7

SYSTEM MODEL AND DATA ACQUISITION OF SAR IMAGE

The two-dimensional target locations (or function) of radar image can be obtained by processing the target-reflected signals in both the cross-range (azimuth) and range directions. Range migration correction may be required for some applications. The principles and basic characteristics of radar image processing have been covered in a previous chapter in terms of Doppler frequency. This chapter describes radar image processing in the wavenumber domain. Section 7.1 describes the range radar imaging in the wavenumber domain; Section 7.2 discusses the cross-range radar imaging. Both the broadside SAR and the squint SAR are covered. Section 7.3 reviews data acquisition and the frequency spectrum of radar images.

7.1 SYSTEM MODEL OF RANGE RADAR IMAGING

7.1.1 System Model

Figure 7.1 shows a radar system configuration for range processing. The radar beam is assumed to be narrow enough in the azimuth direction that all targets are located at y = 0 and randomly distributed along the range (or x axis) direction.

Figure 7.1a is a 3D display of the radar system. The radar, moving at speed V along the y axis, is located at (0, 0, H) and emits a pulse at t = 0. The ground area covered by the radar pulse in the range direction will reflect the pulse wave after a short delay. The shortest distance to the ground is R_N, and the longest distance is R_F. The echo delay time due to the target at R_N is t₀ = (2R_N/c) = (2x₀/c sin θ). The echo delay time due to the target at R_F is t_N−1 = (2R_F/c) = (2x_N−1/c sin θ). The corresponding ground ranges of t₀ and t_N−1 are x₀ = [(ct₀ sin θ)/2] and x_N−1 = [(ct_N−1 sin θ)/2]. The swath D_x, which ranges from x₀ to x_N−1, is related to the radar beamwidth as D_x θ_V (R_N + R_F)/2 = λ (R_N + R_F)/(2W).

FIGURE 7.1 Geometry of a range imaging radar.

Figure 7.1b shows the relationship between radar pulse and returned echoes. Here the horizontal axis indicates the echo return time. The radar pulse emits at t = 0 and the echo returns at time t₀ through t_N−1. Since parameters t and x are linearly related by t = 2x/c, they are used interchangeably in the following discussion.

Consider a set of N discrete point targets, each located at x_n with a reflectivity σ_n, where n = 0, 1, 2,..., N − 1. The ideal target function, which identifies the location of each target along the x axis, is defined as follows:

Figure 7.2 is an example of the ideal target function f₀(x). The targets are randomly located along the x axis between x₀ and x_N−1. The magnitude of each target denotes reflectivity σ_n, and eventually reflects as a bright or dark dot on the radar image.

FIGURE 7.2 An ideal target function.

Let p(t) be the radar signal illuminated on a one-dimensional target area in the range (x) direction as shown in Fig. 7.1. The returned echo signal s(t) becomes

The Fourier transform of Eq. (7.2), with t_n = 2x_n/c, can be expressed as

where P(ω) is the Fourier transform of p(t).

Since the delay time t_n and range x_n are linearly related and interchangeable, the Fourier transform of Eq. (7.1) can be expressed as

From Eqs. (7.3) and (7.4), one obtains

The corresponding equation in time t or range x domain becomes

s(t) = p(t) * f₀(t)

or

where the asterisk denotes convolution.

7.1.2 Reconstruction of Range Target Function

Mathematically, given the received signal spectrum S(ω), the ideal target function f₀(x) can be derived from Eq. (7.5) as

or

However, for Eq. (7.7) to work properly, the function P(ω) must satisfy two conditions: (1) the bandwidth must be infinite, and (2) no zeros can exist in P(ω). Since P(ω) is the frequency spectrum of a radar pulse, its envelope has the sinc-like function. Therefore P(ω) cannot meet conditions 1 and 2 (above) to make Eq. (7.7) work correctly.

One practical method of finding the target function f(x) for range imaging is to utilize the concept of a matched filter as discussed in Chapter 5. Let the matched filter function P*(ω) be chosen as the complex conjugate of the transmitted pulse function P(ω), with the time-domain representation of p*(ω) as p*(− t). The returned echo signal s(t) can be considered as the output of a system (or target) function f₀(t) with input p(t). This output signal s(t) is then applied to the matched filter to obtain the target function f(t). The block diagram for solving the target function f(t) is shown in Fig. 7.3.

