This paper reports a general on-wafer noise figure (NF) de-embedding technique with the analysis of two gain definitions. As implemented in this work, all elements involved in NF measurement were determined and classified as a multi-stage network, and the well-known Friis law is applied to correct the noise contributions coming from other stages. With the two gain definitions, the effects of impedance match on NF are investigated. The result shows an NF of 3.80dB obtained with the de-embedding method and 6.06dB without the de-embedding method. This result is for vector measurement using the available gain, which gives 0.18dB improvement in NF as opposed to scalar measurement utilizing the insertion gain. Furthermore, NF was also measured at 5.63dB, 5.76dB, and 4.75dB under three different source impedances, namely, short, open, and load, respectively.