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Convolution can be used to translate a function f(x.y) to a point (Xo,yo). Given two functions A(x.y) and B(x.y), the 2-D convolution can be expressed as: A(x, y) ® B(x, y) = || A(t,n)B(x – t, y – n) drdn -00 Use the 2-D convolution integral and the point impulse properties to prove the following: a) f(x,y)* 8(x,y) = f(x,y) b) f(x,y) * 8(x – x,,y – Yo) = f(x – x,,y – Yo)

Convolution can be used to translate a function f(x.y) to a point (Xo,yo). Given two functions A(x.y) and B(x.y), the 2-D convolution can be expressed as: A(x, y) ® B(x, y) = || A(t,n)B(x – t, y – n) drdn -00 Use the 2-D convolution integral and the point impulse properties to prove the following: a) f(x,y)* 8(x,y) = f(x,y) b) f(x,y) * 8(x – x,,y – Yo) = f(x – x,,y – Yo)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Use the 2-D convolution integral and the point impulse properties to prove the following:

Convolution can be used to translate a function f(x,y) to a point (Xo,yo). Given two functions A(x,y) and B(x,y), the 2-D convolution can be expressed as: A(x, y) ® B(x, y) = || A(T,n)B(x – T, y – n) drdŋ Use the 2-D convolution integral and the point impulse properties to prove the following: a) f(x,y) * 8(x, y) = f(x,y) b) f(x, y) * 8(x – Xo,y – Yo) = f(x – Xo,y – Yo)
Transcribed Image Text:Convolution can be used to translate a function f(x,y) to a point (Xo,yo). Given two functions A(x,y) and B(x,y), the 2-D convolution can be expressed as: A(x, y) ® B(x, y) = || A(T,n)B(x – T, y – n) drdŋ Use the 2-D convolution integral and the point impulse properties to prove the following: a) f(x,y) * 8(x, y) = f(x,y) b) f(x, y) * 8(x – Xo,y – Yo) = f(x – Xo,y – Yo)
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