8

RANGE–DOPPLER PROCESSING ON SAR IMAGES

In Chapters 6 and 7, we discussed the basics of radar images and provided a system model for analyzing and processing them. The M × N array of 2D radar image raw data is formed by M bursts of radar pulses with N samples each. The radar signal itself is a pulsed LFM waveform, which can be used to measure the target's range or identify its location through the matched filtering. The phase variation for various pulsed LFM waveforms in the azimuth direction leads to the concept that another LFM type of signal exists in the azimuth direction. This signal is not directly generated from the radar, but derived indirectly from the many LFM pulses sent and received by the radar within the synthetic aperture length. This azimuth-direction-based LFM signal plays a key role in fine-tuning the target function in the azimuth direction, and greatly improves the radar image quality.

Also mentioned previously is that radar image processing can be independently processed in the range and azimuth direction if range migration can be neglected. However, as we described in Section 6.6 (of Chapter 6), the range migration could become severe for radar with large synthetic aperture length (or wide antenna beamwidth), or when the radar beam has a nonzero squint angle toward the target. Therefore, the range cell migration correction is needed to decouple the azimuth process from that of the range.

The SAR image processing therefore can be summarized as three major tasks: (1) range compression, (2) range cell migration correction, and (3) azimuth compression. Figure 8.1 shows the block diagram for SAR image processing. Here the raw data refer to the demodulated in-phase–quadrature-phase (I–Q) baseband data.

FIGURE 8.1 Major tasks of SAR radar image processing.

The raw data, after range compression, provide valuable information on the targets' locations in the range direction but do present two problems: (1) the range-compressed signal is spread in the azimuth direction, and (2) the signal migrates to other range cells. The purpose of SAR processing, ideally, is to convert the raw data into a single pixel in the final processed image. Several techniques are available to deal with the three tasks listed in the preceding paragraph. Each technique has its advantages in either computation efficiency or high-quality imaging, and has its own beamwidth limitation and/or bandwidth restriction. In this chapter and Chapter 9, two SAR image processing techniques will be presented: the range–Doppler and the Stolt interpolation. The former technique processes the SAR image based on Doppler frequency shifts, while the latter is based on the wavefront reconstruction theory. The range–Doppler algorithm, developed in the early 1980s, has been the most commonly used algorithm for processing SAR data images. It is computationally efficient and is an accurate approximation for processing radar images. The Stolt interpolation technique was applied to SAR image processing in the early 1990s. It is computation-intensive, but provides some advantages over the range–Doppler method. The range–Doppler algorithm is described in this chapter, and the Stolt interpolation technique is covered in Chapter 9.

We start the SAR image data generation in Section 8.1, where the principles of radar image formation and the method for synthesizing radar image data are reviewed. Section 8.2 provides a description and examples of how broadside SAR data are synthesized. Both single target and multiple targets are covered. Section 8.3 provides the same discussion and examples as in Section 8.2, but for the squint SAR case. The algorithm of range–Doppler processing on an SAR image is reviewed in Section 8.4, and the simulation results for both broadside and squint SAR are described in Section 8.5.

We will focus only on airborne-based radar image data processing; therefore, the estimation of Doppler centroid and Doppler frequency ambiguity, which is critical for satellite-based squint SAR systems, is not addressed here.

8.1 SAR IMAGE DATA GENERATION

The pulsed LFM radar transmitted waveform can be written as follows, from Eq. (5.11)

where A₀ is the amplitude of the transmitter signal and Rect(t/T_p) is a rectangular gate function with T_p as the pulse duration time. The symbol f_c is the carrier frequency, and α is the LFM pulse chirp rate. Equation (8.1) states that the signal p(t) starts at t = 0 and ends at t = T_p; therefore, for simplicity, we will use a(t) to substitute the gate function and let A₀ = 1 in the following discussion.

The echo from K reflectors with reflection coefficient σ_k at ranges R_k, for k = 1, 2,..., K can be represented by a sum of delayed signals as

where τ_k = 2R_k/c is the echo delay time due to the kth target. For a target located at (x_k, y_k) and with the radar position at (0, 0) and assuming the radar height H = 0, the delay time can be computed as follows:

If the delay time τ_k were known, the range R_k could be computed as

The baseband signal s_b(t) can be obtained by removing the carrier frequency followed by a lowpass filter through the guadrature demodulation process. The demodulated baseband signal can be represented as:

Letting n be the time index of the received and digitized LFM signal, one can then express s_b(t_n) as

where n = 0, 1, 2,..., N – 1. Here N is the total number of received range samples, which is a function of the pulse duration time, sampling frequency, and number of targets.

Derivation of Eq. (8.4b) is based on one short duration pulse transmitted from one radar position. If the radar moves along a direction perpendicular to or at some angle from the radar beam, and emits signal at the rate of f_PRF, then a 2D received data array can be generated.

FIGURE 8.2 System model of radar image generation.

Let u_i be the moving radar position along the flight path or y axis, with i = 1, 2,..., M. Here M is the total number of processed azimuth lines, which must be equal to or greater than the number of azimuth lines (Naz) within the synthetic aperture length Ls.

Equation (8.4b) can then be modified as

where τ_uik = (2R_uik/c) is the received signal delay time from the kth target with the radar located at position u_i.

A system model used to generate a broadside SAR image is shown in Fig. 8.2, where the radar is at height H above the ground (or x–y plane) and moves at speed V along the y-axis direction. The symbol θ_H is the radar beamwidth along the y-axis direction, while θ_V is the radar beamwidth along the x-axis direction. The radar center beam is always perpendicular to the flight path in the broadside SAR system. The squint SAR system is similar to that shown in Fig. 8.2, except that the radar center beam is not perpendicular to the flight path, but has a squint angle θ_q.

The following assumptions are made to simplify the system model of Fig. 8.2:

• Ignore the background objects, so that the received signal is noise-free.
• The target is an ideal reflector and stationary; therefore, motion compensation is not required.
• Ignore the effect of the inclination angle. The slant range is considered equal to the ground range.

FIGURE 8.3 A simplified broadside SAR system for radar image generation.

A simplified 2D version of Fig. 8.2 is shown in Fig. 8.3, where the radar height H = 0 and the y axis and flight path are merged.

Three targets are shown in Fig. 8.3, and they are located at (R_0a, y₁), (R_0b, y₂), and (R_0a, y₃), respectively. The vertical axis represents the radar moving direction, while the horizontal axis corresponds to the range, which is perpendicular to the flight path. The 3-dB radar beamwidth is represented as θ_H. The three radar positions y₁, y₂, and y₃ are the positions when the corresponding targets are illuminated under the center beam of the radar. The synthetic aperture length for the targets at range R_0a is Lsa = R_0aθ_H, and for the target at range R_0b this length is Lsb = R_0bθ_H. The total synthetic aperture length of the three targets is L_total = Lsa + (y₃ – y₁).

Figure 8.4 is a simplified 2D version of Fig. 8.2 for a squint SAR system. Here a squint angle θ_q is shown as the angle between the radar center beam to the target and the range axis. The three radar positions with the center beam on the corresponding targets are u₁, u₂, and u₃. The three targets are located at (R_0a, y₁), (R_0b, y₂), and (R_0a, y₃), respectively. The synthetic aperture length for the targets located at range R_0a and R_0b can be computed as follows:

The total synthetic aperture length in this case is

L_total = Lsa + (y₃ – y₁)

Figure 8.3 will now be used to synthesize the broadside SAR image data. The single target case will first be described, followed by a case with three targets.

FIGURE 8.4 A simplified squint SAR system for radar image generation.

8.2 SYNTHESIS OF A BROADSIDE SAR IMAGE DATA ARRAY

8.2.1 Single-Target Case

The radar image generated from a single-target response can be modeled as shown in Fig. 8.5 with a target located at (R₀, y₂) and y₂ = u₂. The three radar positions u₁, u₂, and u₃, are the positions where the radar beam is starting, in the center, and ending to illuminate the target, respectively. The symbols R₁ and R₃ denote the slant ranges when the radar is at locations (0, u₁) and (0, u₃), respectively. The slant range R₂ equals R₀, which is the shortest distance between target and radar.

FIGURE 8.5 Single-target broadside SAR system for radar image generation.

FIGURE 8.6 Received signal array from Fig. 8.5.

