In $d$ dimensions, with conventions \begin{align}\tilde\phi(k) &= \int d^dx\:e^{ik\cdot x}\phi(x),\\\phi(x) &= \frac{1}{(2\pi)^d}\int d^dk\:e^{-ik\cdot x}\tilde\phi(k),\\\left\langle\tilde\phi(k)\tilde\phi^*(k')\right\rangle&=(2\pi)^dP(k)\delta(k-k').\end{align} For white noise, $\langle\phi(x)\phi(x')\rangle = A\delta(x-x'),$ $$\left\langle\tilde\phi(k)\tilde\phi^*(k')\right\rangle=\int d^dx\: d^dx' e^{ik\cdot x -ik'\cdot x'}\langle\phi(x)\phi(x')\rangle=A\int d^dx\: e^{i(k-k')\cdot x}=A(2\pi)^d\delta(k'-k).$$ Thus $P(k)=A$, i.e. __is flat__. Now, the contribution to the pdf is $$\left\langle \phi(x)^2\right\rangle = \frac{1}{(2\pi)^{2d}}\int d^dk\:d^dk'\:e^{i(k+k')\cdot x}\left\langle \tilde\phi(k)\tilde\phi^*(-k')\right\rangle = \frac{1}{(2\pi)^d}\int d^dk\:P(k) = \frac{S_{d-1}}{(2\pi)^d} \int d|k|\: |k|^{d-1}P(|k|)=\frac{S_{d-1}}{(2\pi)^d}\int d\log|k|\: k^dP(|k|),$$ where $S_n$ is the surface area of the unit $n$-sphere. However this $k^d$ term is not part of the power spectrum. Maybe you're thinking of the scale-invariant power spectrum $P(k)=k^{-d}$? 

If we used the convention $$\left\langle\tilde\phi(k)\tilde\phi^*(k')\right\rangle=\frac{(2\pi)^dP(k)\delta(k-k')}{k^d},$$ then white noise is $P(k)\propto k^d$ (__increasing__ with $k$) and the scale-invariant power spectrum is flat. Whichever convention we use, for white noise, the total power (summed over different $k$ with the same $|k|$) increases like $k^{d-1}$, and the average power (averaged over different $k$ with the same $|k|$) is flat.


```python
from numpy import *
from matplotlib import pyplot as plt
%matplotlib inline
```


```python
x = random.randn(256,256)
```


```python
plt.imshow(x)
plt.colorbar()
```


```python
f = fft.fft2(x)
```


```python
plt.imshow(fft.fftshift(abs(f)))
plt.colorbar()
```


```python

```

ec543e79371a422fadc8830d914a2881a437ffc8
ipynb
Jupyter Notebook
notebooks/white noise has a flat power spectrum.ipynb
avirukt/lensing_lfi
1f3b351b690f39488066a06d94335b265fae818a
MIT
2019-05-30T21:27:30.000Z
2019-05-30T21:27:30.000Z
notebooks/white noise has a flat power spectrum.ipynb
avirukt/lensing_lfi
1f3b351b690f39488066a06d94335b265fae818a
MIT
notebooks/white noise has a flat power spectrum.ipynb
avirukt/lensing_lfi
1f3b351b690f39488066a06d94335b265fae818a
MIT
Qwen/Qwen-72B
1. YES
2. YES