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The essential signal-to-noise characteristics of an FM system are studied theoretically, assuming that the transmitted wave is frequency-modulated by a discontinuous signal, which is made by sampling from the original information signal at every sampling point equally spaced by a Nyquist interval and by boxcarring the sampled value over the following interval. Studies are made with a simple model of an FM demodulator of a new type, which may be called the band-dividing FM demodulator. As a result of these studies a limit to the degree of improvement in the threshold of an FM system is obtained. The result obtained here again suggests that it is possible to improve the threshold of the system beyond that of a so-called conventional FM demodulator. It is also shown that the value of the intrinsic signal-to-noise ratio (the ratio of the incoming signal power to the incoming noise power in the baseband bandwidth) at the threshold is not constant but increases with the value of the modulation index. Brief discussions on the threshold effects in the so-called phase-lock and frequency-lock FM demodulators are also given.