Way to extrapolate scarce signal

Firstly apologise if I use the wrong terminology, I don't have formal experience in signal processing, hence I would appreciate the help a lot. I have a time-domain photon counting signal which due to experimental limitation has 256 time bins. This is considered quite 'scarce' in my field, and analysis is difficult. Signals with at least 1024 time bins would be a lot easier to analyse as they will also obviously have more counts.

Is there any method in signal processing where a known "scarce" signal can be extrapolated to a more "dense" signal? I don't mean extrapolating along the time axis, more like "filling in the gaps" either by being provided an expected model/shape/fit, or by guessing the shape itself?

Any suggestions what to look for in terms of terminology would still be very useful!

Edit(more info): the signal is an exponential decay characterised by a Poisson distribution. The technique is referred to as time-correlated single photon counting (TCSPC) and essentially measures the number of excited photons in a series of time bins. More time bins, more information on the decay behaviour of the photons.

• Welcome to SE.SP! It's not clear (to me) quite what you mean? Do you mean taking measurements at 0,1,2,3... and creating measurements at 0,0.5,1,1.5,2,2.5,3,3.5 ? Or something else?
– Peter K.
Jan 24 at 16:20
• I might simply be lacking the experimental background here: what is a photon-counting signal? What information does it convey, at which points in time? (This might seem very basic to you, but I guess for me, I don't have an idea what it does, where it comes from, whether it's sampled at discrete times or whenever an event happens… Please give background!) Jan 24 at 16:26
• @PeterK. thank you! Yes this is what I need. The actual signal is an exponential decay which follows Poisson distribution / Poisson noise. Jan 24 at 16:29
• @MarcusMüller many thanks for the reply, apologies for the vagueness. The signal is an exponential decay following a Poisson distribution, it is essentially showing the amount of photons in an excited state in a given time (which, as I mentioned, decays exponentially with time). The more bins you have, the more information on the photon decays you have and hence the easier to fit the data to extract relevant parameters. I know the model this decay should in principle obey, so I imagine it might be possible to extrapolate the missing info (the way PeterK. has explained above). Jan 24 at 16:38
• what is a bin? How and at what times do you observe the signal? Jan 24 at 16:59

The Poisson distribution is defined as $$f(k; \lambda)=\frac{\lambda^ke^{-k}}{k!}$$ where $$k$$ is your bin index. You can estimate $$\lambda$$ using the maximum likelihood estimator, which is simply the mean of the data: $$\lambda=\frac{1}{N}\sum_{n=0}^{N-1}x_n$$ where $$x_n$$ will be the time bin associated with photon $$n$$ and $$N$$ is the total number of photons or the histogram counts in all bins.