Block convolution with arbitrary delays using fast Fourier transform

In digital signal processing applications, the convolution with a very long finite impulse response (FIR) filters is often required. In order to perform a fast high order convolution, the block convolution using fast Fourier transform (FFT) can be considered, which entails the time delays proportional to the block length. But since the delays due to the block convolution is generally set to 2ⁿ, we do not have much choice in selecting the delays. In this paper, we propose a block convolution with an arbitrary delay, which can be implemented by overlap-save (OLS) or overlap-add (OLA) algorithms. Experimental results show that even though the OLS-based algorithm creates some redundancies in output data due to more overlapped samples, the cost is nearly the same as the previous works. And the OLA-based algorithm has exactly the same complexity at the expense of an additional buffer.