A Quadrature Carrier Multiplexing (QCM) or Quadrature Amplitude Modulation (QAM) method enables two DSBSC modulated waves, resulting from two different message signals to occupy the same transmission band width and two message signals can be separated at the receiver. The transmitter and receiver for QCM are as shown in figure 3.1.
Figure 3.1: QCM transmitter and receiver
The transmitter involves the use of two separate product modulators that are supplied with two carrier waves of the same frequency but differing in phase by -90o. The multiplexed signal s(t) consists of the sum of the two product modulator outputs given by the equation 3.1.
where m1(t) and m2(t) are two different message signals applied to the product modulators. Thus, the multiplexed signal s(t) occupies a transmission band width of 2W, centered at the carrier frequency fc where W is the band width of message signal m1(t) or m2(t), whichever is larger.
At the receiver, the multiplexed signal s(t) is applied simultaneously to two separate coherent detectors that are supplied with two local carriers of the same frequency but differing in phase by -90o. The output of the top detector is bottom detector is 12 Ac m2 (t ).
12 Ac m1 (t ) and that of the For the QCM system to operate satisfactorily, it is important to maintain correct phase and frequency relationships between the local oscillators used in the transmitter and receiver parts of the system.
The Fourier transform is useful for evaluating the frequency content of an energy signal, or in a limiting case that of a power signal. It provides mathematical basis for analyzing and designing the frequency selective filters for the separation of signals on the basis of their frequency content. Another method of separating the signals is based on phase selectivity, which uses phase shifts between the appropriate signals (components) to achieve the desired separation.
In case of a sinusoidal signal, the simplest phase shift of 180o is obtained by “Ideal transformer” (polarity reversal). When the phase angles of all the components of a given signal are shifted by 90o, the resulting function of time is called the “Hilbert transform” of the signal.
Consider an LTI system with transfer function defined by equation 3.2.
The device which possesses such a property is called Hilbert transformer. When ever a signal is applied to the Hilbert transformer, the amplitudes of all frequency components of the input signal remain unaffected. It produces a phase shift of -90o for all positive frequencies, while a phase shift of 90o for all negative frequencies of the signal.
If x(t) is an input signal, then its Hilbert transformer is denoted by shown in the following diagram.
To find impulse response h(t) of Hilbert transformer with transfer function H(f). Consider the relation between Signum function and the unit step function.
Now consider any input x(t) to the Hilbert transformer, which is an LTI system. Let the impulse response of the Hilbert transformer is obtained by convolving the input x(t) and impulse response h(t) of the system.
Applications of Hilbert transform
1. It is used to realize phase selectivity in the generation of special kind of modulation called Single Side Band modulation.
2. It provides mathematical basis for the representation of band pass signals.
Note: Hilbert transform applies to any signal that is Fourier transformable.
This represents Fourier transform of the sine function. Therefore the Hilbert transform of cosine function is sin function given by
Pre-envelope
Consider a real valued signal x(t). The pre-envelope x+ (t )for positive frequencies of the signal x(t) is defined as the complex valued function given by equation 3.8.
The spectrum of the pre-envelope x+ (t ) is nonzero only for positive frequencies as emphasized in equation 3.9. Hence plus sign is used as a subscript. In contrast, the spectrum of the other pre-envelope is x− (t ) is nonzero only for negative frequencies. That
Properties of Hilbert transform
1. “A signal x(t) and its Hilbert transform xˆ(t )have the same amplitude spectrum”.
The magnitude of –jsgn(f) is equal to 1 for all frequencies f. Therefore x(t) and the same amplitude spectrum.
2. IF xˆ(t ) is the Hilbert transform of x(t), then the Hilbert transform of xˆ(t ), is –x(t)”.
To obtain its Hilbert transform of x(t), x(t) is passed through a LTI system with a transfer function equal to –jsgn(f). A double Hilbert transformation is equivalent to passing x(t) through a cascade of two such devices. The overall transfer function of such a cascade is equal to
The resulting output is –x(t). That is the Hilbert transform of xˆ(t ) is equal to –x(t).
Equation 3.11 is the basis of definition for complex envelope ~x (t ) in terms of pre-envelope x+ (t ). The spectrum of
Hence, except for scaling factors, they may be derived from the band pass signal x(t) using the block diagram shown 3.7.
Single Side Band Suppressed Carrier modulation
Standard AM and DSBSC require transmission bandwidth equal to twice the message bandwidth. In both the cases spectrum contains two side bands of width W Hz, each. But the upper and lower sides are uniquely related to each other by the virtue of their symmetry about the carrier frequency. That is, given the amplitude and phase spectra of either side band, the other can be uniquely determined. Thus if only one side band is transmitted, and if both the carrier and the other side band are suppressed at the transmitter, no information is lost. This kind of modulation is called SSBSC and spectral comparison between DSBSC and SSBSC is shown in the figures 3.8 and 3.9.
Frequency-domain description: -
Consider a message signal m(t) with a spectrum M(f) band limited to the interval − w < f < w as shown in figure 3.10, the DSBSC wave obtained by multiplexing m(t) by the carrier wave c(t ) = Ac cos(2πfct)and is also shown, in figure 3.11. The upper side band is represented in duplicate by the frequencies above fc and those below -fc, and when only upper
side band is transmitted; the resulting SSB modulated wave has the spectrum shown in figure 3.13. Similarly, the lower side band is represented in duplicate by the frequencies below fc and those above -fc and when only the lower side band is transmitted, the spectrum of the corresponding SSB modulated wave shown in figure 3.12. Thus the essential function of the SSB modulation is to translate the spectrum of the modulating wave, either with or without inversion, to a new location in the frequency domain.
