What is the Maximum Rate of Change of a BW Limited Signal

Question

If I have a signal that has a max amplitude of 1 and is BW limited by filters to 10Hz-1kHz spectrum. I do not know what the fft/spectrum shape of the frequency looks like and it can change. Is it a guarantee that the fastest possible rate of change of that signal will be equal to the highest rate of change of the sinusoid corresponding to it's highest frequency component?

So for the example above in a randomly chosen slice of 100us am I guaranteed that max change possible will be 0.62 of full scale of the signal?

Thought Process:

1kHz = 1ms period

100us = 10% of period. We can take the fastest rate of change to be around pi since slope is largest.

Sin(0.9pi)-sin(1.1pi) = 0.62

Does this answer your question? Bounds of the derivative of a bounded band-limited function. I vote to close this question. — Dilip Sarwate, Aug 21 '21 at 01:34

score 2 · Answer 1 · answered Aug 21 '21 at 01:32

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Suppose that $x(t)$ is a signal such that $|x(t)| \leq 1$ and $X(f)$ is such that $X(f) = 0$ for $|f| > f_0$. Then, a result due to Bernstein says that $$\max \left| \frac{\mathrm dx}{\mathrm dt}\right| \leq 2\pi f_0.$$ More generally, we have for bounded $x(t)$, $$\max \left| \frac{\mathrm dx}{\mathrm dt}\right| \leq 2\pi f_0 \max |x(t)|.$$

answered Aug 21 '21 at 01:32

Dilip Sarwate

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What kind of bandlimited signals approach that limit? Intuitively, one might expect that a bandlimited square wave is a good candidate for maximum steepness within the constraints? – Knut Inge Aug 21 '21 at 07:15
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@KnutInge A sinusoidal waveform at $f_o$ would be the waveform with the highest slope. A bandlimited square wave with a frequency of $f_o$ would be a sinusoidal waveform, but if the square wave had a max voltage of 1, then it would have less slope than the sinusoidal waveform with voltage of 1. – Dan Boschen Aug 21 '21 at 13:41

score 1 · Accepted Answer · answered Aug 21 '21 at 00:30

This would be a reasonable assumption given we are monitoring the output of a fixed amplitude bandwidth limited system and assuming the dominant signal energy is within the bandwidth of the filter. In general, in order to achieve what the OP has suggested, the input amplitude and system gain must be limited. As a counter example, consider the slope (rate of change) for $5000 * \sin(1.1\pi)$. Further to guarantee no higher signal components, we would need to have an unrealistic brick-wall filter as the typical bandwidth limited filter still has a finite response at higher frequencies.

What is the Maximum Rate of Change of a BW Limited Signal

2 Answers2