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# l9 - CMPEN/EE455 – Lect 9 8 The 2-D pulse one more time...

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1 CMPEN/EE455 Lecture 9 Topics today: 1. Separation of variables in 2-D 2. 2-D impulse and pulse examples Separation of Variables: If f ( x, y ) = product of 1-D functions in x and y : f ( x, y ) = f 1 ( x ) f 2 ( y ) Then, F ( u, v ) = product of 1-D Fourier transforms of f 1 and f 2 : F ( u, v ) = F [ f ( x, y )] = F u [ f 1 ( x )] · F v [ f 2 ( y )] = F 1 ( u ) · F 2 ( v ) Often simplifies work.

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CMPEN/EE455 – Lect 9 2 3 Example Let f ( x, y ) be the following 2-D pulse function: f ( x, y ) = A, if 0 x X and 0 y Y 0 , otherwise
CMPEN/EE455 – Lect 9 3 F [ f ( x, y )] = A Z Y 0 Z X 0 e - j 2 π ( ux + vy ) dxdy = A Z X 0 e - j 2 πux dx · Z Y 0 e - j 2 πvy dy separability in x and y : f ( x, y ) = A · f 1 ( x ) · f 2 ( y ) = A · F 1 ( u ) · F 2 ( v ) = A · { X sin( πuX ) πuX e - jπuX } · { Y sin( πvY ) πvY · e - jπvY } use 1-D Fourier transform of pulse, Lecture 6, pg 5 = AXY sinc( πuX )sinc( πvY ) e - ( uX + vY ) (1) See pictures on next two pages. 3

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CMPEN/EE455 – Lect 9 4 Picture of f ( x, y ) = A · f 1 ( x ) · f 2 ( y ):
CMPEN/EE455 – Lect 9 5 Fourier transform of 2-D pulse.

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CMPEN/EE455 – Lect 9 6 3 Example f ( x, y ) = 1 , if - 1 2 x 1 2 and - 10 y 10 0 , otherwise f ( x, y ) = f 1 ( x ) · f 2 ( y )
CMPEN/EE455 – Lect 9 7 F ( u, v ) = F 1 ( u ) · F 2 ( v ) See Figure 3.3b of | F ( u, v ) | on page 9-12.

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Unformatted text preview: CMPEN/EE455 – Lect 9 8 The 2-D pulse one more time! CMPEN/EE455 – Lect 9 9 3 Example Let f ( x,y ) = δ ( x-a,y-b ) , where a and b are real constants. F [ f ( x,y )] = Z ∞-∞ Z ∞-∞ δ ( x-a,y-b ) e-j 2 π ( ux + vy ) dxdy = ↑ impulse sift at ( x = ,y = ) = | F ( u,v ) | = for all ( u,v )! CMPEN/EE455 – Lect 9 10 We can use separation of variables also: δ ( x-a,y-b ) = δ ( x-a ) · δ ( y-b ) And use known 1-D transforms: F ( u,v ) = F 1 ( u ) · F 2 ( v ) = 3 In general, to plot | F ( u,v ) | , use log (1 + | F ( u,v ) | ) → F ( u,v ) usually concentrated near (0 , 0). → See Figure on next page. CMPEN/EE455 – Lect 9 11 Space- and Fourier-Domain Looks at Saturn CMPEN/EE455 – Lect 9 12 Figure 3.3, part 1. F ( u,v )...
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l9 - CMPEN/EE455 – Lect 9 8 The 2-D pulse one more time...

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