These are good questions, and while I'd take a different approach from most of the other replies to them, I instead want to clear up a misconception about the c

These are good questions, and while I'd take a different approach from most of the other replies to them, I instead want to clear up a misconception about the choice of signal that I notice throughout the thread. What's being tracked on Earth is not the speed of light or the radial distance to each pulsar (which is in any case not known to good precision) but rather the rotation rate of each pulsar. That's known to very good precision for each of the selection of pulsars by each pulsar timing array (PTA) collaboration. Each collaboration selects a number of pulsars with very stable rotation rates. The PTA antennae listen to the beam from the pulsar in convenient radio frequencies; they're not especially interested in the spectral structure of the pulsar signal, and are not deliberately looking for a gravitational redshift of the beam's radio frequencies as opposed to a gravitational redshift of the beam's amplitude. The amplitude modulation of the listened-to frequencies is driven by the very much slower than light rotation of the pulsar which drags a beam around so that we see it very strongly once or twice per rotation. At the top of this page <https://www.astron.nl/pulsars/animations/ https://www.astron.nl/pulsars/animations/> there is an animation of a pulsar; we're interested interested in the bright flashes when the beam fills the video window. The evolution of timing of those bright flashes are what PTAs track. Notice that the flash is sharp rather than smeared out, waxing and waning slowly over the course of a rotation. This is a useful property and distinguishes a pulsar from ordinary astronomical radio soruces (and the essentially constant light from ordinary stars). Here's a typical "chirp" from eight different millsecond pulsars observed at Arecibo <https://scx2.b-cdn.net/gfx/news/2019/palfasurveyr.jpg https://scx2.b-cdn.net/gfx/news/2019/palfasurveyr.jpg>. The first line in each box is the name of the pulsar; the lines below the names are the rotation rate (the other two lines are details about the observation and processing rather than the pulsar). Another useful property is that the "chirp" is seen in an enormous range of frequencies, so two immediately adjacent radiotelescopes listening to the pulsar in two very different radio frequencies will notice the pulse essentially simultaneously. Some millisecond pulsars even flash at high frequencies (up to gamma rays) although in practice PTAs aren't particularly interested in that part of the spectrum (although it is certainly watched by other astronomers!). In many millisecond pulsars the interval between bright flashes is very very very stable (they don't have "starquakes" and don't have significant amounts of matter accreting on them) and thus predictable. That means (with a good frequency standard on our side) we can tell if the gap between a pair of pulses is short or long compared to the long term average. The gravitational waves the pulsar timing array (PTA) experimentalists are interested in will induce a gradual (over the course of months to years) increase and decrease of the gap between pulses. The change is quite small; if we have a "true" millisecond pulsar where the long term average has us seeing one pulse every millisecond exactly, a nanohertz (say, a cycle time of one year and of moderate amplitude) gravitational wave over the course of the year will stretch the millisecond timing to a maximum of one millisecond plus low tens of nanoseconds and a minimum of one milliecond minus low tens of nanoseconds. Most of the time it's between those outer bounds and close to the long term once-per-millisecond-exactly average. The hard work for the experimentalists is in signal processing and spotting sources of noise in the pulse timings.