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An interpolation kernel is required in a wide variety of signal processing applications such as image interpolation and timing adjustment in digital communications. This article presents a general-form interpolation kernel called 3-3-3 interpolation kernel and derives its frequency response in a closed-form by using a simple derivation method. This closed-form formula is preliminary to designing various 3-3-3 interpolation kernels subject to a set of design constraints. The 3-3-3 interpolation kernel is formed through utilising the third-degree piecewise polynomials, and it is an even-symmetric function. Thus, it will suffice to consider only its right-hand side when deriving its frequency response. Since the right-hand side of the interpolation kernel contains three piecewise polynomials of the third degree, i.e. the degrees of the three piecewise polynomials are (3,3,3), we call it the 3-3-3 interpolation kernel. Once the general-form frequency-response formula is derived, we can systematically formulate the design of various 3-3-3 interpolation kernels subject to a set of design constraints, which are targeted for different interpolation applications. Therefore, the closed-form frequency-response expression is preliminary to the optimal design of various 3-3-3 interpolation kernels. We will use an example to show the optimal design of a 3-3-3 interpolation kernel based on the closed-form frequency-response expression. [ABSTRACT FROM PUBLISHER]

This paper proposes a closed-form solution for designing variable one-dimensional (1-D) finite-impulse-response (FIR) digital filters with simultaneously tunable magnitude and tunable fractional phase-delay responses. First, each coefficient of a variable FIR filter is expressed as a two-dimensional (2-D) polynomial of a pair of parameters called spectral parameters; one is for independently tuning the cutoff frequency of the magnitude response, and the other is for independently tuning fractional phase-delay. Then, the closed-form error function between the desired and actual variable frequency responses is derived without discretizing any design parameters such as the frequency and the two spectral parameters. Finally, the optimal solution for the 2-D polynomial coefficients can be easily determined through minimizing the closed-form error function. We also show that the resulting variable FIR filter can be efficiently implemented by generalizing Farrow structure to our two-parameter case. The generalized Farrow structure requires only a small number of multiplications and additions for obtaining any new frequency characteristic, which is particularly suitable for high-speed tuning. [ABSTRACT FROM AUTHOR]