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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
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