Radar ...............................................

Prediction of Range Performance
• The simple form of the radar equation expressed the maximum radar range
Rmax, in terms of radar and target parameters:
• All the parameters are to some extent under the control of the radar
designer, except for the target cross section σ.
• The radar equation states that if long ranges are desired, the transmitted
power must be large.
• the radiated energy must be concentrated into a narrow beam the
received echo energy must be collected with a large antenna aperture,
and the receiver must be sensitive to weak signals.

• In practice, the simple radar equation does not predict the range
performance of actual radar equipments to a satisfactory degree of
accuracy.
• The predicted values of radar range are usually optimistic. In some cases the
actual range might be only half that predicted.
The failure of the simple form of the radar equ is due to
• Statistical nature of the min detectable signal (usually receiver noise)
• Fluctuations and uncertainties in the targets radar cross section
• The losses experienced throughout a radar system
• Propagation effects caused by the earths surface and atmosphere.
• Receiver noise and target cross section requires that the max radar range
should be described probabilistically rather than single parameter.
• Range of radar will be function of the probability of detection Pb and the
probability of false alarm Pfa

Minimum Detectable Signal
• The ability of a radar receiver to detect a weak echo signal is limited by the
noise energy that occupies the same portion of the frequency spectrum
as does the signal energy.
• The weakest signal the receiver can detect is called the minimum
detectable signal.
• The specification of the minimum detectable signal is sometimes difficult
because of its statistical nature and because the criterion for deciding
whether a target is present or not may not be too well defined
• Detection is based on establishing a threshold level at the output of the
receiver.
• If the receiver output exceeds the threshold, a signal is assumed to be
present. This is called threshold detection.
• From fig its clear that The envelope has a fluctuating appearance caused by
the random nature of noise.

• If a large signal is present such as at A in Fig., it is greater than the
surrounding noise peaks and can be recognized on the basis of its
amplitude.
• If the threshold level were set sufficiently high, the envelope would not
generally exceed the threshold if noise alone were present, but would
exceed it if a strong signal were present.
• If the signal were small, however, it would be more difficult to
recognize its presence.
• The threshold level must be low if weak signals are to be detected, but
it cannot be so low that noise peaks cross the threshold and give a false
indication of the presence of targets.
• A matched filter is one designed to maximize the output peak signal to
average noise (power) ratio.

• The ideal matched-filter receiver cannot always be exactly realized in
practice, but it is possible to approach it with practical receiver circuits
• The output of a matched-filter receiver is the cross correlation
between the received waveform and a replica of the transmitted
waveform.
• Hence it does not preserve the shape of the input waveform.

Radar ...............................................

Receiver Noise
Receiver Noise:
• Since noise is the chief factor limiting receiver sensitivity, it is
necessary to obtain some means of describing it quantitatively.
Noise is unwanted electromagnetic energy which interferes with
the ability of the receiver to detect the wanted signal.
• It may originate within the receiver itself, or it may enter via the
receiving antenna along with the desired signal.
• If the radar were to operate in a perfectly noise-free environment
so that no external sources of noise accompanied the desired signal,
and if the receiver itself were so perfect that it did not generate
any excess noise
• still exist an unavoidable component of noise generated by the
thermal motion of the conduction electrons in the ohmic
portions of the receiver input stages.

• This is called thermal noise, or Johnson noise, and is directly
proportional to the temperature of the ohmic portions of the circuit
and the receiver bandwidth.
• The available thermal-noise power generated by a receiver of bandwidth
Bn, (in hertz) at a temperature T (degrees Kelvin) is equal to
Where , k = Boltzmann's constant = 1.38 x 10-23
J/deg. If the temperature T is
taken to be 290 K, (62˚F), the factor kT is 4 x 10-21
W/Hz of bandwidth.
• If the receiver circuitry were at some other temperature, the thermal-noise
power would be correspondingly different.
• A receiver with a reactance input such as a parametric amplifier need not
have any significant ohmic loss. The limitation in this case is the
thermal noise seen by the antenna and the ohmic losses in the
transmission line.

