# Spectrum of WBFM signal for triangular message signal spectrum

We Know:
Frequency Modulated signal is given by: $$s_{FM} (t)=A_{c} \sin \{ w_c t + 2 \pi k_f \int m(t) \, dt \}$$ Now,
if $$\quad m(t)= A_m \cos(w_m t)$$ ,
i.e, its spectrum $$M(f)$$:

then, $$s_{FM} (t)=A_{c} \sin \{ w_c t + 2 \pi k_f \int A_m \cos(w_m t ) \, dt \}$$ $$= A_{c} \sin \{ w_c t + 2 \pi k_f \frac{ A_m \sin(w_m t ) }{w_m} \}$$ $$= A_{c} \sin \{ w_c t + \frac{ k_f A_m }{f_m} \sin(w_m t ) \}$$ $$=A_{c} \sin \{ w_c t + m_f \sin(w_m t ) \}$$ $$= J_0(m_f) A_c \sin(w_c t) + \sum_{n= 1}^{\infty} [ \{ J_n(m_f) A_c \sin(w_c t + n w_m t) \} + \{ J_{-n}(m_f) A_c\sin(w_c t - n w_m t)\}]$$ $$\implies$$ the corresponding FM (Frequency Modulation) signal spectrum [$$S(f)$$] will be:

Now, if the spectrum of message signal is triangular,
i.e,

then, what will be its corresponding WBFM (Wide Band Frequency Modulation) signal spectrum?
i.e, $$m_f > 1$$

for e.g:
For $$m_f \leq 1$$ , the corresponding NBFM signal spectrum is: