The gravity waves produced by two orbiting neutron stars would be very low in amplitude. The energy lost through the gravity waves would be far too low to detect until the pair got to the point of merger.
Physicists created computer simulations to calculate the signatures expected to see from mergers of compact object mergers. This image from a simulation depicts the gravitational field generated by two black holes as a smaller black hole merges with a more massive black hole, just at the stage where they begin to share a common event horizon.
Image courtesy of LIGO
This video shows data from a simulation of merging black holes. Notice that as they get closer together, their orbital frequency increases. In other words, they orbit faster and faster as they approach each other. There is also an increase in the amplitude of the gravity waves. This produces a characteristic "chirp" in the gravity wave signal, that could be compared to a sound that increased in pitch and got louder. When the black holes merge, and share a common event horizon, the chirp rings down and ends.
The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a giant interferometer built to detect gravity waves. A laser beam in the main building is split into two parts by a partially reflecting mirror, and each half is sent down a tunnel four kilometers long. The light beams bounce back and forth many times. When the beams finally recombine, a phase shift between the two beams will indicate if one arm is longer than the other. A passing gravity wave would warp spacetime, changing the relative lengths of the arms. The above photo shows the LIGO facility in eastern Washington state. Another facility in Lousiana simultaneously gathers data for comparison.
This photograph shows the junction of the two arms inside the main building, where the light from the laser is emitted, split and recombined.
This photo shows technicians performing maintenance on the laser assembly inside the junction of the two arms. Gravity waves are so weak that extreme sensitivity is needed to detect them. This facility is in eastern Washington state, hundreds of miles from the ocean, and is sensitive enough to detect waves crashing on the shore.
LIGO was built and improved upon for many years before the sensitivity was high enough for gravity wave detection. The first gravity wave event was detected on Sept. 14, 2015. These plots show the gravity waves detected at both of the LIGO detectors. Note the correlation between the signals, showing the characteristics of increasing amplitude and frequency.
Inspiraling binary black hole simulation by Caltech IGO
The above computer simulation was based on data acquired Sept. 14, 2015 by the LIGO detectors of gravitational waves from merging black holes. This was the first time in history that gravity waves were detected.
chirp recording: https://www.youtube.com/watch?v=TWqhUANNFXw
This is the video released by the LIGO team, discussing events surrounding the gravity wave detection, released shortly after the discovery was announced. Data was carefully checked between the two detectors and verified. The source of the gravity wave signal was determined to be the merger of two black holes of about 30 solar masses each, with the signal emanating from a distant galaxy. This kind of event was very unexpected, as it was believed that not many intermediate-mass black holes existed. Scientists at LIGO thought it was much more probable to get a signal from a binary neutron star merger in our own galaxy.
The scientific paper regarding the detection of the first gravity waves, published on Feb 12, 2016, can be found here: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102
This computer simulation shows the merger of two black holes, and the effect of the warping of the surrounding spacetime. The black holes act as gravity lenses, bending the light from behind the system.
Einsteins's theory of General Relativity introduced the concept that spacetime can be warped by the presence of matter/energy, which implied that gravity waves could exist as ripples in spacetime. Two dense compact objects like neutron stars would produce a specific gravity wave signature as they spiraled inward to merge.