• For radar receivers of the super heterodyne type ,the receiver bandwidth is
approximately that Of the intermediate-frequency stages. It should be
cautioned that the bandwidth B, of Eq. is not the 3-dB, or half-power,
bandwidth commonly employed by electronic engineers. It is an integrated
bandwidth and is given by,
Where, H(f) = frequency-response characteristic of IF amplifier (filter) and fo =
frequency of maximum response (usually occurs at midband). When H (f) is
normalized to unity at midband (maximum-response frequency), H (fo) = 1.
The bandwidth Bn is called the noise bandwidth and is the bandwidth of an
equivalent rectangular filter whose noise-power output is the same as the
filter with characteristic H (f).
• The 3-dB bandwidth is defined as the separation in hertz between the points
on the frequency-response characteristic where the response is reduced to
0.707 (3 dB) from its maximum value.

• The noise power in practical receivers is often greater than can be
accounted for by thermal noise alone. The additional noise components
are due to mechanisms other than the thermal agitation of the conduction
electrons
• The total noise at the output of the receiver may be considered to be equal
to the thermal-noise power obtained from an" ideal” receiver
multiplied by a factor called the noise figure.
• The noise figure Fn of a receiver is defined by the equation
Where, No = noise output from receiver, and Ga = available gain. The standard
temperature T is taken to be 290 K

• The noise No is measured over the linear portion of the receiver input-output
characteristic, usually at the output of the IF amplifier before the
nonlinear second detector.
• The receiver bandwidth Bn is that of the IF amplifier in most receivers. The
available gain Ga is the ratio of the signal out So to the signal in Si, and
kToBn is the input noise Ni in an ideal receiver. Equation of noise figure
may be rewritten as
• The noise figure may be interpreted, therefore, as a measure of the
degradation of signal-to-noise-ratio as the signal passes through the receiver.
• Rearranging Eq. above the input signal may be expressed as

• If the minimum detectable signal Smin, is that value of Si corresponding to
the minimum ratio of output (IF) signal-to-noise ratio (So /No)min necessary
for detection, then
• Substituting Eq. discussed above into Eq. earlier results in the following
form of the radar equation:

Detection Criteria:
Detection of signals is equivalent to deciding whether the receiver
output is due to noise alone or to signal +Noise
This type of information made by a human operator from the
information presented on a radar display
When detection process is carried out automatically by electronic
means without the aid of an operator, the detection criterion must be
carefully and built into the decision making device

Threshold detection:
If the envelope of the receiver output exceeds a pre-established
threshold a signal is said to be present
Threshold level divides the output into a region of no detection
and region of detection.
The radar engineer selects the threshold that divides these two
regions so as to archive a specified probability of false alarm
The other parameters of the radar needed to obtain the signal-to-
noise for the desired probability of detection.

To find Minimum signal to noise ratio required to achieve a
specified probability of detection and probability of false alarm
Signal to noise ratio is needed in order to calculate the maximum
range of a radar .

Figure shows a portion of a superheterodyne radar receiver with IF amplifier
of bandwidth BIF second detector, video amplifier with bandwidth Bv and a
threshold where the detection is made.
the IF filter, second detector and video filter form an envelope detector
in that the output of the video amplifier is the envelope or modulation of the
IF signal.
bandwidth of the radar receiver is the bandwidth of the IF amplifier.
Then envelope of the IF amplifier output is the signal applied to the threshold
detector.
When the receiver output crosses the threshold, a signal is declared to be
Envelope Detector:

• The noise entering the IF filter (the terms filter and amplifier are used
interchangeably) is assumed to be gaussian, with probability-density
function given by
• where p(v) dv is the probability of finding the noise voltage v between the
values of and v+ dv, Ψo is the variance, or mean-square value of the
noise voltage.
• If gaussian noise were passed through a narrowband IF , the probability
density function of the envelope R is given by a form of Rayleigh pdf

. The probability that the envelope of the noise voltage exceed the voltage threshold
Vt is the integral of P( R) evaluated from Vt to ∞
• The probability that the noise voltage envelope will exceed the voltage threshold
VT is
• Thus the probability of false alarm denoted by Pfa is
• Whenever the voltage envelope exceeds the threshold, target detection is
considered to have occurred, by definition.
• Since the probability of a false alarm is the probability that noise will cross the
threshold, Eq. above gives the probability of a false alarm, denoted Pfa

Figure shows the occurrence of false alarms. The average time between
crossings of the decision threshold when noise alone is present is called the
false alarm time Tfa
Where Tk is the time between crossings of the threshold VT by the noise
envelope.

