What is the relationship between an amplifier's step input and the frequency sweep response?
The frequency response is the Fourier transform of the impulse response in the time domain. This is the response when the input is a Dirac-distribution, which is defined by the following properties:

$$\delta(t) = \begin{cases}
+\infty &amp;\mbox{if }t = 0 \\
0 &amp;\mbox{else}
\end{cases}$$
and
$$\int_{-\infty}^{+\infty}\delta(t)dt = 1$$

Integrating this Dirac distribution (as per the second property), will give you the step function, which is equivalent in the Fourier-domain to multiplying with \$\frac{1}{j\omega}\$. Also notable is that the Fourier transform of this Dirac distribution is exactly 1 over all frequencies.

So you can use these properties (\$u(t)\$ is the step function) for any linear system \$f\$:

$$\begin{align}
\mathcal{F}\{f\circ u\} &amp;= \mathcal{F}\{f\}\cdot\mathcal{F}\{u\} \\
&amp;= \mathcal{F}\{f\}\cdot\frac{1}{j\omega}\mathcal{F}\{\delta\} \\
&amp;= \frac{1}{j\omega}\mathcal{F}\{f\}
\end{align}$$

So if you ever measured a step response and want to derive the frequency response of the system from that, you can simply multiply the Fourier transform of the step response by \$j\omega\$.