FIGURE 7.3 Matched filtering for range imaging.

The output of the matched filter can be expressed, in the frequency domain, as follows:

Let psf(t) be a point spread function; that is

psf(t) = F⁻¹[|P(ω)|²]

or

where psf(ω) is the Fourier transform of psf(t); then

Alternatively, the target function can be expressed in terms of range x as

The point spread function psf (t) depends on the spectral shape of the transmitted radar signal p(t). As an example, let |P(ω)| = 1 for ω [−ω₀, ω₀]; then psf(t) becomes

An example of a matched filter output, or reconstructed target function f(x), is shown in Fig. 7.4. As can be seen, each target appears at a different location x_n with different magnitude, which corresponds to the reflection coefficient σ_n. When comparing Fig. 7.2 with Fig. 7.4, one can see that the nth target location changed from a point δ(x − x_n) in the ideal target function f₀(x) to a segment occupied by psf(x − x_n) in the real target function f(x). The point spread function psf(t) becomes sharper, from Eq. (7.9), if the bandwidth of p(t) increases. A sharper psf(x) improves the range resolution, and therefore the quality of the radar image.

The discussion above shows that range image processing detects the returned echo signal on the basis of a given transmitted signal. It depends only on the target distance or range, and is independent of the squint angle of the radar. Therefore, the prior discussion applies to both broadside SAR and squint SAR systems.

FIGURE 7.4 A reconstructed target function f(x).

7.2 SYSTEM MODEL OF CROSS-RANGE RADAR IMAGING

This section discusses two cases of cross-range radar imaging. The broadside SAR is addressed first, followed by squint SAR.

7.2.1 Broadside Radar Case

7.2.1.1 System Model.

The geometry of a typical radar system for cross-range image processing is shown in Fig. 7.5. Similar to but somewhat different from range image processing, the radar pulse is assumed to be narrow enough along the range direction so that all targets are located at the same range x = X_c and randomly distributed along the y axis direction. It is assumed that all targets are located within the radar beamwidth.

Figure 7.5a shows a broadside SAR imaging system with M ground targets located along the cross-range (or y axis) direction. Each target has a reflection coefficient σ_m, where m = 0, 1, 2,..., M − 1. The imaging radar is located at (0, u, H) and moves at speed V along the y axis. The radar illuminates the target area with beamwidth θ_H. The triangular shaded area shows a 2D plane where both radar and ground targets are in the same plane. Certainly the assumption that all targets have a range distance X_c is not practical in the real world. However, as will be discussed in Chapter 8, the radar image can be processed in range and cross-range directions independently if the range migration problems can be ignored. Therefore, the assumption that all targets in the cross-range direction are located at the same range X_c will not lose any generality in our future discussions.

Figure 7.5b is a simplified model based on the triangular shaded area of Fig. 7.5a. The x′ axis is perpendicular to the flight path. The y direction is the cross-range direction, and the targets are all located within the beamwidth of the radar signal. All M targets are located at x′ = X_c′, where X_c′ = (H² + X_c²)^1/2, and y = y_m for m = 0, 1,..., M − 1. We will use x and X_c instead of x′ and X_c′ in the following discussion. With the radar located at y = u, the M targets are represented by their reflection coefficients as σ₀, σ₁,..., σ_M−1.

FIGURE 7.5 (a) A typical cross-range radar imaging system; (b) a simplified system.

Let f₀(y) be an ideal target function in the cross-range domain, which identifies a group of M targets located along the y axis. Then f₀(y) can be defined as follows, similar to the ideal target function described in range domain:

Cross-range image processing is based on the phase history of returned signals from the targets, which are located along the y axis and illuminated under the radar beam. Consider a single target at distance R from the radar, which has a 3-dB beamwidth θ_H. The single-target phase history is computed during the time interval T₁ = Ls/V, where Ls = Rθ_H is the synthetic aperture length and V is the velocity of the moving radar. For a group of M targets spread over a range of length D_y, the time interval for computing phase history becomes T_a = (Ls + D_y) /V.

Figure 7.6 shows the relationship between the radar beam and the targets. The M targets are assumed to be distributed between –D_y/2 and D_y/2. The radar moves between –L_u and L_u along the y axis and continuously illuminate the targets, where L_u = (Ls + D_y)/2.

