The expression for rk in is often used to approximate the phase variation due to noise when the phase fluctuations are much larger than, which is normally the case. The integral is known to have a closed-form solution, but typically results in the estimation value being in the form of a series of terms. Some of these terms have to be computed at a previous time in order to obtain new estimates of the terms for the next time. In the past years, some researchers have begun to look at the properties of the terms and the rate of convergence in order to create algorithms for PLLs. The PLL concept is in fact even older than television broadcasting having been used in the early 20th century to synchronize the step-wise frequency-modulation outputs of acoustic signal generators. The development of this early work was the first important discovery in digital signal processing, helping to accelerate the widespread use of the concept in the late 20th century. A historic account of the early work was described in an article in the Journal of the Institution of Electrical Engineers, volume 91 (1951).
"Communications scientists have always realized that it is important to know the phase of the carrier signal, but in the past they did not realize that the PLL is a natural and powerful synchronizing means for carrier recovery. The first publication to describe PLL for synchronizing the carrier phase is de Bellescize, "Etude Technique De l'Automatisme Phase Laplac. Phase Locked Loops and Their Applications, "IEEE Trans. Commun. Tech. 1959, volume 1, pages 455 through 474."4 It followed a simple model of frequency division by a constant factor. The first practical application of the PLL was for synchronizing the horizontal, phase-modulated output of a video AM transmitter, just as a clock would clock out a tone synchronized with an unmodulated AM carrier. The paper by de Bellescize in 1959 was one of the first to show that PLLs could be used to drive multiple modulations simultaneously, which is important for radio receivers working with a frequency-hopping spread spectrum signal." d2c66b5586