n(t) is white Gaussian process.
T1 and T2 are the periods at which n(t) is sampled,
N1=n(m \cdot T1), N2=n(m \cdot T2).
S1(f) and S2(f) are the spectrum of N1 and N2 respectively.
What is the relation or difference between S1(f) and S2(f)? They are identical?
A continuous time white noise is a stationary process which autocovariance function is the Dirac delta multiplied by the variance of the process. Its spectral density is a uniform density over the whole real line of frequencies. Both autocovariance and spectral density do not exist in the ordinary sense but in a distributional sense. When you sample a continuous time process at times t_i = i T (i integer), you get a discrete time process which autocovariance is also defined at discrete time lags. The resulting autocovariance is a Kroneker delta: sigma^2 at lag 0 and 0 for any non-zero lag. Now the question is what you mean by "spectrum". It can be the spectral density of the discrete time process computed as the discrete Fourier transform of the auto-covariance (the Kroneker delta sequence). In this case, the discrete Fourier transform is a continuous function defined in the interval -1/2T, 1/2T (Nyquist frequencies, frequencies as inverse of period, units Hz), and it is a constant function in this interval. Any spectral density of a second order stationary process can be extrapolated periodically but all information is in the Nyquist interval -1/2T, 1/2T.
Now if you sample at two different periods T_1, T_2, you obtain uniform spectral densities in two different intervals (-1/2T_1, 1/2T_1) and (-1/2T_2, 1/2T_2) although the constant value of the densities is exactly the same (\sigma^2). They are not equal because they are defined in different intervals. However, if you perform the periodic extrapolations of the two spectral densities they are equal.
There are frequent misinterpretations of the above described spectral densities. Apart of changing some constants in the definitions of the Fourier transforms (continuous and discrete), the most common is to conceive the spectral density only defined at discrete frequencies. This misleading idea comes from the systematic application of the FFT (Fast Fourier Transform) to limited time series. In this way, when changing from T_1 to T_2, the frequency points where the discrete Fourier transform are computed also change, and it seems that the two spectral densities cannot be compared.
Hope this is useful.
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