Also shown in Fig. 7.6 are the aspect angles of radar with respect to a particular target. The aspect angle is defined as the angle between the x axis and the line that connects the radar to the target. With radar located at u, the aspect angle with respect to the mth target is defined as

FIGURE 7.6 Relationship between radar beams and targets.

where x_m = X_c in our discussion.

Consider a system with radar located at (0, u), where u is within the range −L_u ≤ u ≤ L_u, and the M targets are located at {(X_c, y)} for y = y₀, y₁,..., y_M−1. Let p(t) be the transmitted signal and R_m(u) be the slant range between the radar and the mth target:

The radar echo signal from M targets can then be expressed as

For purposes of simplicity and to preserve generality, the radar signal p(t) is assumed to be a single-frequency signal. Thus, p(t) = exp(jωt), and Eq. (7.14) becomes

where k = ω/c = 2π/λ. After a baseband conversion, Eq. (7.15) becomes

where

To simplify the notation, s(u) and s_m(u) will be used to represent s_b(t, u) and s_bm (t, u) in the following discussion.

The instantaneous spatial frequency of s_m(u) for u in the synthetic aperture region, −L_u ≤ u ≤ L_u, is

where θ_m(u) is the aspect angle of radar for the mth target when radar is located at (0, u) and can be computed as follows, as defined in Eq. (7.13):

Let an ideal reflector be located at (X_c, 0) with reflection coefficient σ_r = 1. Also let s_r0(u) be a reference signal, which is the echo signal from this ideal reflector when radar is located at (0, u):

From Eq. (7.12) with y = u, the ideal target function becomes

The convolution of s_r0(u) with f₀(u) turns out to be

where s(u) is the baseband signal shown in Eq. (7.16).

FIGURE 7.7 Relationship between received signal and reference signal.

Equation (7.21) states that the baseband echo signal s(u) can be considered as the output function of a system, where f₀(u) and s_r0(u) are the system function and input function, respectively. Note that since the radar location u is a variable along the y axis, the parameters y and u are therefore interchangeable. Figure 7.7 shows the relationship between s_r0(u), f₀(u), and s(u) for the cross-range target (image) processing, which is similar to that of the range target processing discussed in the previous section.

The purpose of cross-range target processing is to find the target location and its corresponding reflection coefficient. Equation (7.21) states that the target-reflected signals are formulated as the convolution of the ideal target function with the reference function.

The spatial frequency (or spatial Fourier) transform converts the signal from the spatial (y or u) domain to the spatial frequency (k_y or k_u) domain. In other words, given a signal g(y), the spatial frequency transform of g(y) is defined as

The spatial frequency transform of s_m(u), expressed in Eq. (7.17), can then be expressed as

As discussed previously, the radar movement is limited from u = −L_u to u = L_u for a group of M targets. It is difficult to obtain a closed form of the integration shown in Eq. (7.23). Instead, the integration will be approximated by the “principle of stationary phase.” This method ignores the integration regions at the far ends and is concerned only with some limited narrow regions, which are explained in the next section.

7.2.1.2 Principle of Stationary Phase.

Consider a general waveform

where a(t) is a slow time-varying amplitude function relative to the phase function (t).

The Fourier transform of s(t) is

Since a(t) is a slow varying function relative to the phase function ωt − (t), the contribution to the integral value from regions of fast fluctuation of ωt − (t) will nearly cancel. The integral expressed above is therefore approximately equal to the integrand at the points where the phase function is nearly constant, or the instantaneous frequency of the phase function is zero. In other words, the time ranges around t = t_s that satisfy

will contribute to the integration in Eq. (7.25). Therefore, one can expand the integrand of Eq. (7.25) as a Taylor series around t_s:

Because a(t) is a slow varying function, only the zeroth-order term of a(t) will be retained. However, both the zeroth- and second-order terms of phase function ωt − (t) will be used for integration. Assume that only one stationary point exists, Equation (7.25) then becomes

Let

where ± sign depends on the sign of ″(t_s). Substituting Eq. (7.29) into Eq. (7.28), one obtains

Consider the following two cases of exp (± js²);

1. Let exp(js²) = exp(−u²); then

   