Figure 8.6 displays the model of a received radar signal array. The vertical axis m represents the azimuth samples. The horizontal axis t represents the delay time of the radar pulse returned from the target. Each azimuth sample (or line) corresponds to the radar position u as shown in Fig. 8.5. There are Naz number of azimuth lines within the synthetic aperture length Ls, and Naz = integer of (Ls/A_s), where A_s is the sample spacing between the azimuth lines. The azimuth location m = 1 is the position where the radar beam starts to illuminate the target, and m = Naz is the last position when the radar beam ceases to illuminate the target. At radar position m = Naz/2, the target is under the center beam of the radar. This is also the position when the target has the shortest distance from the radar. The return time of the radar pulse starts at t₁ and ends at t₁ + T_p + δt. Here t₁ = 2R₀/c = 2R₂/c is the closest target reflection time after the radar beam is transmitted. T_p is the time duration of the radar transmitting signal, and δt = 2(R₁ – R₂)/c = 2(R₃ – R₂)/c is the echo time difference between the center radar beam and the two edges of the 3-dB radar beamwidth.

The curved lines inside the two vertical dotted lines, which occur at both t = t₁ and t = t₁ + T_p, correspond to the echo time for different slant ranges when the radar moves within the synthetic aperture length Ls. It is also the target trajectory seen by radar during the radar movement within Ls. For most airborne-based broadside SARs, the time difference δt is negligible (within ¼ range resolution time limit); therefore, a straight line will be used instead. However, the phase differences among various slant ranges must be preserved.

The time sample n and the slant range sample n′ are defined as follows:

FIGURE 8.7 A simplified and digitized received signal array from Fig. 8.6.

In general, time t and slant range R are related by t = R/c; therefore n = n′. However, when time t refers to the target echo time, the relationship of the time and slant range becomes t = 2R/c. Therefore the relation between the time sample and the slant range sample becomes n = 2n′. In the equation displayed above, f_s is the sampling frequency and R_s is the slant range sample spacing.

For purposes of illustration, the time sample will be used for discussions on SAR image processing in this chapter, and the slant range sample will be used in Chapter 9.

Figure 8.7 shows a simplified and digitized plot of Fig. 8.6. The horizontal time axis is replaced with time sample n. For simplicity, the received time sample n = 1 starts from t = t₁ and ends at n = N_r = integer of (T_pf_s), where T_p is the pulse duration time.

The following airborne-based SAR radar parameters are used for synthesizing the received signal array. The same set of parameters, with minor differences, will be used for both the single target and multiple target cases, and serves both the broadside and squint SAR systems:

Radar beam carrier frequency f_c = 1 × 10¹⁰ Hz

Speed of wave c = 3 × 10⁸ m/s

Radar beam carrier wavelength λ = c/f_c = 3 cm

Symmetric LFM time duration T_p = 6.033 μs

Range FM rate α = 4 × 10¹² Hz/s

Closest range of target R₀ = 7500 m

Range sampling frequency f_s = 30 MHz

Range bandwidth = αT_p = 24.13 MHz

Radar moving speed V = 200 m

Azimuth FM rate = −2V²/(R₀λ) = −355.56 Hz/s

Pulse repetition frequency f_PRF = 500 Hz

Antenna length L = 1m

On the basis of these parameters, the following data can be computed:

Slant range sample spacing R_s = c/f_s = 10 m

Radar 3-dB beamwidth θ_H = λ/L = 0.03 radian

Azimuth sample spacing A_s = V/f_PRF = 0.4 m

Synthetic aperture length Ls = R₀ θ_H = 225 m

Doppler frequency bandwidth B_Dop = 2V/L = 400 Hz

Number of azimuth samples = integer of (Ls/A_s) = 563

Number of time samples within the LFM pulse N_r = (f_sT_p) = 181

Number of time samples N_tot = N_r = 181

The slant range R₁ = R₃ = R₀/cos(0.5θ_H) = 7500.8 m

The slant range difference, namely, R₁ – R₀ = 7500.8 – 7500 = 0.8 m, is less than ¼ of the range resolution, which is (c/2B)/4 = 1.56 m in this setup. Therefore, range migration correction is not needed in this broadside SAR case.

Let both the reflection coefficient and the amplitude of the transmitted signal be equal to one. The baseband signal returned from a single target can then be expressed as follows, from Eq. (8.4c):

For a target located at (R₀, y₁) with the radar position at (0, u_i), the echo delay time τ_ui can be computed as

Figure 8.8 shows the baseband signal waveform of LFM radar with the parameters and assumptions listed above. Figure 8.8a displays the real part of the signal and Fig. 8.8b displays the imaginary part of the signal.

The received signal array based on Eq. (8.5) is displayed in Fig. 8.9; Fig. 8.9a shows the real part of the received signal array, while Fig. 8.9b displays the imaginary part of the signal array. The vertical axis (ordinate) shows the number of azimuth lines (or cross-range samples), where one out of every 30 azimuth lines was sequentially selected throughout the synthetic aperture length. The azimuth lines run from line 1 to Naz, with Naz = integer of (Ls/A_s) = 563, where A_s is the azimuth sample spacing. The horizontal axis (abscissa) shows the time samples, with sampling rate at 30 MHz. The total number of time samples N_r equals the sampling frequency times the pulse duration, that is, f_sT_p = 181 time samples for each azimuth line. Although there is no range migration, the phase changes due to the slant range difference among various azimuth lines are obvious.

FIGURE 8.8 Waveforms of the real and imaginary parts of a baseband symmetric LFM signal.

8.2.2 Multiple-Target Case

A 2D broadside SAR signal array based on Fig. 8.3 is depicted in Fig. 8.10, with three targets located at (R_0a, y₁), (R_0b, y₂), and (R_0a, y₃), respectively. Here the vertical axis (ordinate) represents the azimuth lines (or cross-range samples), while the horizontal axis (abscissa) shows time. Again, the curved lines occur at the beginning and end of the pulse duration time, which can be neglected as described before. Lsa and Lsb are the synthetic aperture lengths corresponding to range R_0a and R_0b, while <r1>, <r2>, and <r3> are the received signal arrays corresponding to targets 1, 2, and 3, respectively. The time variables t₁ and t₂ are the target echo time, with t₁ = 2R_0a/c and t₂ = 2R_0b/c. T_p is the transmitted LFM pulse duration time.

FIGURE 8.9 Waveforms of received baseband signal from Fig. 8.5.

FIGURE 8.10 Received signal arrays from Fig. 8.3.

A simplified version of Fig. 8.10 is shown in Fig. 8.11, where the time axis is digitized to become a time sample axis. Naza and Nazb are the number of azimuth lines corresponding to Lsa and Lsb, respectively. Namely, Naza = integer of (Lsa/A_s) and Nazb = integer of (Lsb/A_s) with A_s as the azimuth sample spacing. D₁ is the sample number of overlapped segment in azimuth direction between signal arrays <r1> and <r3>; that is, D₁ = integer of ((Lsa − (y₃ − y₁))/A_s). D₂ is the azimuth line difference between the two starting lines of <r1> and <r2>; that is D₂ = integer of ((y₂ − y₁) − (Lsb2 − Lsa2))/A_s. N_Rb is the range sample difference between R_0a and R_0b, which is defined later.

The size of the received signal array is M × N, with M = 2 × Naza − D₁ and N = integer of (f_sT_p) + 2N_Rb.

FIGURE 8.11 A simplified and digitized signal array from Fig. 8.10.

Using the same radar parameters used in the single target broadside case, the additional data of target locations and associated data are listed below:

Target 1 location: (R_0a, y₁)

Target 2 location: (R_0b, y₂)

Target 3 location: (R_0a, y₃)

Synthetic aperture length corresponding to R_0a:

Lsa = R_0a θ_H

Synthetic aperture length corresponding to R_0b:

Lsb = R_0b θ_H

Naza = integer of (Lsa/A_s)

Nazb = integer of (Lsb/A_s)

Number of range sample difference between R_0a and R_0b

Again, let both the reflection coefficient and the amplitude of the transmitted signal be equal to one; the baseband signal returned from the three targets can be expressed as follows, from Eq. (8.4c):

Given the three targets located at (R_0a, y₁), (R_0b, y₂), and (R_0a, y₃), and the radar position (0, u_i), the time delay amounts τ_ui1, τ_ui2, and τ_ui3 can be computed as follows:

The three targets locations are listed below.

y₁ = 0m

y₂ = 100 m

y₃ = 150 m

R_0a = 7500 m

R_0b = 7650 m

Accordingly, the following data can be computed.

