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Chapter 3
CONTINUOUS WAVE AND
FREQUENCY MODULATED
RADAR
Keywords. CW radar, Doppler frequency shift
3.1 THE DOPPLER EFFECT
A radar detects the presence of objects and locates their position in space by transmitting electromagnetic energy and observing the returned echo. A pulse radar transmits a relatively short burst of electromagnetic energy, after which the receiver is turned on to listen for the echo. The echo not only indicates that a target is present, but the time that elapses between the transmission of the pulse and the reception of its echo is a measure of the distance to the target. Separation of the echo signal from the transmitted signal is made on the basis of differences in time.
The radar transmitter may be operated continuously rather than pulsed if it is possible to separate the strong transmitted signal from the weak echo.
The received echo-signal power is considerably smaller than the transmitter power (as low as 10−^18 times the transmitter power - or sometimes even less). Separate antennas for transmission and reception help isolate the weak echo from the strong leakage signal, but this isolation is usually not sufficient. A feasible technique for separating the received signal from the transmitted signal, when there is relative motion between radar and target, is based on recognizing the change in the echo-signal frequency caused by what is known as the doppler effect.
It is well known in the field of optics and acoustics that if there is relative motion between the source of a signal and the observer of the signal, along the line joining the two, then an apparent shift in frequency will result. This is the doppler effect and is the basis of CW (Continuous Wave) radars.
Consider Fig.3.1 above in which a CW radar and a target are placed at a distance of R from each other. The target is moving with a speed Vr relative to the radar and along the line joining the radar and the target (also known as the line-of-sight - LOS). Note that the transmitted signal is not in the form of a train of pulses (as in Chapter 2) but a continuous wave with frequency fo. Let the total number of wavelengths (given by λ) contained in the to-and -fro path between the radar and the target be denoted by n. Then,
n =^2 λR (3.1)
One wavelength corresponds to an angular excursion of 2π radians.Thus, the total angular excursion φ made by the electromagnetic wave during its transit to the target and back to the radar is
φ =^2 λ R. 2 π =^4 πRλ (3.2)
When the target is in motion, both R and φ are changing. Now a change
Where,
fd = doppler frequency shift, in Hz c = velocity of propagation = 3 × 108 m/s
Vr = relative velocity of the target with respect to the radar along the line-of-sight.
For a stationary radar and a moving target the relative velocity may be written as
Vr = V cos θ (3.5)
where, V is the target speed and θ is the angle made by the target velocity vector with the LOS. When θ = 0, the doppler frequency is a maximum. The doppler frequency is zero when the trajectory is perpendicular to the radar-target line-of-sight (that is, θ = π 2 = 90o). Also note that the doppler frequency shift positive for an approaching target (that is, Vr is considered to be positive) and negative for a receding target (that is, Vr is considered to be negative).
EXAMPLE 3.1: Positions of the two aircraft, A and B, are as shown in the figure below. Aircraft A has a speed of 600 m/sec and carries a CW radar transmitting at 300 MHz frequency and tracking aircraft B which has a speed of 800 m/sec.
(a) What is the doppler frequency shift recorded by the radar in aircraft A?
(b) Is this shift positive or negative?
(c) What should be the flight direction of aircraft B for the doppler fre- quency shift to be zero?
ANSWER
(a) The transmitted frequency = f 0 = 300M Hz = 300 × 106 Hz.
The relative velocity of aircraft A with respect to aircraft B along the LOS is given by,
vr = 600 cos 45^0 + 800 cos 30^0 = 1117. 08 m/sec. (3.6)
The doppler frequency shift
= fd =^2 vr c f^0 =^2 ×^1117.^08 ×^300 ×^10
6 3 × 108 = 2234.^16 Hz^ (3.7)
(b) Note that aircraft B is actually moving towards aircraft A in a relative sense and hence it is an approaching target, that is, the LOS between A and B is shrinking with time. Thus, the doppler shift is positive, which means
mitted signal. Apart from this, the CW radar also provides a measurement of relative velocity which may be used to distinguish moving targets from stationary objects and clutter.
The expression for doppler frequency shift given in (3.4) is somewhat approximate, though it serves quite well for most practical purposes. The correct expression for the frequency f ∗^ of the echo signal from a target, moving with relative velocity Vr, when the transmitted frequency is fo, is given by
f ∗^ = f 0. 1 + 1 −^ VVr/c r/c^
which, on expansion by Taylor’s series and truncation beyond the first order term, reduces to
f ∗^ = f 0 (1 + 2Vr/c) (3.11)
Truncation beyond the first order terms is justified when vr c (which is usually the case and implies that vr/c is a very small quantity). This, in turn, yields the expression for doppler frequency shift given in (3.4). The phase shift associated with the return signal is
(4πf 0 R)/c (1 − Vr/c) (3.12)
3.2 THE CW RADAR
In Fig. 3.2 we give the block diagram of a simple CW radar.
