# Variance of spectral density is related to the gradient of signal?

Define the frequency variance as: $$\sigma^2 = \int^\infty_{-\infty}\omega^2 P(\omega) d\omega$$ Where $$P(\omega)$$ is the spectral density function, which is the same as normalized power. Therefore, $$\sigma^2 = \frac{\int^\infty_{-\infty}\omega^2 X(\omega)\bar{X}(\omega) d\omega}{\int^\infty_{-\infty} X(\omega)\bar{X}(\omega) d\omega}$$

$$X(\omega)$$ is the Fourier transform of the signal $$x(t)$$. We can rewrite the numerator ($$v$$) as:

$$v = \int^{\infty}_{-\infty}(i \omega X(\omega)e^{i\omega t})(-i \omega \bar{X}(\omega)e^{-i\omega t}) d\omega$$ $$= \int^{\infty}_{-\infty}|i \omega X(\omega)e^{i\omega t}|^2 d\omega$$

I am trying to relate $$v$$ to the following expression of the gradient of the signal $$x(t)$$: $$\frac{dx(t)}{dt}=\int^{\infty}_{-\infty} i \omega X(\omega)e^{i\omega t} d\omega$$

However, all I can come up with is the following inequality:

$$v \geq \left(\frac{dx(t)}{dt} \right)^2$$

which doesn't make sense since $$v$$ is independent of time (and frequency) but $$\left(\frac{dx(t)}{dt} \right)^2$$ is dependent on time.

What is the best way to express $$v$$ in terms of $$\frac{dx(t)}{dt}$$?

• Variance of Fourier transform is related to the second derivative of the signal, not to the first derivative. – Alexandre Eremenko Mar 2 at 0:27
• @AlexandreEremenko I see. However, in that case, the "probability" value would be the Fourier coefficients themselves, not their magnitude, so I am not sure how the expression of the second derivative can be used for further steps. – Chanwoo Chun Mar 2 at 3:16
• @AlexandreEremenko The relationship to the first derivative is posted as an answer to this post if you want to check it out..! – Chanwoo Chun Mar 2 at 16:26

It turns out that the relationship is obvious when I use Parseval's Theorem. First I am rewriting my $$v$$ (variance term) here:

$$v = \int^{\infty}_{-\infty}|i \omega X(\omega))|^2 d\omega$$

$$i \omega X(\omega)$$ is Fourier transform of $$\frac{dx(t)}{dt}$$. Using Perseval's Theorem,

$$\int^{\infty}_{-\infty}|i \omega X(\omega))|^2 d\omega = \int^{\infty}_{-\infty} | \frac{dx(t)}{dt} |^2 dt$$

Therefore,

$$\sigma^2 = \frac{\int^{\infty}_{-\infty} | \frac{dx(t)}{dt} |^2 dt}{\int^{\infty}_{-\infty} | x(t) |^2 dt}$$

This makes sense, because the variance of the frequency, $$\sigma^2$$, is basically a metric that tells us how smooth a function is, which can be summarized with L2 of the derivative of the signal. I am confident with this, but please leave a comment if this can be improved or needs correction.