Convolution Algorithms ∗

can be calculated with the FFT and bring the order of arithmetic operations down to Nlog (N) which can be signi cant for large N . This approach, which is called fast convolutions", is a form of block processing since a whole block or segment of x (n) must be available to calculate even one output value, y (n). So, a time delay of one block length is always required. Another problem is the ltering use of convolution is usually non-cyclic and the convolution implemented with the DFT is cyclic. This is dealt with by appending zeros to x (n) and h (n) such that the output of the cyclic convolution gives one block of the output of the desired non-cyclic convolution. For ltering and some other applications, one wants on going" convolution where the lter response h (n) may be nite in length or duration, but the input x (n) is of arbitrary length. Two methods have traditionally used to break the input into blocks and use the FFT to convolve the block so that the output that would have been calculated by directly implementing (1) or (2) can be constructed e ciently. These are called overlap-add" and over-lap save".