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. VSBis 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
Fig illustrates the spectrum of VSB modulated wave s(t) with respect to the message m(t)( bandlimited )
M(f)
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 Upper side band is modified into the vestigial side band then,
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.
VSB modulated wave is obtained by passing DSBSC through a sideband shaping filter as shown in fig below.
The exact design of this filter depends on the spectrum of the VSB waves. The relation b/n filter transfer function H(f) and the spectrum of VSB waves is given by
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 cos2πfct, which is synchronous with the carrier wave Accos2πfct in both frequency and phase, as in fig below,
For a distortion less reproduction of the original signal m(t), Vo(f) to be a scaled version of M(f). Therefore, the transfer function H(f) must satisfy the condition
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
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 H(f) at any two frequencies equally displaced above and below is Unity.
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,
Note: H(f) is Shown for positive frequencies only.
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,
The DSBSC Modulated wave is
SDSBSC(t) = Ac m(t) cos2πfct ----------------------(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[s-(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
Fig (2) Low pass equivalent to H(f)
Similarly If VSB containing a vestige of the Upper sideband, then s(t) is given by S(t) = Ac/2
