Principles of Modern Radar - Volume 2 1891121537

11.9 Problems 52711.9.2 Equivalence: holography – SAR – Tomography(1) Show that holographic imaging, as described in Paragraph 13.2.3, is strictly equivalentto Synthetic Aperture Radar, when the trajectory of the radar is a circle centered onthe target. (Hint: Consider stepped frequency SAR).(2) Interpret the sampling relations in ⃗k domain (Figure 11-14) as non-ambiguity conditionsin SAR image: δf giving the range ambiguity, and δθ the Doppler ambiguity,with x related to the range gate, and y related to the beamwidth. Give the formalexpressions for these relationships.(3) Show that holographic imaging is a complex version of tomographic imaging. (Hint:use the “projection slice theorem” stating that the Fourier transform of the projectionof an image I (⃗x) is a slice in the Fourier transform of the image, H(⃗k)).11.9.3 Circulating codesAs stated in Paragraph 13.2.2.4, if there is a delay t between the codes transmittedthrough adjacent elements of a uniform linear array (N elements, spacing d = λ/2between adjacent elements), that translates to a differential phase shift between adjacentarray elements at frequency f (in baseband):φ = 2π f tThis is a regular phase shift, from one radiating element to the next: the effect is to steerthe array in the direction θ (relative to the normal to the array):φ = 2π f t = 2π(d/λ) sin θ, with λ = c/( f + f 0 ) ∼ c/f 0f = (d/λ)(1/t) sin θ,linear relation between f and sin θ(1) Another interpretation: synthetic moving array. Show that this process can be interpretedas an array moving with a velocity d/t. What is the “synthetic Doppler effect”in direction θ? What is the total synthetic Doppler bandwidth f ? What is the resolutionrequired in frequency, to recover the directivity of the antenna array? Compare thenumber of independent directions, f/δ f , with the number of independent samplesin the code, Ttot/t.(2) Code selection. What are the requirements on the code, in terms of length and numberof moments, in relation with the antenna length?(3) Diagram quality. The transmission diagram is obtained by Fourier transform of thesignals received on the array, at each time sample (and then eventually digital beamforming on receive, if an array of receivers is available). Show that a one bit PSK code(e.g. Barker code, or constant amplitude zero autocorrelation code) may thus providean acceptable angular diagram.