Principles of Modern Radar - Volume 2 1891121537

6.2 Mathematical Background 215andf (t) = 1 ∫ ∞F(ω)e jωt dω (6.2)2π −∞For a given frequency ω, the complex scalar value F(ω) of the forward transform (6.1)can be thought of as the inner product of the function f (t) and a complex sinusoid offrequency ω, represented by e jωt . The negative sign in the exponential of (6.1) is presentbecause of the definition of the inner product for complex-valued functions [10,11]. Thecollection of values F(ω) for all ω is called the spectrum of the function f (t). The timeand frequency notation used in this section are not the only possible Fourier domains. Wewill encounter others, and t and ω can be thought of as dummy variables whose purposeis to distinguish the two domains bridged by the Fourier transform.Any two complex sinusoids are orthogonal (their inner product is zero) as long astheir frequencies are not identical. This property is used in the analysis (decomposition)and synthesis (creation) of signals using complex sinusoids as the elemental buildingblocks. The spectrum F(ω) represents the amount of each e jωt present in f (t). Theinverse transform is the reconstitution of the original function f (t) by summing over allthe complex sinusoids properly weighted and phased by F(ω). Understanding the Fouriertransform and its basic properties is absolutely essential to SAR processing. An excellentreference is the classic book by Bracewell [9].We next use the definition of the Fourier transform to derive one of its key properties.We will discover what happens to the transform of the function f (t) when it is shifted byt 0 . First substitute f (t − t 0 ) into (6.1) and then change the variable of integration from tto t + t 0 :F{ f (t − t 0 )}==∫ ∞−∞∫ ∞−∞= e − jωt 0f (t − t 0 )e − jωt dtf (t)e − jω(t+t 0) dt∫ ∞−∞f (t)e − jωt dt= e − jωt 0F(ω) (6.3)Thus, the Fourier transform of a shifted function is the transform of the unshifted functionmultiplied by a complex exponential with a linear phase proportional to the amount of theshift. This and other important facts relevant to SAR are collected in Table 6-1.6.2.2 The Sinc FunctionAs an introductory example that we’ll use extensively later, we find the Fourier transformof the rectangle function defined as:( ) trect =T⎧⎪⎨ 1 if |t| ≤ T 2⎪⎩ 0 if |t| > T 2(6.4)