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Block Convolution: overlap-save method

CIRCULAR CONVOLUTION. LINEAR CONVOLUTION + ALIASING. Block Convolution: overlap-save method. Input Signal x [ n ] : arbitrary length Impulse response of the filter h [ n ] : lenght P Block Size: N  we take N samples of x [ n ]

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Block Convolution: overlap-save method

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  1. CIRCULAR CONVOLUTION LINEAR CONVOLUTION+ ALIASING Block Convolution: overlap-save method • Input Signal x[n]: arbitrary length • Impulse response of the filter h[n]: lenght P • Block Size: N  we take N samples of x[n] • There’s ALIASING! right samples: L = N- (P - 1)

  2. Signal x[n] The input signal x[n] is splitted into blocks of length = L...

  3. P - 1 zero padding Lenght L Lenght FFT = N Signal x[n] The entry signal x[n] is splitted in blocks of lenght = N... The Impulse response lenght = P, so we aggregate P - 1 zeros to the signal beggining Then when we compute the circular convolution, only L = N - (P - 1) samples match the linear convolution.

  4. Lenght FFT = N Lenght P N - P zero padding Signal x[n] Signal h[n] To complete the lenght of the NFFT, we aggregate N-P zeros to the impulse response h[n] (lenght P)...

  5. Length FFT = N Signal x[n] Signal h[n] x1[n]*h[n] We compute the first segment of the output performing a circular convolution of x1[n] and h[n] It HAS “aliasing” of P- 1 samples Circular convolutionDOESN’T match the linear convolution  we discard P - 1 samples

  6. Lenght FFT = N Signal x[n] Signal h[n] x1[n]*h[n] We compute the first segment of the output performing a circular convolution of x1[n] and h[n] It HAS “aliasing” of P- 1 samples x1[n]*h[n] = IFFT{X1[k]xH[k]}

  7. Sucesión x[n] Sucesión h[n] x1[n]*h[n] We “copy” the result of the circular convolution of x1[n] and h[n] To the system output, discarding the wrong samples

  8. Signal x[n] Signal h[n] x1[n]*h[n] We “copy” the result of the circular convolution of x1[n] and h[n] to the system output, discarding the wrong samples

  9. Signal x[n] Signal h[n] Signal x2[n] x1[n]*h[n] We process the second blockx2[n] of the input x[n]... (overlapping P - 1 samples with the previous block)

  10. Signal x[n] Signal h[n] x1[n]*h[n] We process the second blockx2[n] of the input x[n]... (solapando P - 1 muestras con el bloque previo) with the impulse response h[n]

  11. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] We process the second blockx2[n] of the input x[n]... (overlapping P - 1 samples with the previous block) with the impulse response h[n] and we obtain the second segmentx2[n]*h[n] Again, we have to discard P - 1 samples of the segment, that are wrong (due to aliasing)

  12. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] We “copy” the result of the secondcircular convolution ofx1[n] and h[n] (discarding the wrong samples)

  13. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] We “copy” the result of the secondcircular convolution ofx1[n] and h[n] (discarding the wrong samples)

  14. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] we process the third block x2[n] of the input x[n]... (overlapping P - 1 samples with the previous block)

  15. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] we process the third block x2[n] of the input x[n]... (overlapping P - 1 samples with the previous block) with the impulse response h[n]

  16. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] we obtain the third segment of the output x3[n]*h[n] discarding the P - 1first samples

  17. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] we copy it to the output...

  18. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] we copy it to the output...

  19. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] we process the fourth block of the input x[n]

  20. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] we process the fourth block of the input x[n] with the impluse response h[n]

  21. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] x4[n]*h[n] we obtain the fourth segment of the output x4[n]*h[n]

  22. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] x4[n]*h[n] We discard the first P - 1samples...

  23. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] x4[n]*h[n] …we copy it to the output

  24. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] x4[n]*h[n] …we copy it to the output

  25. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] x4[n]*h[n] BLOCK convolution

  26. Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] x4[n]*h[n] BLOCK convolution = LINEAR convolution

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