2. Let exp(− js²) = exp(−u²); then

   

Therefore

Equation (7.30) then becomes

Example 7.1 Fourier Transform of Pulsed LFM Signal Consider a pulsed LFM signal, which is represented in Eq. (5.9) as follows:

Comparing it with Eq. (7.24), one obtains

From Eq. (7.26), one can compute the time parameter t_s as

or

and

Therefore, the Fourier transform of p(t) becomes

This equation can be approximated as follows, from Eq. (7.32):

If the constants of exp (jπ/4) and α^−1/2 are ignored, the Fourier transform of p(t) becomes

This equation shown is an approximate form derived from the principle of stationary phase, which differs from that shown in Eq. (5.12). However, it is reasonably accurate as long as the time–bandwidth product of p(t) is greater than 100, which normally is true in practical radar applications.

7.2.1.3 Spatial Fourier Transform of Cross-Range Target Response.

The computation of spatial Fourier transform of Eq. (7.23) will now be approximated by using the principle of stationary phase. For illustration purpose, we will use k_um, instead of k_u, to represent the spatial frequency corresponding to the mth target. Thus, to compute

based on Eqs. (7.25) and (7.32), one can change t to u, ω to k_um. In addition, letting

one can then obtain the corresponding u_s in k_u domain by using Eq. (7.26), that is

or

Therefore

By solving Eq. (7.34) for u_s, which corresponds to t_s in the time domain, one obtains

One can then compute the corresponding (u_s) and (d²/du²)|_u=us as follows:

The symbol k_um is the spatial wavenumber related to the spatial frequency along the cross-range direction, which is quite small. The parameter k = 2π/λ is the wavenumber related to the carrier frequency and is much larger than k_um. Therefore, for k_um k, one obtains

The spatial frequency transform of s_m(u), from Eq. (7.23), then becomes

The term

is a known and slowly-fluctuating amplitude function. It can be ignored in the cross-range imaging process. The spatial frequency spectrum of the echo signal due to the mth target located at (X_c, y_m) can then be expressed as

The characteristics of the spatial frequency k_um are examined next.

Figure 7.8 shows the configuration of radar beams to be used for computing the spatial frequency band limitation. Figure 7.8a displays radar moving along the u- or y-axis direction with two radar beams at different locations, and one ground target located at (X_c, y_m). Radar beam 1, originating from (0, y_m − Ls/2) and shown as the shaded area with beamwidth θ_H, has its upper edge as the first beam illuminating the target. Beam 2, originating from (0, y_m + Ls/2), has its lower edge as the last beam illuminating the same target. Figure 7.8b displays the geometric relationship between the beam angle θ_H, X_c, and Ls. Notice that Ls is the synthetic aperture length with beamwidth θ_H.

FIGURE 7.8 Computation of spatial frequency band limitation.

From Eq. (7.18), the spatial frequency k_um is expressed as

where θ_m(u) is the aspect angle of the mth target located at (X_c, y_m). For a group of M targets occupying an area with length D_y, the radar movement is within the range [−L_u, L_u], where L_u = (Ls + D_y)/2. However, for any single target located at (X_c, y_m), the radar movement is within [y_m−Ls/2, y_m + Ls/2]. The spatial frequency k_um corresponding to a single target m is therefore band limited to

Refer to Fig. 7.8b, with R = [X_c² + (Ls/2)²]^1/2 and θ_H = λ/L and Ls = X_cθ_H, then sin(θ_H/2) = Ls/(2R). Therefore

The bandwidth B_ku and the center of the spatial frequency k_uc can be computed as follows:

Equation (7.38c) states that the bandwidth of spatial frequency k_um is constant and independent of the target location. In other words, all targets occupying an area of length D_y in the cross-range direction will have the same spatial frequency spectrum, but with different phases. Therefore k_u, will be used instead of k_um for future discussion.

For a small beamwidth of θ_H, sin(θ_H/2) θ_H/2 or R X_c. Equations (7.38a) and (7.38b) then become

The bandwidth B_ku and the center of the spatial wavenumber k_uc therefore become

One can compute the changing rate of the spatial frequency k_u, from Eq. (7.34):

Normally X_c (y_m − u); therefore

The slope of the spatial frequency is a constant, which implies that k_u varies linearly with the two ends of the k_u equal to 2π/L and −2π/L, respectively. From Eqs. (7.39a)–(7.39d), it is clear that the spatial frequency band limitation, bandwidth, center of band, and rate of change are all identical for all targets with the same slant range (x = X_c in this case).