The false alarm probability can be expressed in terms of false alarm time by noting the
false alarm probability Pfa is the ratio of the time the envelope is actually above the
threshold to the total time it could have been above the threshold
The average duration of a noise pulse is approximately the reciprocal of the IF
bandwidth , which in the case of the envelope detector is BIF. The average of Tk is the
false alarm time Tfa

• Consider sine-wave signals of amplitude A to be present along with
noise at the input to the IF filter. The frequency of the signal is the same
as the IF mid band frequency fIF. The output or the envelope detector has a
probability-density function given by
• When the signal is absent, A = 0, the probability-density
function for noise alone. Equation above is sometimes called
the Rice probability-density function.

• The probability that the signal will be detected (which is the probability of
detection) is the same as the probability that the envelope R will exceed the
predetermined threshold VT. The probability of detection Pd, is therefore
The expression for Pd along with equ is a function of the signal
amplitude A, threshold VT, and mean noise power . In radar systems
analysis it is more convenient to use signal to noise ratio S/N than

Integration of Radar Pulses
• Where Wr= revolutions per minute (rpm) if a 360 rotating antenna. The
number of pulses received n is usually called hits per scan or pulses per
scan
• Typical parameters for a ground-based search radar might be pulse
repetition frequency 300 Hz, 1.5˚ beam width, and antenna scan rate 5 rpm
(30˚/s).
• These parameters result in 15 hits from a point target on each scan. The
process of summing all the radar echo pulses for the purpose of
improving detection is called integration. Many techniques might be
employed for accomplishing integration.
The Number of pulses returned from a point target by a scanning radar with
pulse repetition rate of fp Hz, an antenna beamwidth degrees, and which
scans at a rate of degress per second is

• Integration may be accomplished in the radar receiver either before the
second detector (in the IF) or after the second detector (in the video).
• Integration before the detector is called pre detection, or coherent,
integration, while integration after the detector is called post detection,
or non coherent, integration.
• Pre detection integration is theoretically lossless, but it requires it requires
the phase of the echo signal pulses to be known and preserved so as
combine the sinewave pulses in phase without loss.
• If n pulses, all of the same signal-to-noise ratio, were integrated by an ideal
pre detection integrator, the resultant, or integrated, signal-to-noise (power)
ratio would be exactly n times that of a single pulse.
• If the same n pulses were integrated by an ideal post detection device, the
resultant signal-to-noise ratio would be less than n times that of a single
pulse.

• This loss in integration efficiency is caused by the nonlinear action of the
second detector, which converts some of the signal energy to noise
energy in the rectification process.
• The comparison of pre detection and post detection integration may be
briefly summarized by stating that although post detection integration is
not as efficient as pre detection integration, it is easier to implement in
most applications.
• Post detection integration is therefore preferred, even though the
integrated signal-to-noise ratio may not be as great.
• An alert, trained operator viewing a properly designed cathode-ray tube
display is a close approximation to the theoretical post detection integrator.

• The efficiency of post detection integration relative to ideal pre detection
integration has been computed by Marcum when all pulses are of equal
amplitude. The integration efficiency may be defined as follows:

Radar Cross section of Targets
Radar Cross Section of Targets:
• The radar cross section of a target is the (fictional) area intercepting that
amount of power which when scattered equally in all directions, produces
an echo at the radar equal to that from the target; or in other terms,

• For most common types of radar targets such as aircraft, ships, and
terrain, the radar cross section does not necessarily bear a simple
relationship to the physical area, except that the larger the target size, the
larger the cross section is likely to be.
• Scattering and diffraction are variations of the same physical process.
• When an object scatters an electromagnetic wave, the scattered field is
defined as the difference between the total field in the presence of the
object and the field that would exist if the object were absent.
• On the other hand, the diffracted field is the total field in the presence of
the object. With radar backscatter, the two fields are the same, and one may
talk about scattering and diffraction interchangeably.
• In theory, the scattered field, the radar cross section, can be determined by
solving Maxwell's equations with the proper boundary conditions applied.

