Overlap–save method

In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal ${\displaystyle x[n]}$ and a finite impulse response (FIR) filter ${\displaystyle h[n]}$:

${\displaystyle y[n]=x[n]*h[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }h[m]\cdot x[n-m]=\sum _{m=1}^{M}h[m]\cdot x[n-m],}$

(Eq.1)

where h[m] = 0 for m outside the region [1, M]. This article uses common abstract notations, such as ${\textstyle y(t)=x(t)*h(t),}$ or ${\textstyle y(t)={\mathcal {H}}\{x(t)\},}$ in which it is understood that the functions should be thought of in their totality, rather than at specific instants ${\textstyle t}$ (see Convolution#Notation).

The concept is to compute short segments of y[n] of an arbitrary length L, and concatenate the segments together. That requires longer input seqments that overlap the next input segment. The overlapped data gets "saved" and used a second time.^[1] First we describe that process with just conventional convolution for each output segment. Then we describe how to replace that convolution with a more efficient method.

Consider a segment that begins at n = kL + M, for any integer k, and define:

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

${\displaystyle y_{k}[n]\ \triangleq \ x_{k}[n]*h[n]=\sum _{m=1}^{M}h[m]\cdot x_{k}[n-m].}$

Then, for ${\displaystyle kL+M+1\leq n\leq kL+L+M}$, and equivalently ${\displaystyle M+1\leq n-kL\leq L+M}$, we can write:

${\displaystyle y[n]=\sum _{m=1}^{M}h[m]\cdot x_{k}[n-kL-m]\ \ \triangleq \ \ y_{k}[n-kL].}$

With the substitution ${\displaystyle j=n-kL}$, the task is reduced to computing ${\displaystyle y_{k}[j]}$ for ${\displaystyle M+1\leq j\leq L+M}$. 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.^[B]

If we periodically extend x_k[n] with period N  ≥  L + M − 1, according to:

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

the convolutions  ${\displaystyle (x_{k,N})*h\,}$  and  ${\displaystyle x_{k}*h\,}$  are equivalent in the region ${\displaystyle M+1\leq n\leq L+M}$. It is therefore sufficient to compute the N-point circular (or cyclic) convolution of ${\displaystyle x_{k}[n]\,}$ with ${\displaystyle h[n]\,}$  in the region [1, N].  The subregion [M + 1, L + M] 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:

${\displaystyle y_{k}[n]\ =\ \scriptstyle {\text{IDFT}}_{N}\displaystyle (\ \scriptstyle {\text{DFT}}_{N}\displaystyle (x_{k}[n])\cdot \ \scriptstyle {\text{DFT}}_{N}\displaystyle (h[n])\ ),}$

(Eq.2)

where:

• DFT_N and IDFT_N 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,^[C] and subsequently discarded.^[D]

(Overlap-save algorithm for linear convolution) h = FIR_impulse_response M = length(h) overlap = M − 1 N = 8 × overlap    (see next section for a better choice) step_size = N − overlap H = DFT(h, N) position = 0  while position + N ≤ length(x)     yt = IDFT(DFT(x(position+(1:N))) × H)     y(position+(1:step_size)) = yt(M : N)    (discard M−1 y-values)     position = position + step_size end 

When the DFT and IDFT are implemented by the FFT algorithm, the pseudocode above requires about N (log₂(N) + 1) complex multiplications for the FFT, product of arrays, and IFFT.^[E] Each iteration produces N-M+1 output samples, so the number of complex multiplications per output sample is about:

${\displaystyle {\frac {N(\log _{2}(N)+1)}{N-M+1}}.\,}$

(Eq.3)

For example, when ${\displaystyle M=201}$ and ${\displaystyle N=1024,}$ Eq.3 equals ${\displaystyle 13.67,}$ whereas direct evaluation of Eq.1 would require up to ${\displaystyle 201}$ complex multiplications per output sample, the worst case being when both ${\displaystyle x}$ and ${\displaystyle h}$ are complex-valued. Also note that for any given ${\displaystyle M,}$ Eq.3 has a minimum with respect to ${\displaystyle N.}$ Figure 2 is a graph of the values of ${\displaystyle N}$ that minimize Eq.3 for a range of filter lengths (${\displaystyle M}$).

Instead of Eq.1, we can also consider applying Eq.2 to a long sequence of length ${\displaystyle N_{x}}$ samples. The total number of complex multiplications would be:

${\displaystyle N_{x}\cdot (\log _{2}(N_{x})+1).}$

Comparatively, the number of complex multiplications required by the pseudocode algorithm is:

${\displaystyle N_{x}\cdot (\log _{2}(N)+1)\cdot {\frac {N}{N-M+1}}.}$

Hence the cost of the overlap–save method scales almost as ${\displaystyle O\left(N_{x}\log _{2}N\right)}$ while the cost of a single, large circular convolution is almost ${\displaystyle O\left(N_{x}\log _{2}N_{x}\right)}$.

Overlap–discard^[2] and Overlap–scrap^[3] are less commonly used labels for the same method described here. However, these labels are actually better (than overlap–save) to distinguish from overlap–add, because both methods "save", but only one discards. "Save" merely refers to the fact that M − 1 input (or output) samples from segment k are needed to process segment k + 1.

The overlap–save algorithm can be extended to include other common operations of a system:^[F][4]

• additional IFFT channels can be processed more cheaply than the first by reusing the forward FFT
• sampling rates can be changed by using different sized forward and inverse FFTs
• frequency translation (mixing) can be accomplished by rearranging frequency bins

• Overlap–add method
• Circular_convolution#Example

• Dr. Deepa Kundur, Overlap Add and Overlap Save, University of Toronto