Lsa = 225

Lsb = 229.5

Naza = Lsa/A_s = 563

Nazb = Lsb/A_s = 573

D₁ = integer of [(Lsa − (y₃ − y₁))/A_s] = 187

D₂ = integer of {[(y₂ − y₁) − (Lsb2 − Lsa2)]/A_s} = 244

N_Rb = integer of [(R_0b − R_0a)/R_s] = 15

Number of azimuth lines M_tot = 2 × Naza − D₁ = 939

Number of time samples N_tot = c × T_p + 2N_Rb = 211.

The received signal arrays corresponding to targets 1–3, in terms of time samples, are shown in Fig. 8.12: Figs. 8.12a and 8.12b show the real and imaginary parts of the signal array <r1>, Figs. 8.12c and 8.12d show the real and imaginary parts of the signal array <r2>, and Figs. 8.12e and 8.12f show the real and imaginary parts of the signal array <r3>. The vertical axis shows the number of azimuth lines, where one out of every 30 azimuth lines was sequentially selected throughout the synthetic aperture length. The horizontal axis shows the time samples, with a 30 MHz sampling rate. The size of signal arrays <r1> is 1–563 (1 to Naza) in the azimuth direction and 1–181 (1 to N_r) in the time sample direction. The size of the signal array <r2> is 244–817 (D₂ to D₂ +Nazb) in the azimuth direction and 30–211 (30 to N_tot) in the time sample direction. The size of signal array <r3> is 377–939 (Naza − D₁ + 1 to 2Naza − D₁) in azimuth direction and 1–181 in the time sample direction.

FIGURE 8.12 Waveforms of the individual received signal from Fig. 8.10.

The overall received signal array, which is the sum of the three signal arrays <r1>, <r2> and <r3>, is displayed in Fig. 8.13. The real part is shown in Fig. 8.13a; the imaginary part, in Fig. 8.13b. Again, the vertical axis (ordinate) shows the number of azimuth lines, where one out of every 30 azimuth lines was sequentially selected throughout the total synthetic aperture length. The horizontal axis (abscissa) shows the time samples. The size of the overall received signal array is from rows 1–939 (2Naza − D₁) in the azimuth direction, and from column 1 to column 211 in the time sample direction.

FIGURE 8.13 Waveforms of the received signals from Fig. 8.10.

8.3 SYNTHESIS OF A SQUINT SAR IMAGE DATA ARRAY

8.3.1 Single-Target Case

The squint radar image generated from a single target response can be modeled as shown in Fig. 8.14, with the target located at (R₀, 0). The three radar positions, u₁, u₂, and u₃, are the locations where the radar beam is beginning, in the center, and ceasing to illuminate the target. The symbols R₁ and R₃ denote the slant ranges when the radar beam is at locations (0, u₁) and (0, u₃), respectively. The symbol R₂ denotes the slant range when the radar beam is at location (0, u₂). The symbol θ_q is the squint angle between the range axis and the radar center beam when the radar is at position (0, u₂).

Figure 8.15 displays the time–azimuth relationship of the received signal array. The vertical axis shows the positions of the radar, while the horizontal axis shows the return time of the echo signal. The received signal array appears as a curved parallelogram. The longest time for echo to arrive occurs at radar position u₁, with t₁ = 2R₁/c; and the shortest time to arrive occurs at radar position u₃, with t₃ = 2R₃/c. At radar position u₂, the echo arrival time is in between t₁ and t₃.

Figure 8.16 shows a simplified plot of Fig. 8.15. The time is replaced with time samples, and the radar position u is replaced with azimuth sample number. As can be seen, the slant range difference causes significant range migration. The maximum range migration amount is ΔN = (R₁ − R₃)/R_s.

FIGURE 8.14 System model of a single-target squint SAR.

FIGURE 8.15 A received signal array from Fig. 8.14.

FIGURE 8.16 A digitized signal array from Fig. 8.15.

The same radar parameters used in the single-target broadside case, together with the squint angle θ_q = 6°, are used to compute the following data:

The maximum range migration amount, namely, (R₁ − R₃) = 7554.1 − 7530.3 = 23.8 m, is larger than ¼ of the sample spacing, range resolution, which is (c/2B)/4 = 1.56 m) in this setup. Therefore, range migration occurs, and range migration correction is needed in this squint SAR case.

FIGURE 8.17 Waveforms of a received baseband signal from Fig. 8.14.

With the same baseband transmitting signal shown in Fig. 8.8, the received baseband signal array can be derived, from Eq. (8.5), and is shown in Fig. 8.17: Fig. 8.17a shows the real part of the received signal array; Fig. 8.17b, the imaginary part of the signal array. The vertical axis shows the number of azimuth lines, where one out of every 30 azimuth lines was sequentially selected throughout the synthetic aperture length. The azimuth line runs from line 1 to line Naz, with Naz = integer of (Ls/A_s) = 569. The horizontal axis represents the time samples. The total number of time samples N_r = 181 for each azimuth line.

As can be seen in Fig. 8.17, the range migration exists and range cell migration correction is therefore needed to process this type of SAR signal.

8.3.2 Multiple-Target Case

A three-target squint SAR system based on Fig. 8.4 is shown in Fig. 8.18, where the three targets are located at (R_0a, y₁), (R_0b, y₂), and (R_0a, y₃), respectively. The radar position of u₁ and u₃ corresponds to the place where the radar beam starts and ends its illumination of the target areas. Position u₂ is where the radar center beam illuminates the target 2. The vertical axis represents the azimuth lines; the horizontal axis, the range. Lsa and Lsb are the synthetic aperture lengths corresponding to the targets located at range R_0a and R_0b. R_1a and R_3a are the longest and shortest slant ranges from the radar to the targets located at range R_0a, while R_1b and R_3b are the longest and shortest slant ranges to the target located at range R_0b. The squint angle is represented as θ_q.

FIGURE 8.18 System model of a three-target squint SAR.

From Fig. 8.18, the following parameters can be derived:

FIGURE 8.19 The received signal arrays from Fig. 8.18.

In addition, the echo arrival time can be computed as follows:

On the basis of the three-target squint SAR system and parameters, the time–azimuth relationship of the received signal array can be plotted as shown in Fig. 8.19.

Figure 8.20 shows the simplified and digitized signal array based on Fig. 8.19 with abscissa (horizontal axis) changes from time to discrete time sample and ordinate (vertical axis) changes from radar position to azimuth line. The data array size is M × N, with M = 2 Naza − D₁ and N = ΔN_b + N_r + ΔN_r.

The parameters ΔN_a, ΔN_b, ΔN_r, D₁, and D₂ are defined below:

ΔN_a = integer of [2(R_1a − R_3a)/R_s]

ΔN_b = integer of [2(R_1b − R_3b)/R_s]

D₁ = integer of [Lsa − (y₃ − y₁)]

D₂ = integer of [y₂ − R_0b tan θ_q]

ΔN_r = integer of [2Δr/R_s]

By applying all data and parameters used in multiple targets for the broadside SAR system, with the additional squint angle θ_q = 6°, the following parameters can be computed:

FIGURE 8.20 The digitized signal arrays from Fig. 8.19.

Lsa = 227.5

Lsb = 232

Naza = Lsa/A_s = 569

Nazb = Lsb/A_s = 581

R_1a = R_0a/cos(θ_q + 0.5θ_H) = 7554.1

R_3a = R_0a/cos(θ_q − 0.5θ_H) = 7530.3

R_1b = R_0b/cos(θ_q + 0.5θ_H) = 7705.2

R_3b = R_0b/cos(θ_q− 0.5θ_H) = 7680.9

D₁ = integer of ((Lsa − (y₃ − y₁))/A_s) = 193

D₂ = integer of [y₂ − R_0b tanθ_q − (Nazb − Naza)/2] = 204

ΔN_a = integer of [2(R_1a − R_3a)/R_s] = 5

ΔN_b = integer of [2(R_1b − R_3b) /R_s] = 5

N_r = integer of (f_s T_p) = 181

ΔN_r = integer of [2Δr/R_s] = 30

Number of azimuth lines M = 2Naza − D₁ = 945

Number of time samples N = ΔN_b + N_r + ΔN_r = 216

Using the data presented above, the received signal array waveforms corresponding to targets 1–3 can be synthesized and are shown in Fig. 8.21: Figs. 8.21a and 8.21b show the real and imaginary parts of signal array <r1>; Figs. 8.21c and 8.21d, the real and imaginary parts of signal array <r2>; Figs. 8.21e and 8.21f, the real and imaginary parts of signal array <r3>. The dimensions of signal arrays <r1> and <r3> are the same, namely, from row 1 to row 569 (Naza) and from column 1 to column 181 (N_r). Signal array <r2> ranges from row 244 to row 817 (D₂ + Nazb) and from column 30 [2Δr/R_s] to column 216.