The transmitter generates a continuous (unmodulated) oscillation of fre- quency f 0 , which is radiated by the antenna. A portion of the radiated energy
is intercepted by the target and is scattered. some of it in the direction of the radar, where it is collected by the receiving antenna. If the target is in motion with a velocity Vr relative to the radar, the received signal frequency
Figure 3.2: A simple CW radar block diagram
will be shifted from the transmitted signal frequency f 0 by an amount ±fd. The plus sign applies if the distance between the radar and the target is decreasing (that is, an approaching target) and the minus sign applies when this distance is increasing (that is, a receding target). The received echo signal at a frequency f 0 ± fd enters the radar via the antenna and is hetero-
3.3 ISOLATION BETWEEN TRANSMITTER
AND RECEIVER
A single antenna serves the purpose of both transmission and reception in the simple CW radar described above. Though, in principle, a single antenna is sufficient as the necessary isolation is obtained by the separation in frequency (as a result of doppler effect), in practice there is considerable transmitter leakage. But this leakage is beneficial too since it supplies the reference fre- quency necessary for the detection of the doppler frequency shift. Otherwise a sample of the transmitted signal must be made available at the receiver. However, there are two reasons why the amount of transmitter leakage power should be kept at a low value.
- The maximum power the receiver input circuitry can withstand, with- out being physically damaged or having its sensitivity reduced, is quite low.
- The transmitter noise which enters the receiver from the transmitter reduces receiver sensitivity.
The amount of isolation required depends on the transmitter power and the accompanying transmitter noise as well as the ruggedness and sensitivity of the receiver. If the safe value of power which might be applied to a receiver were 10mw and if the transmitter power were 1 kw, the isolation between transmitter and receiver must be at least 50 dB.
In long range CW applications, it is the level of the noise accompanying the transmitter leakage signal, rather than the damage this leakage might cause to the receiver circuitry, which determines the amount of isolation required. For example, suppose the isolation between the transmitter and receiver were such that 10mw of leakage signal appeared at the receiver. If
the minimum detectable signal were 10−^13 watt, the transmitter noise must be at least 110 dB below the transmitted carrier.
3.4 SIGN OF THE RADIAL VELOCITY
In many applications of CW radar it is of interest to know if the target is ap- proaching or receding. This might be determined with separate filters located on either side of the intermediate frequency.If the echo-signal frequency lies below the carrier, then the target is receding; whereas if the echo frequency is greater that the carrier, then the target is approaching. This is shown in Fig. 3.3 given below. However, the doppler-frequency spectrum ”folds over” in the video because of the action of the detector, and hence the information about whether the doppler shift is positive or negative is lost. But it is pos- sible to determine its sign from a technique borrowed from single-sideband communication. If the transmitter signal is given by,
Et = Eocos wot
The echo signal from the moving target will be,
Er = K 1 E 0 cos [(wo + wd)t + φ]
where, E 0 = amplitude of the transmitted signal K 1 = a constant determined from the radar equation representing the reduction in power of the echo signal wo = angular frequency of transmitted signal, rad/sec wd = dopper angular frequency shift, rad/sec φ = a constant phase shift, which depends upon the range of initial detection (i.e., distance between the radar and the target)
Figure 3.4: Determination of the sign of the Doppler frequency
The channel B has π 2 phase delay introduced in the reference signal. The output of the channel B mixer is,
EB = K 2 E 0 cos(±wdt + φ + π 2 ) (3.13)
If the target is approaching (positive doppler),the outputs from the two chan- nels are, EA = K 2 E 0 cos(wdt + φ) (3.14) EB = K 2 E 0 cos(wdt + φ + π 2 ) (3.15)
on the other hand, if the target is receding (negative doppler),
EA(−) = K 2 E 0 cos(wdt + φ) (3.16)
EB (−) = K 2 E 0 cos(wdt + φ + π 2 ) (3.17)
the sign of wd and the direction of the target’s motion may be determined according to whether the output of channel B leads or lags the output of channel A. One method of determining the relative phase relationship be- tween the two channels is to apply the outputs of the two channels to a synchronous two-phase motor. The direction of the motor’s rotation is an indication of the direction of the target’s motion.
