In our development of DSB, we saw that the USB and LSB have even amplitude and odd phase symmetry about the carrier frequency. Thus, transmission of both sidebands is not necessary, since either sideband contains sufficient information to reconstruct the message signal 𝑚(𝑡). Elimination of one of the sidebands prior to transmission results in single- sideband (SSB), which reduces the bandwidth of the modulator output from 2𝑊 to 𝑊, where 𝑊 is the bandwidth of𝑚(𝑡). However, this bandwidth savings is accompanied by a considerable increase in complexity.
On the following pages, two different methods are used to derive the time-domain ex- pression for the signal at the output of an SSB modulator. Although the two methods are equivalent, they do present different viewpoints. In the first method, the transfer function of the filter used to generate an SSB signal from a DSB signal is derived using the Hilbert transform. The second method derives the SSB signal directly from 𝑚(𝑡)using the results illustrated in Figure 2.29 and the frequency-translation theorem.
The generation of an SSB signal by sideband filtering is illustrated in Figure 3.9. First, a DSB signal,𝑥DSB(𝑡), is formed. Sideband filtering of the DSB signal then yields an upper- sideband or a lower-sideband SSB signal, depending on the filter passband selected.
f
f f
f
fc–W
fc–W
fc+W fc
fc fc fc+W
0
0 XDSB(f) xDSB (t) xSSB (t)
XSSB(f); LSB XSSB(f); USB
M(f)
m(t)
W 0
0
(a)
(b) Ac cosωct
Sideband f ilter
×
Figure 3.9
Generation of SSB by sideband filtering. (a) SSB modulator. (b) Spectra (single-sided).
The filtering process that yields lower-sideband SSB is illustrated in detail in Figure 3.10.
A lower-sideband SSB signal can be generated by passing a DSB signal through an ideal filter that passes the LSB and rejects the USB. It follows from Figure 3.10(b) that the transfer function of this filter is
𝐻𝐿(𝑓) = 1
2[sgn(𝑓 + 𝑓𝑐) − sgn(𝑓 − 𝑓𝑐)] (3.30) Since the Fourier transform of a DSB signal is
𝑋DSB(𝑓) = 1
2 𝐴𝑐𝑀(𝑓 + 𝑓𝑐) + 1
2 𝐴𝐶𝑀(𝑓 − 𝑓𝑐) (3.31)
the transform of the lower-sideband SSB signal is 𝑋𝑐(𝑓) = 1
4 𝐴𝑐[𝑀(𝑓 + 𝑓𝑐)sgn(𝑓 + 𝑓𝑐) + 𝑀(𝑓 − 𝑓𝑐)sgn(𝑓 + 𝑓𝑐)]
− 14 𝐴𝑐[𝑀(𝑓 + 𝑓𝑐)sgn(𝑓 − 𝑓𝑐) + 𝑀(𝑓 − 𝑓𝑐)sgn(𝑓 − 𝑓𝑐)] (3.32) which is
𝑋𝑐(𝑓) = 1
4 𝐴𝑐[𝑀(𝑓 + 𝑓𝑐) + 𝑀(𝑓 − 𝑓𝑐)]
+ 14 𝐴𝑐[𝑀(𝑓 + 𝑓𝑐)sgn(𝑓 + 𝑓𝑐) − 𝑀(𝑓 − 𝑓𝑐)sgn(𝑓 − 𝑓𝑐)] (3.33)
–fc fc
f 0
DSB spectrum
HL(f)
HL(f)
–fc fc f
0
–fc 0 fc
SSB spectrum (a)
(b)
f sgn (f + fc)
f –sgn (f–fc)
f
Figure 3.10
Generation of lower-sideband SSB.
(a) Sideband filtering process. (b) Generation of lower-sideband filter.
