Narrow Results

A simple method of construction of a pair of orthogonal wavelet frames in L₂(ℝ^d) is presented. This is a generalization of one-dimensional case to higher dimension. The construction is based on the well-known Unitary Extension Principle (UEP). The presented method produces the polyphase components of the filters of the wavelet functions, and hence the filters. A pair of orthogonal wavelet frames can be constructed with an extra condition. In the construction, the polyphase matrix is used as opposed to the modulation matrix. This is less restrictive and yields a fewer wavelet functions in the system than in the previously known constructions.

Frequency channelization is a fundamental signal processing operation employed across various domains, including communications and radio astronomy. The polyphase filterbank (PFB) represents an efficient and versatile means of channelization. When strict constraints are placed on the allowable spectral leakage between neighboring channels, an oversampled PFB design is advantageous. A helpful consequence of the oversampling is that inversion of the PFB to recover high temporal resolution is simplified and can be accomplished accurately using Fourier transforms. We describe this inversion approach and identify key design considerations. We examine the residual error and spectral/temporal leakage behavior when a channelizer and its corresponding inverter are cascaded, concluding that near-perfect reconstruction can be approached with appropriate selection of PFB and inverter design parameters.

Signal channelization enables efficient frequency domain processing and is a mainstay of astronomical signal processing, but applications that require high time resolution necessitate reconstruction of the original wide band signal. In a previous paper, a near-perfect method of reconstructing a time-limited input signal from the output of an oversampled polyphase filterbank (PFB) was described. Here, we consider the case where continuous signals are processed. We show that the most simplistic approach, which utilizes non-overlapping windows and a fast Fourier transform (FFT) channelizer, introduces large errors whose magnitude can equal the signal. The ringing introduced by truncation at the end of a block, combined with the cyclic nature of FFTs, leads to errors that are concentrated at block boundaries. These localized errors can be heavily suppressed by utilizing overlapping windows and nearly completely eliminated by apodizing the data blocks with a Tukey window. After these improvements, the much smaller residual error is concentrated at the PFB channel boundaries and is due to adjacent channels having different gain slopes at the channel boundary. Increasing the channel passband equalizes the gain slope at the channel boundary, and the error is reduced further. With these changes, errors as low as −100dB are achieved, and the method of reconstructing the channelized data meets the stringent signal purity requirement for astronomical applications such as radio pulsar timing.

A system, the synthesis convolver, is described that can process the output of an analysis filter bank and:

• •Reconstruct the analysis filter bank input signal,
• •Implement continuous convolution on the reconstructed signal, and
• •Resample the reconstructed signal to different sample rates.

The synthesis convolver combines the capabilities of a synthesis filter bank and a continuous convolver. The synthesis convolver is based on earlier work but improves upon it, adding convolution and resampling. As well as reconstructing filterbank data, convolution allows dechirping of pulsar signals and resampling allows synthesized data to conform to the VLBI VDIF standard. The spectral overlap-add approach described, compared to earlier work, reduces errors and is more robust to channel gain errors. The system uses windows with smoothed or apodized edges, with the classical Tukey window being used previously. Here the Tukey window is generalized leading to a class of apodized windows. This class of windows is explored and one is found that is close to optimal in all conditions and can reduce errors by up to 40dB compared to an equivalent Tukey window. Achievable aliasing errors are lower than those of a standard polyphase synthesis filter bank. The synthesis convolver provides a high quality and versatile replacement for polyphase synthesis filter banks.