Note #03 Symmetry in Convolution Problems

Note #03 Symmetry in Convolution Problems - a t = . This...

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Note #3. Symmetry in Convolution Problems Spring 2007 This note extends the notion of “even symmetry” to “even symmetry about a line, a t = .” This concept is useful in convolution problems. For example, we know that if the signals ) ( ), ( t h t x are even, then the product, ) ( ) ( ) ( t h t x t y = is even. But what about convolution under these conditions? If ) ( ), ( t h t x are even, is the convolution ) ( ) ( ) ( t h t x t y = even? The answer is not obvious, because convolution does not, in general, preserve the time-reversal operation. That is, if ) ( ) ( ) ( t y t h t x = , then ) ( ) ( ) ( t y t h t x - - . 1. Flip operation about a line, a t = . Consider Then ) 2 ( ) ( 1 a t v t v + - = represents a flip of signal ) ( t v about the line a t = . 2. Notation. The signal can be expressed analytically as + - = else t t t x , 0 1 , 1 ) ( . Recall that < - = 0 , 0 , t t t t t , or graphically, 3. Even symmetry about a line If the signal ) ( t x is even, i.e., if ) ( ) ( t x t x - = , then the signal, When flipped about the line a t = , displays an “even symmetry about the line,
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Unformatted text preview: a t = . This symmetry is is expressed as ) 2 ( ) ( a t w t w +-= . Note #3. p.1 ) ( t v a t a ) ( t v t ) ( t x t 1-1 1 t t ) ( t x t 1-1 1 ) ( ) ( a t x t w-= t a +1 a-1 1 a 4. Property. Derive the following property. If ) ( ), ( t h t x are even, then the convolution ) ( ) ( ) ( t h t x t y = is even. 5. Example 1. Show that the convolution, ) ( ) ( ) ( t h t x t y = , where t e t x-= ) ( and &lt; = else t t h , , 1 ) ( 2 1 , is even. In this case we get, 6. Example 2. In this example, shift the pulse, ) ( t h to the right so that ( 29 2 1 1 ) (-= t h t h . Show that the convolution, ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 1 1-=-= = t y t h t x t h t x t y is symmetrical about the line, 2 1 = t , i.e. , Note #3. p.2 ) ( t y 2 1 t 2 1-) ( 1 t y 2 1 t...
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Note #03 Symmetry in Convolution Problems - a t = . This...

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