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2/6/01 Wave-Based Surveillance Professor David H. Staelin Massachusetts Institute of Technology A1 Lec18.5 - 12/6/01 Antenna Aperture Transform Relations and Resolution A2 ( )x y, for i i iϕϕ ( ) ( ) ( )x y 2j x y x y4 E , e d π + ϕ + ϕλ π ≅ ϕϕ Ω∫ ] 1 1vm− −⎡ ⎤⎣ ⎦ ( ) i ly i ii ,θϕ xϕ yϕ x y z ( ) ( ) ( )x y 2j x y x y x y2 A 1E , , 2 π− ϕ + ϕ λ π⎛ ⎞ϕϕ ≅ ϕ ϕ⎜ ⎟λ ⎝ ⎠∫ l l / = x ; y/ = y ∆ ∆ λ λλ λ Lec18.5 - 2 Define angular spectrum E ncom ng monochromatic s gnals Ex, y [aperturester Not to be confused w th the radialexpanding and dim nish ng waves character zed by E ,R E x, y e dxdy For<< Equiva ently we et x2/6/01 Antenna Aperture Transform Relations and Resolution A3 ( ) ( ) ( ) ( ) ( ) ( ) x y x y j2 x y x y4 j2 x y x y A E , e d E , λ λ λ λ +π λ λ π −π λ λ ≅ ϕϕ Ω∫ ∫∫ ( )i( ) ( ) () ()( ) () () ( ) x y 2 E x y E -E , R E( , ) G i = E r λ λ λ ∞ ∗∆ λ λλ λ λ λ ∞ ↔ ↓ ↓ τ ↔ ∝ ϕ τ ∫∫   l / = x ; y/ = y ∆ ∆ λ λλ λ Lec18.5 - 3 Ex , y E x, y e dx dy ϕ+ ϕ ϕ+ ϕ ϕϕ ≅ Note : E s not stochastic Thus: E x , y transm tting where R E r dx dy ϕ ϕ ϕ ϕ −τ Equiva ently we let x2/6/01 Single Aperture Resolution Limits A4 () () ()A Bi G Tϕ= ϕ f l∆ ϕ () () ()A BTf T fϕ ϕ ϕ= •     () ()Ai f ϕ ϕ = () () () () ()E EG so Rϕλ ϕτ↔ Lec18.5 - 4 Source mage = T ϕ ∗ =cyc es per radian (angle) G f If G f 0, there is no response n the image spectrum T Note : R G f G f ϕ↔ τ≅2/6/01 Single Aperture Resolution Limits A5 mwi l i D / l= λ ( )λ λ 0 Dλyλ xλ () () () () ()E EG so Rϕλ ϕτ↔ ()ER λτ yλτ xλτ D/ λ ()Gf ϕ y fϕ x fϕ lf = λ ≅ Lec18.5 - 5 Note : Zero response to source angular spectral componentsth spatia frequenc es beyond f cyc es/radian Ex , y Example : Note : R G f G f ϕ↔ τ≅ Thus : max Dcyces radian2/6/01 Antenna Responses for Stochastic Signals A6 Ai il i i i i ). ( )( ) ( ){ }1 1 j te− − ω λ λ λ λ = l i ignal ↑ ( ) ( ) () x y E E ,λ λ λ ↔ ↓↓ ⎡ ⎤τ⎣ ⎦ ( i ii ibili( ) ( )2 x y x yE 21 , E E , vm λ λ − ⎡ ⎤ϕτ τ ↔ ⎢ ⎥⎣ ⎦ ↑ ⎡ ⎤⎣ ⎦ ↓↓ ∆ = Lec18.5 - 6 ssume stochastic s gnals from different direct ons are uncorre ated (so no systematic ntens ty var ations n apertureLet E x , y , t vm ster Re E t, x , y Slow y vary ng, narrowband random sThen: E x , y , t , t E R ϕ ϕ Double arrow mpl es rrevers ty for two reasons: expectation and magnitude operators used) , t ϕ ϕ2/6/01 ( ) ( ) 2 x y x y o E E , ,2B ⎡ ⎤ϕϕ⎢ ⎥⎣ ⎦η 2 1 1W m[ i ] − − − Antenna Responses for Stochastic Signals A7 ( )x yE o , 2B λ λ φτ τ ↔η ( )377↑ Ω ( )21 1 1 2 1 v m W m− − − − −= ( )21v m− ( ) ( ) () x y E E ,λ λ λ ↔ ↓↓ ⎡ ⎤τ⎣ ⎦ ( i ii ibili↓↓ ( ) ( )2 x y x yE , E E ,λ λ ⎡ ⎤ϕτ τ ↔ ⎢ ⎥⎣ ⎦ ↑ 21vm−⎡ ⎤⎣ ⎦ ( ) ( )2 1 1 x y, ( )− − − λ λ⎡ ⎤⎣ ⎦ ∆ = Lec18.5 - 7 , f , f =Ι ϕ ϕ Hz ster ster not a phys cal unit, f ohm Hz Hz -1 rad ohm Hz Then: E x , y , t , t E R ϕ ϕ Double arrow mpl es rrevers ty for two reasons: expectation and magnitude operators used) , t ϕ ϕ Can we deduce Wm ster Hz from E x , y , t ? YesΙϕ ϕ2/6/01 Aperture Field Correlations for a Thermal Source A8 ()B K ( )2 4 ssay−Ω= ≅ xϕ yϕ ( ) ( ){ }x o 2 B s o2E S ( ) kTE ⎛ ⎞φ = = Ω η⎜ ⎟λ⎝ ⎠ x y aperture ( ) 2 x x y B /= λ ixϕ → ( )2 x y B o2 EE , kT 2B ⎡ ⎤ϕϕ⎢ ⎥⎣ ⎦ = λ 0 xϕ B i ignal ( )x yE ,λ λλ φ ττ xλ τ s 4B o2 kT 2B Ω −•η • λ x1 0.01 τ = λ ↔ Lec18.5 - 8 square source @ T 0.01 rad 10 ster wattsNote: 0, 0 E x, y, t B 2 as expected. ,,f kT Ιϕϕ 0.01 rad single polar zation •η 0.01 rad s the sbandwidth of interest 102/6/01 Time and Space Field Correlations (3D) A9 ( ) ( ) ( ) ( )x y x yx x 2 2 1 o oE E, , / 2 , , / 2 Hf z− − − λ λ λ λη ↔ φ ητ ττ / ( )λτ↔ϕ ( ) ( )B 2 1 1 x 2 kT ,f ( i− − − ϕ λ Lec18.5 - 9 Wm Wm φ ττ time frequency ,f Wm Hz sterone polar zation) Ιϕ =2/6/01 Aperture Synthesis C1 x y Ai by lλ λτ τ () () ( ) EE i r; i l∗φτ= τ () () () () ()  E 2 2 i E E i iλ λτ•ϕ∗ ϕ ϕ yλτ yλτ A∗ B∗ B A yx2L2 λτ = λ i i follλ ∗ ∗ τ xλτ xλτ ()W λτ Lec18.5 - 10 max max ssume field of s ze within which two smalantennas can be moved independently Note f stationary w.r.t. .e., true angular decorre ation φ −We observe W and retr eve W where s the des red image φτ max max Therefore, in space, we need to measure comb nations n only two quadrants, e.g., A, B because the conjugates A , B ow. max2/6/01 Maximum Antenna Separation Limits Resolution C2 () ()W …


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MIT 6 661 - Wave-Based Surveillance

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