Amplitude Modulation

In amplitude modulation, the amplitude of the carrier signal is varied in accordance to the message signal, while it's frequency and phase remain constant. While this is the case, the expression of the modulated signal shows two new frequency components other than the carrier frequency. The two additional frequency components are the Upper Side Band (USB) and the Lower Side Band (LSB) as commonly known.

\[ e(t) = E_c \cos{\omega_c t} + \frac{mE_c}{2} \cos(\omega_c + \omega_s)t + \frac{mE_c}{2} \cos(\omega_c - \omega_s)t \] where \(e(t)\) is the instantaneous voltage of the modulated signal. \(E_c\) and \(\omega_c\) represent the amplitude and angular frequency of the carrier signal respectively, \(\omega_s\) represents the angular frequency of the message signal and \(m\) represents the amplitude modulation factor.

Now, the question is, how did the two frequency components evolve when the message signal did nothing to the frequency of the carrier. To answer this question, one only needs to know Fourier Series. Any periodic signal can be decomposed as a Fourier sum of sine and cosine waves. To explain this, let us consider a simple square wave. A square wave is a non-sinusoidal periodic wave generated by an infinite sum of sinusoidal waves. The figure below shows the first four partial sums of the Fourier series for a square wave. Thus, in general, any periodic signal can be represented as the sum of sinusoidal waves.

file:///home/arun/my_blog/publish_images/320/fourier_series.png

The above discussion explains the presence of new frequency components in the modulated signal. The modulated signal is a periodic wave which decomposes to the sum of a carrier component, USB and LSB. Or in other words, the Fourier Series of the modulated signal will yield the three frequency components. The three decomposed frequency components can be thought of as the spectral frequencies of the modulated signal, while the frequency of the modulated signal as a whole as its instantaneous frequency.