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Effective
receiving
Area Aeff                                                                             PT .GT
Power Density =
4π .R 2

Tx                     PT                                                                              Sphere x gain
of
antenna

GT
Fraction of incident power
density
of source
PT .G T        1
Power Density =           . δ.
4π .R 2      4π .R 2

R

PT G T σ
Output power from the receive antenna                     PR =
(4πR )  2 2
Aeff

4π
General antenna relationship                 G=           Aeff
λ2

PT G T λ σ
2    2
Output power from receive antenna                         PR =
( 4 π ) R4
3

Maximum range                 Rmax occurs when PR = S min (minimum detectable signal)
1

⎡ PT GT λ2 σ ⎤ 4
2
Rmax = ⎢             ⎥
⎣ (4π ) S min ⎦
3

Rmax proportional to (PT ) 4 - to double Rmax need 16 x PT
1
Note:       i)

ii) apparent dependence on            λ2      can be misleading. For a fixed antenna
aperture

1
G∝                         ∴ G 2 λ 2 ∝ 1 λ2
T
λ   2

suggests short wavelengths for maximum range
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The radar equation above over-estimates the maximum range because it does not include

•        effects of the propagating medium and multipath effects eg atmospheric absorption, ducting
etc
•        atmospheric noise
•        system losses - in antenna feeds, etc
•        signal processing noise
•        target fluctuations
•        clutter - radar scattering from the region around the target which is illuminated by the radar
beam

The received radar signal is superimposed on a 'noise' signal. The noise can arise from

•        atmospheric noise - fluctuations in the meteorological conditions
•        target fluctuations - as the target orientation with respect to the radar beam changes its radar
cross-section (RCS) changes
•        clutter - reflections from areas of the ground or sea surface around the target which are

The word noise was put in inverted commas above because noise is strictly a random and
spontaneous process whereas the target fluctuations and clutter may be systematic to some extent,
though as far as the qualitative radar return signal is concerned their effects appear to be quite
random.
return signals to the radar that may combine with different phases to increase or decrease the direct-
path echo.

A typical echo + noise radar receiver output is shown below. (Note that the signal shown is the
envelope of the microwave signal - the GHz frequency microwave signal is fed through an envelope
detector). It is clear that the correct setting of a threshold is vital if targets are to be correctly identified
without false alarms. If the thresh-hold is set too high genuine targets will be missed, if it is too low a
peak in the noise signal can give a false alarm.
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Threshold

Output
Voltage

Time

No
Noise In
Rx                                      (Si/Ni )
Noise Figure (F) =                F. ≥ 1 F ≥ 0dB
So                          (So/No )
Signal In
G
powers

Fig 15 Signal + noise at receiver output, showing the importance of setting the thresh-
hold to avoid missing targets or false alarms.

Si     So
F=
Ni     No

Here             N i = kTo B is the available noise power at the input

⎛ So   ⎞
If the minimum acceptable signal to noise ratio at the radar receiver output is ⎜
⎜                                    ⎟
⎟
⎝ No   ⎠ min

this requires a minimum signal at the receiver input of S min which is given by

⎛S           ⎞        ⎛S      ⎞
S i = S min = F ⎜ o
⎜N           ⎟N i = F ⎜ o
⎟        ⎜N      ⎟kTo B
⎟
⎝ o          ⎠        ⎝ o     ⎠

Using this value of S min in the equation derived earlier for Rmax we obtain

1

⎡        PT GT λ2 σ
2              ⎤4
Rmax = ⎢                             ⎥
⎢ (4π ) kT0 B F (S 0 N 0 )min ⎥
3
⎣                             ⎦

B is the IF bandwidth of the receiver. Usually we make B ≈ 1/ τ where                                 τ    is the width of the radar
pulse.
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From the above formula it would appear that we can increase the maximum range by decreasing the
radar receiver bandwidth, but the effect is to distort and diminish the echo pulse, thereby reducing the
sensitivity of the radar. If the pulse width (ie duration) is increased to accommodate a decrease in the
bandwidth B this reduces the radar resolution.

The most effective way of increasing the maximum range is to decrease the receiver noise figure F by
using low noise devices in the front end of the receiver - eg a HEMT.

Target fluctuations

•       radar targets are complex shapes with dimensions which are usually large compared with the
radar wavelength so that there is considerable scope for constructive and destructive
interference between the waves reflected from different parts of the target
•       the target may change its orientation with respect to the incident radar beam during the time it
is being observed
•       the reflectivity of a target is specified by its RCS (radar cross-section) s which is a measure
of the fraction of the incident energy that is scattered in the direction of the receiving antenna
•       it is difficult to calculate the RCS of real targets. RCS values are usually determined by
measurement as a statistical average of the signal reflected by the target. RCS values
depend upon the radar wavelength and the orientation of the target. The figure below shows
the variation of RCS with the direction of illumination for an aircraft. The RCS can vary by
20dB or more. These variations are superimposed on variations due to atmospheric

RCS target fluctuations – an         Typical vertical coverage pattern for a
aircraft illuminated                 surveillance radar, showing the lobes and
from different directions.           nulls caused by ground reflections.
Ci l            i      l
Fig.16 Radar cross-section of an aircraft illuminated from different directions (from P
A Lynn)
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•         a range of models - Swerling models - are used to describe the statistical variations in the
RCS s in terms of probability distributions p (σ )dσ which gives the probability that the
cross-section lies between σ and σ + dσ . The modells are summarised below.

