SUMMARY
The equivalent noise bandwidth (ENBW) for a window function is the bandwidth in bins of a brickwall filter that would result in the same noise power from a white noise source as the DFT "filter" (when viewing, appropriately, each bin of the DFT as a bandpass filter). The ENBW for the rectangular window (no further windowing) is 1 bin as demonstrated in the first plot below. The ENBW for any window can be determined from the following equation:
$$\text{ENBW} = N\frac{\sum (w[n])^2}{(\sum w[n])^2} \tag{1} \label{1}$$
Where ENBW is Equivalent Noise Bandwidth (in bins), and $w[n]$ is the window samples.
The ENBW is a useful metric for window comparison and an indication of the resolution bandwidth of the window.
FURTHER DETAILS FOR THE VERY INTERESTED
The ENBW is derived from the processing gain (also called windowing loss) which is the change in signal-to-noise ratio (SNR) due to the effects of windowing (always negative compared to the rectangular window which has no loss).
The processing gain of the window is related to the ENBW as follows:
$$PG = -10\log_{10}(\text{ENBW}) \tag{2} \label{2}$$
This makes perfect sense intuitively: if the ENBW was 2 bins, then we would overestimate the total noise power by a factor of 2 (+3 dB more noise) when we sum the noise power in each bin, while the “signal” power if it only occupied one bin would not be modified relative to the noise thus resulting in a 3 dB degradation in SNR. This is detailed further below.
The processing gain is specifically due to the difference between the coherent gain that applies to the signal (when it occupies only one bin) to the non-coherent gain for the noise, from which we can derive the formula for ENBW as follows:
The coherent gain of the window, meaning the gain that would occur if all the samples of the windowed signal were in phase, would simply be the direct summation of the window, normalized by the number of samples, as follows:
$$G_c = \frac{\sum w[n]}{N} \tag{3} \label{3}$$
For example, in fred harris' classic paper http://web.mit.edu/xiphmont/Public/windows.pdf the coherent gain for the Blackman window is listed as $0.42$ which is the gain we would get as $N$ approaches $\infty$, or $20\log_{10}(0.42)= -7.54 \text{ dB}$. If we use the formula above we predict the actual coherent gain for any size $N$, such as with $N=30$ the predicted gain is 0.406 (or -7.83 dB).
>> sum(blackman(30))/30
0.406
The result of this, which provides further intuition on "coherent gain" is given in the plot below.
Likewise, non-coherent gain refers to the change in DFT output level of non coherent samples (such as white noise) due to the window function. Non coherent samples sum in power, resulting in an rms result given as:
$$G_{nc} = \sqrt{\frac{\sum w[n]^2}{N}} \tag{4} \label{4}$$
So we see that noise and signal components will change differently due to windowing and the ratio of this difference is the Windowing Loss, also called the "Processing Gain" as defined earlier and here given as:
$$PG = 20\log_{10}\bigg(\frac{G_c}{G_{nc}}\bigg) \tag{5} \label{5}$$
By equating $\ref{2}$ with $\ref{5}$ we get:
$$PG = -10\log_{10}(\text{ENBW}) = 20\log_{10}\bigg(\frac{G_c}{G_{nc}}\bigg)$$
$$= -10\log_{10}(\text{ENBW}) = 10\log_{10}\bigg(\frac{G_c}{G_{nc}}\bigg)^2$$
$$= -10\log_{10}(\text{ENBW}) = -10\log_{10}\bigg(\frac{G_{nc}}{G_c}\bigg)^2 \tag{6}$$
So
$$\text{ENBW} = \bigg(\frac{G_{nc}}{G_c}\bigg)^2 \tag{7} \label{7}$$
which by substituting $\ref{3}$ and $\ref{4}$ into $\ref{7}$ results in $\ref{1}$.
What may be confusing at first is how bins related to correlated signals can have a different power result than bins related to noise components after windowing, resulting in the change in SNR. This is explained intuitively by the ENBW: Each bin is reporting the power in its own bin plus some or all of the adjacent bins due to the spectral widening from the window. Thus in the case of white noise where all the bins are at or close to the same power level, when you sum all the bins you will over-report that actual power since power in adjacent bins is getting double-counted. In the case of a single tone (with no other tones present), it's power occupies one bin so can't get double counted (but would of course effect multiple tones due to the spectral leakage). Without windowing (meaning using a rectangular window), the ENBW is 1 bin, so for noise the power sum of all the bins would be equal to the total power which is consistent with Parseval's theorem. This is NOT the case after windowing as explained above. We also see from this that the SNR for a waveform that itself occupies several DFT bins would also be affected differently as the waveform would thereby approach the result we get with noise, meaning reduced windowing loss since the signal and the noise would approach being affected equally.
ENBW and PG are useful metrics when comparing window functions.
Update: I just saw this related article posted on Linked-In, I only read it over quickly but appears to be much more detailed and relevant to this post so will link here:
https://www.gaussianwaves.com/2020/09/equivalent-noise-bandwidth-enbw-of-window-functions/
At the risk of confusing the original question, I have added a couple of graphs showing my window (black trace) vs Blackman Harris (green trace). My experiments are audio-based so a 48kHz sample rate, 1Hz bin width, 32-bit integer precision for the data.
The first plot is a synthesised 1kHz sine and totally shows the precision limitation you mentioned. The second picture is trying to emulate something more real, 1000.5Hz sine wave with added noise. As you can see, the BH window doesn't manage to show the noise floor.
– Richard Sep 13 '20 at 03:19When the sine wave is synchronised to the sampling block and an integer number of waves fit the block exactly (like the 1000Hz example), then at only -180dB the peak is one bin wide. With this asynchronous 1000.5Hz the peak is a fat 15 bins wide down at -180dB. I have ambitions though to tune some of that out and make the window more useful.
I'll have a go at the zero padding thing. It may take me some while to code that but sounds interesting. Thanks.
– Richard Sep 13 '20 at 04:12If it is relevant Dan; I can use any value - I just tried a few random ones including 1,013.666Hz for example and 1,328.776486712549098Hz. The width, using my window at -180dB, is never more than 15 bins - though as I said I would rather it were thinner. By contrast, the Blackman Harris window only seems to be good to about -98dB after which spectral leakage blocks out all detail.
– Richard Sep 13 '20 at 04:53