FIGURE 8.21 Waveforms of the individual received signal from Fig. 8.18.

The waveform of received signal array, which is the sum of three targets, is shown in Fig. 8.22. The real and imaginary parts of the waveform are shown in Figs. 8.22a and 8.22b, respectively.

8.4 RANGE–DOPPLER PROCESSING OF SAR DATA

The synthesis of 2D SAR data discussed in the previous section shows that the received data from a point target have a waveform with size M × N in both time and azimuth directions. The spread in time is caused by the time duration of the transmitted LFM pulse. The spread in azimuth is because the target is under the radar illumination, which causes the radar to receive the echo signal for the period during which the radar moves through the whole path of synthetic aperture length.

The three major functions of SAR data processing—namely, range compression, range cell migration correction, and azimuth compression—can be implemented by a typical range–Doppler processor shown as a block diagram in Fig. 8.23.

FIGURE 8.22 Waveforms of the received signals from Fig. 8.18.

FIGURE 8.23 Flow diagram of the range–Doppler algorithm.

The 2D raw data in Fig. 8.23 refer to the received baseband signal, which is a complex number after in-phase–quadrature-phase (I–Q) demodulation. The range compression is performed by the blocks of range FFT (on 2D raw data), FFT (on range reference function), range window function, multiplier, and inverse range FFT. The “corner turn” operation is a data management scheme, that converts the range compressed 2D data from row-based to column-based data for the subsequent operation of azimuth FFT. After azimuth FFT, the 2D data will be in the range–Doppler frequency domain. range cell migration correction, if required, is performed at this stage. Azimuth compression is performed by the blocks of FFT (on azimuth reference function), azimuth window function, multiplier, and inverse azimuth FFT. The output of azimuth compression is the reconstructed image of focused targets.

A discussion of the range–Doppler processor now follows.

8.4.1 Range Compression

Figure 8.24 displays an M × N raw data array together with the mth row of data. Each row consists of N digitized data, which serve as 1D range signal for image processing.

Range compression performs the matched filtering on every row of range samples. The matched filter h(t) is designed with h(t) = s(−t)*, where s(t) equals the transmitted signal, which has been explored in detail in Chapter 5 and is not repeated here. The matched filter serves as a reference function and is operated on every row of data during the range compression process. A window function, such as the Hanning window, is normally applied to the range reference function in order to reduce the sidelobe effect. After matched filtering on every row of 2D data, which is implemented in the frequency domain using the FFT technique, the inverse FFT is then applied to obtain the 2D time-domain range compressed signal again.

FIGURE 8.24 (a) An M × N 2D data array; (b) mth row of 2D data array.

8.4.2 Corner Turn

Range compression processes the M × N data array on a row basis, namely, 1 × N, and the output remains the same quantitatively. Appropriate zero padding on the data array and the selection of FFT size is needed to reduce the edge effect of circular convolution. After range compression, the corner turn is required to shuffle the row-based data to become column-based data, which is needed for the azimuth FFT operation.

Figure 8.25 demonstrates the “corner turn” operation. The nth column data, which are shown in Fig. 8.25a as an M × 1 array, are rotated horizontally and appears as a 1 × M row data array in Fig. 8.25b.

FIGURE 8.25 Operation of a corner turn.

The fast, modular, and multiple memory banks addressing schemes and mass memory storage capability due to the rapid evolution of digital signal processor (DSP) and integrated circuit (IC) technology, have made the corner turn much easier and faster to operate on column-based data.

8.4.3 Range Cell Migration Correction

Range cell migration correction operates on the range-compressed data in the time or range domain. The amount of range migration can be corrected either in range–Doppler frequency or range–spatial (or slow time) domain. However, as discussed in Chapter 6 and shown in Fig. 6.17, the multiple equal slant range targets and their trajectories appear randomly along the azimuth direction in the range–spatial domain, but show up as one only in the range–Doppler frequency domain. Therefore, only the range–Doppler frequency domain is discussed here.

8.4.3.1 Computation of Range Migration Amount.

The range (which is different from slant range) axis is the line passing the target and perpendicular to the radar flight path. Let R₀ be the closest distance between radar and target and r be the slant range between the radar and the target. The slant range difference can be computed as follows:

Here is the angle between the range axis and the line connecting the radar to the target. Equation (8.7a) shows the amount of range migration required for correction with respect to R₀. From Eq. (6.3a), one obtains

Therefore

Let N be the sample length of FFT and f_PRF be the pulse repetition frequency along the azimuth direction; then the kth bin of Doppler frequency can be expressed as

and

For a high-frequency LFM signal, λkf_PRF 2VN; therefore

The number of range cells required for correction becomes

Equation (8.7d) is then computed for k = 1, 2...N with respect to the target-dependent closest distance R₀. It works well for broadside SAR systems, where f_Dc = 0. Modifications of Eqs. (8.7) are required for the squint SAR system to make the range cell migration correction with respect to the slant range distance R₃. Notice that R₃ is the shortest slant range between the radar and target, which corresponds to the Doppler frequency f_DL. Therefore

and

After range migration correction, the range differences among slant ranges are corrected and aligned along either with the reference range R₀ or slant range R₃. If R₀ is chosen, as is true for the broadside SAR system, the range cell migration correction is equivalently converts the slant range (not perpendicular to the flight path) to the range (perpendicular to the flight path). Notice that f_DL in Eq. (8.8b) refers to the true, not folded, low band edge of Doppler frequency.

Since the arrival time of the target-reflected signal varies while the radar moves through the whole synthetic aperture length, the peak of the range-compressed pulse may not be sampled accurately by the A/D converter with sampling frequency f_s. In other words, the peak of the range-compressed pulse may appear as a fraction of the sample away from the the sample digitized by the A/D converter, and may become unavailable for image processing.

Two major steps are used to correct the range cell migration: (1) fractional range sample interpolation and (2) range sample shift. These two steps can be performed in one process; however, for illustration purposes, they will be described separately.

Figure 8.26 shows an example of a 2D data array before range migration correction. This 2D data array resembles the range-compressed radar signal from a single-target squint SAR case. As can be seen, the range-compressed pulsed data along the Doppler frequency direction do not line up but spread along the range direction, and the magnitude of the pulse peaks varies because the data are not correctly sampled at the peak points. The correction process first computes the range migration amount on all row data corresponding to each Doppler frequency bin. The fractional part of the range migration amount is used to interpolate the digitized row samples. The integer part of the range migration amount is then used to shift the row data with respect to a reference sample. Further examples dealing with the squint SAR system are given later.

FIGURE 8.26 A range-compressed signal array in range–Doppler frequency domain.

8.4.3.2 Fractional Range Sample Interpolation.

The fractional part of the range sample migration amount, which is computed from Eq. (8.8b) in this example, is used to interpolate the digitized row samples. The set of 16 sinc filters with 8-tap coefficients, as discussed in Section 1.5 of Chapter 1, serves as a good example of this interpolation process. Each computed fractional sample is first quantized with a 16-step quantizer. On the basis of this quantized amount, a corresponding sinc filter is then chosen. The entire row of the data is then convolved with this sinc filter, and the output of convolution is the interpolated signal with the peak positioned appropriately. This process continues for every row of the data array along the Doppler frequency axis. Notice that for multiple targets, multiple pulses will appear in the row-based data. However, in the range–Doppler frequency domain, they all share the same amount of range migration. Therefore, one sinc filter works for all the fractional sample interpolation processes.

