Total Variation of a Signal - Is It Proportional to Signal Energy? In an audio application, I found it very useful to measure the total variation of a signal $y[n]$
$$\sum_{n=n_0}^{n_0+N} |y[n]-y[n-1]|$$
over a window of time length $N$ (discrete analogous to total variation of a function).
I've noticed that:

during &quot;background noise only&quot; parts of the signal, this total variation is low

during &quot;background noise + musical sound&quot; parts of the signal, the total variation is strictly higher.


Thus, it worked well in my application for envelope detection, etc. After doing my application, I heard about total variation denoising, and it seems to confirm why it works:

It is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the absolute gradient of the signal is high.
This noise removal technique has advantages [...] total variation denoising is remarkably effective at simultaneously preserving edges whilst smoothing away noise in flat regions, even at low signal-to-noise ratios

The total variation of the signal over a time-window is in fact the distance traveled on the y-axis by the 1D-curve $n \mapsto y[n]$, so we understand why it works:

when the signal is noise only, the waveform of the signal &quot;travels&quot; at a nearly-constant rate (see left of the following image)

when the signal is noise + musical sound, the waveform of the signal &quot;travels&quot; more! (see right of the image)



Now the question:
Question: It seems that the total variation of the signal over a time-window is more or less proportional to the energy present in the signal during this window. Is this true? At least for signals with a zero mean?
No it is not.
Total Variation is like the amount of changes in the signal.
Though changes require energy it doesn't mean they are proportional.
For instance, imagine that during a Window we see a constant signal of high value.
Clearly this high energy signal (Unless energy for you is the Variance, usually it is the 2nd moment) yet its Total Variation is zero.
For instance:

While the green signal has low TV but high energy the red signal has higher TV yet smaller energy.