Many audio signal processing methods are formulated in the time-frequency (T-F) domain which is obtained by the short-time Fourier transform (STFT). The properties of the STFT are fully characterized by window function, number of frequency channels, and time-shift. Thus, designing a better window is important for improving the performance of the processing especially when a less redundant T-F representation is desirable. While many window functions have been proposed in the literature, they are designed to have a good frequency response for analysis, which may not perform well in terms of signal processing. The window design must take the effect of the reconstruction (from the T-F domain into the time domain) into account for improving the performance. In this paper, an optimization-based design method of a nearly tight window is proposed to obtain a window performing well for the T-F domain signal processing.
For designing a low-condition-numbered window, design methods of tight windows have been proposed @cite_28 @cite_20 . These methods aim to find a tight window with better frequency responses. However, since the constraint to the tight window greatly limits the set of variables, desired characteristics may not be obtained.
@cite_28 @cite_20
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We consider the problem of maintaining e-approximate counts and quantiles over a stream sliding window using limited space. We consider two types of sliding windows depending on whether the number of elements N in the window is fixed (fixed-size sliding window) or variable (variable-size sliding window). In a fixed-size sliding window, both the ends of the window slide synchronously over the stream. In a variable-size sliding window, an adversary slides the window ends independently, and therefore has the ability to vary the number of elements N in the window.We present various deterministic and randomized algorithms for approximate counts and quantiles. All of our algorithms require O(1 e polylog(1 e, N)) space. For quantiles, this space requirement is an improvement over the previous best bound of O(1 e2 polylog(1 e, N)). We believe that no previous work on space-efficient approximate counts over sliding windows exists. Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time–frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler–Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found. We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T in O (n^ 3 2 S^ 1 2 ), smaller than previous lower bounds for comparison-based algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and space efficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows, where the task is to take an input of length 2n-1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a time-space tradeoff lower bound of T in Omega(n^2 S) for randomized branching programs to compute the number of distinct elements over sliding windows. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k, k neq 1. This shows that those frequency moments and the decision problem F_0 mod 2 are strictly harder than element distinctness. We complement this lower bound with a T in O(n^2 S) comparison-based deterministic RAM algorithm for exactly computing F_k over sliding windows, nearly matching both our lower bound for the sliding-window version and the comparison-based lower bounds for the single-window version. We further exhibit a quantum algorithm for F_0 over sliding windows with T in O(n^ 3 2 S^ 1 2 ). Finally, we consider the computations of order statistics over sliding windows. In this paper, we provide the performance analysis of unequal error protection (UEP) random linear coding (RLC) strategies designed for transmission of source messages containing packets of different importance over lossy packet erasure links. By introducing the probabilistic encoding framework, we first derive the general performance limits for the packet-level UEP coding strategies that encode the packets of each importance class of the source message independently (non-overlapping windowing strategy) or jointly (expanding windowing strategy). Then, we demonstrate that the general performance limits of both strategies are achievable by the probabilistic encoding over non-overlapping and expanding windows based on RLC and the Gaussian Elimination (GE) decoding. Throughout the paper, we present a number of examples that investigate the performance and optimization of code design parameters of the expanding window RLC strategy and compare it with the non-overlapping RLC strategy selected as a reference.