Figure 8.27 displays the 2D data array after the fractional range sample interpolation process. By comparing the plots between Figs. 8.26 and 8.27, one can see that the peaks of the compressed and interpolated pulses in Fig. 8.27 appear to be aligned along time samples 181–185.

8.4.3.3 Range Sample Shift.

The integer part of ΔN_Rk, shown in Eq. (8.8b), serves as the base for the range sample shift. In this example, range sample 181 is chosen as the reference sample R₃. The peaks previously aligned on different range samples are shifted left, based on integer part of ΔN_Rk, to align with sample 181. Figure 8.28 displays the compressed pulse array after the sample shift.

FIGURE 8.27 A range-compressed signal array after fractional interpolation.

FIGURE 8.28 A range-compressed signal array after sample shift.

8.4.4 Azimuth Compression

On completion of range cell migration correction, a matched filter or reference function in the azimuth direction is applied to further compress the 2D data array. As discussed in Section 6.4, the received signal along the azimuth direction also appears as a LFM waveform. This LFM signal serves as a reference function or matched filter in the same way as in range compression.

Similar to the ordinary LFM signal, the LFM signal in the azimuth direction is specified by three parameters: the Doppler frequency centroid f_Dc, the Doppler frequency-changing rate β, and the pulse duration time T_a. These are discussed next.

8.4.4.1 Doppler Frequency Centroid.

The Doppler frequency centroid f_Dc equals zero for a broadside SAR system. For a squint SAR system with given squint angle θ_q, f_Dc is nonzero and can be computed as follows, from Eq. (6.13a) or (6.16c):

8.4.4.2 Doppler Frequency Change Rate β.

The Doppler frequency changing rate β in terms of slow time s can be computed as, from Eq. (6.16a):

Since only small radar beamwidth and small squint angle are considered, the value of cos³ θ_u 1. Therefore

8.4.4.3 Pulse Duration Time T_a.

The pulse duration time of azimuth LFM signal can be computed as

where Ls is the synthetic aperture length and V is the velocity of radar. Given the radar beamwidth θ_H, the synthetic aperture length Ls can be computed as

1. Broadside SAR:

   Ls = 2R₀ tan(0.5θ_H)

2. Squint SAR with squint angle θ_q:

   Ls = R₀ tan(θ_q + 0.5θ_H) − R₀ tan(θ_q − 0.5θ_H).

Since R₀ is dependent on target location, the synthetic aperture length Ls and consequently the pulse duration time T_a are range-dependent. Therefore, the azimuth matched filter is also range-dependent, and the azimuth compression must be performed with a different matched filter, which is very time-consuming. In practical applications, only a limited number of azimuth matched filters are used, with each filter shared by a small segment of range samples.

8.5 SIMULATION RESULTS

The radar image data synthesized in Section 8.2 will now be processed by the range–Doppler algorithm. The broadside SAR system with one and three targets is described first, followed by the squint SAR system with one and three targets.

8.5.1 Broadside SAR with Single Target

The parameters and data array described in Section 8.2.1 for a single-target broadside SAR will be adopted for target reconstruction. Since the baseband signal of an LFM radar is represented as

p_b(t) = exp(jπαt²),

where α = 4 × 10¹² Hz/s and T_p = 6 × 10⁻⁶ s, the range matched filter is chosen as

The azimuth matched filter can be designed as

h_a (s) = exp(− jπβs²).

The parameters Ls, T_a and β can be computed as follows:

FIGURE 8.29 Waveforms of transmitter baseband signal, range reference function, and azimuth reference function.

Figure 8.29 displays the waveforms of baseband signal p_b(t) together with the range reference function h_r(t) and the azimuth reference function h_a(s). The real part of the signal is shown in Figs. 8.29a, 8.29c, and 8.29e; the imaginary part, in Figs. 8.29b, 8.29d, and 8.29f. The p_b(t), h_r(t) and h_a(s) are shown in Figs. 8.29a and 8.29b, 8.29c and 8.29d, and 8.29e and 8.29f, respectively. The sampling frequency is 30 MHz for range (or time) signal and 500 Hz for azimuth signal. Notice that no windowing function is used in generating the reference functions.

As can be seen, there are 181 samples in the range reference function, which corresponds to the sample numbers in the 6-μs pulse duration time. The real part of the range reference function h_r(t) is identical to that of the transmitting baseband signal p_b(t), while the imaginary part of h_r(t) is negative to that of p_b(t). The sample number of azimuth reference function is Naz = Ls/A_s = 563. The azimuth pulse duration time is T_a = 1.125 s.

Figure 8.30 shows the frequency spectrum of both the range and azimuth reference functions. The spectrum of the range reference function is based on a 256-point DFT with origin at bin 128 and sampling frequency of 30 MHz. The Doppler frequency spectrum of azimuth reference function is based on a 1024-point DFT with origin at bin 512 and sampling frequency of 500 Hz.

FIGURE 8.30 Frequency spectra of range and azimuth matched filters.

The synthesized single-target received data array shown in Fig. 8.9 will be used for range compression. The operation of range compression transforms every row of the time sample data into the frequency domain using a 256-point DFT. The frequency-domain data are then multiplied by the 256-point Fourier-transformed range reference function. The result is then inverse-Fourier-transformed back to the time-domain. The time-domain range-compressed data are shown in two displays, the 3D view in Fig. 8.31 and the 2D view in Fig. 8.32.

Figure 8.31 shows the magnitude of range-compressed data in the spatiotemporal domain. All compressed pulses are aligned along time sample 181 with equal magnitude and span through the whole synthetic aperture length Ls. The number 181 is the sample length of transmitter pulse.

Figure 8.32 is another view of the range-compressed data array. The horizontal axis (abscissa) shows the time samples; and the vertical axis (ordinate), the azimuth lines, together with magnitudes of the compressed pulses for each azimuth line. There are 563 rows along the vertical axis, and only 1 out of every 30 rows is displayed here. As can be seen, all compressed pulses align at time sample 181.

From Fig. 8.32, one can see that the range migration problem does not occur. Therefore, no range cell migration correction is required in this case.

FIGURE 8.31 3D view of a range-compressed signal array based on Fig. 8.5.

The number of azimuth samples within the synthetic aperture length is Naz = Ls/A_s = 563. Transformation of the range-compressed signal array into Doppler frequency domain is performed by applying a 1024-point DFT on every column of the range-compressed signal array. Since Naz = 563, zero padding is required for the 1024-point DFT. The magnitude of the center-shifted Doppler frequency spectrum, with origin at bin 512, is shown as a 3D view in Fig. 8.33. The two perpendicular axes of the base plane are the time samples and the Doppler frequency bins, respectively. As can be seen, the Doppler frequency spectrum aligns along time sample 181, and the center of the Doppler frequency spectrum is at bin 512 with a bandwidth of ~800 frequency bins. Given the azimuth sampling frequency f_PRF = 500 Hz, the Doppler frequency bandwidth can be computed as 500 × 800/1024 400 Hz.

FIGURE 8.32 2D view of a range-compressed signal array based on Fig. 8.31.

FIGURE 8.33 3D view of a range–Doppler frequency spectrum based on Fig. 8.31.

Another way to view the display of the center-shifted Doppler frequency spectrum is shown in Fig. 8.34. Here the horizontal axis shows time samples; the vertical axis, Doppler frequency bins. There are 1024 rows along the vertical axis, and only 1 out of every 30 rows is displayed here. Similar to that of Fig. 8.32, the Doppler frequency spectrum aligns at time sample 181, and the center of the Doppler frequency spectrum is at bin 512 with ~400 Hz bandwidth.

FIGURE 8.34 2D view of a range–Doppler frequency spectrum based on Fig. 8.33.

The azimuth compression is performed by applying the range-dependent azimuth matched filter to every column of the Doppler frequency data array. Only one azimuth matched filter is used in this single-target example. The azimuth matched filter h_az(s) is first transformed into the Doppler frequency domain, using a 1024-point DFT, to become H_az(ω_D). Without using any windowing functions, the azimuth matched filter H_az(ω_D) is then multiplied with every column of the data array in the range–Doppler frequency domain. A 1024-point IDFT is then applied on every column of the azimuth-compressed data array to obtain the reconstructed target.

