A CLASS OF CONVOLUTIONAL TIME‐VARYING FILTERS

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Abstract
A linear time‐varying filtering process usually is realized by applying a number of time‐invariant filters to overlapping time regions of a trace and transitionally merging these different regions. We describe the design of a different type of linear time‐varying filter where the frequency spectrum of the filter is shifted along the frequency axis as a function of time without appreciable change in the spectrum shape. The design is based on a given time‐invariant filter with the desired spectrum shape. For long filter length, the design procedure is rather complicated. Furthermore, only a constant rate of frequency shift is possible. However, for many practical situations where the frequency shift over the filter length is much less than the bandwidth of the filter, the process of time‐varying filtering can be further simplified without appreciable frequency error. In fact, time‐varying filtering is achieved through modifying the complex signal representation of the original time invariant filter by a preapplication of a specified time‐varying frequency‐shifting operator and a post‐application of a corresponding frequency restoration operator. Under this short filter approximation, the more general time‐varying filter in which the rate of frequency shift is nonuniform can be realized. Behavior of the convolutional time‐varying filters is demonstrated by examples of filtering of sampled sinusoidal signals.

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