Windows of Opportunity – ENBW

When you search online for ENBW you’ll like­ly find a list of sites relat­ed to EnBW AG, a Ger­man ener­gy giant (which, on a total­ly unre­lat­ed note, we were a cus­tomer of until recent­ly). While ener­gy is quite rel­e­vant in the con­text of this post, EnBW has noth­ing to do with it. This post is about ENBW as in equiv­a­lent noise band­width and was sup­posed to be an update for the Win­dows of Oppor­tu­ni­ty post a while ago. It turned out a bit too exten­sive and so I made an extra post.

The equiv­a­lent noise band­width (in the con­text of win­dow­ing) is defined as the band­width of an ide­al rec­tan­gu­lar instru­ment func­tion which has the same ampli­tude as the max­i­mum of the instru­ment func­tion of the win­dow of inter­est (which usu­al­ly occurs at zero fre­quen­cy) and which pass­es the same total ener­gy. This def­i­n­i­tion is hard­ly intel­li­gi­ble, but I find those of oth­ers to be not much bet­ter (e.g. [1]). Fur­ther­more, such a def­i­n­i­tion does not at all make clear what the ENBW is sup­posed to quan­ti­fy. After all, there are no win­dow func­tions with rec­tan­gu­lar instru­ment func­tions.

Fig­ure 1: Some prop­er­ties of the Hann win­dow and its instru­ment func­tion. The win­dow func­tion (top left) is nor­mal­ized to T = 1; graph also shows the win­dow ener­gy rel­a­tive to the box win­dow. Instru­ment func­tion (top right) is shown for the spec­tral region cov­er­ing the first 10 DFT bins. Close-up (bot­tom) shows main lobe height, max­i­mum scal­lop­ing loss (see below), rel­a­tive height of first side­lobe, and frac­tion of ener­gy con­tained in main lobe, E0, to total ener­gy E.

The real prob­lem that the ENBW can quan­ti­fy is that of a win­dow-depen­dent noise lev­el and hence a win­dow-depen­dent sig­nal-to-noise ratio (SNR) obtained in spec­tral mea­sure­ments. To illus­trate how this comes to be we’ll start with a white noise sig­nal, i.e. one that has a con­stant pow­er spec­tral den­si­ty (PSD). The PSD of the win­dowed noise sig­nal is also con­stant, but its val­ue is low­er since the ener­gy of the win­dow with which the noise sig­nal is mul­ti­plied is less than uni­ty (except for the box win­dow)1 – this is some­times called the inco­her­ent pow­er gain. The win­dow ener­gy is shown in the top left graph of the win­dow info­graph­ics that were also used in the orig­i­nal post – it is -4.26dB for the Hann win­dow.

If there is also some use­able sig­nal spec­tral com­po­nent apart from back­ground noise, its ener­gy is dis­trib­uted through­out the spec­trum by con­vo­lu­tion with the par­tic­u­lar instru­ment func­tion (result­ing in leak­age). Gen­er­al­ly, the greater the leak­age, the less scal­lop­ing occurs, because the main lobe of the instru­ment func­tion is wider and hence flat­ter near zero fre­quen­cy. But at the same time the main lobe and thus the result­ing height of the cor­re­spond­ing spec­tral peaks must be low­er, since the total ener­gy in the instru­ment func­tion is fixed (and equal to total win­dow ener­gy) – this is some­times called the coher­ent pow­er gain since it applies to dis­crete spec­tral lines only. For the Hann win­dow the main lobe height is -6.02dB and there­fore the height of any sin­gle spec­tral line will be 6.02dB below its real val­ue. To obtain cor­rect ener­gy lev­els of the spec­tral peaks (in the absence of scal­lop­ing), the main lobe height (in dB units) is usu­al­ly sub­tract­ed from the PSD. How­ev­er, this over­com­pen­sates the win­dow-spe­cif­ic reduc­tion of the noise floor – for the Hann win­dow, peak com­pen­sa­tion is 6.02dB while the noise floor is only 4.26dB below its true val­ue. This decreas­es the observed SNR by 1.76dB or a fac­tor of 1.5.

If the total ener­gy of the win­dow func­tion is put into a rec­tan­gu­lar spec­tral slice with the same height as the main lobe, the width of that rec­tan­gle (mea­sured in units of DFT bins) is the same as the ratio of total win­dow ener­gy to main lobe height (or the ratio of inco­her­ent pow­er gain to coher­ent pow­er gain) and thus the same as the SNR decrease. Hence, the def­i­n­i­tion of ENBW is cor­rect, but quite abstract.

[1] Noise Band­width [Wikipedia]

1 We’ll try to show that the PSD of a white noise sig­nal is unaf­fect­ed by win­dow­ing, except for a mul­ti­plica­tive fac­tor – the inco­her­ent pow­er gain – spe­cif­ic to the win­dow func­tion $w(t)$.
The PSD is the Fouri­er trans­form of the auto­cor­re­la­tion func­tion which for a white noise sig­nal $n(t)$ is

$$\mathcal{R}(\tau) = \mathcal{E}\Bigl[n(t) n^*(t-\tau) \Bigr] = \frac{N_0}{2} \delta\left(\tau\right)$$

where $\mathcal{E} \bigl[\cdot\bigr]$ is the expec­ta­tion val­ue, $N_0/2$ is the PSD and $\delta(\cdot)$ is the Dirac Delta func­tion. For the win­dowed noise sig­nal $n’(t) = n(t) w(t)$ we have

$$\mathcal{R}’(\tau) = \mathcal{E}\Bigl[n(t) w(t) \cdot n^*(t - \tau) w^*(t - \tau) \Bigr]$$

Since the white noise sig­nal and the win­dow func­tion are sta­tis­ti­cal­ly inde­pen­dent, we can write

\begin{align} \mathcal{R}’(\tau) &= \mathcal{E}\Bigl[n(t) n^*(t - \tau)\Bigr] \cdot \mathcal{E}\Bigl[w(t) w^*(t - \tau) \Bigr]\vphantom{\frac{1}{1}}\\ &= \mathcal{R}(\tau) \cdot \mathcal{E}\Bigl[w(t) w^*(t - \tau) \Bigr]\\ &= \frac{N_0}{2} \delta\left(\tau\right) \cdot \mathcal{E}\Bigl[w(t) w^*(t - \tau) \Bigr] \end{align}

The auto­cor­re­la­tion $\mathcal{R}’(\tau)$ is unequal to zero only for $\tau = 0$ for which the expec­ta­tion in the last line sim­ply describes the win­dow ener­gy. Hence, the PSD of the win­dowed process is equal to that of the unwin­dowed process mul­ti­plied by the win­dow ener­gy. The box win­dow, which has unit ener­gy, does not affect the noise PSD.

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