Vestigial Side band Modulation

Vestigial Sideband Modulation (VSB) is a technique used in television broadcasting and other communication systems to reduce the bandwidth required to transmit a signal. In traditional amplitude modulation (AM), both sidebands are transmitted, which effectively doubles the bandwidth required. However, VSB only transmits a small portion of one sideband, resulting in a bandwidth reduction of up to 50%.

 

Vestigial Side band Modulation

In VSB modulation, the carrier signal is modulated with the baseband signal to produce a double-sideband signal. However, before transmission, one of the sidebands is filtered out and a portion of the other sideband is retained. The retained portion of the other sideband is known as the vestigial sideband.

The size of the vestigial sideband is typically around one-third the width of the original sideband, and it is usually the lower sideband that is retained. This is because the upper sideband contains higher-frequency information that is less critical for image and sound quality in television broadcasting.

At the receiving end, a special type of filter known as a vestigial sideband filter is used to remove the vestigial sideband and reconstruct the original signal. This filter is designed to have a flat frequency response in the main part of the retained sideband and to have a steep cutoff at the edges of the vestigial sideband.

Vestigial sideband is a type of Amplitude modulation in which one side band is completely passed along with trace or tail or vestige of the other side band. VSB is a compromise between SSB and DSBSC modulation. In SSB, we send only one side band, the Bandwidth required to send SSB wave is w. SSB is not appropriate way of modulation when the message signal contains significant components at extremely low frequencies. To overcome this VSB is used.

Frequency Domain Description

The following Fig illustrates the spectrum of VSB modulated wave s (t) with respect to the message m (t) (band limited) 

Spectrum of VSB wave containing the vestige of the LSB

Assume that the Lower side band is modified into the vestigial side band. The vestige of the lower sideband compensates for the amount removed from the upper sideband. The bandwidth required to send VSB wave is

𝐵 = 𝑤 + 𝑓𝑣

where 𝑓𝑣 is the width of the vestigial side band.

Similarly, if USB is modified into the VSB then

Spectrum of VSB wave containing the vestige of the USB

The vestige of the Upper sideband compensates for the amount removed from the Lower sideband. The bandwidth required to send VSB wave is 𝐵 = 𝑤 + 𝑓𝑣 , where fv is the width of the vestigial side band.
Therefore, VSB has the virtue of conserving bandwidth almost as efficiently as SSB modulation, while retaining the excellent low-frequency base band characteristics of DSBSC and it is standard for the transmission of TV signals.

Generation of VSB Modulated Wave

VSB modulated wave is obtained by passing DSBSC through a sideband shaping filter as shown in below fig.

Block Diagram of VSB Modulator

The exact design of this filter depends on the spectrum of the VSB waves. The relation between filter transfer function H (f) and the spectrum of VSB waves is given by

𝑆(𝑓) =𝐴𝑐[𝑀(𝑓 − 𝑓 ) + 𝑀(𝑓 + 𝑓 )]𝐻(𝑓) … … … … … … … (1)

Where M(f) is the spectrum of Message Signal. Now, we have to determine the specification for the filter transfer function H(f) It can be obtained by passing s(t) to a coherent detector and determining the necessary condition for undistorted version of the message signal m(t). Thus, s (t) is multiplied by a locally generated sinusoidal wave cos (2πfct) which is synchronous with the carrier wave Accos(2πfct) in both frequency and phase, as in the below fig.,

Block Diagram of VSB Demodulator

Then,

The spectrum of 𝑉(𝑓) as shown in fig below

Spectrum of the product modulator output

Pass 𝑣(𝑡) to a LPF to eliminate VSB wave corresponding to 2𝑓𝑐
𝑉o (𝑓) =𝐴𝑐 /4𝑀(𝑓)[𝐻(𝑓 − 𝑓c) + 𝐻(𝑓 + 𝑓c )] … … … … … . (5)
The spectrum of 𝑉 𝑜(𝑓) is in fig below,

Spectrum of demodulated signal 𝑣𝑜(𝑡)

For a distortion less reproduction of the original signal 𝑚(𝑡), 𝑉 𝑜(𝑓) is to be a scaled version of 𝑀(𝑓). Therefore, the transfer function 𝐻(𝑓) must satisfy the condition
𝐻(𝑓 − 𝑓𝑐 ) + 𝐻(𝑓 + 𝑓𝑐) = 2𝐻(𝑓𝑐 ) … … … … … … … (6)

Where 𝐻(𝑓𝑐 ) is a constant
Since m(t) is a band limited signal, we need to satisfy eqn. (6) in the interval - w < f < w. The requirement of eqn. (6) is satisfied by using a filter whose transfer function is shown below.

Frequency response of sideband shaping filter

Note: H ( f ) is shown for positive frequencies only The response is normalized so that H ( f ) at 𝑓𝑐 is 0.5. Inside this interval 𝑓𝑐 − 𝑓𝑣 ≤ 𝑓 ≤ 𝑓𝑐+ 𝑓𝑣response exhibits odd symmetry .i.e., sum of the values of
frequencies equally displaced above & below is Unity H ( f ) at any two Similarly, the transfer function H( f ) of the filter for sending Lower sideband along with the vestige of the Upper sideband is shown in fig below,

Frequency response of sideband shaping filter

Note: H ( f ) is shown for positive frequencies only

 Time Domain Description:

Time domain representation of VSB modulated wave, procedure is similar to SSB Modulated waves. Let s(t) denote a VSB modulated wave and assuming that s(t) containing Upper sideband along with the Vestige of the Lower sideband. VSB modulated wave s(t) is the output from Sideband shaping filter, whose input is DSBSC wave. The filter transfer function H(f) is of the form as in fig below,