From our study of DSB, we know that 1
2 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡)↔ 1
4 𝐴𝑐[𝑀(𝑓 + 𝑓𝑐) + 𝑀(𝑓 − 𝑓𝑐)] (3.34) and from our study of Hilbert transforms in Chapter 2, we recall that
̂
𝑚(𝑡)↔−𝑗(sgn𝑓)𝑀(𝑓) By the frequency-translation theorem, we have
𝑚(𝑡)𝑒±𝑗2𝜋𝑓𝑐𝑡↔𝑀(𝑓 ∓ 𝑓𝑐) (3.35) Replacing𝑚(𝑡)by𝑚(𝑡)̂ in the previous equation yields
̂
𝑚(𝑡)𝑒±𝑗2𝜋𝑓𝑐𝑡↔−𝑗𝑀(𝑓 ∓ 𝑓𝑐)sgn(𝑓 ∓ 𝑓𝑐) (3.36) Thus,
ℑ−1{1
4 𝐴𝑐[𝑀(𝑓 + 𝑓𝑐)sgn(𝑓 + 𝑓𝑐) − 𝑀(𝑓 − 𝑓𝑐)sgn(𝑓 − 𝑓𝑐)]}
= −𝐴𝑐 1
4𝑗𝑚(𝑡)𝑒̂ −𝑗2𝜋𝑓𝑐𝑡+ 𝐴𝑐 1
4𝑗𝑚(𝑡)𝑒̂ +𝑗2𝜋𝑓𝑐𝑡= 1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡) (3.37)
+
+
−
+
Ac/2
Ac/2 xc(t) USB/SSB
xc(t) LSB/SSB
m(t) m(t) cos ct
m(t)
ω
m(t) m(t) sinωct cosωct
sinωct
› ›
Carrier oscillator
Σ Σ
H(f) = −jsgn (f)
×
×
Figure 3.11
Phase-shift modulator.
Combining (3.34) and (3.37), we get the general form of a lower-sideband SSB signal:
𝑥𝑐(𝑡) = 1
2 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) +1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡) (3.38) A similar development can be carried out for upper-sideband SSB. The result is
𝑥𝑐(𝑡) = 1
2 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) −1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡) (3.39) which shows that LSB and USB modulators have the same defining equations except for the sign of the term representing the Hilbert transform of the modulation. Observation of the spectrum of an SSB signal illustrates that SSB systems do not have DC response.
The generation of SSB by the method of sideband filtering the output of DSB modulators requires the use of filters that are very nearly ideal if low-frequency information is contained in𝑚(𝑡). Another method for generating an SSB signal, known asphase-shift modulation, is illustrated in Figure 3.11. This system is a term-by-term realization of (3.38) or (3.39). Like the ideal filters required for sideband filtering, the ideal wideband phase shifter, which performs the Hilbert transforming operation, is impossible to implement exactly. However, since the frequency at which the discontinuity occurs is 𝑓 = 0 instead of 𝑓 = 𝑓𝑐, ideal phase-shift devices can be closely approximated.
An alternative derivation of𝑥𝑐(𝑡)for an SSB signal is based on the concept of the analytic signal. As shown in Figure 3.12(a), the positive-frequency portion of𝑀(𝑓)is given by
𝑀𝑝(𝑓) = 1
2ℑ{𝑚(𝑡) + 𝑗 ̂𝑚(𝑡)} (3.40) and the negative-frequency portion of𝑀(𝑓)is given by
𝑀𝑛(𝑓) = 1
2ℑ{𝑚(𝑡) − 𝑗 ̂𝑚(𝑡)} (3.41) By definition, an upper-sideband SSB signal is given in the frequency domain by
𝑋𝑐(𝑓) = 1
2 𝐴𝑐𝑀𝑝(𝑓 − 𝑓𝑐) + 1
2 𝐴𝑐𝑀𝑛(𝑓 + 𝑓𝑐) (3.42)
Inverse Fourier-transforming yields 𝑥𝑐(𝑡) = 1
4 𝐴𝑐[𝑚(𝑡) + 𝑗 ̂𝑚(𝑡)]𝑒𝑗2𝜋𝑓𝑐𝑡+ 1
4 𝐴𝑐[𝑚(𝑡) − 𝑗 ̂𝑚(𝑡)]𝑒−𝑗2𝜋𝑓𝑐𝑡 (3.43) which is
𝑥𝑐(𝑡) = 1
4 𝐴𝑐𝑚(𝑡)[𝑒𝑗2𝜋𝑓𝑐𝑡+ 𝑒−𝑗2𝜋𝑓𝑐𝑡] + 𝑗1
4 𝐴𝑐𝑚(𝑡)[𝑒̂ 𝑗2𝜋𝑓𝑐𝑡− 𝑒−𝑗2𝜋𝑓𝑐𝑡]
= 12 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) −1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡) (3.44) The preceding expression is clearly equivalent to (3.39).