Since k_u in the spatial frequency domain corresponds to ω_D in the Doppler frequency domain, it is interesting to see that the parameters derived in the spatial frequency domain match those derived from the Doppler frequency discussed in Chapter 6. In other words, the spatial frequency k_u, the bandwidth B_ku and k_uc (the centroid of k_u) derived from Eqs. (7.37c), (7.39a), and (7.39b) match ω_D, B_D, and ω_Dc, respectively, derived in Eq. (6.8). This match comes from the conditions set by assigning R₀ = X_c, r = R_m, and k_u = ω_D/V.

7.2.1.4 Reconstruction of Cross-Range Target Function.

Once the spatial frequency spectrum of the mth target located at (X_c, y_m) is known, the spatial frequency spectrum for all target responses [s_m(u), m = 0, 1,..., M −1] can be obtained:

The spatial frequency spectrum of reference signal s_r0(u), defined in Eq. (7.19), can be obtained by letting m = 0 and y_m = 0 in Eq. (7.40):

The spatial frequency transform of an ideal target function, defined in Eq. (7.20), is

From Eqs. (7.40)–(7.42), one obtains

Again, the ideal target function f₀(u) can be derived as the inverse spatial frequency transform of S(k_u)/S_r0(k_u), where S(k_u) and S_r0(k_u) are the spatial frequency transforms of s(u) and s_r0(u), respectively. However, because of the bandwidth limitation, it cannot be solved in this way as discussed in the range imaging case, and a practical reconstruction method is again via the matched filter, which is defined as the complex conjugate of the reference function S_r0(k_u). Multiplication of the spatial frequency function S(k_u) by the matched filter function S*_r0(k_u), yields the following product:

Since the spatial frequency transform of each target is band-limited, one can rewrite Eq. (7.37b) as

where

Equation (7.44) then becomes

Assuming a small beamwidth, the inverse spatial frequency transform of I_m(k_u) is

Since u and y are interchangeable, the inverse spatial frequency transform of F(k_y) becomes

Let psf_m(y) = i_m(y) be difined as the point spread function of the mth target, similar to the point spread function used in range imaging shown in Eq. (7.11), the reconstructed cross-range target function becomes

Comparing Eqs. (7.49) and (7.50) with Eqs. (7.10) and (7.11) in Section 7.1.2, one obtains the following observations:

• In both cases, the target functions are a group of sinc functions in the range x and cross-range y directions, respectively.

  

  FIGURE 7.9 Matched filtering for cross-range imaging.

• The function i_m(y) in the cross-range y direction can be considered as the point spread function psf_m(y), while the point spread function psf(x) in the range x direction is computed as F⁻¹{|P(ω)|²}.
• In both cases, a reference function is needed for reconstruction purposes. A matched filter defined as the complex conjugate of the time-reversed transmitted signal is chosen as the reference function in range image processing. For the case of cross-range image processing, a matched filter is chosen as the complex conjugate of the reversed slow time of the echo signal from an ideal reflector located at (X_c, 0) (or at the center of the cross-range targets).

On the basis of the discussion above, the cross-range image processing under the broadside case can be represented in a system block diagram as shown in Fig. 7.9.

7.2.2 Squint Radar Case

7.2.2.1 System Model.

Figure 7.10 displays the geometry of a squint-mode radar system for cross-range imaging. Similar to the broadside mode, all targets are located at x = X_c and spread along the y axis with all targets located within the radar beamwidth. Each target is represented by its reflection coefficient σ_m, m = 1, 2,..., M. The radar is located at (0, 0, H) and moves at speed V along the y axis. However, in contrast to the broadside-mode radar, the center beam of the squint mode radar aims at the target area with a squint angle θ_q, which is the angle between the range axis and the center beam of the radar.

FIGURE 7.10 A squint mode cross-range imaging system.

