Sub-Landau sampling of Multiband signals and designing Periodic Nonuniform Sampling Andrzej Tarczynski 1 , Jan Samsonowicz 2 , Yoon Mi Hong * 3 and Kostadin Tzvetkov 1 1) Center for System Anaylsis, University of Westminster, 115 New Cavendish Street, London, W1W 6UW, UK 2) Faculty of Mathematics and Information science, Warsaw University of Technology, Pl. Politechniki 1, 00-661, Warsaw, Poland 3) Department of Mathematical Sciences, KAIST, Daejeon, 373-1, South Korea ABSTRACT In this paper we explore possibilities of using sub-Landau sampling (SLS) for the purpose of constructing digital filters. From the practical point of view, an interesting and important case occurs when the processed signal is to be passed through a linear filter characterized by one or more stopbands. In this situation there is no need to reconstruct signal components whose fre- quencies coincide with the stopbands. However, the signal still needs to be reconstructed in the remaining frequency ranges. The signal needs reconstructability but only partial. This implies that sub-Landau sampling could possibly be used here, but the sampling rates still cannot be arbitrarily low. The sampling scheme we exploit here is periodic nonuniform sampling (PNS). We show that many, but unfortunately not all, digital filtering problems could be effectively tackled with the use of SLS. OVERVIEW OF THE ANALYSIS Here we superficially describe the main points of the proposed approach. 1. PNS is a sampling scheme where the positions of the sampling instants are repeated every T seconds. That is, if t n is a sampling instant then t n+kL = t n + kT is also a sampling instant. Here T is the period of the sampling scheme, L is the number of samples placed in one period and k is an arbitrary integer. Note that PNS is fully defined by T and the layout of the sampling instants in the first period: 0 ≤ t 1 <t 2 < ··· <t L <T . In our analysis we propose to use period T long enough so that its inverse f T =1/T provides satisfactory resolution in frequency domain. 2. We divide the frequency axis into an infinite number of the following intervals I k I k :=  (2k - 1)π T , (2k + 1)π T  . Each interval has length f T =2π/T . We assume that the processed signal is bandlimited and occupies only a finite number of not necessarily adjacent intervals. We denote these intervals by I n 1 ,I n 2 , ··· ,I n N ; N is the number of occupied intervals. The numbers n k do not have to be put in a monotonic order. However, we assume that the first P intervals represent those bands of the input signal that have to be passed through the filter. For simplicity we also assume that