Sliding fourier transform with generalized triangular windows

In this paper, we propose an algorithm to compute time-dependent Fourier transform (TDFT) with generalized triangular window. The algorithm recursively computes the TDFT at arbitrary frequency with integrated triangular windowing. A common way to compute TDFT with window is to firstly multiply the input data by the window function and then carry out the TDFT on the windowed data. In this paper, we show that the TDFT with triangular window at a given frequency satisfies a complex-coefficient second-order difference equation. To compute the TDFT, only seven complex multiplications and six additions are needed. The window can be a family of triangular functions, including the Bartlett window and the Welch window of the first order. Our result is useful for estimating single, few, or unevenly distributed frequencies.