lecture4 - Continuous-Time Convolution EE 313 Linear...

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Unformatted text preview: Continuous-Time Convolution EE 313 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 4 - 2 ( 29 ( 29 ( 29 ( 29 τ τ τ d t f f t f t f 2 1 2 1- ∫ ≡ ∞ ∞- Convolution Integral • Commonly used in engineering, science, math • Convolution properties – Commutative: f 1 ( t ) * f 2 ( t ) = f 2 ( t ) * f 1 ( t ) – Distributive: f 1 ( t ) * [ f 2 ( t ) + f 3 ( t )] = f 1 ( t ) * f 2 ( t ) + f 1 ( t ) * f 3 ( t ) – Associative: f 1 ( t ) * [ f 2 ( t ) * f 3 ( t )] = [ f 1 ( t ) * f 2 ( t )] * f 3 ( t ) – Shift: If f 1 ( t ) * f 2 ( t ) = c ( t ), then f 1 ( t ) * f 2 ( t - T ) = f 1 ( t - T ) * f 2 ( t ) = c ( t - T ). – Convolution with impulse, f ( t ) * δ ( t ) = f ( t ) – Convolution with shifted impulse, f ( t ) * δ ( t-T ) = f ( t-T ) important later in modulation 4 - 3 Graphical Convolution Methods • From the convolution integral, convolution is equivalent to – Rotating one of the functions about the y axis – Shifting it by t – Multiplying this flipped, shifted function with the other function – Calculating the area under this product – Assigning this value to f 1 ( t ) * f 2 ( t ) at t ( 29 ( 29 ( 29 ( 29 τ τ τ d t f f t f t f- ≡ ∫ ∞ ∞- 2 1 2 1 4 - 4 3 τ 2 f ( τ ) 2-2 + t 2 + t g ( t- τ ) * 2 2 t f ( t )-2 2 3 t g ( t ) Graphical Convolution Example...
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lecture4 - Continuous-Time Convolution EE 313 Linear...

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