FIR Fractional Hilbert Transformers With Raised-Cosine Magnitude Response

Fractional Hilbert transformers find applications in communications and image processing. Various methods have been developed for their design. Some of them start from previously designed conventional Hilbert transformer, whereas others perform the design directly. In this paper, we present a closed-form method for the design of FIR fractional Hilbert transformers, which is based on well-known Fourier series method. The presented method results in transformers whose transfer functions approximate raised-cosine magnitude response with fractional phase shift in the least-squares sense. We used such frequency response because the corresponding impulse response is well localized in time, what enables the use of the Fourier series method without additional window function. The features of the proposed transformers are illustrated by examples which include the design of fractional and conventional transformers as well as complex Hilbert filters.