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I have a question on frequency modulation. For example, the following signal is frequency modulated:

y(n)=sin(2*pi*(400+cos(2*pi*4*t))*t).

Since the maximum of cos function is from -1 to 1, so I should expect to see that, y(n) has frequency components between 399 to 401. However, if I take the fft, the result is not so. Can some one please explain why?

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You may think that the bandwidth of an FM signal is \$2\Delta f\$, where \$\Delta f\$ is the frequency deviation (the maximum difference between the instantaneonus frequency \$f(t)\$ and the carrier frequency \$f_c\$).

However, the frequency of a signal cannot be changed in an instant, therefore when frequency modulating a carrier, you will introduce additional frequencies below \$f_c-\Delta f\$ and above \$f_c+\Delta f\$.

The bandwidth of a frequency modulated signal is theoretically infinite, but it can be approximated with the help of Carson's bandwidth rule (http://en.wikipedia.org/wiki/Carson_bandwidth_rule). It's an approximation based on the highest frequency in the modulating signal (\$f_m\$) and the frequency deviation. In your case, the deviation is 1 Hz and \$f_m\$ is 4 Hz, so the approximated bandwidth around the carrier is 10 Hz.

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