l7 - 1 CMPEN/EE455 Lecture 7 Topics today: 1. Delta...

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1 CMPEN/EE455 Lecture 7 Topics today: 1. Delta (impulse) function 2. Convolution and correlation — basic linear filtering Note on Notation for Space and Frequency Domains: Space Domain Frequency Domain f ( x ), f ( x,y ) F ( u ), F ( u,v ) f ( t ), f ( t,z ) F ( μ ), F ( μ,ν ) Used for continuous-space only for 1-D/2-D DFT
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CMPEN/EE455 – Lect 7 2 Delta Function: Z x 1 x 0 δ ( x - x i ) dx = 1 , if x 0 x i x 1 0 , otherwise Also, the impulse-sift property: Z -∞ f ( x ) δ ( x - x o ) dx = f ( x o ) Fourier transform of δ ( x - a ) ( a = some real constant): F [ δ ( x - a )] = Z -∞ δ ( x - a ) e - j 2 πux dx = e - j 2 πua
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CMPEN/EE455 – Lect 7 3 and ( σ = some real constant) F - 1 [ δ ( u - σ )] = Z -∞ δ ( u - σ ) e j 2 πux dx = e j 2 πσx Particular cases when a = 0 and σ = 0: F [ δ ( x )] = F - 1 [ δ ( u )] =
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l7 - 1 CMPEN/EE455 Lecture 7 Topics today: 1. Delta...

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