Case1. Echoes have a constant amplitude for all hits on one scan but there is no correlation
between the echoes from one scan to the next.
This form of distribution applies to complex shapes with many similar reflecting
surfaces eg aircraft
Case2. Echoes vary from pulse to pulse as well as scan to scan - this may arise with a
rapidly fluctuating target

⎛ −σ ⎞
The PDF for both cases 1 and 2 is p (σ ) =
1
exp⎜       ⎟ with σ ≥ 0
σ (av ) ⎜ σ (av ) ⎟
⎝       ⎠
s(av) is the long-term average RCS.

Cases 3 and 4. These are the same as cases 1 and 2, respectively, but they apply to
targets which have one dominant reflecting surface. Then

4σ          ⎛ − 2σ ⎞
p(σ ) =                       ⎜ σ (av ) ⎟ with σ ≥ 0
exp⎜         ⎟
σ (av )    2
⎝         ⎠
Multipath effects

Multipath arises when the signal reflected by the target is returned to the receive antenna by more
than one path. Then the total received signal depends upon the phase relationships between the
signals which travel by the different paths. Usually the most important paths are the direct signal and
the ground reflected signal - particularly if the 'ground' is a water surface because at microwave
frequencies water has a reflection coefficient which is close to unity and so the amplitudes of the
direct and the ground waves will be almost equal. Their sum can vary between almost zero
(antiphase) and twice each amplitude.

‘in phase’ or
180 degrees out of
phase
Ground reflected

Receiving
signal

range

Fig 17     Multipath
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Multipath depends upon the environment of the receive antenna - ie adjacent scatterers - and
the polar pattern of the antenna. It may be possible to point nulls of the antenna radiation
pattern in the direction of strong multipath scatterers.

Clutter

Clutter refers to the scattering of a radar beam by the ground, sea etc around the target. Because the
target may occupy only a small fraction of the total area illuminated by the radar beam its echo may
be lost in the clutter. In some cases the clutter signal may be random and noise-like in nature, but it
may also have systematic features - for example, scattering from the sea may be particularly strong at
frequencies where there is some sort of match between the radar wavelength and the wavelength of
the water waves - either the main waves which are immediately apparent, or the 'fine-structure' waves
that are superimposed on the main waves. These in turn will depend upon the wind strength and
direction and the depth of the water, so that there will be correlations between the clutter
characteristics and the meteorological conditions.
On land, similarly, clutter depends upon the nature and topology of the surface, the moisture content
of the ground, the siting of the radar antenna etc.
Various steps can be taken to diminish the effects of clutter on the echo signal, such as
(i) filter out echo signals that are do not have a Doppler frequency shift - this permits the
removal of the clutter component from echoes from moving targets
(ii) de-sensitise the receiver for a short time after the transmission of a radar pulse, so that
you will reduce the signal from the receiver due to clutter from the close environment,
which is likely to be dominant. This is done to avoid saturating the receiver. It does           not
improve the signal to clutter ratio for the receiver

Avoids
Tx                                                     saturation

sensitivity

time

Fig 18 Receiver sensitivity (gain) reduction for short ranges to reduce clutter signal

(iii) shape the antenna beam to reduce clutter

Sea surface                                  Sea surface

Fig 19 Narrowing of radar beam to reduce clutter
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(iv) if the statistical distribution of the clutter (its PDF) is known it may be possible to
reduce its effect by appropriate signal processing.

Pulse compression
• the sensitivity of a radar - ie its ability to detect weak echo signals - depends upon the energy
which is contained in the transmitted pulse, not the power transmitted. Thus, long medium power
level pulses are as effective as short high power pulses if they contain the same amount of
energy. Lower power pulses avoid electrical breakdown in waveguide etc and are more suitable
for semiconductor sources.
• a solution is to use pulse compression in which the short radar pulse is lengthened before
transmission to reduce its avarage power level, but coded so that its frequency changes linearly
during the duration of the pulse (a chirp pulse). The echo chirp pulse is compressed (pulse
compression) as soon as it is received so that the range resolution of the original short pulse is
restored. Pulse expansion and compression can be achieved using matching pairs of SAW
(surface acoustic wave).

P(out)

f

Fig. 20 A chirp pulse (linear FM pulse)
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• a large effective antenna aperture is obtained by using an antenna on a moving platform (an
aircraft or an orbiting satellite) and combining the returned signals from radar pulses transmitted
at many successive positions. The antenna may be considered as a large linear array

•   the large effective aperture gives a very large improvement in the angular resolution of the radar

•   SARs are usually sideways-looking radars that illuminate a swathe of the earth's surface. The
resolution is often specified in terms of the along-track or azimuthal resolution and the range
resolution perpendicular to the swathe. The along track resolution is improved by the application
of the SAR technique, but the range resolution is determined by the pulse duration in the same
way as for a conventional radar. Range resolution can be improved by standard pulse
comprression techniques

•   SARs have many applications in remote sensing The advantage of microwave remote sensing
is that it can be carried out at night and through rain and fog ie under conditions where optical
methods cannot be used. Polar regions which have continuous cloud cover for long periods can
always be monitored. This information is invaluable for the understanding and prediction of a
range of environmental coditions - weather forecasting, global warming and sea level changes.

Remote sensing applications of SAR include

•   measurement of ocean currents
•   measurements of ocean temperatures
•   monitoring of vegetation, use of pesticides, fertilizers etc
•   surveys of natural resources - minerals etc
•   imaging of the earth's surface
•   monitoring of the atmosphere - CFCs, temperature distributions etc

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