Figure 7.10a shows the configuration of a cross-range imaging system, while Fig. 7.10b shows a simplified 2D model of Fig. 7.10a; Fig. 7.10b can also be considered as the overhead view of Fig. 7.10a. The center of the target area along the y axis is at (X_c′, Y_c) in Fig. 7.10b, with R₂ as the distance from the radar to the center of the target area. The distance R₂ equals (X_c² + Y_c² + H²)^1/2 in Fig. 7.10a and equals (X_c′² + Y_c²)^1/2 in Fig. 7.10b, where X_c′ = (X_c² + H²)^1/2. To simplify the discussion, X_c will be used instead of X_c′ in the following discussion. R₁ and R₃ are the shortest and the longest distances from the radar to the target, respectively. The squint angle can be computed as θ_q = tan⁻¹(Y_c/X_c).

Figure 7.11 displays the relationship between the target area and radar beam. All targets are located at x = X_c, and spread along the y axis with the center of the target area at y = Y_c. The length of targets spreading around Y_c is D_y. The radar moves along the y axis from y = −L_u to y = L_u, where it continuously illuminates the targets located inside the region of D_y. The length 2L_u = 2Ls + D_y, where Ls is the synthetic aperture length. Here the length D_y is assumed to be less than Ls.

Consider the case when the radar is located at y = u, −L_u ≤ u ≤ L_u. Let the targets be located at {(X_c, y_m)}, where m = 0, 1,..., M −1, and

FIGURE 7.11 Relationship between targets and squint radar beam.

The received echo signal due to the target located at (X_c, y_m) is

The received echo signal due to a group of M targets can then be expressed as

where

The pulsed LFM signal is normally used for radar image processing. Again, we will assume that the transmitted signal is a single frequency; that is, p(t) = exp(jωt). The baseband received signal becomes

which is identical to Eq. (7.16) for the broadside case.

By defining an ideal reflector with reflection coefficient σ_r = 1, located at (X_c, Y_c), one can express the reference signal s_r0(u) received by a radar located at y = u as

Notice that the value of y_m [Y_c − (D_y/2), Y_c + (D_y/2)], where Y_c is nonzero in the squint radar case. Similar to the broadside radar case, the ideal target function f₀(u) for squint radar is also defined with Y_c = 0. Therefore, by defining y′_m = y_m – Y_c, one can see that all targets are located at y′_m [−D_y/2, D_y/2] and the ideal target function becomes

The convolution of s_ro(u) with f₀(u) turns out to be

By comparing Eqs. (7.51) and (7.54), one can see that the baseband signal s_b(t, u) is identical to s(u), which is the output of a system with input signal s_r0(u) and system function f₀(u). Therefore, the same system block diagram used for broadside-mode radar, as shown in Fig. 7.7, can also be applied to squint-mode radar.

7.2.2.2 Spatial Fourier Transform of Cross-Range Target Response.

From Eq. (7.23), the spatial Fourier transform of s_m(u) can be expressed as

Based on Eqs. (7.33) and (7.34), and following the same steps as used in the broadside case, one obtains

and

or

k_u = 2k sinθ_m(u),

where θ_m(u) is the aspect angle of the target at (X_c, y_m). The spatial Fourier transform S_m(k_u) can then be approximated as follows:

Given a group of targets that occupies a length D_y and centers on y = Y_c, the radar movement must be within [−L_u, L_u] with L_u = Ls + D_y/2 for the targets to be under radar beam illumination. However, for a single target located at (X_c, y_m), the radar movement is in the range [−Ls/2+y_m−Y_c, Ls/2+y_m−Y_c]. The spatial frequency k_um for the mth target is therefore band-limited to

Figure 7.12 shows the configuration of radar beams used for computing the spatial frequency band limitation based on a single target located at (X_c, y_m), where the radar moves upward along the y axis. Radar beam 1, shown as the shaded area, has its upper edge as the first beam illuminating the target. Beam 2, also shown as the shaded area, has its lower edge as the last beam illuminating the same target. The squint angle θ_q is the angle between the x axis and the line from the origin (0, 0) to the center of targets (X_c, Y_c). The squint angle can be computed as θ_q = tan⁻¹(Y_c/X_c). The corresponding sin θ_m(y_m−Y_c+Ls/2) and sin θ_m(y_m−Y_c−Ls/2) can be computed as

FIGURE 7.12 Computation of spatial frequency band limitation for squint radar.

