"Broadband spectroscopy" and "Time-domain spectroscopy" are generic terms whose interesting details only emerge in narrower context of specific frequency ranges (NMR, NIR, IR, VIS, UV, ...) and the capabilities and limitation of the instrumentation used to cover each case.

Speaking in generic terms, a "spectroscopy" measures an in-phase (adsorpion) and/or out-of-phase (dispersion) response of a system to a harmonic perturbation as a function of the frequency of the latter.

This can be done in two ways:

1) Scanning the frequency over a certain range where any interesting phenomena happen. This is called continuous-wave (CW) spectroscopy. If the scanned range is rather narrow with respect to its center value, we say that the system is narrow-band. In the opposite case we talk about broad-band. The distinction is usually dictated by the technical capabilities of the instrumentation, but in many cases it can be also due to nature of the sample.

2) Acquiring the transient signal "response" S(t) after an impulsive "perturbation" of the measured system. This is called time-domain (TD) spectroscopy. A very general thorem states that in [nearly] linear systems the TD response is the Fourier transform of the CW spectrum (and vice versa). So, in principle at least, the two approaches are equivalent.

However, in practice, both CW and TD spectroscopies are subject to a number of technological constraints (for example, speed of ADC converters or the TD excitation capabilities) and comparative pros and cons (S/N ratio, artifacts, ...). Consequently, there are many contexts where CW prevails over TD and vice versa.

Without further specification of what exactly you are talking about, it is hard to tell more.

Please FIRST try a Keyword search on the web. Next review the information found.

You can find these information form Analytical Chemistry textbooks such as Skoog West Holler and Harris

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