## Finding Pulsars

At first, you might think that finding pulsars would be fairly easy. All one should have to do is point a radio telescope at the sky and look for little pulses of radiation. But there are several reasons why that isn't possible. To start with, the strength of any one individual pulse is usually very weak. In fact, for most pulsars, a single pulse is not even strong enough to detect &mdash it gets swamped by the background radio waves (what astronomers refer to as noise). It is not all that different from trying to see stars in the day time. The bright light from the Sun makes the stars impossible to see. In the case of pulsars, the noise that is inherent in our instruments makes a single pulse very difficult to see, except in the case of the brightest pulsars.

 Figure 3 &mdash Here we illustrate the technique of folding. In the top panel, we place ticks ever 1 second in our data. In the middle panel, we fold over at these tick marks. In the bottom panel, we add up the signal in each layer of our fold. In this way, we can find a weak pulsar whose would otherwise be lost in the noise. However, we must know the period ahead of time. In this example, we used a period of 1 second.

In order to solve this problem, astronomers must combine many pulses together in order to build up a usable signal. This is commonly referred to as folding. To get an idea for how it works, imagine having a long strip of paper. On this paper we make a mark that indicates the strength of a signal detected with our radio telescope &mdash the higher the mark is on the paper, the stronger the signal. Now imagine that we make these marks continuously while someone pulls the strip of paper along underneath our pen. What we would have in the end is a record of how strong our signal was during the course of our observation. On the left of the paper is the first mark that we made, and on the right is the last mark. Each mark represents the strength of the signal detected by our telescope at some point in time. The paper might look something like Figure 3. Now this imagine just looks like a bunch of random marks &mdash that is the noise, or the background. Somewhere buried in that noise is the signal from our pulsar. Let's say that we know ahead of time that our pulsar has a period of one second. Then we could make a mark on our paper that represents one second of time, i.e. one period of the pulsar. We can then imagine folding our paper on-top of itself, so that these marks line up. If we then had a way of adding the signals from different layers of this fold together, we would eventually see the signal from the pulsar emerge.

But this is a difficult task when you can't actually tell where the individual pulses are. In other words, this technique only works efficiently if we know the period of the pulsar ahead of time. If we don't have that knowledge, we would have to fold every observation at every possible period and look to see if a pulsar signal emerged. This is an impossible task, so we have to find another way.

### Fourier Transforms

Luckily, there is a tool called the Fourier transform that can help us. Fourier transforms are mathematical operations that are perfectly suited for finding periodic signals (that is, something that repeats in a given set of data). The important thing for you to know is that the Fourier transform does what its name suggests &mdash it transforms the data from one form into another. In the case of pulsar searches, the transformation takes data that is in a form of signal vs. time, and turns it into something in the form of signal vs. frequency. Frequency is related to time as $$\mathrm{frequency} = \frac{1}{\mathrm{time}}$$ Stated differently, the frequency tells us how many times something repeats in a given second. If something spins with a period of exactly one second, then its frequency is once per second. If it spins faster, say going around 20 times per second, then the frequency is 20 per second. Scientists define a unit called the Hertz, abbreviated as Hz, that is equal to one cycle per second. So in the second example above, our frequency would have been 20 Hz.

What would the frequency be of something that has a period of 0.01 seconds? Well, 0.01 second is one-hundredth of a second, meaning that in a full second the object would make 100 full rotations. So our frequency is 100 Hz. Mathematically, we could also have divided by the period--1/(0.01 seconds) = 100 Hz. We can make a general formula to illustrate this point. $$f_{\mathrm{spin}} = \frac{1}{P_{\mathrm{spin}}}$$ where $f_{\mathrm{spin}}$ is the spin frequency and $P_{\mathrm{spin}}$ is the spin period.

When we Fourier transform a set of pulsar data, we can look for signals in the frequency domain. If we find one, we can then go back and fold our data in the time domain and determine if our signal came from a real pulsar. For example, suppose we see a signal at a frequency of 100 Hz in our Fourier transformed data. We would then fold our data at a spin period of 0.01 seconds. This is a more efficient way of searching for pulsars than doing blind folds.

When we work in the frequency domain, we can also take advantage of harmonics. We say that the spin frequency is the fundamental or first harmonic. The second harmonic occurs at twice the frequency of the first harmonic, the third harmonic occurs at three times the frequency of the first harmonic, and so on. Mathematically, the frequency of the $n^{\mathrm{th}}$ harmonic is $n \times f$, where $f$ is the spin frequency. Harmonics occur naturally almost any time there is a periodic signal. You are probably familiar with harmonics, even if you don't know it. Musical instruments, like violins, pianos, or the human voice, produce sound waves, and waves are periodic. So when a violin player plays a note, say an A-flat, what you hear is actually all the harmonics of an A-flat. These higher harmonics give the instrument a rich, pleasing sound, and cause it to sound like a violin instead of some other instrument, such as a piano, which would have a different harmonic structure. If you have ever heard a computer play a pure tone (i.e., only the first harmonic), you know that it sounds hollow and not very pleasing.

Harmonics add a richness to pulsar signals as well (although keep in mind that radio waves are not sound waves). Some, and often quite a bit, of the power from a pulsar signal is in the higher harmonics. If we only used the first harmonic (the spin frequency), then we would lose precious sensitivity, especially to weak pulsars.

Modern computers are very good at taking the Fourier transform of a set of data. Still, the transformed data set is pretty big, and it would be very difficult, if not impossible, for a person to look for every possible pulsar signal. Instead, we have written computer programs that look at the Fourier transformed data and try to find the signals from pulsars. These computer programs are very good at what they do. Unfortunately, it isn't as simple as hitting Enter on the keyboard and getting out all the information about a pulsar. There are many things that may look like a pulsar at first glance, but which are in fact something entirely different (and usually much more ordinary). Separating the true pulsars from these impostors requires a human touch, and this is one of the most important tasks that you will be undertaking. But before we talk more about that, we should talk about these impostors, and why we need to worry about them.