As we see in case of DSB-SC only sidebands are transmitted as they bear all informations about the signal. On the other hand, the two sidebands are identical and they carry same information. So, why not just send a single sideband and construct the other sideband from that.
SSB-SC Modulator using Hilbert Transform
Single-sideband has the mathematical form of quadrature amplitude modulation (QAM) in the special case where one of the baseband waveforms is derived from the other, instead of being independent messages:
Sssb(t) = s(t).cos(2πf0t) - 
(t).sin(2πf0t)
Where s(t) is the message (real valued), 
(t) is the Hilbert transform, and f0 is the radio carrier frequency.
To understand this formula, we may express s(t) as real part of a complex valued function, with no loss of information.
s(t) = Re{sa(t)} = Re{s(t) + j.
(t)}
where j represents the imaginary unit. sa(t) is the analytical representation of s(t), which means that it comprises only the positive-frequency components of s(t)
Sa(f) = S(f) for f > 0
= 0 for f < 0
Where Sa(f) and S(f) are the respective Fourier transforms of sa(t) and s(t).
Therefore the frequency-translated function Sa(f – f0) contains only one side of S(f). Since it also has only positive-frequency components, its inverse Fourier transform is the analytical representation of sssb(t).
sssb(t) + 
ssb(t) = F -1 {Sa(f – f0)} = sa(t).e2πf0t,
and again the real part of this expression causes no loss of information. With Eular’s formula to expand e2πf0t , we obtain
sssb(t) = Re{sa(t).e2πf0t}
= Re {[ s(t) + j.(t)] . [cos(2πf0t) + j.sin(2πf0t)]}
= s(t). cos(2πf0t) - (t). sin(2πf0t)
Coherent demodulation of sssb(t) to recover s(t) is the same as AM: multiply by cos(2πf0t), and lowpass to remove the “double frequency” components around frequency 2f0. If the demodulating carrier is not is not in the correct phase (cosine phase here), then the demodulated signal will be some linear combination of s(t) and (t), which is usually acceptable in voice communications.
Lower sideband
s(t) can also be recovered as the real part of the complex-conjugate, sa*(t), which represents the negative frequency portion of S(f), when f0 is large enough that S(f - f0) has no negative frequencies, the product sa*(t).e2πf0t is another analytical signal, whose real part is the actual lower-sideband transmission.
sa*(t).e2πf0t
= slsb(t) + lsb(t)
slsb(t) = Re{ sa*(t). e2πf0t}
= Re {[ s(t) + j.(t)] . [cos(2πf0t) + j.sin(2πf0t)]}
= s(t). cos(2πf0t) + (t). sin(2πf0t)
The sum of the two sideband signals is:
susb(t) + slsb(t) = 2s(t).cos(2πf0t)
which is the classic model of suppressed carrier double sideband AM.
One important characteristic of the analytical signal is that its spectral content lies in the positive Nyquist interval. This is because if we shift the imaginary part of our analytic (complex) signal by 90 degrees (+j) and add it to the real part, the negative frequencies will cancel while the positive frequencies will add. This results in a signal with no negative frequencies. Also, the magnitude of the frequency component in the complex signal is twice the magnitude of the frequency component in the real signal. This is similar to a one-sided spectrum, which contains the total signal power in the positive frequencies.
Other SSB-SC Modulators
Bandpass filtering
One method of producing an SSB-SC signal is to remove one of its sidebands via filtering, leaving only either the upper sideband (USB), and the sideband with the higher frequency, or less commonly the lower sideband (LSB), the sideband with lower frequency. Most often, the carrier is reduced or removed entirely (suppressed), being referred to in full as single sideband suppressed carrier (SSBSC).
Assuming both sidebands are symmetric, which is the case for a normal AM signal, no information is lost in the process.
Hartley modulator
Hartley modulator uses phasing to suppress the unwanted sideband. To generate an SSB signal with this method, two versions of the original signal are generated, mutually 900 out of phase for any single frequency within the operating bandwidth. Each one of these signals then modulates carrier waves (of one frequency) that are also 900 out of phase with each other. By either adding or subtracting the resulting signals, a lower or upper sideband signal results. A benefit is to allow an analytical expression for SSB signals, which can be used to understand effects such as synchronous detection of SSB.
