parameter listed in Table 1. In this case, Δ dfm is about 1 m. Calculated by (17), the mismatch introduced by this RCM is about 4 × 10 −5 rad, which is negligible. Here, Δ = 5 m is assumed. On the other side, Δ dfm is about 10 times of the range resolution. This RCM must be compensated according to SAR imaging theory [17]. In a word, for DBF FMCW SAR, the impact of continuous motion antenna is negligible during multichannel azimuth reconstructing [15]; however, it must be considered during RCM correction.

Signal Processing
Diagram. The first part of signal processing for DBF FMCW SAR is the azimuth ambiguity suppression. All the recorded subaperture signals whose azimuth frequencies are aliasing become an equivalent single channel signal without azimuth ambiguity. To focus the signal, the second part of the diagram is the classic single channel SAR imaging.
In the first part, multichannel azimuth reconstruction algorithm [15] is chosen because the algorithm can relax the imposed constraints on system design and the algorithm performance is rarely affected by the continuous antenna motion. In the second part, frequency scaling algorithm [16,[18][19][20] is combined with the DBF technology. This algorithm has not only efficient and accurate processing but also realtime integrated motion compensation [19] and easy sweep frequency nonlinear correction [16,20]. The entire signal processing block diagram of DBF FMCW SAR is shown in Figure 3 including the following steps.
Step 1. Apply Azimuth Fast Fourier Transform (FFT) to the received signal of each channel. Then, the Doppler domain signal ( , ; 0 ) is obtained, where is the azimuth frequency.
Step 2. Design the azimuth reconstruction filter P( ) by using (14) and (15). The reconstruction filter of channel can be written as Step 3. Multiply each channel signal to the corresponding reconstruction filter. At this time, the signal becomes AR, ( , ; 0 ) = ( , ; 0 ) ⋅ ( ) .
Step 4. Add each channel signal after passing the corresponding reconstruction filter. Then, we can obtain the equivalent single channel signal ( , ; 0 ), which has nonazimuth frequency aliasing. And it can be written in convolution [16] as follows: where is the RCM factor defined by where src( , ; 0 ) is the secondary range compression as follows: ] .

(22)
Step 5 (frequency scaling transform). Multiplying the signal ( , ; 0 ) with the frequency scaling factor, we have the following: Step 6 (continuous antenna motion correction). Multiplying the signal with the Doppler frequency correction factor, we have the following: In actual implementation, Steps 5 and 6 can be completed at the same time. The signal FS ( , ; 0 ) is obtained as follows: After an equivalent transform, (25) can be expressed as Step 7 (RVP correction). The correction is applied in 2D frequency domain. It is to eliminate the last exponential term in (26). After FFT in range dimension and multiplying with the RVP correction factor RVPC ( , ), IFFT in range domain is performed. The signal RVPC ( , ; 0 ) is obtained as follows: The RVP correction factor is expressed as Step 8 (inverse frequency scaling transform). Multiply RVPC ( , ; 0 ) with the inverse frequency scaling factor IFS ( , ) which is Step 9. Secondary range compression: multiply the signal with the secondary range compression factor SRC ( , ; ) which is described as where the approximation ≈ 0 is required.
Mathematical Problems in Engineering Step 10. Bulk range shift: multiply the signal with the bulk range shift factor BV ( , ; ) which is described as In actual implementation, Step 8 to Step 10 can also be completed at the same time. We obtain Step 11. Apply FFT to RCMC ( , ; 0 ) in range dimension. At this time, the range compression is completed as follows: Step 12. The 2D focused image of DBF FMCW SAR is obtained after azimuth matched filtering [18].

Simulation Results
The parameters of an example FMCW SAR system using dual channels combined with DBF are listed in Table 1. The corresponding parameters of single channel FMCW SAR, which is very similar to MicroBSAR described in [6], are also given. There are two parts in the simulation. One is imaging simulation, the other is performance simulation.

