  In this paper, new insights in frequency-domain implementations of digital
finite-length impulse response filtering (linear convolution) using overlap-add
and overlap-save techniques are provided. It is shown that, in practical
finite-wordlength implementations, the overall system corresponds to a
time-varying system that can be represented in essentially two different ways.
One way is to represent the system with a distortion function and aliasing
functions, which in this paper is derived from multirate filter bank
representations. The other way is to use a periodically time-varying
impulse-response representation or, equivalently, a set of time-invariant
impulse responses and the corresponding frequency responses. The paper provides
systematic derivations and analyses of these representations along with filter
impulse response properties and design examples. The representations are
particularly useful when analyzing the effect of coefficient quantizations as
well as the use of shorter DFT lengths than theoretically required. A
comprehensive computational-complexity analysis is also provided, and accurate
formulas for estimating the optimal DFT lengths for given filter lengths are
derived. Using optimal DFT lengths, it is shown that the frequency-domain
implementations have lower computational complexities (multiplication rates)
than the corresponding time-domain implementations for filter lengths that are
shorter than those reported earlier in the literature. In particular, for
general (unsymmetric) filters, the frequency-domain implementations are shown
to be more efficient for all filter lengths. This opens up for new
considerations when comparing complexities of different filter implementations.
