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We study the instantaneous frequency of a signal which is formed by the summation of a sinewave and a bandlimited signal. We refer to the composite signal as a sinewave squelched or simply a squelched signal. We study the behavior of the squelched signal at its real zeros and extrema. In particular when the sinewave frequency is equal to the largest frequency in the signal's spectrum and its amplitude is greater than the signal's maximum amplitude, the analytic signal corresponding to this resultant signal (hereafter abbreviated as RZ signal) has only real zeros which contain all of the bandlimited signal's information. FM systems often use zero crossing techniques in the demodulation process. For a process with a normal amplitude distribution, we derive a relationship between the zero crossing density of an RZ converted signal and that of the original signal. One application of these analyses is in ultrasonic FM imaging. We give an explanation for the behavior of the FM imaging system as a function of amplitude and frequency of the added cosine.