Generalized Sweep Equation

The sweep equation in its simplest form is written : \[S(t) = A_0(t)sin(\varphi(t))\]

Where \(A_0(t)\) is the envelope and \(\varphi\) the instantaneous phase, both as a function of time. This is the expression used in traditional vibrator electronics.

The instantaneous frequency is the time derivative of the instantaneous phase : \[f(t) = \frac{1}{2\pi}\frac{d\varphi}{dt}\]

The full sweep equation has the instantaneous sweep frequency computed by integrating the time-domain sweep rate \(S_r\) : \[f(t) = f_{min} + \int_0^TS_r(t)dt\]

Which is then integrated again over the sweep length to get the instantaneous phase : \[\varphi(t) = \varphi_0 + 2\pi\int_0^Tf(t)dt\]

The complete sweep equation becomes: \[S(t)=\cos\Bigg[\varphi_0 + 2\pi\int_0^T \bigg(f_{min} + \int_0^TS_r(t)dt \bigg) dt \Bigg]\]

References / Further Reading

Julien Meunier and Thomas Bianchi. How long should the sweep be ?, pages 1–5. 2012. doi : 10.1190/segam2012-0182.1. URL