The lower-sideband SSB signal is derived in a similar manner. By definition, for a lower- sideband SSB signal,
𝑋𝑐(𝑓) = 1
2 𝐴𝑐𝑀𝑝(𝑓 + 𝑓𝑐) + 1
2 𝐴𝑐𝑀𝑛(𝑓 − 𝑓𝑐) (3.45) This becomes, after inverse Fourier-transforming,
𝑥𝑐(𝑡) = 1
4 𝐴𝑐[𝑚(𝑡) + 𝑗 ̂𝑚(𝑡)]𝑒−𝑗2𝜋𝑓𝑐𝑡+ 1
4 𝐴𝑐[𝑚(𝑡) − 𝑗 ̂𝑚(𝑡)]𝑒𝑗2𝜋𝑓𝑐𝑡 (3.46) which can be written as
𝑥𝑐(𝑡) = 1
4 𝐴𝑐𝑚(𝑡)[𝑒𝑗2𝜋𝑓𝑐𝑡+ 𝑒−𝑗2𝜋𝑓𝑐𝑡] − 𝑗1
4 𝐴𝑐𝑚(𝑡)[𝑒̂ 𝑗2𝜋𝑓𝑐𝑡− 𝑒−𝑗2𝜋𝑓𝑐𝑡]
= 12 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) +1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡)
This expression is clearly equivalent to (3.38). Figures 3.12(b) and (c) show the four signal spectra used in this development:𝑀𝑝(𝑓 + 𝑓𝑐), 𝑀𝑝(𝑓 − 𝑓𝑐), 𝑀𝑛(𝑓 + 𝑓𝑐), and𝑀𝑛(𝑓 − 𝑓𝑐).
There are several methods that can be employed to demodulate SSB. The simplest tech- nique is to multiply𝑥𝑐(𝑡)by a demodulation carrier and lowpass filter the result, as illustrated in Figure 3.1(a). We assume a demodulation carrier having a phase error𝜃(𝑡)that yields
𝑑(𝑡) =[1
2 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) ±1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡)]
{4 cos[2𝜋𝑓𝑐𝑡 + 𝜃(𝑡)]} (3.47) where the factor of 4 is chosen for mathematical convenience. The preceding expression can be written as
𝑑(𝑡) = 𝐴𝑐𝑚(𝑡) cos 𝜃(𝑡) + 𝐴𝑐𝑚(𝑡) cos[4𝜋𝑓𝑐𝑡 + 𝜃(𝑡)]
∓𝐴𝑐𝑚(𝑡) sin 𝜃(𝑡) ± 𝐴̂ 𝑐𝑚(𝑡) sin[4𝜋𝑓̂ 𝑐𝑡 + 𝜃(𝑡)] (3.48) Lowpass filtering and amplitude scaling yield
𝑦𝐷(𝑡) = 𝑚(𝑡) cos 𝜃(𝑡) ∓ ̂𝑚(𝑡) sin 𝜃(𝑡) (3.49)
M(f) Mp(f)
Xc(f)
Xc(f)
Mn(f)
Mn(f + fc)
Mp(f + fc) Mn(f –fc)
Mp(f–fc)
–fc fc
–fc fc
f f
f
f
f W
W
–W 0 0 –W 0
(a)
(b)
(c) Figure 3.12
Alternative derivation of SSB signals. (a)𝑀(𝑓),𝑀𝑝(𝑓), and𝑀𝑛(𝑓). (b) Upper-sideband SSB signal.
