ch 10 summary - Summary Continuous-Time Fourier Transform...

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Summary – Continuous-Time Fourier Transform (CTFT) (Chapter 10) 1. The CTFT of a CT signal x t ( ) is defined as X f ( ) = x t ( ) −∞ e j 2 π f t dt , − ∞ < t < ( f is a continuous variable, called CT frequency (units: Hz = sec -1 .) ) 2. The inverse CTFT is given by x t ( ) = X f ( ) e j 2 f t −∞ df , − ∞ < t < . ( Examples: (i) If x t ( ) = e at u t ( ) where a is a real constant, a > 0 , then X f ( ) = e a + j 2 f ( ) t dt 0 = 1 a + j 2 f . (ii) Let B > 0 and define H f ( ) = 1, f B ; H f ( ) = 0, f > B (that is, H f ( ) = rect f 2 B Λ Ν Μ Ξ Π Ο ). Then h t ( ) = e j 2 f t df = B B 2 B sinc 2 B t ( ) .) 3. Useful CTFT properties: (i) Time-shift: x 1 t ( ) = x t T ( ) X 1 f ( ) = e j 2 f T X f ( ) (ii) Frequency-shift: x 1 t ( ) = e j 2 f 0 t x t ( ) X 1 f ( ) = X f f 0 ( ) (iii) Modulation: x 1 t ( ) = x t ( ) cos 2 f 0 t ( ) X 1 f ( ) = 1 2 X f f 0 ( ) + X f + f 0 ( ) { } (iv) Convolution: y t ( ) = x t ( ) * h t ( ) Y f ( ) = X f ( ) H f ( ) Key application: This says that if x t ( ) is input to a LTI system having impulse response h t ( ) , resulting in output y t ( ) , then in the frequency domain we have the input-output relationship Y f ( ) = X f ( ) H f ( ) . H f ( ) is the system’s frequency response , given by H f ( ) = h t ( ) −∞ e j 2 f t dt = H s ( ) | s = j 2
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This note was uploaded on 10/27/2011 for the course ECE 313 taught by Professor Muschinski during the Fall '08 term at UMass (Amherst).

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ch 10 summary - Summary Continuous-Time Fourier Transform...

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