digital signal processing
Windows for harmonic analysis are considered which have characteristic property of simple representation in the form of trigonometrical series with small number of terms m (low-order windows). The ratio P of the main lobe amplitude of the window spectrum to the weighted level of maximum sidelobe at the given frequency interval is accepted to be an efficiency function. The paper solves the problem of window synthesis maximizing above-mentioned efficiency function.
First, the properties of the efficiency function are examined. It is shown that the problem has unique solution, and the efficiency function is continuous and convex in the space of window coefficients but nondifferentiable and has narrow and deep ravines.
Based on these properties, numerical method to locate optimal solution is suggested, namely the algorithm of random search with constant radius and random direction. The algorithm includes verification procedure to guarantee optimality of synthesized window.
Using this algorithm the family of low-order windows (m = 14) is synthesized which minimize sidelobe suppression level P at the specified frequency interval for the given main lobe width. The calculated windows are tabulated for the range of sidelobe levels P = 18…125 dB.
The obtained window family is infinite and includes as discrete points well-known Hamming (m = 1, P = 43,2 dB), Blackman-Harris (m = 2, P = 71,5 dB), Nuttall (m = 3, P = 98.2 dB) and Albrecht (m = 4, P = 125.4 dB) windows. For the given value of P the synthesized windows provide smaller main-lobe width and smaller signal-to-noise loss in comparison with known low-order windows.
Finally, the paper discusses problems of the windows usage for digital harmonic analysis both in time and frequency domains.