Why are there two sets of ENBW correction factors?

Question

For calculating the equivalent noise bandwidth of a non-brickwall filter, I can find two different sets of numbers, both of which claim they are similar things:

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Order   EqNBW
1   1.5708
2   1.1107
3   1.0472
4   1.0262
5   1.0166
6   1.0115
7   1.0084
8   1.0065
9   1.0051
10  1.0041

• RF Cafe Filter Equivalent Noise Bandwidth "Butterworth (fco = 3 dB)")
• ADC Noise Figure by Walt Kester
  • Figure 2: Relationship Between Noise Bandwidth and 3-dB Bandwidth for a Butterworth Filter
  • "will pass the same noise power as the non-ideal filter"
• Linear Circuit Design Handbook by Hank Zumbahlen
  • Figure 6-147 : "Relationship between noise bandwidth and 3 dB bandwidth for Butterworth filter"
• The Concepts of Noise Bandwidth and Cumulative Noise

 

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1 1.57
2 1.22
3 1.16
4 1.13
5 1.12

• Equivalent Noise Bandwidth by Radio Geek
  • "have same integrated noise power"
• Equivalent Noise Bandwidth by Tim J. Sobering
  • "a noise power equivalent to the original transfer function"
• PGA309 Noise Filtering
• Operational Amplifier Noise by Art Kay
  • "brick wall correction filter"

Which is correct?

Or are they both correct; just used in different calculations?

Update

After figuring this out, I made a chart of the different factors and the types of filters they work for: ENBW Filter correction factors vs order

Answer 1

The effective noise bandwidth depends on the shape of transfer function. It's easy to calculate it numerically.

See my Matlab script below that calculates the ENBW for a Butterworth lowpass filter. You can adapt it to your needs.

for N=1:10
  [b,a] = butter(N, 1, 's');
  f = @(x) (abs(freqs(b,a,x)).^2);
  bw = integral(f, 0, 1e6);
  fprintf('Order: %d, ENBW: %g\n',N, bw);
end 

In case you don't have Matlab, the output is given below

Order: 1, ENBW: 1.5708
Order: 2, ENBW: 1.11072
Order: 3, ENBW: 1.0472
Order: 4, ENBW: 1.02617
Order: 5, ENBW: 1.01664
Order: 6, ENBW: 1.01152
Order: 7, ENBW: 1.00844
Order: 8, ENBW: 1.00645
Order: 9, ENBW: 1.0051
Order: 10, ENBW: 1.00412