# What's the distribution of the DFT of a real-valued, zero-mean, normally distributed random vector?

Suppose $$X$$ is a real-valued N-dimensional Gaussian vector, $$X \sim \mathcal{N}(\mathbf{0}, C_X)$$. The discrete Fourier transform can be obtained by left-multiplying with the unitary DFT matrix, i.e. $$\hat{X} = W X$$ (where $$W$$ is defined as in this page). What is the distribution of $$\hat{X}$$?

My attempt to answer this is as follows: Since the DFT is linear, $$\hat{X}$$ will also be Gaussian, and thus $$\hat{X} \sim \mathcal{N}(\mathbf{0}, WC_XW^\top)$$... BUT that doesn't make sense to me; $$\hat{X}$$ is inevitably complex valued!

On the other hand, saying $$\hat{X} \sim \mathcal{CN}(\mathbf{0}, WC_XW^H)$$ doesn't make sense either; even if it is correct, what does it mean to have complex valued variances? Do we not have to account for the fact that $$X$$ is purely real?

• The closest I could find to an answer only mentions the element-wise variances, not the entire covariance matrix, and only addresses the specific case of zero-mean i.i.d noise. Nov 11, 2023 at 17:30

As the DFT of real $$X$$ is conjugate symmetric, $$\hat{X}$$ is not N-dimensional jointly Gaussian and neither your two distributions is correct.

Representing the N-dimensional DFT by a $$2N$$ dimensional linear transformation $$Y =\begin{bmatrix}\hat{X}_{re}\\\hat{X}_{im}\end{bmatrix}=\begin{bmatrix}W_{re}&-W_{im}\\W_{im}&W_{re}\end{bmatrix}\begin{bmatrix}X\\0\end{bmatrix}=\begin{bmatrix}W_{re}\\W_{im}\end{bmatrix}X=\tilde{W}X$$

The covariance matrix $$\tilde{W}C_X\tilde{W}^T$$ is singular and the distribution is degenerate. You can find the subspace to which $$\hat{X}$$ is mapped by exploiting the conjugate symmetric property as follows.

Exclude the last $$N/2$$ lines of both $$W_{re}$$ and $$W_{im}$$, and the line corresponding to the imaginary part of DC from the matrix $$\tilde{W}$$ to get $$\tilde{W}_{sub}$$ (by $$\tilde{W}_{sub}=S\tilde{W}$$ with $$S$$ is a diagonal matrix with corrresponding 1 and 0 in its diagonal). Hence the dimensions of $$\tilde{W}_{sub}$$ is $$(N-1)\times N$$.

$$\tilde{W}_{sub}C_X\tilde{W}_{sub}^T$$ is non-singular and the $$SY$$ is a $$(N-1)$$-dimensional zero mean multivariate Normal with covariance $$\tilde{W}_{sub}C_X\tilde{W}_{sub}^T$$.

We know that the distribution of $$SY$$ fully characterizes $$Y$$, and $$\hat{X}$$, because the conjugate symmetric property implies that the dimension of $$\hat{X}$$ is at most $$N-1$$.

The $$2N$$-dimensional real $$Y$$, and the $$N$$-dimensional complex $$\hat{X}$$, has zero probability outside of the space of $$SY$$ and is impulsive from the view of such higher-than-$$(N-1)$$-real-dimensional density.

• Thanks!! Could you please suggest a resource for understanding why $\hat{X}$ won't be Gaussian, and how to deal with degenerate Gaussian distributions? The wiki article goes into measure theory stuff, surely there must be a simpler way to understand it? Nov 15, 2023 at 6:34
• My understanding is still shaky, but what if we instead excluded the last N/2 lines from the original complex matrix $W$ and set the imaginary DC part to zero - is this equivalent to your method? Can we then say $\hat{X} \sim \mathcal{CN}(0, \bar{W}C_X\bar{W}^H$? Intuitively I'm trying to get at a DFT matrix that corresponds to something like what numpy.fft.rfft does. Nov 15, 2023 at 6:59
• @DangerousTim To understand why the elements of $\hat{X}$ are not jointly Gaussian, you can read Chapter 3 of "STOCHASTIC PROCESSES: THEORY FOR APPLICATIONS" of Robert Gallager. Specifically, 3.4.3 for real vectors and 3.7 for the complex model. Nov 15, 2023 at 10:36
• For the degenerate distributions, you might have a simpler way if you can exploit the structure of the non-singular covariance matrix induced by the DFT. I don't know how. Nov 15, 2023 at 10:39
• @DangerousTim For the question about manipulating directly the complex DFT matrix $W$, you can but this does not mean that the remaining part of $\hat{X}$ is circularly symmetric complex Gaussian like your notation because the resulting vector cannot be written as a linear transform of iid circularly symmetric gaussian random variables (note that the definition of complex gaussian vector is not standardized and I am using the definition of the aforementionned Gallager book and the Wikipedia article). Nov 15, 2023 at 10:51