# White noise gain of uniform spline interpolation

Uniform spline interpolation is fully defined by its continuous-domain impulse response. I noticed that the integral of the square of the impulse response over its support gives the average amplification of the power of white noise error in the samples.

What is this figure or its square root called? Is "white noise gain" $(\operatorname{WNG})$ appropriate for the square root?

For example for linear interpolation: Figure 1. Linear interpolation impulse response

$$\operatorname{WNG} = \sqrt{\int_{-1}^1\left(\left\{\begin{array}x+1&\text{if }x<0\\1-x&\text{otherwise}\end{array}\right.\right)^2dx} = \sqrt{\frac{2}{3}}\approx0.8164965809$$

This figure is the squared norm or energy of the impulse response:

$$\|h(t)\|^2=\int_{\mathrm{R}} |h(t)|^2 dt$$

As you point out correctly, this is the noise amplification. Considering a signal

$$r(t) = x(t) + n(t)$$ with $x(t)$ is the signal of interest and $n(t)$ is AWGN with variance $\sigma^2$. Then, after filtering

\begin{align} y(t) &= h(t)*r(t)\\ &=h(t)*x(t) + \int_{\mathrm{R}} h(t-\tau)n(\tau)d\tau\\ &=h(t)*x(t) + \tilde{n}(t) \end{align}

with $\tilde{n}(t)$ being the filtered noise. Now, considering

\begin{align} E[\tilde{n}(t)\tilde{n}^*(t)] &= \int_\mathrm{R}\int_\mathrm{R} h(t-\tau_1)h^*(t-\tau_2)E[n(\tau_1)n(\tau_2)]d\tau_1 d\tau_2\\ &=\sigma^2\int_{\mathrm{R}^2}h(t-\tau_1)h^*(t-\tau_2)\delta(\tau_1-\tau_2)d\tau_1d\tau_2\\ &=\sigma^2\int_\mathrm{R}h(t-\tau_1)h^*(t-\tau_1)d\tau_1\\ &=\sigma^2\|h(t)\|^2 \end{align}

where we have used that the noise is white, i.e. its autocorrelation is a delta-function. Note that after filtering generally the noise is not white anymore, but colored.