Confusion with applying incremental bandwidth to measure noise power spectral density

Question

From a text, regarding the following paragraph about random noise:

Being not very informed on this topic, I don't understand what meant by the following statement:

"If such a measurement is made over a specific incremental bandwidth Δf at a specific center frequency f_k, the output noise voltage can be expressed in terms of..."

Does that mean a band-pass filter applied for each Δf? And is so will it be applied at the input or output? And how is this done by analog way in reality? Or is it more convenient to measure the entire signal and apply digital methods?

Answer 1

The sentence you're referring to describes a mathematical model; so, no, there's no bandpass filter for each \$\Delta f\$; that would be impossible, since \$\Delta f \to 0\$.

Describing it like this is necessary to mathematically capture the idea that if you measure something in arbitrarily small regions, the thing you observe converges to a density; in your case, for any continuous spectrum, the idea is that your "power in measurable (hence, non-infinitesimally-narrow) bandwidth" converges to "power spectral density".