# Why isn't this a right expression for FM?

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.

• That looks like phase modulation. – Chu May 6 '17 at 19:55
• are you kidding me? This doesn't looks like anything – Chirag Arora May 6 '17 at 19:57
• Why do you think that this is not a correct expression for FM? – Dave Tweed May 6 '17 at 21:06
• @Chu: Phase modulation would be $A_c \cos(2\pi f_c t + Km(t))$. – Dave Tweed May 6 '17 at 21:08
• 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. – Chirag Arora May 6 '17 at 21:19

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 $$A_c \cos\left(\varphi_0 + f_c t + K \int_0^t m(u) \mathrm{d}u\right)$$