0
$\begingroup$

The link of the statement is evidenced in this book:

enter image description here enter image description here

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.