Principles of Modern Radar - Volume 2 1891121537

352 CHAPTER 8 Interferometric SAR and Coherent Exploitationexample, the 340 km range is about 1,133,333 wavelengths at an RF of 1 GHz (λ = 0.3 m),so the phase shift is about 4,533,333π radians. An airborne X band (10 GHz, λ = 3 cm)radar at a range of 20 km experiences a phase shift of about 2,666,667π radians.The phase φ a of a complex signal sample Ae jφ a= I a + jQ a , whether a time-domainsample or an image pixel, is measured in practice using the four-quadrant arctangentfunction atan 4 (Q a /I a ) that returns not φ a but its principal value ˜φ a , which satisfies˜φ a = 〈φ a 〉 2π = φ a − k · 2π (8.21)The notation 〈·〉 2π means modulo 2π. Here k is an unknown integer such that ˜φ a is in therange [−π,π). ˜φ a is called the wrapped phase.Given a wrapped phase measurement ˜φ a , can we recover the original phase valueφ a ? In the absence of additional information, the answer for a single phase value is no.However, given a one-dimensional phase function φ a (t), and assuming that φ a (t) hasfinite bandwidth and is sampled at the Nyquist rate or above, it is possible to unwrap thesampled, wrapped measurements ˜φ a [n] to recover φ a [n] [18]. Although there are practicaldifficulties, it is often possible to do so for the 2-D phase functions of InSAR, as willbe discussed in Section 8.5.3. Thus, if we can compute the wrapped IPD ˜φ ab from thewrapped measurements ˜φ a and ˜φ b , it should be possible to recover the unwrapped IPDφ ab , which can then be used to determine the elevation profile.One way to obtain ˜φ ab is to compute it as˜φ ab = 〈 ˜φ a − ˜φ b (8.22)〉2πNote that ˜φ a − ˜φ b can range from −2π to 2π, so equation (8.22) performs an additionalwrapping operation on that result to wrap it into the range [−π,π). To see that this producesthe desired result, consider˜φ a − ˜φ b = (φ a − 2πk a ) − (φ b − 2πk b )= φ a − φ b + 2π(k a − k b )= φ a − φ b + 2πk (8.23)for some k. Clearly 〈φ + 2πk〉 2π = 〈φ〉 2π = ˜φ for any phase φ. Therefore〈 〉 ˜φ a − ˜φ b 2π = 〈φ a − φ b 〉 2π = ˜φ ab (8.24)Thus, the wrapped IPD can be computed by wrapping the difference between the wrappedphases at the individual apertures.Another way to compute the wrapped IPD is to form the interferogram. Represent thetwo individual SAR images as a(x,y) = A a (x,y)e j ˜φ a (x,y) and b(x,y) = A b (x,y)e j ˜φ b (x,y) .The interferogram is defined asI ab (x,y) ≡ a(x,y)b ∗ (x,y)= A a A b exp [ j ( ˜φ a − ˜φ b)]where b ∗ represents the complex conjugate of b. Consequently,arg {I ab (x,y)} = atan 4( ˜φ a − ˜φ b)(8.25)= ˜φ ab (8.26)Note that the phase of the interferogram does not have to be explicitly wrapped again,as suggested by equation (8.22), because the only means we have to measure that phase