We investigated the influence of amplitude modulation (AM) noise and phase modulation (PM) noise of a laser source on the frequency stability in frequency stabilization systems. We estimated the frequency stability and evaluated the efficacy of a noise reduction technique (the Doppler-trend subtraction method) of a laser diode frequency stabilization system, where enhanced intensity noise arising from PM-to-AM noise conversion through a reference gas cell is reduced using the technique employed in modulation transfer spectroscopy. To evaluate the relationship between the laser's intrinsic noise and its frequency stability, we performed noise spectrum measurements and formulated frequency stability in addition to measuring Allan standard deviation. As a result, it is found that the extra noise generated in PM-to-AM conversion is efficiently removed by the Doppler-trend subtraction method and that within the feedback bandwidth, the frequency stability becomes 1 order of magnitude better than that without the method.