Question:
Published on: 20 June, 2022

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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