The spectral density and autocorrelation function of white noise is, respectively;

The spectral density of white noise is Uniform and the autocorrelation function of White noise is the Delta function.

Explanation:

The power spectral density is basically the Fourier transform of the autocorrelation function of the power signal, i.e.

\({S_x}\left( f \right) = F.T.\left\{ {{R_x}\left( \tau \right)} \right\}\)

Also, the inverse Fourier transform of a constant function is a unit impulse.

The power spectral density of white noise is given by:

\(\rm{S_{X}\left(f\right)=\frac{\eta}{2}}\) for all frequency 'f', i.e.

Now auto-correlation is inverse Fourier transform (IFT) of power spectral density function.

\(\rm{R_x\left(\tau\right)\mathop \to \limits^{IFT}S_X\left(f\right)}\)

The inverse Fourier transform of the power spectrum of white noise will be an impulse as shown:

\(\rm{R_x\left(\tau\right)=\frac{\eta}{2}\delta\left(\tau\right)}\)