• The radar cross section of a simple sphere is shown in Fig. as a function of
its circumference measured in wavelengths (2Πa /λ, where a is the radius
of the sphere and λ is the wavelength).
• The region where the size of the sphere is small compared with the
wavelength (2Πa/λ << 1) is called the Rayleigh region, after Lord
Rayleigh who, in the early 1870’s first studied scattering by small particles.
• Lord Rayleigh was interested in the scattering of light by microscopic
particles, rather than in radar. His work preceded the original
electromagnetic echo experiments of Hertz by about fifteen years.
• The Rayleigh scattering region is of interest to the radar engineer because
the cross sections of raindrops and other meteorological particles fall
within this region at the usual radar frequencies.

• The radar cross section in the Rayleigh region is proportional to the
fourth power of the frequency , and is determined more by the volume
of the scatter than by its shape (λ-4
)
• rain and clouds are essentially invisible to radars which operate at relatively
long wavelengths (low frequencies).
• The usual radar targets are much larger than raindrops or cloud particles, and
lowering the radar frequency to the point where rain or cloud echoes are
negligibly small will not seriously reduce the cross section of the larger
desired targets.

• At the other extreme from the Rayleigh region is the optical region,
where the dimensions of the sphere are large compared with the
wavelength (2Πa/λ >> 1).
• Here radar scattering from a complex object such as an aircraft is
characterized by significant changes in the cross section when there is a
change in a frequency or aspect angel at which the object is viewed.
• Scattering from aircrafts or ships at microwave frequencies generally in
the optical region.
• For large 2Πa/λ, the radar cross section approaches the optical cross
section.
• In between the optical and the Rayleigh region is the Mie, or resonance
region, where the wavelength is comparable to the object dimensions.

Transmitter Power
Transmitter Power:
• The power Pt in the radar equation is called by the radar engineer the peak
power.
• The peak pulse power as used in the radar equation, it is not the
instantaneous peak power of a pulse of sine wave, but one half the
instantaneous peak value
• The average radar power Pav, is also of interest in radar and is defined as
the average transmitter power over the duration of total transmission
(pulse-repetition period).
• If the transmitted waveform is a train of rectangular pulses of width τ and
pulse-repetition period Tp = l/fp, the average power is related to the peak
power by

• The Radar duty cycle can be expressed as Pav/Pt, τ/Tp, or τfp, A Pulse radar
might typically have duty cycles of from 0.001 to 0.5 more or less. while a
CW radar which transmits continuously has a duty cycle of unity.
• Writing the radar equation in terms of the average power rather than the
peak power, we get

• The bandwidth and the pulse width are grouped together since the product
of the two is usually of the order of unity in most pulse-radar applications.
• definition of duty cycle given above, the energy per pulse and
also equal to
• Substitute this in the above radar equ.
• where ET is the total energy of the n pulses, which equal nEP

PRF and Range Ambiguities
• The “Pulse Repetition Frequency (PRF)” is determined primarily by the
maximum unambiguous range at which targets are not expected.
• From equ unambiguous range
• Is given by
• Echoes which appear from beyond the maximum unambiguous range,
especially from some large target or clutter source or when anomalous
conditions occur at extend the normal range of the radar beyond the horizon

• Echo signals arrived at the time later than the pulse-repetition period are
called multiple-time-around echoes.
• The apparent range of these ambiguous echoes can result in error and
confusion
• Another problem with multiple around echoes is that clutter echoes
from ranges grater than Run can mask target echoes at the short ranges
• Consider the three targets labeled A. B, and C in Fig.
• Target A is located within the maximum unambiguous range Runamb of the
radar, target B is at a distance greater than Runamb but less than 2Runamb while
target C is greater than 2Runamb but less than 3Runamb. The appearance of the
three targets on an A-scope is sketched in Fig

• The multiple-time-around echoes on the A-scope cannot be
distinguished from proper target echoes actually within the maximum
unambiguous range. Only the range measured for target A is correct;
those for B and C are not.
• One method of distinguishing multiple-time-around echoes from
unambiguous echoes is to operate with a varying pulse repetition
frequency. The echo signal from an unambiguous range target will appear
at the same place on the A-scope.
• However, echoes from multiple-time-around targets will be spread
over a finite range as shown in Fig.9 c. The prf may be changed
continuously within prescribed limits or it may be changed discretely
among several predetermined values.
• The number of separate pulse repetition frequencies will depend upon the
degree of the multiple-time targets. Second-time targets need only two
separate repetition frequencies in order to be resolved.

