lecture13 - EE313 Linear Systems and Signals Fall 2010...

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EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Fourier Transform Properties
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13 - 2 ( 29 ( 29 - - = dt e t f F t j ϖ ϖ ( 29 ( 29 - = ϖ ϖ π ϖ d e F t f t j 2 1 ( 29 ( 29 ϖ F t f ( 29 ( 29 ϖ π - f t F 2 0 1 τ /2 - τ /2 t f ( t ) 0 τ ϖ F ( ϖ ) -6 π τ -4 π τ - 2 π τ 2 π τ 4 π τ 6 π τ Duality Forward/inverse transforms are similar Example: rect(t/ τ ) τ sinc( ϖ τ / 2) Apply duality τ sinc(t τ /2) 2 π rect(- ϖ / τ ) rect(·) is even τ sinc(t τ /2) 2 π rect( ϖ / τ )
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13 - 3 Scaling Given and that a 0 | a | > 1: compress time axis, expand frequency axis | a | < 1: expand time axis, compress frequency axis Extent in time domain is inversely proportional to extent in frequency domain (a.k.a bandwidth) f ( t ) is wider spectrum is narrower f ( t ) is narrower spectrum is wider ( 29 ( 29 ϖ F t f ( 29 a F a at f ϖ 1
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