Window functions obtained by convolution integrals of rectangular windows

Abstract

The rectangular, triangular, and Parzen windows, which have been defined independently of each other, are obtained by repeating convolution integrals of the rectangular window with itself. They are extended to make a series of window functions. This series has the advantage that the sidelobe fall-off is variable though it has the disadvantage that the mainlobe band-width for the same highest sidelobe level is 1–1·3 times those of the gaussian, Kaiser, Dolph–Chebyshev or Blackman–Harris windows. Moreover, this series has a desirable mainlobe bandwidth of 0·85–1 times that of cos^x x windows for the same highest sidelobe level and sidelobe fall-off. Thus, this series is useful in designing window functions whose mainlobe bandwidth is in between that of the gaussian or Kaiser windows and that of cos^x x windows and whose sidelobe fall-off is as good as that of the cos^x x windows.