Non-Russian scientists are ahead of Russian in many branches of science, because since 1990 year Russian science slowed down. But hurt in some areas remained. In particular, in signal processing. That is why I managed to get a general formula for the signal envelope. I looked at the research on this site concerning the signal envelope and did not find a general analytical formula for the signal envelope. Calculated using the mathlab envelope of the high-frequency signal with sinusoidal modulation, but there is no general formula.
It must be said that in the theory of signals, the envelope is searched for using a conjugate signal $\sigma(t)$ if there is a high-frequency carrier. This conjugate signal is practically defined as a magnitude $\sigma(t)=\frac{ds(t)}{d\Omega t};\sigma(t)=s(\Omega t-\pi/2)$. It turns out the approximate formula for the envelope. 768
With the exact hard to count formula of <PERSON> transform $\sigma(t)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{s(\tau)}{t-\tau}d\tau$. The result is an envelope $s^2(t)+\sigma^2(t)$.It turns out that the formula is valid for a high-frequency carrier and a smooth modulation function. The envelope of the low frequency signal has no physical meaning. I use a complex conjugate signal to find the envelope. $$\sigma(t)=s^(t);|s(t)|^2=a(t)exp(i\Omega t)a^(t)exp(-i\Omega t)=|a(t)|^2$$. The envelope is a smooth positive function, but should smoothly touch zero. But in the case of the envelope for solving the wave equation has its own characteristics. The line of the stationary phase of the signal is determined by the formula $$\frac{\partial z}{\partial t}=c(\omega)=(\Omega_0+\omega)/k(\omega)$$ Then the received signal is determined by the formula, where the high frequency of the signal for which we are looking for the envelope $$s(t,z)=[\int_{0}^{\infty}a[t-z/c(\omega)]exp[i(\Omega_0+\omega)(t-z/c(\omega))]d\omega+\int_{-\infty}^{0}a[t-z/c(\omega)]exp[i(-\Omega_0+\omega)(t-z/c(\omega))]d\omega]$$ Multiply by the complex conjugate signal and integrate over time, we get $$|s(t,z)|^2=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} a^[t-z/c(\omega_1)] a[t-z/c(\omega)]exp[i(\pm\Omega_0+\omega_1)z/c(\omega_1)-i(\pm \Omega_0+\omega)z/c(\omega))]\delta(\pm\Omega_0+\omega_1-\omega\mp\Omega_0) d\omega d\omega_1/2\pi; (1)$$ Where we choose the sign of the frequency $\Omega_0$ equal to the sign of the frequency $\omega$. Where used the formula $$\delta(\omega_1-\omega)=2\pi\int_{-\infty}^{\infty} exp[i(\omega_1-\omega)t]dt$$ Formula (1) is converted to $$|s(t,z)|^2=\int_{-\infty}^{\infty}a^[t-z/c(\omega)] a[t-z/c(\omega)]d\omega /2\pi$$ We take into account the smooth dependence of time $a(t)$, we get an envelope that depends on time, but the formula becomes approximate, since the delta function is determined approximately. 281