The advantage of SSB modulation is reduced bandwidth and the elimination of high power carrier wave. The main disadvantage is the cost and complexity of its implementation.
Frequency Discrimination Method for generating an SSBSC modulated wave
Consider the generation of SSB modulated signal containing the upper side band only. From a practical point of view, the most severe requirement of SSB generation arises from the unwanted sideband, the nearest component of which is separated from the desired side band by twice the lowest frequency component of the message signal. It implies that, for the generation of an SSB wave to be possible, the message spectrum must have an energy gap centered at the origin as shown in figure 3.14. This requirement is naturally satisfied by voice signals, whose energy gap is about 600Hz wide.
Figure 3.14: Message spectrum with energy gap at the origin
The frequency discrimination or filter method of SSB generation consists of a product modulator, which produces DSBSC signal and a band-pass filter to extract the desired side band and reject the other and is shown in the figure 3.15.
Figure 3.15: Frequency discriminator to generate SSBSC wave
Application of this method requires that the message signal satisfies two conditions:
1. The message signal m(t) has no low-frequency content. Example: - speech, audio, music.
2. The highest frequency component W of the message signal m(t) is much less than the carrier frequency fc.
Then, under these conditions, the desired side band will appear in a non-overlapping
interval in the spectrum in such a way that it may be selected by an appropriate filter. In designing the band pass filter, the following requirements should be satisfied:
1) The pass band of the filter occupies the same frequency range as the spectrum of the desired SSB modulated wave.
2. The width of the guard band of the filter, separating the pass band from the stop band, where the unwanted sideband of the filter input lies, is twice the lowest frequency component of the message signal.
When it is necessary to generate an SSB modulated wave occupying a frequency band that is much higher than that of the message signal, it becomes very difficult to design an appropriate filter that will pass the desired side band and reject the other. In such a situation it is necessary to resort to a multiple-modulation process so as to ease the filtering requirement. This approach is illustrated in the following figure 3.16 involving two stages of modulation.
The SSB modulated wave at the first filter output is used as the modulating wave for the second product modulator, which produces a DSBSC modulated wave with a spectrum that is symmetrically spaced about the second carrier frequency f2. The frequency separation between the side bands of this DSBSC modulated wave is effectively twice the first carrier frequency f1, there by permitting the second filter to remove the unwanted side band.
Time-domain description
The time domain description of an SSB wave s(t) in the canonical form is given by the equation 3.15.
where SI(t) is the in-phase component of the SSB wave and SQ(t) is its quadrature component. The in-phase component SI(t) except for a scaling factor, may be derived from S(t) by first multiplying S(t) by cos(2πf c t ) and then passing the product through a low-pass filter. Similarly, the quadrature component SQ(t), except for a scaling factor, may be derived from s(t) by first multiplying s(t) by sin(2πfc t ) and then passing the product through an identical filter.
The Fourier transformation of SI(t) and SQ(t) are related to that of SSB wave as follows, respectively.
Consider the SSB wave that is obtained by transmitting only the upper side band, shown in figure 3.11. Two frequency shifted spectraS ( f − fc ) and S( f + fc ) are shown in figure 3.12 and figure 3.13 respectively. Therefore, from equations 3.16 and 3.17, it follows that the corresponding spectra of the in- phase component SI(t) and the quadrature component SQ(t) are as shown in figure 3.14 and 3.15 respectively.
From the figure 3.14, it is found that
where M(f) is the Fourier transform of the message signal m(t). Accordingly in-phase component SI(t) is defined by equation 3.18.
Following the same procedure, we can find the canonical representation for an SSB wave s(t) obtained by transmitting only the lower side band is given by
Phase discrimination method of SSB generation
Time domain description of SSB modulation leads to another method of SSB generation using the equations (3.23) or (3.24). The block diagram of phase discriminator is as shown in figure 3.16.
The phase discriminator consists of two product modulators I and Q, supplied with carrier waves in-phase quadrature to each other. The incoming base band signal m(t) is applied to product modulator I, producing a DSBSC modulated wave that contains reference phase sidebands symmetrically spaced about carrier frequency fc. The Hilbert transform mˆ (t ) of m(t) is applied to product modulator Q, producing a DSBSC modulated that contains side bands having identical amplitude spectra to those of modulator I, but with phase spectra such that vector addition or subtraction of the two modulator outputs results in cancellation of one set of side bands and reinforcement of the other set. The use of a plus sign at the summing junction yields an SSB wave with only the lower side band, whereas the use of a minus sign yields an SSB wave with only the upper side band. This modulator circuit is called Hartley modulator.
Single –tone SSB-LSB modulation
Consider a single-tone message signal m(t ) = Am cos(2πf m t ) and its Hilbert transform mˆ (t ) = Am sin(2πfmt ). Substituting these in equation (3.24), we get
Therefore the single tone SSBSC wave is a sinusoidal wave of frequency equal to sum/ difference of carrier and message frequencies for USB/LSB.
Demodulation (coherent detection) of SSBSC wave
Demodulation of SSBSC wave using coherent detection is as shown in 3.17. The SSB wave s(t) together with a locally generated carrier c(t ) = A 1 cos(2πf t + φ) is applied to a product modulator and then low-pass filtering of the modulator output yields the message signal.
The first term in the above equation 3.26 is desired message signal. The other term represents an SSB wave with a carrier frequency of 2fc as such; it is an unwanted component, which is removed by low-pass filter.
