Theory of the Design, and Operation of Digital Filters

Chapter 16 present the theoretical basis for digital filters, including issues related to their design. The basic characteristics and structures of finite impulse response and infinite impulse response filters are presented and discussed. In addition, the ideal and practical transfer characteristics of the digital filters are defined. The basic advantage of finite impulse response filters is that they can be designed to have an exact linear phase. However, infinite impulse response filters are generally more efficient computationally. The methods for filters design and related algorithms, which are based on the bilinear transformation method, windowed Fourier series, and algorithms based on iterative optimization, are also presented.

Multi-Rate Discrete-Time Signal Processing

Chapter 17 presents the multi-rate signal process, starting with explanations of the up-sampling and down-sampling procedures on a discrete signal in the time domain. The operations of a down-sampler (decimator) and an up-sampler (interpolator) are analysed in the frequency domain, emphasizing the problem of possible aliasing. Complex systems that include both up-sampling and down-sampling are analysed and the problem of complexity reduction is mentioned. The operation of systems that combine an interpolator and an interpolation filter, and a decimator and a decimation lowpass filter, is presented in the time and frequency domains. In particular, the problem of reducing the complexity of a multi-rate system is addressed, and a polyphase decomposition for both finite impulse response filters and infinite impulse response filters is offered as an efficient solution.

Finite Impulse Response Filters for Stagger-Period Signals, their Designs and Applications

Those theories of conventional filters for uniform-period signals do not apply to the analysis and design of the finite impulse response (FIR) filters for stagger-period signals. In this paper, we defined the fundamental concepts related to the stagger-period signals, derived the calculating equations, and described the time-variant property of the stagger-period filter; we proposed the Fourier transform pair between the frequency and impulse responses of this type filter, and proved the inverse of each other. Then, we discussed the design methods of stagger-period frequency-selective FIR filters, including lowpass, bandpass, and high-pass, presented the staggered windowing philosophies, illustrated different windows’ effectiveness, and described the principles and designs of optimized stagger-period high-pass filters with the match algorithm. As applications, we introduced three staggered optimization algorithms: eigenvalue, match, and linear prediction; and discussed performances of the filters designed for a moving target indication (MTI) radar. The stagger-period MTI filters not only extended the blind speed of flying targets, but also had an optimized improvement factor. Finally, we proposed a mathematical programming to search the best period code, which makes this type filter’s velocity response flattened. Meanwhile, we compared properties of the stagger-period to uniform-period filters, and provided with some examples to illustrate the theories and designs.

Finite Impulse Response Filters for Stagger-Period Signals, their Designs and Applications

Those theories of conventional filters for uniform-period signals do not apply to the analysis and design of the finite impulse response (FIR) filters for stagger-period signals. In this paper, we defined the fundamental concepts related to the stagger-period signals, derived the calculating equations, and described the time-variant property of the stagger-period filter; we proposed the Fourier transform pair between the frequency and impulse responses of this type filter, and proved the inverse of each other. Then, we discussed the design methods of stagger-period frequency-selective FIR filters, including lowpass, bandpass, and high-pass, presented the staggered windowing philosophies, illustrated different windows’ effectiveness, and described the principles and designs of optimized stagger-period high-pass filters with the match algorithm. As applications, we introduced three staggered optimization algorithms: eigenvalue, match, and linear prediction; and discussed performances of the filters designed for a moving target indication (MTI) radar. The stagger-period MTI filters not only extended the blind speed of flying targets, but also had an optimized improvement factor. Finally, we proposed a mathematical programming to search the best period code, which makes this type filter’s velocity response flattened. Meanwhile, we compared properties of the stagger-period to uniform-period filters, and provided with some examples to illustrate the theories and designs.