Multiplier-less Broadband and Linear Phase Digital Hilbert Transformers

Abstract

The Hilbert transformation for generating the analytic signal or signal envelope is widely used in modern communication receivers, in radar and sonar systems. It introduces a 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^{\\circ }$$\\end{document} phase shift of the input signal. Since the impulse response of the ideal Hilbert transformer is non-causal, it must be approximated by an FIR or IIR filter. This paper shows results of novel algorithms for designing broadband digital IIR Hilbert transformers and its implementation. The designs employ Galerkin or collocation techniques. The transfer function of the Hilbert transformer is a rational polynomial of low order and exhibits approximately linear phase. The filters match the 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^{\\circ }$$\\end{document} phase shift requirement of Hilbert transformers almost perfectly and exhibit approximately constant group delay in the passband. The achieved image rejection ratio is typically larger than 50 dB. The quantization of the filter coefficients is realized by a Canonical Signed Digit (CSD) representation, reducing the hardware resources compared with two’s complement. The resulting filters are multiplier-less, which is crucial for high-speed signal processing and low power consumption. The design techniques and the CSD representation are realized in a MATLAB toolbox. The filters were moreover implemented in VHDL and SystemC. Additionally, a MATLAB tool for automatically generating a VHDL package containing the filter parameters has been implemented.