ANALYSIS OF CHARACTERISTICS OF A CROSS-PRODUCT FREQUENCY-DISCRIMINATOR IN 4-PHASE-SHIFT KEYING SYSTEMS

Characteristics of a cross-product frequency discriminator and the res ponses of an automatic frequency control (AFC) closed loop using the d iscriminator are analyzed theoretically in this paper for practical fo ur-phase-shift keying (4PSK) transmission systems that have band-limit ing filters in their transmitters and receivers. The general expressio ns for the characteristics and the responses are derived when these fi lters are assumed to have arbitrary characteristics and three types of 4PSK, i.e., QPSK, offset QPSK, and pi/4-shift QPSK, are assumed. The expression for a deterministic component and the autocorrelation funct ions for random components are obtained first in the case of an open l oop. Those results then are used to obtain the power spectral densitie s, and the expression for variance of frequency fluctuation is shown. The variance is used to calculate frequency jitter in an AFC closed lo op. The variance of frequency fluctuation consists of three components . The components yielded From a product of noise and noise and a produ ct of noise and signal are independent of the modulations. The compone nt from a product of signal and signal, on the other hand, depends on the modulations. The components for quadra-phase-shift keying (QPSK) a nd for pi/4-shift QPSK are equal, but the component for offset QPSK is larger than that for the others.