Usual and unusual concepts
2.4 POLYPHASE MODULATION AND DEMODULATION
Simple amplitude modulation stays in the alternating current domain, namely,
where A is the carrier frequency amplitude;
fc is the carrier frequency;
Nc is the carrier phase;
B is the modulation frequency amplitude;
f is the modulation frequency; m Nm is the modulation phase;
t is the time in seconds.
The modulated wave consists of two sidebands at f – f and f + f . Notice that the phases of the modulation of thesec m c m sidebands have opposite signs. Thus, one is the complex conjugate of the other.
A signal returned from a scatterer contains the original modified by the movement of the scatterer. This modification is the change of phase per second between the transmitter and the receiver. The phase of the transmitted pulse changes at 2B radians along each wavelength, 8. If the scatterer moves at v m/s towards the radar, then the change of distance to and from the scatterer per second is 2v, and the change of phase is 2v 2B/8 radians per second or 2v/8 Hz. For an S- band radar, wavelength 0.1 m, a transmitted pulse at 3 GHz is returned at 3 000 001 000 Hz by a scatterer moving at 50 m/s towards the radar.
In practice, the echoes are a mixture of fixed and moving echoes. This is equivalent to single-sideband modulation in communications. In Figure 2.6, the fixed echo vector or clutter is drawn, by convention, vertically. The moving echo changes its phase with time so that the combined vector is amplitude and phase modulated. Mathematically, this may be represented by
Carrier or fixed echo vector Modulation or
moving echo vector
Sum of carrier and modulation or fixed and
moving echo vectors
m
1
_____
1 + m
Resultant vector ' 1 % m exp(jN)
-1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
2 4 6 8 10 12 14 16 18 20
phi
-0.1 0 0.1 0.2
phi 1.6
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
6.2 6.3 6.4
phi
Phase difference
A Modulated waveform
B Reference waveform C Phase modulation
A
B C
A B Figure 2.6 Vectors representing fixed and moving echoes.
(2.10)
Figure 2.7 A single-sideband modulated carrier.
where N = 2 B fDoppler.
A representation of how this looks on an oscillograph with a greatly reduced ratio of carrier to modulation frequency is shown in Figure 2.7.
The phase differences in Figure 2.7 are too small to be seen without magnification. Figure 2.8 shows the same waveform but with the phase angle transferred to the complex plane so as to look like the vectors in Figure 2.6. In Figure
-1 -0.5 0 0.5 1
5 10 15
20 25
-0.5 1
Modulation envelope,
a spiral
Carrier in amplitude and phase Time
Amplitude Phase
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2 4 6 8 10 12 14 16 18 20
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2 4 6 8 10 12 14 16 18 20
Reference directions for three- phase power systems
Rotating vector
Reference directions for two- phase systems
Three-phase waveforms Two-phase waveforms
Cartesian reference axes
Figure 2.8 A single-sideband modulated carrier with the phase shown in the complex plane.
Figure 2.9 Three-phase and two-phase representations of a rotating vector.
2.8 the axes have been rotated to give the best view of the waveform.
Ideally a vector detector would remove the carrier and leave the helix as a signal for further processing. Because it is impossible to pass ideal vector videos on a single wire, such components do not exist and another solution must be found. This is the same as the voltage in a three-phase power circuit or a circuit with any number of phases. Three-phase and two-phase systems are shown in Figure 2.9. The two-phase system has Cartesian reference axes.
The number of phases for power systems was settled early in the 20th century, although a number of two-phase cables were still in use many, many years later. Le Blanc transformers perform the transformation for voltage and the number of phases. Counting the return wires and the neutral, four wires are required for transmission in a three-phase system and five for a two-phase system. For balanced circuits, three and four conductors are required. This leaves the three-phase power distribution system as that which transmits polyphase power with the least number of conductors.
Polyphase voltage ' ej 2Bft
phase1 ' V sin(2Bft) ' V1 phase2 ' V sin(2Bft % 120°) ' 8 V1 phase3 ' V sin(2Bft % 240°) ' 82V1
Polyphase voltage ' V exp j2Bft
' V (cos 2Bft % j sin 2Bft) ' Re(V) % Im(V)
-1 -0.5
0 0.5 1
-1
-0.5
0.5
1
0.1
0.2
0.3
Real
Imaginary
Time Sin(2B10t)
Cos(2B10t)
Exp(2B 10t)
8 ' exp j2B
3 ' &1 2 % j 3
2
3 Vphase
(2.11)
(2.12)
Figure 2.10 A 10 Hz polyphase voltage and its two-phase components.
