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Unformatted text preview: EE 462/562 Principles of Medical and Radar Imaging Lecture One EE 462/562 Dr. M. Soumekh System Geometry Target Radiating Source Scattered waves Receiver EE 462/562 Dr. M. Soumekh 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 Example f(t) g(t) LTI system h(t) Here ? ? and ? ? are known Solution: (1) ? ? = ? ? ∗ ? ? (2) ? = ? ∙ ?() Inverse problem: To identify a system (i.e. its properties) from its response to the known sources EE 462/562 Dr. M. Soumekh 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) Example: LTI System(Cont.) EE 462/562 Dr. M. Soumekh Vary the frequency of excitation, and record 2 1 and Example: LTI System(Cont.) ? ? = 1 cos ? + 1 ? ? = 2 cos ? + 2 = 2 cos ? + 2 − 1 + 1 = 2 ??? ? + 2 − 1 + 1 ?? ¡???? 2 − 1 ?? ??? ???¢ • Transfer function £() = 2 1 ∠£ = • By varying ω and recording ¤ ¥ ¤ ¦ , we can reconstruct the system transfer function which is £() EE 462/562 Dr. M. Soumekh Example: LTI System(Cont.) EE 462/562 Dr. M. Soumekh ω 1 H(ω) ω ∠H(ω) ω Mathematic Solution If ? = ? ∙ ? Then ? = ¡(¢) £(¢) ω 2 ω 3 ω 4 ω 1 ω 2 ω 3 ω 4 Review of signal system and Fourier Transform • 1. Inverse Fourier Transform (F.T.) ? ? = 1 2 ?()? ?¡? ¢ ∞ −∞ = £ p(ω i ) ∙ e jΩ c t ∆ω 2π Ωi¤iΔΩ • 2. Forward Fourier Transform ? = ?(?)? −?¡? ¢ ∞ −∞ = £ p(T i ) ∙ e jΩt I Ωi¤iΔΩ ∆T Fourier Transform is an informationpreserving operator We have ?(?) ⇄ () = ℱ (?) ?(?) ? ? = ℱ (¡) −¥ () EE 462/562 Dr. M. Soumekh Review of signal system and Fourier Transform (Cont.) Spatial Signals • Inverse Fourier Integral ? ? = 1 2 ?(?...
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This note was uploaded on 02/24/2011 for the course EE 562 taught by Professor Dr.soumekh during the Spring '11 term at SUNY Buffalo.
 Spring '11
 Dr.Soumekh

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