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.

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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