Considering a message signal \(e_m=E_mcos\omega_mt\) and carrier signal by \(e_c=E_csin\left(\omega_c+\theta\right)t\), find the expression of the resultant FM wave.
The general expression of FM signal is
\(S\left(t\right)=E_csin\left(\omega_ct+\theta\right)\)
\(S\left(t\right)=E_csin\left(\omega_ct+2\pi k_f\int_{0}^{t}{e_mdt}\right)\)
Where
\(\theta=2\pi k_f\int_{0}^{t}{E_mcos\omega_mtdt}\)
\(S\left(t\right)=E_csin\left(\omega_ct+2\pi k_f\int_{0}^{t}{E_mcos\omega_mtdt}\right)\)
\(S\left(t\right)=E_csin\left(\omega_ct+\frac{2\pi E_mk_f}{\omega_m}sin\omega_mt\right)\)
\(S\left(t\right)=E_csin\left(\omega_ct+\frac{2\pi E_mk_f}{2\pi f_m}sin\omega_mt\right)\)
\(S\left(t\right)=E_c\sin\left(\omega_ct+\frac{∆f}{f_m}\sin\omega_mt\right)\)
Where \(E_mk_f=∆f\)
\(k_f\) is the frequency sensitivity.
∆f is the frequency deviation
\(\frac{∆f}{f_m}=\beta\)
\(\beta\) is called modulation index
The resultant FM signal expression is
\(S\left(t\right)=E_csin\left(\omega_ct+\beta sin\omega_mt\right)\)
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Fig. 1
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