Lecture21_BlockConvolution.pdf

2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 a c e g i k b

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? 𝒏 = 𝒌=−∞ ? 𝒌 𝒉[𝒏 − 𝒌] 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 A C E G I K B D F H J L n ? 𝟔 = 𝒌=? 𝟔 ? 𝒌 𝒉[𝟔 − 𝒌] = ? + ½? + ¼? Section 1 Data Section 2 Data Some outputs require a mix of data from adjacent sections
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EE7372 DSP Theory Rationale for Overlap Save Method Question: How do we fix the fact that some outputs require data from adjacent sections? Answer: Overlap data from adjacent sections. 13 0 2 4 4 6 8 8 10 12 1 3 5 5 7 9 9 11 A C C E G G B D D F H H ? ? [𝒏] n ? ? [𝒏] ? ? [𝒏] The OVERLAP in the sections of data are the OVERLAP in the algorithm’s name I J
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EE7372 DSP Theory Overlap Save Summary - Input 14 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 A C E G I K B D F H J L ?[𝒏] n
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EE7372 DSP Theory Overlap Save Summary - Augmenting 15 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 A C E G I K B D F H J L ?[𝒏] n
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EE7372 DSP Theory Overlap Save Summary Sectioning Data 16 0 2 4 4 6 8 8 10 12 1 3 5 5 7 9 9 11 A C C E G G B D D F H H ? ? [𝒏] n ? ? [𝒏] ? ? [𝒏] I J
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EE7372 DSP Theory Overlap Save Summary Computing Circular Convolution 17 ?? ? 𝑠 𝑘 = ? 𝑠 𝑘 ? 𝑘 where 𝑘 = 0,1,2, … 5 𝑡ℎ?𝑛 𝑦 𝑛 = ?𝐷𝐹𝑇 ? 𝑠 [𝑘]
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EE7372 DSP Theory Overlap Save Summary - Saving Data 18 0 2 4 4 6 8 8 10 12 1 3 5 5 7 9 9 11 ? ? [𝒏] ? ? [𝒏] ? ? [𝒏] SAVE SAVE SAVE
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EE7372 DSP Theory Overlap Save Summary - Output 19 0 2 4 4 6 1 3 5 5 7 ?[𝒏] Save non-aliased output samples to form y[n] 0 2 4 4 6 8 8 10 12 1 3 5 5 7 9 9 11 SAVE SAVE SAVE
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EE7372 DSP Theory Overlap Save Algorithm Based on DFT_Length Point Circular Convolution (assume a convenient DFT_Length Point DFT engine is available) and assume h[n] is of length P. Add P-1 zeros to the front of x[n] because the first P-1 samples are aliased when using an DFT_Length point DFT to convolve an DFT_Length point and a P point sequence Add zeros to the end so that there are an integer number of sections Divide x[n] into DFT_Length sections with an overlap of P-1 samples from the previous section Perform circular convolution of section with h[n] Save the un-aliased samples of the convolution (index values P to DFT_Length, DFT_Length-P+1 total samples) 20
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EE7372 DSP Theory OVERLAP ADD ALGORITHM 21
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EE7372 DSP Theory Rationale for Overlap Add Method 22 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 A C E G I K B D F H J L 1 ¼ ½ 𝒉[𝒏] ?[𝒏] n n
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EE7372 DSP Theory Rationale for Overlap Add Method 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 A C E G I K B D F
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  • Fall '18
  • Circular convolution, DFTs, Overlap Save Method

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