On an instantaneous frequency estimator with FIR filters having maximally flat frequency response error magnitude

Abstract

In this paper an instantaneous frequency estimator (IFE) of a discrete-time base band complex signal is considered. The IFE is built around one-band, maximally flat linear-phase FIR filters, which are used for differentiating and delaying Cartesian components of the complex signal. One of the key features of the estimator is that it avoids problems related to the ambiguity of the instantaneous phase waveform. The quality of the estimator is tested. A closed-form formula for the static characteristic of the IFE is derived and expressed as a function of the frequency responses of the filters used. Two representative test signals: a full band complex linear frequency modulated (LFM) chirp and a three-component complex synthetic signal are used to demonstrate the characteristic features of the estimator. If the chirp is sufficiently long in comparison with the length of the filters, the instantaneous frequency (IF) estimation errors are comparable to those obtained by using the static characteristic. For this case, the IF estimation error plots for the practical versus ideal IFE are presented and a design chart showing the dependence of the IF estimation error magnitude on the input signal bandwidth and the FIR filters’ length is given. This chart can be exploited in, e.g., FM-telemetry applications, where the IF carries a very slowly changing telemetric message. The three-component signal chosen allows demonstration of the ability of the estimator to track the IF which extends beyond the signal spectral range, permitting measurement even beyond the Nyquist frequency. Finally, the power of the proposed IFE to measure the stability of highly precise frequency oscillators is shown.