The paper reports the result of our research on graded-gap diodes with a cathode static domain as possible noise sources for different modern applications based on the use of THz wave frequencies. Such diodes represent a two-terminal n + -n  -nn + structure containing a cathode graded-gap layer with a band gap that increases from the cathode contact towards the n-region. We study GaN and AlN based diodes with InzGa1-zN and InzAl1-zN based graded-gap layers, respectively. It is shown that the localization of a strong electric field and low values of impact ionization threshold can be achieved by using a graded-gap layer with a narrow-gap material on the cathode. The diode simulation is performed using ensemble Monte Carlo technique. Noise generation is investigated numerically with time sampling of electric current over the time domain. The influence of scattering mechanisms acting on charge carriers on the noise properties of the diode is explored. Analyzing the noise power spectral density (NPSD), it is found that maximum NPSD can be observed in GaN diodes with InzGa1-zN layer, and the NPSD magnitude depends both on the size of the n --region (at a fixed diode length) and on the position of the graded-gap layer with respect to the end of the high resistance region (n -). It is established that polar phonon scattering and alloy-disorder scattering are the main mechanisms affecting the noise properties of the diodes. It is shown that the diodes demonstrate bias voltage regions where the dependence of NPSD on bias voltage is linear, and the value of NPSD increases by an order of magnitude.