= Paul D. Hale
{c}
\b[Paul D. Hale] is a <metrology> researcher named with <Tracy S. Clement> and <Dylan F. Williams> on a 2001 NIST paper about <optical receiver> <frequency response>. It compares frequency-domain <optical heterodyne detection> with time-domain measurements using an <oscilloscope>, including corrections for the electrical connection and finite optical stimulus.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[Byline, p. 1 introduction and p. 3, Time-domain measurements]{ref}
An <oscilloscope> trace need not be the receiver's isolated <impulse response>. In an original idealized model of <linear time-invariant systems>, let $r(t)$ be the receiver <impulse response>, $p(t)$ the optical pulse envelope and $o(t)$ the readout <impulse response>. Ignoring mismatch for this illustration, the measured waveform is the <convolution>
$$
y(t)=(r*p*o)(t),\qquad Y(f)=R(f)P(f)O(f).
$$
A broader measured pulse can therefore reflect the receiver, the stimulus or the readout. If all three envelopes are normalized <Gaussian functions>, with independent convolution widths, their <variances> add:
$$
\sigma_y^2=\sigma_r^2+\sigma_p^2+\sigma_o^2.
$$
For illustrative widths $\sigma_y=5\,\mathrm{ps}$, $\sigma_p=3\,\mathrm{ps}$ and $\sigma_o=\sqrt{7}\,\mathrm{ps}$, the inferred receiver width is $\sigma_r=3\,\mathrm{ps}$, rather than $5\,\mathrm{ps}$. These are chosen model values, not results reported by Hale.
The example motivates characterizing the stimulus and readout as well as the specimen. Dividing by $P(f)O(f)$ where that product is nearly zero also amplifies measurement noise, so a formal <Fourier transform> inversion does not guarantee a reliable corrected response. This is why <measurement uncertainty> belongs alongside the quoted <frequency response>.
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