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This paper describes the first algorithm for computing the inverse chirp z-transform (ICZT) in O(n log n) time. This matches the computational complexity of the chirp z-transform (CZT) algorithm that was discovered 50 years ago. Despite multiple previous attempts, an efficient ICZT algorithm remained elusive until now. Because the ICZT can be viewed as a generalization of the inverse fast Fourier transform (IFFT) off the unit circle in the complex plane, it has numerous practical applications in a wide variety of disciplines. This generalization enables exponentially growing or exponentially decaying frequency components, which cannot be done with the IFFT. The ICZT algorithm was derived using the properties of structured matrices and its numerical accuracy was evaluated using automated tests. A modification of the CZT algorithm, which improves its numerical stability for a subset of the parameter space, is also described and evaluated.

Abstract-This paper describes a set of experiments in which an upper-torso humanoid robot learned to slide a card through a card reader. The small size and the flexibility of the card presented a number of manipulation challenges for the robot. First, because most of the card is occluded by the card reader and the robot's hand during the sliding process, visual feedback is useless for this task. Second, because the card bends easily, it is difficult to distinguish between bending and hitting an obstacle in order to correct the sliding trajectory. To solve these manipulation challenges this paper proposes a method for constraint detection that uses only proprioceptive data. The method uses dynamic joint torque thresholds that are calibrated using the robot's movements in free space. The experimental results show that using this method, the robot can detect when the movement of the card is constrained and modify the sliding trajectory in real time, which makes solving this task possible.