• One of the fundamental limitations is the fold over of nearby targets; that
is, nearby strong ground targets (clutter) can be quite large and can
mask weak multiple-time-around targets appearing at the same place
on the display.
• more time is required to process the data when resolving ambiguities.
• Ambiguities may theoretically be resolved by observing the variation
of the echo signal with time (range).

System Losses
System Losses:
• One of the important factors omitted from the simple radar equation
was the losses that occur throughout the radar system. The losses reduce
the signal-to-noise ratio at the receiver output.
• predicted with any degree of precision beforehand.
• The antenna beam-shape loss, collapsing loss, and losses in the microwave
plumbing are examples of losses which can be calculated if the system
configuration is known.
• These losses are very real and cannot be ignored in any serious
prediction of radar performance.
• Losses not readily subject to calculation and which are less predictable
include those due to field degradation and to operator fatigue or lack of
operator motivation.
• All the loss mentioned above vary considerably depending on the radar
design and how the radar is maintained.

Plumbing loss:
• There is always some finite loss experienced in the transmission lines
which connect the output of the transmitter to the antenna.
• The losses in decibels per 100 ft for radar transmission lines
• At the lower radar frequencies the transmission line introduces little
loss. At the higher radar frequencies, attenuation may not always be
small and may have to be taken into account.
• In addition to the losses in the transmission line itself, an additional loss
can occur at each connection or bend in the line and at the antenna
rotary joint if used.
• The signal suffers attenuation as it passes through the duplexer. the
greater the isolation required from the duplexer on transmission, the
larger will be the insertion loss.
• By insertion loss is meant the loss introduced when the component, in this
case the duplexer, is inserted into the transmission line.

• The precise value of the insertion loss depends to a large extent on the
particular design. For a typical duplexer it might be of the order of 1 dB.
• In S-band (3000 MHz) radar, for example, the plumbing losses might be as
follows:

• The antenna gain that appears in the radar equation was assumed to be a
constant equal to the maximum value.
• But in reality the train of pulses returned from a target with a scanning
radar is modulated in amplitude by the shape of the antenna beam.
• probability of detection would have to be performed assuming a
modulated train of pulses rather than constant-amplitude pulses.
• Instead a beam-shape loss is added to the radar equation to account for
the fact that the maximum gain is employed in the radar equation
rather than a gain that changes pulse to pulse.
• This is a simpler, accurate method. It is based on calculating the reduction
in signal power and thus does not depend on the probability of detection
Beam-shape loss:

• Let the one-way-power antenna pattern be approximated by the
gaussian expression exp (-2.78θ2
/θB
2
), where θ is the angle measured from
the center of the beam and θB is the half power beam width.
• If nB is the number of pulses received within the half-power beamwidth θB,
and n is the total number of pulses integrated, then the beam-shape loss
relative to a radar that integrates all n pulses with an antenna gain
corresponding to the maximum gain at the beam center is

Limiting loss:
• Limiting in the radar receiver can lower the probability of
detection.
• Although a well-designed and engineered receiver will not limit
the received signal under normal circumstances, intensity
modulated CRT displays such as the PPI or the B-scope have limited
dynamic range
• Some receivers, however, might employ limiting for some special
purpose.

• If the radar were to integrate additional noise samples along with
signal-to-noise pulses, the added noise results in degradation called the
collapsing loss.
• It can occur in displays which collapse the range information, such as the
C-scope which displays elevation vs. azimuth angle.
• The echo signal from a particular range interval must compete in a
collapsed-range C-scope display
• not only with the noise energy contained within that range interval but
with the noise energy from all other range intervals at the same
elevation and azimuth.
• A collapsing loss can occur when the output of a high-resolution radar is
displayed on a device whose resolution is coarser than that inherent in the
radar.
• A collapsing loss also results if the outputs of two (or more) radar
receivers are combined and only one contains signal while the other
contains noise.
Collapsing loss:

• The mathematical derivation of the collapsing loss, assuming a square-
law detector may be carried out as suggested by Marcum who has
shown that
• The integration of m noise pulses, along with n signal-plus-noise pulses
with signal-to-noise ratio per pulse (S/N)n, is equivalent to the integration
of m + n signal-to-noise pulses each with signal-to-noise ratio
n(S/N)n /(m + n).
• The collapsing loss is equal to the ratio of integration loss Li for m+n
pulses to the integration loss for n pulses

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