Reconstruction of Complex Baseband Signals via M-Periodic Nonuniform Bandpass Sampling and Least-Squares Optimal Time-Varying FIR Filters

This paper considers the reconstruction of digital complex baseband signals from M-periodically nonuniformly sampled real bandpass signals. With such a sampling, bandpass signals with arbitrary frequency locations can be sampled and reconstructed, as opposed to uniform sampling which requires the signal to be within one of the Nyquist bands. It is shown how the reconstruction can be carried out via an M-periodic time-varying finite-length impulse response (FIR) filter or, equivalently, a set of M time-invariant FIR filters. Then, a least-squares design method is proposed in which the M filter impulse responses are computed in closed form. This offers minimal filter orders for a given desired bandwidth. This is an advantage over an existing technique where ideal filters are first derived (ensuring perfect reconstruction) and then windowed and truncated, which leads to suboptimal filters and thus higher filter orders and implementation complexity. A design example illustrates the efficiency of the proposed design technique.

On Frequency-Domain Implementation of Digital FIR Filters Using Overlap-Add and Overlap-Save Techniques

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.