In the above figure m(t) represents the message signal. And waveform of the message signal is represented as cos(ꞷmt).
Shifting the baseband signal 900 out of phase cannot be done simply by delaying it, as it contains a large range of frequencies. In analog circuits, a wideband 90 degree phase difference network is used. The method was popular in the days of vacuum radios. Nowadays this method, utilizing the Hilbert transform to phase shift the baseband audio, can be done at low cost with digital circuitry.
Weaver modulator
The Weaver modulator, uses only lowpass filters and quadrature mixers, and is a favoured method in digital implementations. In weaver’s method, the band of interest is first translated to be centered at zero, conceptually by modulating a complex exponential exp(jωt) with frequency in the middle of the voiceband, but implemented by a quadrature pair of sine and cosine modulators at the frequency (e.g. 2KHz). This complex signal or pair of real signals is then lowpass filtered to remove the undesired sideband that is not centered to zero. Then, the single sideband complex signal centered at zero is upconverted to a real signal, by another pair of quadrature mixers, to the desired center frequency.
SSB-SC Detector
The front end of an SSB receiver is similar to that of an AM or FM receiver, consisting of a superheterodyne RF front end that produces a frequency-shifted version of the radio frequency (RF) signal within a standard intermediate frequency (IF) band.
To recover the original signal from the IF SSB signal, the sideband must be frequency shifted to down its original range of baseband frequencies, by using a product detector which mixes it with the output of a beat frequency oscillator (BFO). In other words, it is just another stage of heterodyning. For this work, the BFO must be exactly adjusted.
In communications and electronic engineering, an intermediate frequency (IF) is created by mixing the carrier signal with a local oscillator signal in a process called heterodyning, resulting in a signal at the difference or beat frequency. Intermediate frequencies are used in superheterodyne radio receivers, in which an incoming signal is shifted to an IF for amplification before final detection is done.
A simple example
A receiver is tuned to a Morse code signal, and the receiver’s intermediate frequency (IF) is fIF = 45000 Hz. That means Morse code’s dits (dots) and dahs (dashes) have become pulses of 45000 Hz signal, which is inaudible. To make them audible, the frequency needs to be shifted into audio range, for instance faudio = 1000 Hz. To achieve that, the desired BFO frequency is fBFO = 44000 or 46000. Because it produces new signals at the sum and difference of the two signal frequencies.
Detection of an SSB signal
As an example, consider an IF SSB signal centered at frequency Fif = 45000 Hz. The baseband frequency it needs to be shifted to is Fb = 2000 Hz. The BFO output waveform is cos(2π. fBFO . t). When the signal is multiplied by the BFO waveform, it shifts the signal to (Fif + Fbfo), and to | Fif - Fbfo |, which is known as the beat frequency or image frequency. The objective is to choose an FBFO that results in | Fif - Fbfo | = Fb = 2000 Hz. The unwanted components at (Fif + Fbfo) can be removed by lowpass filter.
Advantages of SSB-SC over DSB-SC
- SSB-SC uses half bandwidth of DSB-SC
Q & A and Summary
1. What is the primary difference between Single Sideband Suppressed Carrier (SSB-SC) modulation and Double Sideband Suppressed Carrier (DSB-SC) modulation?
The key difference between SSB-SC and DSB-SC modulation is in the bandwidth used for transmission. While both modulations suppress the carrier signal, SSB-SC transmits only one sideband (either the upper or the lower), whereas DSB-SC transmits both the upper and lower sidebands. This results in SSB-SC requiring half the bandwidth of DSB-SC, making it more efficient in terms of bandwidth usage.
2. Explain the significance of the Hilbert transform \( \hat{m}(t) \) in the SSB-SC modulation process.
The Hilbert transform \( \hat{m}(t) \) of the modulating signal \( m(t) \) is crucial in SSB-SC modulation because it generates the quadrature component of the modulating signal. In the modulation formula:
\( S(t) = \frac{A_c}{2} \left[ m(t) \cos(2\pi f_c t) \pm \hat{m}(t) \sin(2\pi f_c t) \right] \),
the term \( \hat{m}(t) \sin(2\pi f_c t) \) represents the signal shifted 90 degrees in phase relative to the original \( m(t) \). This phase shift allows the carrier to be suppressed, and the signal is transmitted with only one sideband, either upper or lower, improving bandwidth efficiency.
