noise - Why are there two sets of ENBW correction factors?

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:

---

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

---

1 1.57
2 1.22
3 1.16
4 1.13
5 1.12

Which is correct?

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

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

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