Figure 8.35 is a 3D view of the reconstructed target. The two axes of the base plane correspond to time samples and azimuth lines. The reconstructed target appears as an impulse-like signal with very narrow sidelobes around the peak. The impulse-like target appears at time sample 181 and azimuth line 563. These two numbers correspond to the sample length of range pulse duration and azimuth pulse duration, respectively.

Figure 8.36 shows the cross sectional view of Fig. 8.35: Fig. 8.36a shows the display in terms of the azimuth lines with time sample 181; Fig. 8.36b, in terms of the time samples with azimuth line 563. The magnitude of the pulse is 181, which matches the sample length of the range pulse duration.

FIGURE 8.35 3D view of a reconstructed single-target function based on Fig. 8.33.

FIGURE 8.36 Cross-sectional view of a reconstructed single-target function based on Fig. 8.35.

8.5.2 Broadside SAR with Multiple Targets

The system model for broadside SAR with three targets shown in Fig. 8.3, together with the received data array shown in Fig. 8.13, will be used to reconstruct the targets.

Both the LFM transmitter signal and the matched filter for range compression will remain the same as in the single-target case:

The azimuth matched filter h_az(s) and the Doppler frequency change rate β are range-dependent. There are three targets located at two different ranges in this example: therefore, two sets of h_az(s) and β must be designed:

The azimuth pulse duration times T_a1 and T_a2, corresponding to h_az1(s) and h_az2(s), respectively, are computed as follows:

The synthesized three-target data array, shown in Fig. 8.13, is then range-compressed by first transforming every row of range data into the frequency domain using a 256-point DFT. The frequency-domain data are then multiplied by the 256-point Fourier transformed range reference function. The result is then inverse-Fourier-transformed back to the time domain.

Figure 8.37 is a 3D view of the range-compressed data array. There are two columns of a thin wall-like array. The first array has two magnitudes and lines up along time sample 181; the second array has one magnitude and aligns along time sample 211. The first array corresponds to two targets located at the same range R_0a = 7500 m. The larger magnitude portion of the first array occurs at the overlapped region between the two synthetic aperture lengths Lsa1 and Lsa2. The second array corresponds to the target located at range R_0b = 7650 m.

FIGURE 8.37 3D view of a range-compressed signal array based on Fig. 8.3.

FIGURE 8.38 2D view of a range-compressed signal array based on Fig. 8.37.

Figure 8.38 shows a different view of the same range-compressed data array. The horizontal axis represents time samples; the vertical axis, azimuth lines, together with magnitude of the compressed pulse for each azimuth line. There are 2Naza − D₁ = 939 rows along the vertical axis, and only 1 out of every 30 rows is displayed here. As can be seen, there are two sets of column arrays: one extending through the whole length of azimuth samples (939) and aligning at time sample 181, the other extending with shorter azimuth samples (563) and lining up at time sample 211. Again, the range migration problem does not occur here, and range cell migration correction is not required in this case.

The next step in reconstructing the targets is transformation of the data array from the range–azimuth domain into the range–Doppler frequency domain. The total synthetic aperture length along the azimuth direction due to targets 1 and 3 is 2Naza − D₁ = 949, the sample length of the azimuth matched filter is Naza = 563, and the output sample length of the azimuth matched filter is (949 + 563 − 1) = 1511. Therefore, a 2048-point DFT with zero padding on azimuth samples is adopted. By applying 2048-point DFT to every column of the range-compressed data array, the magnitude of the center-shifted Doppler frequency spectrum, with origin at bin 1024, is displayed in Fig. 8.39. The two perpendicular axes of the base plane are the time samples and the Doppler frequency bins, respectively. As expected, two columns of the Doppler frequency spectrum array exist; one lines up along time sample 181, and the other aligns at time sample 211. The left one has larger magnitude than the right one, due to the existence of multiple targets. Both columns of the Doppler frequency spectrum array are centered on frequency bin 1024 with a bandwidth of ~1600 frequency bins. Since the sampling frequency f_s = f_PRF = 500 Hz, the Doppler frequency bandwidth can be computed as 500 · (1600/2048) 391 Hz.

FIGURE 8.39 3D view of a range–Doppler frequency spectrum based on Fig. 8.37.

FIGURE 8.40 2D view of a range–Doppler frequency spectrum based on Fig. 8.39.

Figure 8.40 shows another view of the center-shifted Doppler frequency spectrum. Here the horizontal axis represents time samples; and the vertical axis, Doppler frequency bins. Similar to that of Fig. 8.38, both columns of the Doppler frequency spectrum array align at two time samples: one at 181 and the other at 211. Both arrays are centered at Doppler frequency bin 1024. The magnitude of the left column array seems to have a smaller value at some part of the frequency bins. This is due to the phase cancellation of two targets located at the same range of R_0a. Actually, both spectrum arrays have about the same bandwidth, but the left one has larger magnitude than the right one because of the existence of two targets.

The azimuth matched filter is then applied on every column of the data array in the Doppler frequency domain to perform the azimuth compression. Because of the existence of two targets, two azimuth matched filters, h_az1(s) and h_az2(s), are used. They are first transformed into the Doppler frequency domain, using a 2048-point DFT with zero padding. One is applied to the region around time sample 181, and then multiplied by every column of the data array in that region. The other is applied to the region around time sample 211, and then multiplied by every column of the data array in that region. A 2048-point IDFT is then applied on every column of the whole data array, and the reconstructed targets appear.

Figure 8.41 is a 3D view of the reconstructed targets. The x and y axes denote time samples and azimuth lines, respectively. The z axis represents magnitude. There are three impulse-like targets with about the same magnitude; two of them are located at (181, 563) and (181, 939), respectively, while the third one is located at (211, 818).

Figure 8.42 displays a cross-sectional view of Fig. 8.41 at two different time samples. Figure 8.42a shows the magnitude of the target distributed along the azimuth axis at time sample 181; Fig. 8.42b shows the magnitude of the target distributed along the azimuth axis at time sample 211. There are two pulses (or targets) appearing in Fig. 8.42a, one at azimuth line 563 and the other at azimuth line 939. Only one pulse (or target) appears in Fig. 8.42b, located at azimuth line 818. The two targets, appearing at azimuth lines 939 and 563, are located at range distance R_0a. They are separated by 376 samples in the azimuth direction and correspond to 376 × 0.4 m = 150.4 m in distance. This number matches closely to the target location specification (y₃ − y₁ = 150 m). The third target is located at range distance R_0b and appears at azimuth line 818. This target is 255 samples away from the first target (818–563) in the azimuth direction, which corresponds to 255 × 0.4 m = 102 m. Since two azimuth matched filters were used at targets located at R_0a and R_0b; therefore, adjustment of the range ratio, R_0a/R_0b = (7500/7650) = 0.98, should be made. The number 102 m should be adjusted to become 102 m × 0.98 = 100 m, which matches well with the target specification (y₂ − y₁ = 100 m).

FIGURE 8.41 3D view of a reconstructed target function based on Fig. 8.39.

FIGURE 8.42 Cross-sectional view of Fig. 8.41 at range samples 181 and 211.

Figure 8.43 displays a cross-sectional view of Fig. 8.41 at three azimuth lines. Figure 8.43a shows the magnitude of the target distributed along the time sample axis at azimuth line 563; Fig. 8.43b, at azimuth line 818; and Fig. 8.43c, at azimuth line 939. In Figs 8.43a, 8.43b, and 8.43c one pulse (or target) is located at range samples 181, 211, and 181, respectively. The sample length of the transmitting pulse is 181, which is where the targets in Figs. 8.43a and 8.43c are located. The target of Fig. 8.43b is located at time sample 211, which is 30 time samples away from targets 1 and 3, and corresponds to 10 × (30/2) = 150 m. It matches well with specification [R_0b − R_0a = (7650–7500) = 150 m].

FIGURE 8.43 Cross-sectional view of Fig. 8.41 at azimuth lines 563, 818, and 939.

8.5.3 Squint SAR with Single Target

The signal array of the single-target squint SAR described in Section 8.3.1 will be used to reconstruct the target. The baseband signal and the matched filter used in the single-target broadside SAR system will also be used in this case, and they are expressed as

Here α = 4 × 10² Hz/s and T_p = 6 × 10⁻⁶ s.

