Add shifted multiplied copies for all τ or discrete

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add shifted, multiplied copies for all τ (or discrete k ) Or: break the signal into each continuous point or discrete sample send each one through individually to produce blurred points add up the blurred points
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Convolution and Linear Systems Convolution Example: Convolution - One Way f ( t ) = [ ] h ( t ) = [ ] f ( 0 ) h ( t - 0 ) = [ ] f ( 1 ) h ( t - 1 ) = [ ] f ( 2 ) h ( t - 2 ) = [ ] f ( 3 ) h ( t - 3 ) = [ ] f ( 4 ) h ( t - 4 ) = [ ] f ( t ) * h ( t ) = k f ( k ) h ( t - k ) = [ ]
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Convolution and Linear Systems Convolution Convolution - Another Way To Look At It f ( t ) * h ( t ) = k f ( k ) h ( t - k ) = k h ( k ) f ( t - k ) Think of it this way: flip the function h around zero shift a copy to output position t point-wise multiply for each position k the value of the function f and the shifted inverted copy of h add for all k and write that value at position t Same as the spatial filtering we’ve already talked about, except the mask is the flipped impulse response.
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Convolution and Linear Systems Convolution Example: Convolution - Another Way f ( t ) = [ ] h ( t ) = [ ] f ( t ) * h ( t ) = k f ( k ) h ( t - k ) = [ ]
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Convolution and Linear Systems Convolution Why Flip the Impulse Function? An input at t produces a response at t + τ of h ( τ ) . How do you determine how the output at t is affected by input at t + τ ? h ( - τ )
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Convolution and Linear Systems Convolution Other Applications of Convolution: Polynomial Multiplication p 1 ( x ) = x 2 + x + p 2 ( x ) = x 2 + x + p 1 ( x ) p 2 ( x ) = x 4 + x 3 + x 2 + x +
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Convolution and Linear Systems Convolution Convolution in Statistics If you add the results from two independent distributions, the distribution of this sum is the convolution of the two independent distributions. Central Limit Theorem : As you convolve any distribution with itself an infinite number of times, in the limit it approaches a normal (Gaussian) distribution.
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Convolution and Linear Systems Convolution Convolution in Two Dimensions In one dimension: f ( t ) * h ( t ) = -∞ f ( τ ) h ( t - τ ) d τ In two dimensions: I ( x , y ) * h ( x , y ) = -∞ -∞ I ( τ x , τ y ) h ( x - τ x , y - τ y ) d τ x d τ y Or in discrete form: I ( x , y ) * h ( x , y ) = j k I ( j , k ) h ( x - j , y - k )
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Convolution and Linear Systems Convolution Properties of Convolution Convolution has the following mathematical properties: Commutative Associative Distributes over addition In simple terms, convolution has the same mathematical properties as multiplication.
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