NEB; mean and AC of WSS signal through LTI

Noise equivalent bandwidth

The noise equivalent bandwidth (NEB) $B_N$ of a filter is the bandwidth of an ideal rectangular filter, with the same peak (or zero-frequency) gain, that passes the same noise power as the actual filter when both are driven by the same white noise.

If white noise of PSD $N_0/2$ is applied, actual output power $= \frac{N_0}{2}\int_{-\infty}^{\infty}|H(f)|^2df$ and ideal output power $= \frac{N_0}{2}|H(0)|^2 \cdot 2B_N$. Equating:

$$B_N = \frac{\int_0^{\infty}|H(f)|^2\,df}{|H(0)|^2}$$

 |H(f)|^2
  |H(0)|^2 +-------+         area of rectangle
           |\      |         = area under |H(f)|^2
           | \     |
           |  \__  |
           |     ~~|~~~___
           +-------+------------> f
           0       BN

Example: RC-LPF has $B_N = 1/(4RC) = 1.57 f_{3dB}$. Then output noise power is simply $N_0 B_N |H(0)|^2$.

Mean and autocorrelation at the output

Let a WSS process $X(t)$ (mean $\mu_X$, autocorrelation $R_X(\tau)$) drive a stable LTI system with impulse response $h(t)$:

$$Y(t) = \int_{-\infty}^{\infty} h(\alpha)X(t-\alpha)\,d\alpha$$

Mean

$$\mu_Y = E[Y(t)] = \int_{-\infty}^{\infty} h(\alpha)E[X(t-\alpha)]\,d\alpha = \mu_X\int_{-\infty}^{\infty}h(\alpha)\,d\alpha = \mu_X H(0)$$

The output mean is the input mean times the DC gain, a constant.

Autocorrelation

$$\begin{aligned} R_Y(t, t+\tau) &= E[Y(t)Y(t+\tau)] \\ &= \int\!\!\int h(\alpha)h(\beta)\,E[X(t-\alpha)X(t+\tau-\beta)]\,d\alpha\,d\beta \\ &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} h(\alpha)h(\beta)\,R_X(\tau+\alpha-\beta)\,d\alpha\,d\beta \end{aligned}$$

This depends only on $\tau$, so $Y(t)$ is also WSS. In compact form:

$$R_Y(\tau) = h(\tau) * h(-\tau) * R_X(\tau)$$

Power spectral density (Fourier transform, using $h(-\tau) \leftrightarrow H^*(f)$):

$$S_Y(f) = H(f)H^*(f)S_X(f) = |H(f)|^2 S_X(f)$$

Output power

$$E[Y^2(t)] = R_Y(0) = \int_{-\infty}^{\infty}|H(f)|^2S_X(f)\,df$$

For white input, $S_X = N_0/2$, this gives $N_0|H(0)|^2B_N$, which links back to the NEB.