The azimuth matched filter with squint angle θ_q can be designed as

h_az (s) = exp(− j2π f_Dcs − jπβs²),

where f_Dc is the centroid of Doppler frequency, and s refers to slow time. The parameters f_Dc, T_a, and β can be computed as follows:

The waveform of azimuth reference function h_az(s) is shown in Fig. 8.44. The waveform was plotted with 5 times the sampling frequency f_PRF for a better view. The real part of the signal is shown in Fig. 8.44a; the imaginary part, in Fig. 8.44b. No windowing function is used in generating the reference function.

From the LFM waveforms described above, one can observe that the signal is no longer symmetric about the center point as it is in the broadside case. The azimuth reference function has a total of Naz = Ls/A_s = 569 azimuth samples, and the azimuth pulse duration time is T_a = 1.1375 s.

FIGURE 8.44 Waveforms of the real and imaginary parts of azimuth reference function.

FIGURE 8.45 Frequency spectrum of azimuth reference function.

The Doppler frequency spectrum of the azimuth reference function h_az(s) is shown in Fig. 8.45. Since the azimuth signal is naturally digitized at pulse repetition frequency f_PRF, only the digitized spectrum is observable, and its replicas are duplicated at the rate of f_PRF (=500 Hz). The Doppler frequency spectrum shown in Fig. 8.45 is the observed spectrum. It is the fallback version of the true spectrum (located at passband) and is obtained by applying a 1024-point DFT on h_az(s) with origin at bin 1. As described in Section 7.3.1.2, and shown as an example in Fig. 7.15, the true spectrum folds from passband to baseband and splits into two portions to become the observable spectrum. The spectrum shown in Fig. 8.45 consists of two parts; one extends from frequency bin 1 to about bin 190 and represents the upper portion of the frequency band, while the other extends from bin 410 to bin 1024 and represents the lower portion of the frequency band.

The synthesized data array with a single target shown in Fig. 8.17 will be used for range compression. Every row of the time sample data is first transformed into the frequency domain using a 256-point DFT. The frequency-domain data are then multiplied by the Fourier-transformed range reference function. The result is then inverse-Fourier-transformed back to the time domain. A 3D view of the time–spatial-domain range-compressed data array is shown in Fig. 8.46.

The range-compressed data shown in Fig. 8.46 differ significantly from those of the broadside SAR shown in Fig. 8.31. The compressed range data array no longer lines up at time sample 181, but migrates to several time samples around it. In addition, they are no longer equal in magnitude as those shown in Fig. 8.31. The differences are due primarily to the range migration in the squint SAR system.

Figure 8.47 shows a different view of the range-compressed data array. Here, the horizontal and vertical axes are again defined in the same way as in Fig. 8.32. However, the data array migrates to several time samples away from time sample 181 for the same reason of range migration. Range cell migration correction is therefore required in this squint SAR case.

FIGURE 8.46 3D view of a range-compressed signal based on Fig. 8.14.

FIGURE 8.47 2D view of a range-compressed signal from Fig. 8.14.

FIGURE 8.48 3D view of a spatial Fourier transformed signal from Fig. 8.46.

To correct the range cell migration problem, the range-compressed data array in the range–azimuth domain is transformed into the range–Doppler frequency domain. Since the number of azimuth samples is 569, the 1024-point FFT is chosen and applied to every column of the range–azimuth data array.

Figure 8.48 is a 3D view of the range-compressed data array in the range–Doppler frequency domain with origin at bin 1. The x axis represents the time sample axis with sample numbers 1–256, the y axis represents the Doppler frequency axis with frequency bins 1–1024, and the z axis represents the magnitude of the data array. For squint SAR system, the data array has a Doppler frequency bandwidth of B_D < f_PRF and a center frequency of f_Dc. Similar to the previous discussion when azimuth reference filter was used for squint SAR, the Doppler frequency spectrum is folded into two parts in the baseband display. The upper frequency part, starting from bin ~400 to bin 1024, is the lower portion of the true frequency spectrum, while the lower part, starting from bin 1 to bin ~200, is in fact the upper portion of the true frequency spectrum. The center frequency f_Dc is located at bin ~800.

A different view of the range-compressed signal in the range–Doppler frequency domain is shown in Fig. 8.49, where 1 out of 30 frequency bins are displayed. The range-compressed data array along the Doppler frequency direction is no longer aligned at time sample 181; instead, it migrates from time samples 181 to 185. A range cell migration correction is therefore required.

For the squint SAR system, the range migration correction with respect to slant range R₃ in the time sample domain equivalently involves aligning the frequency bins with respect to f_DL in the Doppler frequency domain, where the maximum amount of range migration occurs at f_DU. Since the squint angle θ_q is greater than 0.5θ_H, Eq. (6.16d) can be applied to compute the true or unfolded f_DU and f_DL:

FIGURE 8.49 2D view of a spatial Fourier-transformed signal from Fig. 8.46.

Given a 1024-point DFT and sampling frequency f_PRF = 500 Hz, the Doppler frequency bin spacing can be computed as Δf_D = (500/1024) = 0.488 Hz. Therefore the folded f_DL corresponds to bin number 1 + (1195 − 2 × f_PRF)/0.488 = 401, while the folded f_DU corresponds to bin number 1 + (1593 − 3 × f_PRF)/0.488 = 191. The bandwidth of Doppler frequency spectrum then has a total number of frequency bins = (1024 − 401 + 191) = 814.

The maximum amount of range cell migration can be computed from Eq. (8.8b) as

where k = 814, f_DL = 1195, and R_s = 10 were used.

Alternatively, one can also compute the maximum amount of range migration by computing the difference between slant ranges R₁ and R₃, which corresponds to f_DU and f_DL, respectively; that is, from Eq. (6.11b)

The maximum number of range cells that need to be corrected becomes

This matches closely with the result based on Eq. (8.8b).

The fractional range sample interpolation and range sample shift are then applied on the signal array of Fig. 8.48. An 8-tap interpolation sinc filter with sample resolution as described in Section 1.6 (of Chapter 1) is applied here for fractional sample interpolation. A 3D view of the range cell migration corrected Doppler frequency spectrum is shown in Fig. 8.50.

FIGURE 8.50 3D view of Fig. 8.46 after range cell migration correction.

FIGURE 8.51 2D view of Fig. 8.46 after range cell migration correction.

As can be seen, the magnitudes of the spectrum are about equal, and the spectrum consists of two parts: one extending from frequency bins 1 to ~190 and the other, from frequency bins 390 to ~1024. The range cell migration corrected Doppler frequency spectrum now lines up at time sample 181.

A different view of the range migration corrected signal spectrum is shown in Fig. 8.51. Here the horizontal axis represents time samples, and the vertical axis represents Doppler frequency bins with 1 out of 30 bins displayed. Compared to Fig. 8.49, the new plot shows that all Doppler frequency bins are aligned along time sample 181.

The range cell migration corrected data array in the range–Doppler frequency domain is then processed for azimuth compression. Different range-dependent azimuth matched filters h_az(s) should be applied to every column of the signal array. However, only one filter is used in this simulation. Once the azimuth compression is completed, the data array is inverse-Fourier-transformed back to the range–azimuth domain. A 3D view of the final range–azimuth data array is shown in Fig. 8.52, where an impulse-like signal appears at time sample ~181 and azimuth sample ~571.

Two cross-sectional views of Fig. 8.52 are shown in Fig. 8.53. Figure 8.53a is a cross-sectional view at time sample 181. The signal distributed along the azimuth direction appears as an impulse at azimuth sample 571, with a very narrow pulsewidth. Figure 8.53b is a cross-sectional view at azimuth sample 571. The signal distributed along the time sample direction appears as an impulse at time sample 181, with a similarly narrow pulsewidth. Again, the number 181 matches the time sample length of the range matched filter, while the number 571 differs slightly from the sample length used in the azimuth matched filter, which has 569 samples. The reconstructed signal appears quite accurately as expected.

FIGURE 8.52 3D view of a reconstructed target function from Fig. 8.14.

8.5.4 Squint SAR with Multiple Targets

The system model shown in Fig. 8.18 for the multiple-targets squint SAR system, together with the synthesized data array shown in Fig. 8.22, will be used to reconstruct the targets.

Again, the range matched filter remains the same as before:

h_r(t) = exp(− jπαt²).

FIGURE 8.53 Cross-sectional view of Fig. 8.52 at range sample 181 and azimuth line 571, respectively.

