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Unformatted text preview: 6.003: Signals and Systems 6.003: Signals and Systems CT Fourier Transform April 8, 2010 Filtering Notion of a filter. LTI systems cannot create new frequencies. can only scale magnitudes and shift phases of existing components. Lecture 17 April 8, 2010 CT Fourier Transform Representing signals by their frequency content. X ( j )= x ( t ) e jt dt (analysis equation) 1 x ( t )= 2 X ( j ) e jt d (synthesis equation) generalizes Fourier series to represent aperiodic signals. equals Laplace transform X ( s ) | s =j if ROC includes j axis. inherits properties of Laplace transform. complex-valued function of real domain . simple inverse relation more general than table-lookup method for inverse Laplace. duality. filtering . applications in physics . Lowpass Filtering Higher frequency square wave: &lt; 1 /RC . 1 2 T t 1 2 1 j kt ; 2 x ( t ) = e = k odd jk T 1 . 01 Example: Low-Pass Filtering with an RC circuit | H ( j ) | . 1 R + + v i C v o . 01 . 1 1 10 100 1 /RC H ( j ) | 2 . 01 . 1 1 10 100 1 /RC Source-Filter Model of Speech Production Vibrations of the vocal cords are filtered by the mouth and nasal cavities to generate speech. Filtering LTI systems filter signals based on their frequency content. Fourier transforms represent signals as sums of complex exponen- tials. 1 x ( t ) = 2 X ( j ) e jt d Complex exponentials are eigenfunctions of LTI systems. e jt H ( j ) e jt LTI systems filter signals by adjusting the amplitudes and phases of each frequency component. 1 1 x ( t ) = 2 X ( j ) e jt d y ( t ) = 2 H ( j ) X ( j ) e jt d buzz from vocal cords throat and nasal cavities speech 1 6.003: Signals and Systems Lecture 17 April 8, 2010 Filtering Example: Electrocardiogram An electrocardiogram is a record of electrical potentials that are generated by the heart and measured on the surface of the chest....
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