Why does increasing FFT length (narrowing bandwidth) not decrease Noise Power per bin?

Question

I was under the impression that zero-mean white guassian noise had a constant Power Spectral Density which means that smaller bandwidths should decrease the power in that band. I'm getting surprising results for average power in each FFT bin, it's staying the same irrespective of how long I make the FFT. Am I not measuring power correctly?

Here is a reduced for of my MATLAB script (should be Octave-compatible). Please excuse my odd habit of time vectors going up-down and frequency and misc ones going left-right.

%pkg load communications
% ^UNCOMMENT^ if using OCTAVE
close all

Q=8;        % Lowest power of 2 to iterate though in FFT loop
R=12;       % Highest power of 2 to iterate though in FFT loop
N=1;        % Linear PSD of noise
c=0;        % Loop counter

K=2.^(Q:R)  % Loop vector

for k=K
    n=sqrt(N).*(randn(k,1)+j.*randn(k,1))./sqrt(2); % complex noise
    P=(abs(fft(n)')).^2./k; % Noise Power in frequency domain
    c=c+1;    
    e(c)=mean(P); % Mean power measurement across bins
    %v(c)=var(P);
    figure
    plot(10.*log10(P))
end
e
%v

I feel like I'm missing something fundamental about this. Does FFT binning not truly band-limit signal power because of the way the algorithm works? I'm basically trying to get an output analogous to a spectrum analyzer in that is shows something proportional to true power.

Answer 1

A spectrum analyzer typically has more than one mode. One mode normalizes the frequency output such that a pure tone of amplitude “one” with frequency centered on a bin center has an amplitude of “one” in the spectrum display.

For a pure tone, nonrandom, centered on a bin frequency of amplitude “one” , the corresponding uniformly weighted N sample DFT bin magnitude will be “N”, so all the bins are reduced by dividing by N. (before magnitude squaring)

An actual Spectrum Analyzer will include a fudge factor correction for nonuniform windows, and average multiple DFTs .

I haven’t seen too many spectrum analyzers that don’t have a DFT under the hood. The last analog swept frequency analyzer I used was a very long time ago. I have seen a few modern versions at frequencies where converters aren’t good enough but DFT versions dominate most of the market.