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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) CIRCULAR CONVOLUTION LINEAR CONVOLUTION + ALIASING Signal x[n] The input signal x[n] is splitted into blocks of length = L... Signal x[n] P - 1 zero padding Lenght L Lenght FFT = 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. Signal x[n] Signal h[n] Lenght P N - P zero padding Lenght FFT = N To complete the lenght of the N FFT, we aggregate N-P zeros to the impulse response h[n] (lenght P)... Signal x[n] Signal h[n] x1[n]*h[n] Length FFT = 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 convolution DOESN’T match the linear convolution we discard P - 1 samples Signal x[n] Signal h[n] x1[n]*h[n] Lenght FFT = 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]} 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 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 Signal x[n] Signal h[n] Signal x2[n] x1[n]*h[n] We process the second block x2[n] of the input x[n]... (overlapping P - 1 samples with the previous block) Signal x[n] Signal h[n] x1[n]*h[n] We process the second block x2[n] of the input x[n]... (solapando P - 1 muestras con el bloque previo) with the impulse response h[n] Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] We process the second block x2[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 segment x2[n]*h[n] Again, we have to discard P - 1 samples of the segment, that are wrong (due to aliasing) Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] We “copy” the result of the second circular convolution of x1[n] and h[n] (discarding the wrong samples) Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] We “copy” the result of the second circular convolution of x1[n] and h[n] (discarding the wrong samples) 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) 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] 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 - 1 first samples Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] we copy it to the output... Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] we copy it to the output... 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] 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] 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] 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 - 1 samples... 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 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 Signal x[n] Signal h[n] x1[n]*h[n] x2[n]*h[n] x3[n]*h[n] x4[n]*h[n] BLOCK convolution 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 =