Windows modify the amplitude of frequency domain functions

The discrete Fourier transform and power spectral density are often used in analyzing data from analog-to-digital converters. These analyses normally apply a window to the data to alleviate the effects of leakage. It is shown how windows modify the magnitude of a discrete Fourier transform and the level of a power spectral density computed by Welch´s method. For white noise, the magnitude of the discrete Fourier transform at a fixed frequency has a Rayleigh probability distribution. For sine waves with an integral number of cycles and quantization noise, the theoretical values of the amplitude of the discrete Fourier transform and power spectral density are calculated. It is demonstrated that the signal-to-noise ratio in a single discrete Fourier transform or power spectral density frequency bin is related to the normal time-domain definition of the signal-to-noise ratio. The answer depends on the discrete Fourier transform length, the window type and the function averaged