Overlap

In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal



x
[
n
]


{\displaystyle x[n]}
and a finite impulse response (FIR) filter



h
[
n
]


{\displaystyle h[n]}
:

where h[m] = 0 for m outside the region [1, M].
The concept is to compute short segments of y[n] of an arbitrary length L, and concatenate the segments together. Consider a segment that begins at n = kL + M, for any integer k, and define:





x

k


[
n
]

≜


{



x
[
n
+
k
L
]
,


1
≤
n
≤
L
+
M
−
1




0
,




otherwise


.








{\displaystyle x_{k}[n]\ \triangleq {\begin{cases}x[n+kL],&1\leq n\leq L+M-1\\0,&{\textrm {otherwise}}.\end{cases}}}





y

k


[
n
]

≜


x

k


[
n
]
∗
h
[
n
]
=

∑

m
=
1


M


h
[
m
]
⋅

x

k


[
n
−
m
]
.


{\displaystyle y_{k}[n]\ \triangleq \ x_{k}[n]*h[n]=\sum _{m=1}^{M}h[m]\cdot x_{k}[n-m].}
Then, for kL + M ≤ n ≤ kL + L + M − 1, and equivalently M ≤ n − kL ≤ L + M − 1, we can write:




y
[
n
]
=

∑

m
=
1


M


h
[
m
]
⋅

x

k


[
n
−
k
L
−
m
]


≜



y

k


[
n
−
k
L
]
.


{\displaystyle y[n]=\sum _{m=1}^{M}h[m]\cdot x_{k}[n-kL-m]\ \ \triangleq \ \ y_{k}[n-kL].}
With the substitution j ≜ n-kL, the task is reduced to computing yk(j), for M ≤ j ≤ L + M − 1. These steps are illustrated in the first 3 traces of Figure 1, except that the desired portion of the output (third trace) corresponds to 1 ≤ j ≤ L.
If we periodically extend xk[n] with period N ≥ L + M − 1, according to:





x

k
,
N


[
n
]

≜


∑

ℓ
=
−
∞


∞



x

k


[
n
−
ℓ
N
]
,


{\displaystyle x_{k,N}[n]\ \triangleq \ \sum _{\ell =-\infty }^{\infty }x_{k}[n-\ell N],}
the convolutions



(

x

k
,
N


)
∗
h



{\displaystyle (x_{k,N})*h\,}
and




x

k


∗
h



{\displaystyle x_{k}*h\,}
are equivalent in the region M ≤ n ≤ L + M − 1. It is therefore sufficient to compute the N-point circular (or cyclic) convolution of




x

k


[
n
]



{\displaystyle x_{k}[n]\,}
with



h
[
n
]



{\displaystyle h[n]\,}
in the region [1, N]. The subregion [M, L + M − 1] is appended to the output stream, and the other values are discarded. The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem:

where:

DFTN and IDFTN refer to the Discrete Fourier transform and its inverse, evaluated over N discrete points, and
L is customarily chosen such that N = L+M-1 is an integer power-of-2, and the transforms are implemented with the FFT algorithm, for efficiency.
The leading and trailing edge-effects of circular convolution are overlapped and added, and subsequently discarded.

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