3. What role does the carrier suppression play in the SSB-SC modulation scheme?
Carrier suppression in SSB-SC modulation is a critical feature that allows the modulation to be more bandwidth-efficient. By suppressing the carrier, only the sideband (either upper or lower) is transmitted. This eliminates the redundant carrier component present in DSB-SC modulation, reducing the total bandwidth required for transmission by half. This makes SSB-SC particularly advantageous for applications where bandwidth is a limiting factor, such as in long-range radio communications.
4. In the frequency-domain description of SSB-SC, how does the choice of transmitting the upper or lower sideband affect the spectrum of the modulated signal?
In SSB-SC modulation, the modulated signal's spectrum is centered around the carrier frequency \( f_c \), and it consists of either the upper or the lower sideband. When transmitting the upper sideband (USB), the frequency components of the message signal are translated upward by \( f_c \), resulting in a spectrum that ranges from \( f_c \) to \( f_c + \omega_m \) for the positive frequencies. Conversely, transmitting the lower sideband (LSB) involves shifting the frequency components downward, with the spectrum ranging from \( -f_c \) to \( -f_c - \omega_m \) for the negative frequencies. The choice of sideband determines the direction in which the spectrum is translated in the frequency domain.
5. Describe the demodulation process of an SSB-SC signal using coherent detection.
The demodulation of an SSB-SC signal using coherent detection involves multiplying the received SSB signal by a coherent carrier (i.e., a carrier with the same frequency \( f_c \) and phase as the transmitted one). This process is expressed as:
\( S(t) \cdot 2 \cos(2\pi f_c t) \)
This results in a signal that contains both the original message \( m(t) \) and additional high-frequency components (i.e., \( \cos(4\pi f_c t) \) and \( \sin(4\pi f_c t) \)). These high-frequency components are then removed using a low-pass filter, leaving only the scaled version of the original message signal. The final step involves scaling the signal by \( \frac{2}{A_c} \), which retrieves the original message \( m(t) \).
6. How does low-pass filtering contribute to the demodulation of an SSB-SC signal?
Low-pass filtering plays an essential role in removing the high-frequency components from the demodulated signal. After multiplying the received SSB-SC signal by the coherent carrier \( 2 \cos(2\pi f_c t) \), the resulting signal contains both the desired message signal \( m(t) \) and unwanted high-frequency terms like \( \cos(4\pi f_c t) \) and \( \sin(4\pi f_c t) \). These high-frequency terms are effectively removed by the low-pass filter, leaving only the message signal \( m(t) \) scaled by \( \frac{A_c}{2} \). This filtered signal is then scaled back to recover the original message signal.
7. What are the advantages of SSB-SC modulation over traditional AM modulation in terms of bandwidth efficiency?
The primary advantage of SSB-SC modulation over traditional Amplitude Modulation (AM) is its bandwidth efficiency. In AM, both the upper and lower sidebands are transmitted, which effectively doubles the bandwidth required for transmission. In contrast, SSB-SC modulation transmits only one of the sidebands (either the upper or lower sideband), reducing the required bandwidth by half compared to AM. This results in more efficient use of the available spectrum, making SSB-SC ideal for applications where bandwidth is limited or needs to be conserved, such as in long-range radio communications or in satellite communications.
8. What is the significance of using the trigonometric identities \( \cos^2(2\pi f_c t) = \frac{1 + \cos(4\pi f_c t)}{2} \) and \( \sin(2\pi f_c t) \cos(2\pi f_c t) = \frac{1}{2} \sin(4\pi f_c t) \) during the demodulation process?
The use of these trigonometric identities during the demodulation process simplifies the expression for the demodulated signal. By applying these identities, we can rewrite the terms in the product \( 2S(t) \cos(2\pi f_c t) \) in a form that clearly separates the high-frequency components (i.e., terms involving \( \cos(4\pi f_c t) \) and \( \sin(4\pi f_c t) \)) from the baseband message signal \( m(t) \). The high-frequency components are subsequently filtered out using a low-pass filter, leaving only the desired baseband message signal. This step is essential to recover the original signal without interference from the high-frequency components.