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Like AM that can be represented as:

$$\bigl(A_c+Km(t)\bigr)cos\bigl(2\pi f_ct\bigr)$$

Here, simply the amplitude is being varied by a sinusoid.

Therefore similarly, why can't FM be written as:

$$A_ccos\Bigl(2\pi\bigl(f_c+Km(t)\bigr)t\Bigr)$$

Obviously the constants are such that the frequency doesn't go negative.

Isn't the above equation intuitively correct? The frequency is being varied by a sinusoid again.

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  • \$\begingroup\$ That looks like phase modulation. \$\endgroup\$ – Chu May 6 '17 at 19:55
  • \$\begingroup\$ are you kidding me? This doesn't looks like anything \$\endgroup\$ – Chirag Arora May 6 '17 at 19:57
  • \$\begingroup\$ Why do you think that this is not a correct expression for FM? \$\endgroup\$ – Dave Tweed May 6 '17 at 21:06
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    \$\begingroup\$ @Chu: Phase modulation would be \$A_c \cos(2\pi f_c t + Km(t))\$. \$\endgroup\$ – Dave Tweed May 6 '17 at 21:08
  • \$\begingroup\$ Because the derived expression for FM in every source looks like the expression of phase modulation you have mentioned. Both FM and PM have been treated equally. Also, the expression I have mentioned doesn't plot well on the graph. Therefore it's wrong but I wonder why. \$\endgroup\$ – Chirag Arora May 6 '17 at 21:19
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An FM-modulated signal looks like $$A_c \cos\left(\varphi\left(t\right)\right)$$

At time \$t\$, its frequency is $${1 \over 2 \pi}{\mathrm{d}\varphi \over \mathrm{d}t}$$

And you want it to be $$f_c + K m(t)$$

So you must have $$\varphi(t) = \varphi_0 + \int_0^t f_c + K m(u) \mathrm{d}u = \varphi_0 + f_c t + K \int_0^t m(u) \mathrm{d}u$$

Hence, the expression for an FM-modulated signal is $$A_c \cos\left(\varphi_0 + f_c t + K \int_0^t m(u) \mathrm{d}u\right)$$

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