H ( f ) of sideband shaping filter

The DSBSC modulated wave is 𝑆𝐷𝑆𝐵𝑆𝐶(𝑡) = 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡).......................(1)
It is a band pass signal and has in-phase component only. Its low pass complex envelope is given by
𝑆 ̃𝐷𝑆𝐵𝑆𝐶(𝑡) = 𝐴𝑐𝑚(𝑡) … … … … … . (2)
The VSB modulated wave is a band pass signal.
Let the low pass signal ~ s (t) denote the complex envelope of VSB wave s(t), then
𝑠(𝑡) = 𝑅𝑒[𝑠̃(𝑡) exp(𝑗2𝜋𝑓𝑐𝑡)].....................................(3)

To determine ~ s (t) we proceed as follows
1. The side band shaping filter transfer function H ( f ) is replaced by its equivalent complex low pass transfer function denoted by 𝐻 ̃(𝑓) as shown in fig below

Low pass equivalent to H ( f )

We may express 𝐻 ̃(𝑓) as the difference between two components 𝐻 ̃𝑢(𝑓) and 𝐻 ̃𝑣(𝑓) as 𝐻 ̃(𝑓) = 𝐻 ̃𝑢(𝑓) − 𝐻 ̃𝑣(𝑓) … … … … … … … . (4)
These two components are considered individually as follows
i. The transfer function 𝐻 ̃𝑢(𝑓) pertains to a complex low pass filter equivalent to a ban pass filter design to reject the lower side band completely as

where 𝑚𝑄(𝑡) is the response produced by passing the message through a low pass filter of impulse response ℎ𝑄(𝑡). Substitute eqn. 12 in eqn.3 and simplify, we get
𝑆(𝑡) = 𝐴𝑐⁄2 𝑚(𝑡) cos(2𝜋𝑓 𝑐 𝑡) − 𝐴𝑐⁄2 𝑚𝑄(𝑡)sin(2𝜋𝑓 𝑐 𝑡) ......................(13)
Where 𝐴𝑐⁄2 𝑚(𝑡)................𝑖𝑛 − 𝑝ℎ𝑎𝑠𝑒 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡
𝐴𝑐⁄2 𝑚𝑄(𝑡) … … … 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑢𝑟𝑒 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡
Note:
1. If vestigial side band is increased to full side band, VSB becomes DSCSB, i.e., m Q(t) =0.
2. If VSB is reduced to zero, VSB becomes SSB.i.e., mQ(t) =𝑚 ̂(t).Where the 𝑚 ̂ (t) is the Hilbert transform of m(t) Similarly if the VSB containing a vestige of the upper sideband, then s(t) is given by
S(t)= Ac/2 m(t) cos (2πfct) + Ac/2 mQ(t) sin (2πfct)-------(14)

Envelope detection of a VSB Wave plus Carrier

To make demodulation of VSB wave possible by an envelope detector at the receiving end it is necessary to transmit a sizeable carrier together with the modulated wave. The scaled expression of VSB wave by factor ka with the carrier component 𝐴𝑐cos(2𝜋𝑓𝑐 𝑡) can be given by 

The detector output is distorted by the quadrature component 𝑚𝑄(𝑡) as indicated by equation (2)
Methods to reduce distortion
1. Distortion can be reduced by reducing percentage modulation 𝑘𝑎
2. Distortion can be reduced by reducing 𝑚𝑄(𝑡) by increasing the width of the vestigial sideband.

Advantages of Vestigial Sideband Modulation (VSB):

1. Reduced bandwidth: VSB modulation reduces the bandwidth required to transmit a signal by up to 50%, compared to traditional AM where both sidebands are transmitted.
2. Good signal quality: VSB modulation provides good signal quality due to the retention of a portion of one sideband, which allows the receiver to reconstruct the original signal.
3. Compatibility: VSB modulation is compatible with existing analog television broadcasting systems, making it a cost-effective solution for upgrading to digital broadcasting.

Disadvantages of Vestigial Sideband Modulation (VSB):

1. Complexity: The design and implementation of VSB modulation systems can be more complex than traditional AM systems due to the need for vestigial sideband filters.

2. Sensitivity to phase errors: VSB modulation is sensitive to phase errors in the transmission channel, which can lead to distortion in the reconstructed signal.

3. Limited spectral efficiency: Although VSB modulation reduces the bandwidth required for transmission, it is still less spectrally efficient than other modulation techniques such as Quadrature Amplitude Modulation (QAM).

4. Interference: VSB modulation is susceptible to interference from adjacent channels, which can cause ghosting and other image artifacts in television broadcasting

Applications of VSB

Vestigial Sideband Modulation (VSB) is primarily used in analog television broadcasting, where it has been widely adopted as the standard modulation scheme for over-the-air broadcasting in North America. However, it can also be used in other communication systems that require efficient use of bandwidth, such as:

1. Cable television: VSB is used in some cable television systems to transmit digital television signals to subscribers.

2. Radar systems: VSB can be used in radar systems to reduce the bandwidth required for transmission and improve range resolution.

3. Audio transmission: VSB can be used for transmitting high-quality audio signals, particularly in broadcasting systems where bandwidth is limited.

4. Video transmission: VSB can be used for transmitting high-quality video signals, particularly in applications where bandwidth is limited, such as surveillance systems.

Overall, VSB modulation is a versatile technique that can be used in a range of communication systems where bandwidth efficiency is important. However, with the transition to digital broadcasting and the development of more advanced modulation techniques, VSB is becoming less common in newer communication systems..