Principles of Modern Radar - Volume 2 1891121537

216 CHAPTER 6 Spotlight Synthetic Aperture RadarTABLE 6-1 Properties of the Fourier Transform Commonly Used in SAR Imaging. The double arrowsymbol ⇔ is used to indicate that two functions are related via the Fourier transform.Property Description Application to SARLinearity af(t) + bg(t) ⇔ aF(ω) + bG(ω), Superposition of scatterers in an imagewhere a and b are constantsTime scaling f (at) ⇔ |a| 1 a )Deramp scaling of range axisTime shifting f (t − t 0 ) ⇔ F(ω)e − jωt 0Displacement of scatterer from scene centerFrequency shifting e jω0t f (t) ⇔ F(ω − ω 0 ) Modulation and basebandingArea under F(ω) f (0) = ∫ +∞−∞ F(ω)dωPeak value of point scatterer responseMultiplication/convolution f (t) ∗ g(t) ⇔ F(ω)G(ω) Spectral windowing; phase error effecton point scatterer responseCorrelation f (t)⋆g(t) ⇔ F(ω)G ∗ (ω) Matched filtering, or pulse compressionIn this case, the Fourier transform integral is easy to write, as the rect function simplyalters the limits of integration of (6.1), where f (t) = 1:F(ω) =∫ T/2−T/2Next, we carry out the integral and evaluate it at the limits:∫ T/2e − jωt jωtT/2e− dt = − jω ∣−T/2=−T/2(cos ω T 2)− j sin( ) ωsin2 T = T ω2 T( ) T= T sinc2π ωe − jωt dt (6.5)(ω T ) (− cos ω T ) (− j sin ω T )222− jω(6.6)The last line of this expression is the sinc function, which is defined as sinc(x) =sin(π x)/π x. The word sinc is a contraction of the Latin term sinus cardinalis, or cardinalsine. The derivation above begins with a rect function in the time domain and obtains asinc function in the frequency domain. These domains are commonly reversed in radarengineering. The rect function is often used to represent the constant-amplitude spectrumof a transmitted signal. After pulse compression (see the correlation property in Table 6-1as well as Chapter 2), the spectrum is real-valued. Thus, the signal in the time domain isthe sinc function determined by the relationship:( ) ωrect ⇔ 2π Bsinc(Bt) (6.7)2π Bwhere B is the bandwidth of the transmitted signal (in Hz, not rad/s).Figure 6-1 plots the sinc function for several values of B, and we notice two things:First, the maximum value of the sinc function is equal to 2π B as implied by (6.7). This