undefined Solution The convolution for the function ( cos ω t ) ∗ ( cos ω t ) by using integration . The convolution for the function ( cos ω t ) ∗ ( cos ω t ) is 1 2 ω sin ( ω t ) + 1 2 t cos ( ω t ) _ . Definition used: The convolution of the functions f and g , denoted by the standard notation f ∗ g and it is defined by the integral ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ . Calculation: Given that, the function is ( cos ω t ) ∗ ( cos ω t ) . Here, it is observed that the functions f ( t ) = ( cos ω t ) and g ( t ) = ( cos ω t ) . Now compute the convolution for the function ( cos ω t ) ∗ ( cos ω t ) as follows. ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ = ∫ 0 t ( cos ω τ ) ( cos ω ( τ − t ) ) d τ = ∫ 0 t ( cos ω τ ) ( cos ( ω τ − ω t ) ) d τ = 1 2 ∫ 0 t 2 ( cos ω τ ) ( cos ( ω τ − ω t ) ) d τ Use the trigonometry equation 2 cos x cos y = cos ( x + y ) + cos ( x − y ) and simplify further as follows. ( f ∗ g ) ( t ) = 1 2 ∫ 0 t ( cos ( ω τ ) + ( ω τ − ω t ) ) + ( cos ( ω τ ) − ( ω τ − ω t ) ) d τ = 1 2 ∫ 0 t ( cos ( 2 ω τ − ω t ) ) + ( cos ( ω t ) ) d τ = 1 2 [ ∫ 0 t ( cos ( 2 ω τ − ω t ) ) d τ + ∫ 0 t ( cos ( ω t ) ) d τ ] = 1 2 [ ∫ 0 t ( cos ( 2 ω τ − ω t ) ) d τ + ( cos ( ω t ) ) ∫ 0 t d τ ] Simplify further, ( f ∗ g ) ( t ) = 1 2 [ [ sin ( 2 ω τ − ω t ) 2 ω ] 0 t + ( cos ( ω t ) ) [ τ ] 0 t ] = 1 2 [ [ sin ( 2 ω t − ω t ) 2 ω − sin ( − ω t ) 2 ω ] + ( cos ( ω t ) ) [ t − 0 ] 0 t ] = 1 2 [ [ sin ( ω t ) 2 ω + sin ( ω t ) 2 ω ] + t ( cos ( ω t ) ) ] = 1 2 ω sin ( ω t ) + 1 2 t cos ( ω t ) Thus, the convolution for the function ( cos ω t ) ∗ ( cos ω t ) is 1 2 ω sin ( ω t ) + 1 2 t cos ( ω t ) _ .

10th Edition

ISBN: 9781119480150

Author: Kreyszig

Publisher: WILEY C

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