(c) Lower-sideband SSB signal.
for the demodulated output. Observation of (3.49) illustrates that for𝜃(𝑡)equal to zero, the demodulated output is the desired message signal. However, if𝜃(𝑡) is nonzero, the output consists of the sum of two terms. The first term is a time-varying attenuation of the message signal and is the output present in a DSB system operating in a similar manner. The second term is a crosstalk term and can represent serious distortion if𝜃(𝑡)is not small.
Another useful technique for demodulating an SSB signal is carrier reinsertion, which is illustrated in Figure 3.13. The output of a local oscillator is added to the received signal𝑥𝑟(𝑡).
This yields
𝑒(𝑡) =[1
2 𝐴𝑐𝑚(𝑡) + 𝐾]
cos(2𝜋𝑓𝑐𝑡) ±1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡) (3.50) which is the input to the envelope detector. The output of the envelope detector must next be computed. This is slightly more difficult for signals of the form of (3.50) than for signals of the form of (3.10) because both cosine and sine terms are present. In order to derive the
Σ e(t) yD(t)
xr(t)
K cosωct
Envelope detector
Figure 3.13
Demodulation using carrier reinsertion.
Quadrature axis
b(t)
a(t) (t) R(t)
θ Direct
axis
Figure 3.14
Direct-quadrature signal representation.
desired result, consider the signal
𝑥(𝑡) = 𝑎(𝑡) cos(2𝜋𝑓𝑐𝑡) − 𝑏(𝑡) sin(2𝜋𝑓𝑐𝑡) (3.51) which can be represented as illustrated in Figure 3.14. Figure 3.14 shows the amplitude of the direct component𝑎(𝑡), the amplitude of the quadrature component𝑏(𝑡), and the resultant𝑅(𝑡).
It follows from Figure 3.14 that
𝑎(𝑡) = 𝑅(𝑡) cos 𝜃(𝑡) and 𝑏(𝑡) = 𝑅(𝑡) sin 𝜃(𝑡) This yields
𝑥(𝑡) = 𝑅(𝑡)[cos 𝜃(𝑡) cos(2𝜋𝑓𝑐𝑡) − sin 𝜃(𝑡) sin(2𝜋𝑓𝑐𝑡)] (3.52) which is
𝑥(𝑡) = 𝑅(𝑡) cos[2𝜋𝑓𝑐𝑡 + 𝜃(𝑡)] (3.53) where
𝜃(𝑡) = tan−1 (𝑏(𝑡)
𝑎(𝑡) )
(3.54) The instantaneous amplitude𝑅(𝑡), which is the envelope of the signal, is given by
𝑅(𝑡) =√
𝑎2(𝑡) + 𝑏2(𝑡) (3.55)
and will be the output of an envelope detector with𝑥(𝑡)on the input if𝑎(𝑡)and𝑏(𝑡)are slowly varying with respect tocos 𝜔𝑐𝑡.
