Handbook of Fourier analysis & its applications

Abstract

TABLE 2.3. Continuous time Fourier transform (CTFT) theorems. The Fourier transform, typically complex, can be expressed in rectangular (Cartesian) coordinates as X(u) = R(u) + jI(u), or in polar coordinates, X(u) = |X(u)|e j X(u). Convolution is defined by x(t) * h(t) = ∞ −∞ x(τ) h(t − τ) dτ. The deterministic correlation integral is x(t) h(t) = ∞ −∞ x(τ)h * (τ − t)dτ = x(t) * h * (−t). When x = h, this operation is deterministic autocorrelation. T sampling theorem n x n sinc(2Bt − n) ↔ n x n e −jπu/B u 2B

Ultrasonic Signal Time-Expansion Using DAC Frequency Modulation

“…(20) Such a signal is visible in Figure 3 and corresponds to a 200 µs Gaussian pulse oscillating at 100 KHz with a bandwidth of 1 σ 30 KHz. The Fourier transform of (20) can be written as [17,18]…”

“…As y(t) is, by definition, the time-expanded signal of x(t), from the properties of the Fourier Transforms [17], it follows that…”