Characterization and computation of canonical tight windows for Gabor frames

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical $h^0$ is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples

On Lerch's transcendent and the Gaussian random walk

Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-\beta<0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\geq0\}$ as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum $M=\max\{S_n:n\geq0\}.$ These expressions are in terms of Taylor series about $\beta=0$ with coefficients that involve the Riemann zeta function. Our results extend Kingman's first-order approximation [Proc. Symp. on Congestion Theory (1965) 137--169] of the mean for $\beta\downarrow0$. We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802], and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin summation as key ingredients.Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Performance of Hybrid NbTiN-Al Microwave Kinetic Inductance Detectors as Direct Detectors for Sub-millimeter Astronomy

In the next decades millimeter and sub-mm astronomy requires large format imaging arrays and broad-band spectrometers to complement the high spatial and spectral resolution of the Atacama Large Millimeter/sub-millimeter Array. The desired sensors for these instruments should have a background limited sensitivity and a high optical efficiency and enable arrays thousands of pixels in size. Hybrid microwave kinetic inductance detectors consisting of NbTiN and Al have shown to satisfy these requirements. We present the second generation hybrid NbTiN-Al MKIDs, which are photon noise limited in both phase and amplitude readout for loading levels $P_{850GHz} \geq 10$ fW. Thanks to the increased responsivity, the photon noise level achieved in phase allows us to simultaneously read out approximately 8000 pixels using state-of-the-art electronics. In addition, the choice of superconducting materials and the use of a Si lens in combination with a planar antenna gives these resonators the flexibility to operate within the frequency range $0.09 < \nu < 1.1$ THz. Given these specifications, hybrid NbTiN-Al MKIDs will enable astronomically usable kilopixel arrays for sub-mm imaging and moderate resolution spectroscopy.Comment: 7 pages, 3 figures. Presented at SPIE Astronomical Telescopes and Instrumentation 2014: Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VI

Photon noise limited radiation detection with lens-antenna coupled Microwave Kinetic Inductance Detectors

Microwave Kinetic Inductance Detectors (MKIDs) have shown great potential for sub-mm instrumentation because of the high scalability of the technology. Here we demonstrate for the first time in the sub-mm band (0.1...2 mm) a photon noise limited performance of a small antenna coupled MKID detector array and we describe the relation between photon noise and MKID intrinsic generation-recombination noise. Additionally we use the observed photon noise to measure the optical efficiency of detectors to be 0.8+-0.2.Comment: The following article has been submitted to AP