Windowed Fourier transform

The basic idea here is to multiply the data train with a window function and compute the Fourier transform. We use the Welch Window for our Windowed Fourier transforms (WFT). The WFT is defined by the relation,

$$\rho(t,f) = \int_{-\infty}^\infty h(\tau) w(\tau - t) e^{2\pi i f\tau} d\tau,$$

where, $w(t,\tau)$ is given as,

$$w(\tau) = 1.0 - \frac{4.0 \tau^2}{d^2} \ \ \ \ \ \ : \Vert\tau\Vert < d/2$$ (10.1.221)

$$= 0.0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ : \Vert\tau\Vert \geq d/2,$$ (10.1.222)

where $d$ is the width of the window.