Mathematically, for three phases [1],
where
The line voltage for a three-phase system is the voltage between the vector arrow tips. That is , where Vphase is the phase voltage or the length of the arrowed vector. These voltages are commonly as follows:
Phase voltage 110 117 220 240
Three-phase line voltage 191 203 381 416 For two-phase or Cartesian representation this is
This is shown in Figure 2.10.
The power delivered by a three-phase power circuit is the sum of the powers in each of the individual phases. In alternating current and polyphase systems, the power is the voltage times the conjugate of the current. Statistically, this is the correlation function which, for time limited waveforms, one waveform is dragged by the other and the integral of their product is plotted. This is shown in Figure 2.11.
Polyphase power may be converted into large numbers of phases using transformers with star and zig-zag windings.
This is useful for rectifying polyphase alternating currents. Each of the phases is fed to a separate rectifier so that the ripple amplitude is decreased and the ripple frequency is increased. Smaller capacitors are required for smoothing.
The energy in the correlation function is represented by its peak. With constant impedance, the current is proportional to the voltage so that power is proportional to the voltage multiplied by its complex conjugate. The peak of the autocorrelation function is always real. For nonperiodic waveforms, the envelope tends to a steady line. The energy per second gives the power.
-0.5 0 0.5 1
-0.5
0.5 -1.5
-1 -0.5
0.5 1
1.5
0 0.5 1
-1 -0.5
0.5 1 -1.5
-1 -0.5
0.5 1
1.5 -1
-0.5 0 0.5 1
-1 -0.5
0.5 1 -1.5
-1
1 1.5 -0.5
0.5
-1 -0.5 0 0.5 1
-1
-0.5 0.5
1 -1.5
-1 -0.5
1 1.5 0.5
x
Modulus of the cross-correlation
function Cross-correlation function is formed as
this waveform is drawn past the other
Cross-correlation function (as a function of displacement x) x
x
+ 4
I
g(u-x)h(u) du!
!!
!4 g(u-x)
h(u)
.
u
u
Synchronous detector
COHO distribution 0° or reference phase
COHO + 90°
Coherent oscillator
or COHO
In-phase or I video
Quadrature or Q video Synchronous
detector Input
signal
Two-phase or Cartesian detector
I Q
Synchronous detectors have been in use since the late 1930s as channel detectors in multichannel frequency division multiplex (FDM) telephone
1
transmission systems and, by extension, for the detection of single-sideband (SSB) radio signals. Before the Second World War the normally 60 kHz pilot signal for the modulators and demodulators was distributed from one central oscillator. This is the case with the coherent oscillator (COHO) in a radar.
Figure 2.11 Polyphase power representation.
Figure 2.12 A two-phase or Cartesian detector.
In order to remove the carrier and preserve all the amplitude and phase information, the real and imaginary components of the modulation must be determined. The reference phase of the carrier is provided by the coherent oscillator (COHO) of the radar. Synchronous detectors give the Cartesian component with respect to the reference phase1 so that two separate synchronous detectors are required for each of the two Cartesian component outputs. The block diagram of such a detector is shown in Figure 2.12.
The rotating vector on the right of Figure 2.12 may rotate to the right or the left depending on whether the Doppler
Linear amplitude
detector
Coherent oscillator or COHO
Amplitude video, A
Phase video, N Phase
detector Input
signal
Polar detector
A N
Positive phase sequence component ' V% exp%j(2Bft % 2) ; Negative phase sequence component ' V& exp&j(2Bft % 2)
Figure 2.13 Block diagram of a polar detector.
(2.13) frequency of the echo is above or below the carrier. This affects the order, or sequence, of the peaks appearing in the in-phase and quadrature videos, which are called positive and negative sequences.
Theoretically, polar detection is possible. Historically, videos formed from first the modulus (linear detection) or the square of the modulus (square law detection) were used for display on A-scopes and plan position indicators (PPI).
Much theoretical work has been carried out on these signals. Later, moving target indication (MTI) stages were developed using the phase information only from a phase detector. So far, the integrated circuit components used for signal processing are made for Cartesian components, so polar detectors are a curiosity. Figure 2.13 is a block diagram of such a detector.
The term “second detector” comes from superheterodyne radio receivers and this is the stage where the modulation is recovered (see Chapter 9, Detectors).