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Three-Steps of Linear Convolution

For any given n , how to obtain Step 1: time reversal of either signal (e.g., f(k)  f(-k) ) Step 2: shift f(-k) by n samples to obtain f(n-k) Step 3: multiply h(k) and f(n-k) for each k and then take the summation over k. Three-Steps of Linear Convolution. Note

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Three-Steps of Linear Convolution

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  1. For any given n, how to obtain Step 1: time reversal of either signal (e.g., f(k)f(-k) ) Step 2: shift f(-k) by n samples to obtain f(n-k) Step 3: multiply h(k) and f(n-k) for each k and then take the summation over k Three-Steps of Linear Convolution Note You need to change variable n to get the whole sequence. EE465: Introduction to Digital Image Processing

  2. 1D Linear Convolution f(n)=[1 2 3 4] h(n)=[1 –1] 4 1 3 2 1 o -1 o origin EE465: Introduction to Digital Image Processing

  3. Step 1:Time Reversal h(-k)=[–1 1] h(k)=[1 –1] 1 1 o o -1 -1 EE465: Introduction to Digital Image Processing

  4. h(-k) h(1-k) h(2-k) 1 1 1 o o o -1 n=1 -1 n=2 n=0 Step 2: Shift 4 3 2 f(k)=[1 2 3 4] 1 o EE465: Introduction to Digital Image Processing

  5. g(0)=1 g(1)=1 g(2)=1 1 1 1 o o o -1 n=1 -1 n=2 n=0 Step3: Multiply-and-Add 4 3 2 f(k)=[1 2 3 4] 1 o EE465: Introduction to Digital Image Processing

  6. Final Result f(n)=[1 2 3 4] h(n)=[1 –1] g(n)=[1 1 1 1 -4] If the lengths of two input signals are N1 and N2 respectively, the length of the output signal will be N1+N2-1. EE465: Introduction to Digital Image Processing

  7. 2D Linear Convolution n n x(m,n) h(m,n) • 4 1 • 2 5 3 • 1 • 1 -1 m m EE465: Introduction to Digital Image Processing

  8. Step 1:Time Reversal h(m,n) h(m,-n) h(-m,-n) n • 1 • 1 -1 m • -1 • 1 1 -1 1 1 1 EE465: Introduction to Digital Image Processing

  9. h(1-k,-l) h(-k,-l) h(1-k,1-l) -1 1 1 1 -1 1 1 1 -1 1 1 1 Step 2: Shift l • 4 1 • 2 5 3 k x(k,l) EE465: Introduction to Digital Image Processing

  10. y(1,0)=3 y(1,1)=10 y(0,0)=2 -1 1 1 1 -1 1 1 1 -1 1 1 1 Step3: Multiply-and-Add l • 4 1 • 2 5 3 k x(k,l) EE465: Introduction to Digital Image Processing

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