Similar to the multiple targets of the broadside SAR system, two azimuth matched filters will be used in this example:

The parameters f_Dc, T_a1, T_a2 β₁, and β₂ can be computed as follows:

The synthesized data array shown in Fig. 8.22 serves as raw data for range compression. Every row of the range data is first transformed into the frequency domain using a 256-point DFT. The frequency-domain data are then multiplied by the 256-point Fourier-transformed range reference function. The result is then inverse-Fourier-transformed back to the time domain. Figure 8.54 displays a 3D view of the range-compressed data in the time–spatial (range–azimuth) domain. There are two column-like data arrays along the azimuth direction. The left array corresponds to two targets located at range R_0a, while the right array corresponds to one target located at range R_0b. The magnitudes of the two data arrays appear to be unequal and show several peaks. This is due to the range migration that occurs in the squint SAR system, and range migration correction is required.

FIGURE 8.54 3D view of a range-compressed signal from Fig. 8.18.

A different view of the range-compressed data array is shown in Fig. 8.55, where 1 out of 30 azimuth lines are displayed. There are two columns of data arrays in the plot: one located at time sample ~181 and the other, at time sample ~211. The first column of the data array corresponds to two targets located at range R_0a. The dual-magnitude portion of the array is the beamwidth-overlapped region from the two targets. The second column of the data array corresponds to the target located at range R_0b. The first column of the data array extends in the azimuth direction for a total of 2Naza − D₁ = 945 azimuth samples;. the second one extends in the azimuth direction for a total of Nazb = 581 azimuth samples. Range migration is clearly visible from either the first or second column of the data array.

FIGURE 8.55 2D view of a range-compressed signal from Fig. 8.18.

FIGURE 8.56 3D view of spatial Fourier-transformed signal from Fig. 8.54.

To correct the range migration problem, the data array in the range–Azimuth (time–spatial) domain is first transformed into the range–Doppler frequency domain. Since the azimuth sample length of the first column data array is 2Naza − D₁ = 945 and the sample length of the azimuth matched filter is 569, a 2048-point DFT is chosen to transform every column of the range-compressed data array into the Doppler frequency domain. A 3D view of the range–Doppler frequency spectrum is displayed in Fig. 8.56 with Doppler frequency origin at bin 1.

There are two arrays of the Doppler frequency spectrum: one along time sample ~181 and the other, along time sample ~211. Both spectrum arrays are folded into two parts as a result of the natural digitization at the pulse repetition frequency f_PRF, and their magnitudes are unequal because of the range migration problem.

A different view of the Doppler frequency spectrum described above is shown in Fig. 8.57. As can be seen, the two spectrum arrays have the same size along the Doppler frequency direction. Because of phase cancellation from the two targets located at the same range distance R_0a, some portions of the spectrum array in the left column appear to have smaller magnitude than do those in the rest of the spectrum array. The magnitude of the left column array is higher than that of the right one because there are two targets.

FIGURE 8.57 2D view of a spatial Fourier-transformed signal from Fig. 8.54.

Range cell migration correction is then performed on the Doppler frequency spectrum arrays described above. For a squint SAR system, the amount of range migration will be corrected with respect to the slant range R₃. The maximum amount of range migration occurs at f_DU. Both f_DU and f_DL can be computed as follows, from Eq. (6.16d):

The frequency f_DU corresponds to bin number 1 + (1593 − 3 × f_PRF) × (2048/500) = 382. The frequency f_DL corresponds to bin number 1 + (1195 − 2 × f_PRF) × (2048/500) = 802. The Doppler frequency spectrum then has a total number of frequency bins equal to (2048 − 802 + 382) = 1628, and the bandwidth is (1628 × 500)/2048 = 397.5 Hz.

The maximum amount of range cell migration correction for the two column data arrays can be computed from Eq. (8.8b) as

where k = 1628, f_DL = 1195, R_0a = 7500, R_0b = 7650, and R_s = 10 were used.

Alternatively, the maximum amount of range migration can also be obtained by computing the difference between slant ranges R₁ and R₃; that is, from Eq. (6.11b), one obtains

The maximum number of range cells that need to be corrected becomes

The results closely match those based on Eq. (8.8b).

FIGURE 8.58 3D view of Fig. 8.56 after range cell migration correction.

Both fractional range cell interpolation and range sample shift are then applied on the signal array of Fig. 8.56. For simplicity, only ΔN_k1 was used in simulation and the result is shown in Fig. 8.58.

As can be seen, the magnitudes of the range migration corrected spectrum array become equal for both columns of the spectrum array. The magnitude of the left column array doubles that of the right one because there are two targets.

A different view of the range migration corrected spectrum is shown in Fig. 8.59. The horizontal axis shows the time samples, and the vertical axis represents the Doppler frequency bins with 1 of 30 bins shown. The new plot shows that the two spectrum arrays are aligned at two time samples, one at sample 181 and the other at sample 211. The bandwidth of both arrays is about equal, yet the magnitude of the left column is greater than that of the right column because there are two targets. Again, some portions of the left column appear smaller than the rest of the same array because of phase cancellation from the two targets at the same range distance R_0a.

Azimuth compression is then performed on the range migration corrected spectrum array in the range–Doppler frequency domain. The range-dependent azimuth matched filter h_az1(s) is applied to columns of time samples 1–190 of the spectrum array, and h_az2(s) is applied to columns (time sample) 191–256 of the spectrum array. After the azimuth compression, the Doppler frequency spectrum array is inverse-Fourier-transformed back to the range–azimuth (or time–spatial) domain. A 3D view of the final data array in the spatiotemporal domain is shown in Fig. 8.60 with two impulse-like signals appearing at (181,571) and (181,947), respectively, and the third one at (211,786). All three targets appear to have the same magnitude.

FIGURE 8.59 2D view of Fig. 8.56 after range cell migration correction.

Figure 8.61 displays a cross-sectional view of Fig. 8.60 at two time samples: Fig. 8.61a shows the magnitude of the targets distributed along the azimuth axis at time sample 181; Fig. 8.61b, at time sample 211. Two pulses (or targets) appear in Fig. 8.61a: one at azimuth sample 571 and the other at azimuth sample 947. Only one pulse (or target) appears in Fig. 8.61b, located at azimuth sample 786. The difference between samples 947 and 571 along the azimuth direction is 376, which corresponds to 376 × 0.4 m = 150.4 m. This number closely matches the specification (y₃ − y₁ = 150 m). The difference between samples 786 and 571 along the azimuth direction is 215, which corresponds to 215 × 0.4 m = 86 m. This number is 14 m shorter than the specification, y₂ − y₁ = 100 m. This discrepancy is due to two factors: (1) the two targets are located at different range distances, with one located at range R_0a and the other at range R_0b (range differences require the use of different azimuth matched filters and adjustment of the range ratio); and (2) the nonzero squint angle is θ_q = 6°, which shortens the azimuth distance by an amount δy = Δx tan θ_q.

FIGURE 8.60 3D view of a reconstructed target function from Fig. 8.18.

FIGURE 8.61 Cross-sectional view of Fig. 8.60 at range samples 181 and 211.

Therefore, adjustment of the range ratio R_0a/R_0b = (7500/7650) = 0.98 should be applied to the number 86 to become 86 × 0.98 m = 84.28 m. In addition, with Δx = 150 m in the range direction and θ_q = 6°, the azimuth distance Δy is reduced by an amount δy = Δx tan θ_q = 15.76 m. Therefore, the theoretical result should be (100 − 15.76) = 84.24 m. The simulation result, 84.28 m, matches the specification well in this case.

Figure 8.62 shows a cross-sectional view of Fig. 8.60 at three azimuth lines: Fig. 8.62a shows the magnitude of the target distributed along the time sample axis at azimuth line 571; Fig. 8.62b, at azimuth line 786; and Fig. 8.62c, at azimuth line 947. A single target is located at the same time sample (181) in both Figs. 8.62a and 8.62c. The single target in Fig. 8.62b is located at time sample 211. The time sample difference between these two sets of targets is (211−181) = 30, which corresponds to range difference 10 × (30/2) = 150 m. The simulation result closely matches the target specification (R_0b − R_0a = 150 m).

FIGURE 8.62 Cross-sectional view of Fig. 8.60 at azimuth lines 571, 786, and 947.