SSB-SC Modulation

Theory

Single Sideband Suppressed Carrier (SSB-SC) modulation is a bandwidth-efficient form of amplitude modulation in which only one sideband of the modulated signal is transmitted, while the carrier component is completely suppressed. Let the message signal \( m(t) \) be band-limited to a bandwidth \( B \).

The SSB-SC modulated signal \( s(t) \) is expressed as:

\( s(t) = \frac{A_c}{2} \left[ m(t)\cos(2\pi f_c t) \pm \hat{m}(t)\sin(2\pi f_c t) \right] \)

Where:

• \( A_c \) is the amplitude of the carrier signal.
• \( m(t) \) is the baseband (modulating) signal.
• \( f_c \) is the carrier frequency.
• \( \hat{m}(t) \) is the Hilbert transform of \( m(t) \), representing a 90° phase-shifted version of the signal.

The positive sign corresponds to the upper sideband (USB), while the negative sign corresponds to the lower sideband (LSB). Since only one sideband is transmitted, the required transmission bandwidth is equal to \( B \), which is half that of DSB-SC modulation.

Block Diagram

Fig 1: SSB-SC Modulation

Frequency Domain Description

Here, \( \omega_c = 2 \pi f_c \) is the carrier's angular frequency in radians per second. Using \( \omega \) in the frequency-domain representation simplifies the mathematical expressions for Fourier transforms and spectral analysis. The relationship between angular frequency \( \omega \) and linear frequency \( f \) is \( \omega = 2 \pi f \).

Let \( M(j\omega) \) denote the Fourier transform of \( m(t) \). In DSB-SC modulation, the spectrum of the message signal is shifted to \( \pm \omega_c \), producing two symmetric sidebands.

In SSB-SC modulation, one of these sidebands is completely suppressed. Therefore:

• For the upper sideband (USB): \( S(j\omega) = \frac{A_c}{2} M(j(\omega - \omega_c)) \)
• For the lower sideband (LSB): \( S(j\omega) = \frac{A_c}{2} M(j(\omega + \omega_c) \))

Fig 2: Frequency-domain characteristics of SSB modulation. (a) Magnitude spectrum of message signal, with energy gap from −ωₐ to ωₐ. (b) Magnitude spectrum of DSB-SC signal. (c) Magnitude spectrum of SSB modulated wave, containing upper sideband only. (d) Magnitude spectrum of SSB modulated wave, containing lower sideband only.

SSB-SC Demodulation

Block Diagram

Fig 3: SSB-SC Demodulation

Demodulation using Coherent Detection

The recovery of the message signal requires coherent detection, in which the received signal is multiplied by a locally generated carrier that is perfectly synchronized in both phase and frequency.

\( s(t) \cdot 2\cos(2\pi f_c t) \)

Substituting for \( s(t) \):

\( 2 s(t)\cos(2\pi f_c t) = A_c m(t)\cos^2(2\pi f_c t) \pm A_c \hat{m}(t)\sin(2\pi f_c t)\cos(2\pi f_c t) \)

Using trigonometric identities:

• \( \cos^2(2\pi f_c t) = \frac{1}{2}\left[1 + \cos(4\pi f_c t)\right] \)
• \( \sin(2\pi f_c t)\cos(2\pi f_c t) = \frac{1}{2}\sin(4\pi f_c t) \)

This simplifies to:

\( = \frac{A_c}{2} m(t) + \frac{A_c}{2} m(t)\cos(4\pi f_c t) \pm \frac{A_c}{2} \hat{m}(t)\sin(4\pi f_c t) \)

Low-Pass Filtering

The components at frequency \( 2f_c \) are removed using a low-pass filter (LPF), leaving only the baseband term:

\( \hat{m}_{out}(t) = \frac{A_c}{2} m(t) \)

After appropriate scaling, the original message signal \( m(t) \) is recovered. Accurate carrier synchronization is essential; otherwise, phase or frequency errors will introduce distortion in the recovered signal.