A comparison of (3.50) and (3.55) illustrates that the envelope of an SSB signal, after carrier reinsertion, is given by
𝑦𝐷(𝑡) =√[1
2 𝐴𝑐𝑚(𝑡) + 𝐾]2
+[1
2 𝐴𝑐𝑚(𝑡)̂ ]2
(3.56) which is the demodulated output𝑦𝐷(𝑡)in Figure 3.13. If𝐾is chosen large enough such that
[1
2 𝐴𝑐𝑚(𝑡) + 𝐾]2
≫[1
2 𝐴𝑐𝑚(𝑡)̂ ]2
the output of the envelope detector becomes 𝑦𝐷(𝑡) ≅ 1
2 𝐴𝑐𝑚(𝑡) + 𝐾 (3.57)
from which the message signal can easily be extracted. The development shows that carrier reinsertion requires that the locally generated carrier must be phase coherent with the original modulation carrier. This is easily accomplished in speech-transmission systems. The frequency and phase of the demodulation carrier can be manually adjusted until intelligibility of the speech is obtained.
EXAMPLE 3.2
As we saw in the preceding analysis, the concept of single sideband is probably best understood by using frequency-domain analysis. However, the SSB time-domain waveforms are also interesting and are the subject of this example. Assume that the message signal is given by
𝑚(𝑡) = cos(2𝜋𝑓1𝑡) − 0.4 cos(4𝜋𝑓1𝑡) + 0.9 cos(6𝜋𝑓1𝑡) (3.58) The Hilbert transform of𝑚(𝑡)is
̂
𝑚(𝑡) = sin(2𝜋𝑓1𝑡) − 0.4 sin(4𝜋𝑓1𝑡) + 0.9 sin(6𝜋𝑓1𝑡) (3.59) These two waveforms are shown in Figures 3.15(a) and (b).
As we have seen, the SSB signal is given by 𝑥𝑐(𝑡) = 𝐴𝑐
2 [𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) ± ̂𝑚(𝑡) sin(2𝜋𝑓𝑐𝑡)] (3.60) with the choice of sign depending upon the sideband to be used for transmission. Using (3.51) to (3.55), we can place𝑥𝑐(𝑡)in the standard form of (3.1). This gives
𝑥𝑐(𝑡) = 𝑅(𝑡) cos[2𝜋𝑓𝑐𝑡 + 𝜃(𝑡)] (3.61) where the envelope𝑅(𝑡)is
𝑅(𝑡) = 𝐴𝑐 2
√𝑚2(𝑡) + ̂𝑚2(𝑡) (3.62)
and𝜃(𝑡), which is the phase deviation of𝑥𝑐(𝑡), is given by 𝜃(𝑡) = ± tan−1
(𝑚(𝑡)̂ 𝑚(𝑡) )
(3.63) The instantaneous frequency of𝜃(𝑡)is therefore
𝑑
𝑑𝑡[2𝜋𝑓𝑐𝑡 + 𝜃(𝑡)] = 2𝜋𝑓𝑐± 𝑑 𝑑𝑡
[ tan−1
(𝑚(𝑡)̂ 𝑚(𝑡)
)]
(3.64) From (3.62) we see that the envelope of the SSB signal is independent of the choice of the sideband.
The instantaneous frequency, however, is a rather complicated function of the message signal and also depends upon the choice of sideband. We therefore see that the message signal𝑚(𝑡)affects both the envelope and phase of the modulated carrier𝑥𝑐(𝑡). In DSB and AM the message signal affected only the envelope of𝑥𝑐(𝑡).
(a)
(b)
(c)
(d)
(e) t
t t t t m(t)
m(t)
R(t)
xc(t)
xc(t) 0
0
0
0
0
Figure 3.15
Time-domain signals for SSB system. (a) Message signal.
(b) Hilbert transform of the message signal. (c) Envelope of the SSB signal. (d) Upper-sideband SSB signal with message signal.
(e) Lower-sideband SSB signal with message signal.
The envelope of the SSB signal,𝑅(𝑡), is shown in Figure 3.15(c). The upper-sideband SSB signal is illustrated in Figure 3.15(d) and the lower-sideband SSB signal is shown in Figure 3.15(e). It is easily seen that both the upper-sideband and lower-sideband SSB signals have the envelope shown in Figure 3.15(c). The message signal𝑚(𝑡)is also shown in Figures 3.15(d) and (e).
■