An iterative design of constrained recursive digital filters is developed. The designing scheme requires no initial values. The constraints are subjected to degrees of both numerator and denominator, transmission zeros and poles, if any, and passband and stopband shaping. The resulting filter completes a prescribed magnitude of either passband or stopband ripples. The optimality property of the filters is examined in detail with emphasis on specifications. The designing scheme involves the elliptic design as a special case. Illustrative examples are also given.

Based on the cyclotomic polynomials, this paper describes a family of efficient and practical interpolators for interpolated FIR filters. The family can be applied to bandpass filters as well as lowpass/highpass filters without any multiplications. It also mitigates the inconvenience to select a practical interpolation factor, and gains a further saving in computational complexity required. Several examples are given to demonstrate the effectiveness for reducing the computational complexity required.

This paper presents three types of prefiltering for FIR digital filters to decrease the number of multipliers required. The first type is based on cyclotomic polynomials. It can be applied to any types of band-selective filters. The second is a mirror-image quadratic polynomial to make a passband shaping. Both types of the prefilters are used with the interpolation technique, and this improves each primitive characteristic in terms of the sharp transition. In the prefilter-equalizer design approach, these prefilterings bring about the reduction of the number of multipliers required in hardware implementation. The prefiltering efficiency is demonstrated by a few examples.

A systematic synthesis is presented to realize any digital filter into a power-wave digital filter. After three canonical matrix representations are introduced, a set of key concepts which comprises cascade interconnection of digital two-ports, pole localization, and computability is presented for the canonical cascade synthesis of lossless digital two-ports. The synthesis procedure consists of global decomposision and local decomposition. The procedure is so general as to give a unified solution to arbitrary frequency responses realization, and is so useful as to find new circuit structures. The synthesized circuits are of robustness and modularity. An illustrative example is included.

This paper describes a design method of linear phase recursive FIR digital filters. The basic structure consists of a transversal part cascaded with a cyclotomic resonator, which is characterized by cycotomic polynomials and has no multipliers. The digital filters implemented by this method require the short wordlength both for multiplier coefficients and for signals in their transversal part. By introducing integer arithmetic, the filtering operation proceeds fast and exactly. As a bonus, it is possible to employ a multiplier-less implementation in most practical applications. While the stability of this type of structure requires an integer-valued impulse response, a satisfactory procedure assures the requirement. A parameter to control the approximation error is found somewhat predictively rather than tentatively.

A direct procedure to realize pipelinable low-sensitivity digital filters is developed only in the z-domain. It is possible to realize arbitrary digital transfer functions using this procedure. The key concept for the low-sensitivity property lies both in the matching concept in doubly-terminated lossless networks and in the localization of transmission zeros. The synthesis procedure is based on successive extractions of transmission zeros by means of lossless but not always reciprocal transfer scattering matrices. Since a transfer scattering matrix involves the transmission zeros as its poles, such a matrix is suitable for their localization. Furthermore a universal first/second-degree section is derived explicitly.

Fast wavelet transform is presented for realtime processing of wavelet transforms. A processor for the fast wavelet transform is of the frequency sampling structure in architectural level. The fast wavelet transform owes its parallelism both to the frequency sampling structure and parallel tapping of a series of delay elements. Computational burden of the fast transform is hence independent of specific scale values in wavelets and the parallel processing of the fast transform is readily implemented for real-time applications. This point is quite different from the computation of wavelet transforms by convolution. We applied the fast wavelet transform to detecting detonation in a vehicle engine for precise real-time control of ignition advancement. The prototype wavelet for this experiment was the Gaussian wavelet (i.e. Gabor function) which is known to have the least spread both in time and in frequency. The number of complex multiplications needed to compute the fast wavelet transform over 51 scales is 714 in this experiment, which is less than one tenth of that required for the convolution method. Experimental results have shown that detonation is successfully detected from the acoustic vibration signal picked up by a single knock sensor embedded in the outer wall of a V/8 engine and is discriminated from other environmental mechanical vibrations.