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This paper introduces a procedure for constructing convolutional codes using finite-field wavelets. This provides novel insight into the study of convolutional codes and permits the design of the new convolutional codes that is not possible by conventional methods. Exploiting algebraic properties of the wavelet codes, we show that a rate K/L wavelet convolutional encoder is a basic encoder that is noncatastrophic. In addition, we prove that any rate 1/L wavelet convolutional encoder is minimal-basic and that every L-band orthogonal wavelet system generates a rate 1/L self-orthogonal code. As an application of wavelet convolutional codes, we construct time-varying convolutional codes. These codes have unique trellis structures that result in fast and low computational complexity decoding algorithms. As examples, we present some time-varying wavelet convolutional codes that can be decoded faster than comparable lime-invariant convolutional codes. We construct 16- and 32-state time-varying wavelet convolutional codes with minimum-free distances of seven and eight, respectively. These codes have the same minimum-free distance as the best time-invariant codes of the same rate and state complexity, but they can be decoded almost twice as fast. We show that a 32-state time-varying wavelet convolutional code is superior to the Lauer code in performance while having almost the same decoding complexity. Although the 32-state wavelet code is inferior to the 16-state Golay convolutional code as far as computational complexity, it outperforms this code in the performance. We also prove that orthogonal filterbanks generate self-dual time-varying codes. We give a design for doubly even self-dual time-varying convolutional codes by imposing some constraints on the filters that define the L-band orthogonal wavelets. As another application of wavelet convolutional codes, we propose a new scheme for generating rate-adaptive codes. These codes have the property that multiple rates of the code can be decoded on one trellis and its subtrellises. [ABSTRACT FROM AUTHOR]