The link of the statement is evidenced in this book:

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I don't understand how equation 85 come? I try to use Carson rule to see this inequality but turns out I have $2 \Delta f$ instead of $3 \Delta f$.

So someone could help me to understand this?


Checking the book you linked to, this equation refers to FM modulation over a (weakly) non-linear channel, modeled as a memoryless third-order non-linearity (Eq. 82). The third-order non-linearity produces signals at twice and at three times the carrier frequency $f_c$ (Eq. 84). Note that also the frequency deviations of these additional signal components at $2f_c$ and at $3f_c$ are doubled and tripled, respectively.

If $\Delta f$ is the frequency deviation of the original FM signal, the signal component at $2f_c$ has a frequency deviation of $2\Delta f$, and according to Carson's bandwidth rule, that signal component extends down to approximately

$$f_1=2f_c-2\Delta f-W\tag{1}$$

where $W$ is the bandwidth (highest frequency) of the message signal. The original FM signal centered at $f_c$ extends up to approximately

$$f_2=f_c+\Delta f+W\tag{2}$$

In order to be able to separate the original FM signal from the signal component at $2f_c$ created by the non-linearity, we require $f_2<f_1$, i.e.,

$$f_c+\Delta f+W<2f_c-2\Delta f-W\tag{3}$$

or, equivalently,

$$f_c>3\Delta f+2W\tag{4}$$

which is the same as Eq. $(85)$ in the book.


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