v = A sin (ω_ct + m sin ω_mt) is the expression for

Analysis:

Let the carries signal S_c(t) = A_c sin ω_ct

The message signal m(t) = A_m sin ω_m t

So, the expression of FM is:

\({S_{fm}}\left( t \right) = {A_c}\sin \left( {{\omega _c}t + {k_f}\;\mathop \smallint \limits_0^t m\left( t \right)dt} \right]\) 

\({s_{fm}}\left( t \right) = {A_c}\sin \left( {{\omega _c}t + {k_f}{A_m}\mathop \smallint \limits_0^t \sin {\omega _m}t} \right]\) 

\({s_{fm}}\left( t \right) = {A_c}\sin \left[ {{\omega _c}t - \frac{{{k_f}{A_m}}}{{{\omega _m}}}\cos {\omega _m}t} \right]\) 

If m: modulation index then:

\(\therefore m = \frac{{{k_f}{A_m}}}{{{\omega _m}}}\) 

S_fm(t) = A_c sin [ω_ct – m cos ω_m t]

So hence given expression is not FM signal.

Now,

The expression of PM is:

S_pm(t) = A_c sin (ω_ct + k_p m(t)]

= A_c sin (ω_c t + k_p A_m sin ω_m t)

If m is the modulation index, then:

S_pm(t) = A_c sin (ω_c t + m sin ω_m t)      

∴ m = k_p A_m

Hence given expression is of PM signal.