EE562 - EE 462/562 Principles of Medical and Radar Imaging...

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EE 462/562 Principles of Medical and Radar Imaging Lecture One EE 462/562 Dr. M. Soumekh
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System Geometry Target Radiating Source Scattered waves Receiver EE 462/562 Dr. M. Soumekh
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Transmitted signal INPUT SYSTEM OUTPUT Scattered Waves Target Forward problem: to find the output (response) of a known system to a know input EE 462/562 Dr. M. Soumekh
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Example f(t) g(t) LTI system h(t) Here ? ? and ? ? are known Solution: (1) ? ? = ? ? ∗ ? ? (2) ? 𝜔 = ? 𝜔 ∙ ?(𝜔) Inverse problem: To identify a system (i.e. its properties) from its response to the known sources EE 462/562 Dr. M. Soumekh
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Example: LTI System Here ? ? and ? ? are known ?(?) is unknown f(t) g(t) h(t) ch1 ch2 Scope EE 462/562 Dr. M. Soumekh f(t) g(t) LTI system h(t)
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Example: LTI System(Cont.) EE 462/562 Dr. M. Soumekh Vary the frequency of excitation, and record 𝐴 2 𝐴 1 and 𝜏
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Example: LTI System(Cont.) ? ? = 𝐴 1 cos 𝜔? + 𝜃 1 ? ? = 𝐴 2 cos 𝜔? + 𝜃 2 = 𝐴 2 cos 𝜔? + 𝜃 2 − 𝜃 1 + 𝜃 1 = 𝐴 2 ??? 𝜔 ? + 𝜃 2 − 𝜃 1 𝜔 + 𝜃 1 ?? ????? 𝜃 2 − 𝜃 1 𝜔 ?? ??? ???𝑎? Transfer function ?(𝜔) = 𝐴 2 𝐴 1 ∠? 𝜔 = 𝜏𝜔 By varying ω and recording 𝐴 2 𝐴 1 , we can reconstruct the system transfer function which is ?(𝜔) EE 462/562 Dr. M. Soumekh
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Example: LTI System(Cont.) EE 462/562 Dr. M. Soumekh ω 1 |H(ω)| ω ∠H(ω) ω Mathematic Solution If ? 𝜔 = ? 𝜔 ∙ ? 𝜔 Then ? 𝜔 = ?(𝜔) ?(𝜔) ω 2 ω 3 ω 4 ω 1 ω 2 ω 3 ω 4
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Review of signal system and Fourier Transform 1. Inverse Fourier Transform (F.T.) ? ? = 1 2𝜋 ?(𝜔)? ?𝜔? ?𝜔 −∞ = p(ω i ) ∙ e c t ∆ω ωi=iΔω 2. Forward Fourier Transform ? 𝜔 = ?(?)? −?𝜔? ?𝜔 −∞ = p(t i ) ∙ e jωt i ωi=iΔω ∆t Fourier Transform is an information-preserving operator We have ?(?) ⇄ 𝑃(𝜔) 𝑃 𝜔 = ℱ (?) ?(?) ? ? = ℱ (𝜔) −1 𝑃(𝜔) EE 462/562 Dr. M. Soumekh
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Review of signal system and Fourier Transform (Cont.) Spatial Signals Inverse Fourier Integral ? ? = 1 2𝜋 ?(?
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