# Distribution of Gaussian noise in the frequency domain

I have $$N$$ samples of Gaussian noise independently drawn from a normal distribution $$n(t) \sim \mathcal{N}(0, \sigma^2)$$. How would the noise be distributed in the frequency domain? I.e., what is $$n(f)$$? I understand it should also be Gaussian, and centered at $$0$$. Using the unitary DFT matrix,

$$E(n(f)) = \frac{1}{\sqrt{N}}\sum_{t=0}^{N-1}E(n(t))e^{-\frac{2\pi ift}{N}} = 0$$

However, what would the variance of the distribution be?

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. Jul 12, 2023 at 14:01

Suppose we have $$n$$ IID Gaussian random variables $$w_1, w_2, \dots, w_n \sim \mathcal{N} \left( 0, \sigma^2 \right)$$. From independence,

$${\Bbb E} \left( w_i w_j \right) = \sigma^2 \delta_{ij}$$

where $$\delta_{ij}$$ is the Kronecker delta. Let $${\bf w} := \begin{bmatrix} w_1 & w_2 & \dots & w_n\end{bmatrix}^\top$$. Note that $${\bf w} \sim \mathcal{N} \left( {\bf 0}_n, \sigma^2 {\bf I}_n \right)$$. Hence, the noise signal $$w$$ is white (which is why I called it $$w$$).

Let $${\bf F} {\bf w}$$ be the discrete Fourier transform (DFT) of $$\bf w$$, where $$\bf F$$ is the $$n \times n$$ (unitary) DFT matrix. Since the transformation $${\bf w} \mapsto {\bf F} {\bf w}$$ is linear, fortunately, Gaussian-ness is preserved and, quite amusingly, we have $${\bf F} {\bf w} \sim \mathcal{N} \left( {\bf 0}_n, \sigma^2 {\bf I}_n \right)$$ as well because $${\bf F} {\bf F}^* = {\bf I}_n$$.

Take the Fourier transform of the PDF to get

$$\mathcal{F} \left( \frac {1} {\sigma \sqrt{2\pi}} e^{ - \frac {x^2} {2\sigma^2} } \right) = \frac {1} {\sigma \sqrt{2\pi}} \cdot \mathcal{F} \left( e^{ - \frac {x^2} {2\sigma^2} } \right) = \frac {1} {\sigma \sqrt{2\pi}} \cdot \left( \sqrt{ 2\pi \sigma^2} \cdot e^{ -2\pi^2 \sigma^2 f^2 } \right) = e^{ - \frac {f^2} {2 / (2\pi\sigma)^2}}.$$

Thus, the variance of the noise in the frequency domain is $$(2\pi\sigma)^{-2}$$.
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