For a radar with small beamwidth and small squint angle, X_c Ls and X_c Y_c; therefore

The spatial frequency k_um is then band-limited as

which is independent of the location of the mth target. The bandwidth and the center of the spatial frequency k_um or simply k_u can then be computed as follows:

The change rate of k_u is similar to Eq. (7.39d), and can be expressed as

Therefore, all targets at the same range distance (x = X_c in this case) will share the same band limitation, bandwidth, center of band, and rate of spatial frequency change as shown in Eqs. (7.56a)–(7.56d). The spatial frequency bandwidth B_ku and the slope of the spatial frequency k′_u are identical to those in the broadside case. However, the center of the spatial frequency k_uc = 2kθ_q is different from that in the broadside case.

The spatial frequency spectra corresponding to all received signals {s_m(u), m = 0, 1,..., M −1} can therefore be expressed as follows, from Eq. (7.55):

7.2.2.3 Reconstruction of Cross-Range Target Function.

The reference signal received from an ideal reflector is shown in Eq. (7.52) as

The corresponding spatial Fourier transform can be obtained as

The ideal target function for a group of M targets can be expressed as follows, from Eq. (7.53):

The spatial frequency transform of the ideal target function is

From Eqs. (7.57)–(7.59), one obtains

S(k_u) = F₀(k_u)S_r0(k_u),

which is identical to Eq. (7.43) derived in the broadside case.

The matched filter, defined as the complex conjugate of the reference function S_r0(k_u), is again used to reconstruct the target function. The output of the matched filter is

By comparing Eq. (7.60a) with Eq. (7.44), one can see that the only difference is the phase function exp(jk_uY_c). The phase factor exp(jk_uY_c) of F(k_u) implies that in the spatial domain the target function f(u) is shifted downward along the u or y axis by Y_c. This is because of the ideal target function f(u) in the squint case was defined with Y_c = 0, which in turn caused the phase factor exp(jk_uY_c) to appear in the spatial frequency function F(k_u). Since the true squint target area is centered on y = Y_c, therefore, the correct target function in the squint case is to remove the phase factor exp(jk_uY_c) from Eq. (7.60a). Thus, the true target function in the squint radar is obtained by taking the inverse spatial frequency transform of F′(k_u), which is defined as follows:

The spatial frequency transform of each target is band-limited as shown in Eq. (7.56b); therefore one can rewrite Eq. (7.60b) as

where

The band limitation of I_m(k_u) is the same as that shown in Eq. (7.56b). Notice that I_m(k_u) is a bandpass signal with the center of the spatial frequency k_uc = 2kθ_q and bandwidth B_ku = 4π/L. Let I_bm(k_u) be the baseband signal of I_m(k_u):

Then, I_bm(k_u) = I_m(k_u − 2kθ_q), and the inverse spatial frequency transform of I_bm(k_u) can be computed as follows:

Therefore, the inverse spatial frequency transform of I_m(k_u) becomes

i_m(u) = i_bm(u)exp(j2kθ_qu).

The final reconstructed cross-range target function can be obtained as

which is similar to Eq. (7.49), except for an extra phase factor exp(j2kθ_qy).

From the preceding discussion, one can see that the techniques used for cross-range image processing are quite similar for both broadside radar and squint radar. However, there are two major differences: (1) the reference function is defined differently—the broadside radar deals with an ideal reflector located at (X_c, 0), and the squint radar deals with an ideal reflector located at (X_c, Y_c); and (2) the nonzero squint angle θ_q causes an extra phase factor exp(j2kθ_qy) in the squint radar case. Overall, the system block diagram of Fig. 7.9 works for both broadside and squint radar cases.

7.3 DATA ACQUISITION, SAMPLING, AND POWER SPECTRUM OF RADAR IMAGE

The 2D radar imaging data is formed by digitizing the received radar signal at sampling frequency f_s, which satisfies the Nyquist requirement. This signal in general is a pulse or linear FM signal with carrier frequency ranges from tens of megahertz to tens of gigahertz, and its bandwidth could be a few megahertz to hundreds of megahertz. The 2D radar imaging data are arranged in a matrix-like array of complex numbers. Each row of data corresponds to one reflected radar pulse, while each column of data contains information on the same target from successive reflected radar pulses at a constant time interval. Each column of data serves as the along-track direction imaging data, and is equivalently sampled by the pulse repetition frequency f_PRF.