System Parameters Design.
To achieve the required resolution in azimuth, the Doppler bandwidth must be greater than 700 Hz. The transmit antenna length in azimuth should be less than 0.2 m. According to the swath width, the transmit antenna length in elevation can be chosen to be 2.9 cm, which corresponds to 40 ∘ beamwidth. The transmit antenna gain is 19.5 dB with the antenna efficiency 0.5. The receive antenna is the same with the transmit antenna for single channel FMCW SAR. In the two-channel system, the receive antennas are placed in azimuth. The receive antenna gain is still the same. However the overall receive antenna length in azimuth reaches 0.4 m.
According to the Nyquist theorem, the PRF of classic FMCW SAR should be greater than the Doppler bandwidth. However, the PRF of DBF FMCW SAR can be chosen to be 700/2 Hz using two receive channels. To ensure a slant range resolution of 0.1 m, a chirp bandwidth of 1.5 GHz is necessary. The range processing gain 66.3 dB is obtained for DBF FMCW SAR. Nevertheless, that value of single channel FMCW SAR is only 63.3 dB. The sampling frequency of DBF FMCW SAR and single channel system are 16 MHz and 32 MHz, respectively. All the parameters are listed in Table 1.

Imaging
Simulation. The target raw data are simulated using the two-channel system parameters listed in Table 1 by (9). To investigate the method of DBF technology and the effects of the continuous antenna motion, the raw data are processed by using the algorithm described in Figure 3.
The azimuth reconstruction is shown in Figure 4. Figure 4(a) shows the azimuth aliasing spectrum of the first channel signal in the two-channel system before reconstruction filter. After filters of 1 ( ) and 2 ( ), respectively, add the first channel signal and the second channel signal together. Then the azimuth spectrum is completely reconstructed. The reconstruction result is shown in Figure 4(b). The DBF coherently combines the two channel signals in azimuth and reconstructs a wide Doppler spectrum about 700 Hz.
Using the parameters of the two-channel example system, the signal processing presented in Section 3.3 is simulated. Figure 5 plots the 2D contour maps of the impulse response function of a point target. Only the values above −27 dB are plotted. The result of the proposed processing algorithm is plotted at the bottom. Figure 5(a) is the result without azimuth reconstruction. The performance is seriously degraded because the azimuth spectrum is overlapped as Figure 4(a) shows. To explore the characteristic of FMCW SAR distinguished with pulse SAR, Figure 5(b) illustrates the result without Doppler frequency correction factor which is expressed as (24). In terms of the resolution and shape of the response, it is shown that the reconstruction is successful and the impact of the motion within one sweep in FMCW SAR needs to be compensated. Table 1, NESZ for DBF FMCW SAR has been calculated over a range of incidence angles, which is shown in Figure 6. The result for single channel system is also shown in Figure 6. When calculating NESZ, the receiver noise figure is chosen  as 3 dB. The DBF FMCW SAR achieves a NESZ below −10 dB along the entire swath. 5 W of transmitted power is sufficient to image vegetation and farmland. However, the result of classic FMCW SAR in the edge of the swath is only −5 dB, which is not sufficient to obtain good far-range imagery. The reason is that there is higher signal gain for DBF FMCW SAR.

Conclusions
A multichannel receive antenna FMCW SAR system combined with DBF is introduced. DBF FMCW SAR overcomes the limitation of high PRF caused by high azimuth resolution. It has higher receive antenna gain and range processing gain than single channel system. The received signal of DBF FMCW SAR is modeled. The impact of continuous antenna motion which is the main characteristic of FMCW SAR is negligible in multichannel azimuth reconstruction for DBF FMCW SAR, but it must be considered in the process during RCM correction. The whole signal processing diagram is given. Simulated point target experiments have been performed to verify the processing diagram successfully. Future works will include performance analysis thoroughly, moving target indication, and the miniaturization of the system.