In recent years, tree-structured analysis/reconstruction systems have been extensively studied for use in subband coders for speech. In such systems, it is imperative that the individual channel signals be decimated in such a way that the number of samples coded and transmitted do not exceed the number of samples in the original speech signal. Under this constraint, the systems presented in the past have sought to remove the aliasing distortion while minimizing the overall analysis/reconstruction distortion. In this paper, it is shown that it is possible to design tree-structured analysis/reconstruction systems which meet the sampling rate condition and which result in exact reconstruction of the input signal. The conditions for exact reconstruction are developed and presented. Furthermore, it is shown that these conditions are not overly restrictive and high-quality frequency division may be performed in the analysis section. A filter design procedure is presented which allows high-quality filters to be easily designed.

Exact reconstruction techniques for tree-structured subband coders

The concepts of digital signal processing are playing an increasingly important role in the area of multirate signal processing, i.e. signal processing algorithms that involve more than one sampling rate. In this paper we present a tutorial overview of multirate digital signal processing as applied to systems for decimation and interpolation. We first discuss a theoretical model for such systems (based on the sampling theorem) and then show how various structures can be derived to provide efficient implementations of these systems. Design techniques for the linear-time-invariant components of these systems (the digital filter) are discussed, and finally the ideas behind multistage implementations for increased efficiency are presented.

Interpolation and decimation of digital signals&#8212;A tutorial review

If each delay element of a linear phase low-pass digital filter is replaced by M delay elements, an (M + 1) -band filter is produced. The transition-width of this (M + 1) -band filter is 1/M that of the prototype low-pass filter. A complementary filter can be obtained by subtracting the output of the (M + 1) -band filter from a suitably delayed version of the input. The complementary filter is an (M + 1) -band filter whose passbands and stopbands are the stopbands and passbands, respectively, of the original (M + 1) -band filter. If the frequency responses of the original ( M + 1) -band filter and its complementary filter are properly masked and recombined, narrow transition-band filter can be obtained. This technique can be used to design sharp low-pass, high-pass, bandpass, and bandstop filters with arbitrary passband bandwidth.

Frequency-response masking approach for the synthesis of sharp linear phase digital filters

A new approach to implement computationally efficient finite impulse response (FIR) digital filters is presented. The filter structure is a cascade of two sections. The first section generates a sparse set of impulse response samples and the other section generates the remaining samples by using interpolation. The method can be used to implement most practical FIR filters with significant savings in the number of arithmetic operations. Typically 1/2 to 1/8 of the number of multipliers and adders of conventional FIR filters are required in the implementation. The saving is achieved both in the linear phase and the non-linear phase cases. In addition, the new implementation gives smaller coefficient sensitivities and better roundoff noise properties than conventional implementations.

Interpolated finite impulse response filters

The digital filtering process can be achieved by a set of phase shifters with suitable characteristics. A particular set, named polyphase network, is defined and analyzed. It permits the use of recursive devices for efficient sample-rate alteration. The comparison with conventional filters shows that, with the same active memory, a reduction of computation rate approaching a factor of 2 can be achieved when the alteration factor increases. A more substantial gain can be obtained in the direct realization of a uniform bank of recursive filters through combination of the polyphase network with a discrete Fourier transform (DFT) computer; savings in hardware also result from the low sensitivity of the structure to coefficient word lengths.

Digital filtering by polyphase network:Application to sample-rate alteration and filter banks

Any digital filter can be decomposed into two basic subsets, an extrapolator the output of which is sampled at a frequency depending only on the filter bandwidth and an interpolator delivering the filtered signal at the imposed output sampling rate. Redundancy in extrapolator and interpolator is removed by introducing half-band nonrecursive filtering elements for which definition, performance figures and efficient implementation are supplied. They reduce significantly the necessary computation and storage at the cost of a slight group delay increase. A formula is given for the amount of multiplications to be carried out every second in a filter; it depends on the filter bandwidth, signal to distortion ratio, and input-output sampling rate. The method is extended to recursive filters and a comparison is made with existing techniques of implementing digital filters for the needs in computation and storage hardware: a specific example of design underlines the reduction in computation speed achieved in practice through this method, which brings digital filters in a most favorable position for their competition against analog filters in many application fields.

G. Lepagnol

Papers

Interpolation, extrapolation, and reduction of computation speed in digital filters