FIGURE 7.13 I–Q radar signal generation.

The received radar signal is a downconverted intermediate signal. It is first passed through an in-phase–quadrature–phase splitter to produce the in-phase and quadrature-phase (or I–Q) components. These two signals are then lowpass-filtered and digitized to render a complex number pair that serves as one row element of the 2D radar image data. Figure 7.13 displays the generation of I–Q signals from a received radar signal, where the reference signal is dependent on the transmitted signal.

The sampling frequency f_s in the range direction must satisfy the Nyquist requirement; that is, f_s ≥ B, where B is the bandwidth of the transmit signal. Assuming f_s = B, the range sample spacing is therefore

Along the azimuth direction, the radar moves with speed V and transmits a pulse at the time interval PRI = 1/f_PRF. The sampling frequency is therefore equal to f_PRF. In the spatial frequency domain, the wavenumber k_u or k_y equals ω_D/V in the Doppler frequency domain. Given the bandwidth of B_ku = 4π/L, the corresponding Doppler frequency bandwidth B_Dop = 2V/L. The along track sampling frequency f_PRF must satisfy the relationship f_PRF ≥ B_Dop. Assuming f_PRF = B_Dop, the azimuth sample spacing can then be computed as follows:

7.3.1 Digitized Doppler Frequency Power Spectrum

7.3.1.1 Broadside SAR.

For broadside SAR, the Doppler centroid f_Dc = 0, and the bandwidth of Doppler frequency is

Since each point of the azimuth data corresponds to the received signal along the flight path at a fixed-range cell, the Doppler frequency power spectrum is discrete and equivalently sampled at the rate of f_PRF. The observed Doppler frequency power spectrum is therefore centered on the origin with replicas appearing at multiples of the sampling frequency f_PRF. To prevent the Doppler frequency spectrum from aliasing to each other, the pulse repetition frequency f_PRF must be chosen to be greater than B_D. As an example Fig. 7.14 displays the Doppler frequency spectra of the broadside SAR image, where the true or absolute Doppler spectrum is shown inside the dotted box. The maximum Doppler frequencies appear at two sides of the absolute Doppler spectrum but with opposite signs. Their values are − V/L and V/L, respectively.

FIGURE 7.14 Doppler frequency spectra of a broadside SAR.

7.3.1.2 Squint SAR.

For a nonzero squint angle θ_q, the Doppler frequencies corresponding to the two ends of the 3-dB radar beamwidth, namely, f_DU and f_DL, can be computed as follows, from Eq. (6.16d) or (6.16e):

The Doppler frequency bandwidth is then

B_D = f_DU − f_DL.

The Doppler frequency centroid for a nonzero squint angle is as follows, from Eq. (6.16c):

The Doppler frequency centroid computed above is the true or absolute centroid frequency, with f_DU and f_DL located at two sides of f_Dc. Because of the built-in sampling frequency f_PRF, the absolute Doppler frequency power spectrum is then repeated at the multiples of f_PRF. Let f_Dc be the absolute Doppler centroid and f_DC′ be the observed Doppler centroid. The relationship between the absolute Doppler centroid and the observed Doppler centroid is given by

f_Dc = f_Dc′ +M_ambf_PRF,

where M_amb is the Doppler ambiguity and is the smallest integer defined by

As an example, Fig. 7.15 displays both the true (absolute) Doppler frequency spectrum and the digitized and observed spectra of a squint SAR with M_amb = 2. Figure 7.15a shows the true frequency spectrum, where the Doppler frequency centroid f_Dc is nonzero with f_DU and f_DL located at two sides. Figure 7.15b shows the digitized and observed Doppler frequency spectra. The observed Doppler frequency centroid is located away from the origin and labeled as f_DC′. The true Doppler frequency centroid f_Dc differs from the observed Doppler frequency centroid f_Dci by 2 times the sampling frequency f_PRF. The observed spectrum is no longer symmetric around the origin because the Doppler frequency centroid f_Dc is not a multiple integer of f_PRF.

FIGURE 7.15 Doppler frequency spectra of a squint SAR.

Both the true and the observed Doppler frequencies serve as important parameters for radar image reconstruction, which is discussed later.