Analytical expressions are derived for a new set of optical beams, in which the radial dependence is described by a sum of Bessel distributions of different orders, modified by a flat-topped Gaussian function expressed in the form 1 - [1 - exp(-xi2)]M, where xi is a dimensionless parameter and M(> or = 1) is a scalar quantity. The flat-topped Gaussian function can be readily expanded into a series of the lowest-order Gaussian modes with different parameters; this situation makes it possible to express the optical beam as a series of conventional Bessel-Gaussian beams of different orders. The propagation features of this new set of optical beams are investigated to reveal how a windowed Bessel beam passes progressively from a smooth Gaussian window toward the hard-edge limit.