FourierTransform

# FourierTransform - ECE 342 Communication Theory Fall 2005...

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Unformatted text preview: ECE 342 Communication Theory Fall 2005, Class Notes Prof. Tiffany J Li www: http://www.eecs.lehigh.edu/ ∼ jingli/teach email: [email protected] Fourier Transform • A time-domain function x ( t ) (not necessarily periodic) has a frequency-domain spec- ification X ( w ): X ( w ) = Z ∞-∞ x ( t ) e- jwt dt, (1) x ( t ) = 1 2 π Z ∞-∞ X ( w ) e jwt dw (2) • The right hand side of 2, known as Fourier Integral , is of the nature a Fourier Series with fundamental frequency Δ w approaching zero. • We call X ( w ) the direct Fourier Transform of x ( t ), x(t) the inverse Fourier Transform of X ( w ), and x ( t ) and X ( w ) a Fourier Transform pair. Symbolically this is expressed as: X ( w ) = F [ x ( t )] , x ( t ) = F- 1 [ X ( w )] or x ( t ) ⇐⇒ X ( w ) • Instead of using angular frequency Δ w , one can also use frequency Δ f (where w = 2 πf ) the corresponding Fourier Transform pair is: X ( f ) = Z ∞-∞ x ( t ) e- j 2 πft dt, (3) x ( t ) = Z ∞-∞ X ( f ) e j 2 πft df (4) Spectrum • G ( w ) is complex = ⇒ to plot the spectrum G ( w ) as a function of w , we need to plot both the amplitude spectrum , | X ( w ) | vs w , and the phase spectrum , theta x ( w ) vs w , where G ( w ) = | G ( w ) | e jθ x ( w ) . 1 Conjugate Symmetry Property • If x ( t ) is a real function of t , then G ( w ) and G (- w ) are complex conjugates, i.e. G (- w ) = G * ( w ) = ⇒ the amplitude spectrum | G ( w ) | is an even function, and the phase spectrum θ x ( w ) is an odd function | G (- w ) | = | G ( w ) | , θ x (- w ) =- θ x ( w ) • This conjugate symmetry property holds only for real signals. • A similar property also holds for the Fourier series of periodic real signals. Existence of Fourier Transform • Same as the existence of Fourier series: strong Dirichlet conditions and weak Dirichlet conditions....
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FourierTransform - ECE 342 Communication Theory Fall 2005...

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