Binary and Ternary High Resolution Codes Generation Using CHEBYSHEV CHAOTIC METHOD in MATLAB

8/7/2019 Binary and Ternary High Resolution Codes Generation Using CHEBYSHEV CHAOTIC METHOD in MATLAB

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1.INTRODUCTION

Beginning with short pulses, and going on to frequency modulated pulses

and beyond, waveforms have been developed for each of the common task of radar

and sonar-initial detection, localization and classification. Chaotic waveforms which

are generated by non-linear systems offer a new very broad source of signals. They

are deterministic (defined by an iterative map or differential equation), and can

therefore be practically implemented. They are non periodic, which suggests there

are potential advantages in security and can be used as (infinitely) long spreading

sequences.

There has been intense interest in their use in covert communications

systems, and this work provides concepts and results which are useful in long range

sensing by radar and sonar.

Pulse compression schemes using linear FM have seen wide applications.

Linear FM, featuring simple implementation and post processing, has very high peak

side lobe level and the width of the main lobe is relatively large, which limits therange resolution. Windowing techniques are usually applied to suppress the side lobe

level.

The performance of range resolution radar depends on the autocorrelation

pattern of the coded waveform which is nothing but the matched filter output. For best

performance, the autocorrelation pattern of the optimum coded waveform must have a

large peak value for zero shift and zero value for non-zero shifts.

In this work, good binary phase codes and ternary codes are generated

using chebyshev map equation to achieve a low PSL. It is not an exhaust search

method. It is possible to generate infinite number of codes at larger lengths easily , by

changing the initial conditions by very small increment, threshold level and

bifurcation factor.

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2.BASICS OF RADAR

2.1 RADAR IN BRIEF

Radar is an electromagnetic sensor for the detection and location of

reflecting objects. Its operation can be summarized as follows:

The radar radiates electromagnetic energy from an antenna to propagate in

space. Some of the radiated energy is intercepted by a reflecting object, usually called

a target, located at a distance from the radar. The energy intercepted by the target isreradiated in many directions. Some of the reradiated (echo) energy is returned to and

received by the radar antenna.

After amplification by a receiver and with the aid of proper signal

processing, a decision is made at the output of the receiver as to whether or not a

target echo signal is present. At that time, the target location and possibly other

information about the target is acquired.

2.2 Block Diagram of Radar System:

It is a very elementary basic block diagram showing the subsystems usually found in

radar. The transmitter, which is shown here as a power amplifier, generates a suitable

waveform for the particular job the radar is to perform. It might have an average

power as small as milli watts or as large as megawatts. (The average power is a far

better indication of the capability of radars performance than is its peak power.) Most

radars use a short pulse waveform so that a single antenna can be used on a time-shared basis for both transmitting and receiving.

The function of the duplexer is to allow a single antenna to be used by

protecting the sensitive receiver from burning out while the transmitter is on and by

directing the received weak echo signal to the receiver rather than to the transmitter.

The antenna is the device that allows the transmitted energy to be propagated into

space and then collects the echo energy on receive. It is almost always a directive

antenna, one that directs the radiated energy into a narrow beam to concentrate thepower as well as to allow the determination of the direction to the target. An antenna

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that produces a narrow directive beam on transmit usually has a large area on receive

to allow the collection of weak echo signals from the target. The antenna not only

concentrates the energy on transmit and collects the echo energy on receive, but it also

acts as a spatial filter to provide angle resolution and other capabilities.

Figure 2.1 block diagram of radar system.

The receiver amplifies the weak received signal to a level where its

presence can be detected. Because noise is the ultimate limitation on the ability of a

radar to make a reliable detection decision and extract information about the target,

care is taken to insure that the receiver produces very little noise of its own. At themicrowave frequencies, where most radar are found, the noise that affects radar

performance is usually from the first stage of the receiver, as a low-noise amplifier.

For many radar applications where the limitation to detection is the unwanted radar

echoes from the environment (called clutter), the receiver needs to have a large

enough dynamic range so as to avoid having the clutter echoes adversely affect

detection of wanted moving targets by causing the receiver to saturate. The dynamic

range of a receiver, usually expressed in decibels, is defined as the ratio of themaximum to the minimum signal input power levels over which the receiver can

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operate with some specified performance. The maximum signal level might be set by

the nonlinear effects of the receiver response that can be tolerated (for example, the

signal power at which the receiver begins to saturate), and the minimum

Signal might be the minimum detectable signal. The signal processor,which is often in the IF portion of the receiver, might be described as being the part of

the receiver that separates the desired signal from the undesired signals that can

degrade the detection process. Signal processing includes the matched filter that

maximizes the output signal-to-noise ratio. Signal processing also includes the doppler

processing that maximizes the signal-to-clutter ratio of a moving target when clutter is

larger than receiver noise, and it separates one moving target from other moving

targets or from clutter echoes.

The detection decision is made at the output of the receiver, so a target is

declared to be present when the receiver output exceeds a predetermined threshold. If

the threshold is set too low, the receiver noise can cause excessive false alarms. If the

threshold is set too high, detections of some targets might be missed that would

otherwise have been detected. The criterion for determining the level of the decision

threshold is to set the threshold so it produces an acceptable predetermined average

rate of false alarms due to receiver noise.

After the detection decision is made, the track of a target can be

determined, where a track is the locus of target locations measured over time. This is

an example of data processing. The processed target detection information or its track

might be displayed to an operator; or the detection information might be used to

automatically guide a missile to a target; or the radar output might be further

processed to provide other information about the nature of the target. The radar

control insures that the various parts of a radar operate in a coordinated and

cooperative manner, as, for example, providing timing signals to various parts of theradar as required.

The radar engineer has as resources time that allows good Doppler

processing, bandwidth for good range resolution, space that allows a large antenna,

and energy for long range performance and accurate measurements. External factors

affecting radar performance include the target characteristics; external noise that

might enter via the antenna,interference from other electromagnetic radiators; and

propagation effects due to the earths surface and atmosphere.

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2.3 Radar Frequency bands

The traditional band names originated as code-names during World War

II and are still in military and aviation use throughout the world in the 21st century.

They have been adopted in the United States by the IEEE, and internationally by theITU. Most countries have additional regulations to control which parts of each band

are available for civilian or military use.

Other users of the radio spectrum, such as thebroadcasting and electronic

countermeasures (ECM) industries, have replaced the traditional military designations

with their own systems.

Bandname

Frequencyrange

Wavelengthrange

Notes

HF 330MHz 10100 mCoastal radar systems, over-the-horizon

radar(OTH) radars; 'high frequency'

P < 300 MHz 1 m+'P' for 'previous', applied retrospectively

to early radar systems

VHF 30330 MHz 0.96 m Very long range, ground penetrating;'very high frequency'

UHF 3001000 MHz 0.31 m

Very long range (e.g.Tf_2_nprsid3495173ballistic missile

early warning), ground penetrating,

foliage penetrating; 'ultrahigh frequency'

L 12GHz 1530 cmLong range air traffic control and

surveillance; 'L' for 'long'

S 24 GHz 7.515 cmTerminal air traffic control, long-range

weather, marine radar; 'S' for 'short'

C 48 GHz 3.757.5 cm

Satellite transponders; a compromise

(hence 'C') between X and S bands;

weather

X 812 GHz 2.53.75 cm Missile guidance, marine radar, weather,

medium-resolution mapping and ground

surveillance; in the USA the narrow

range 10.525 GHz 25 MHz is used forairport radar. Named X band because the

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frequency was a secret during WW2.

Ku 1218 GHz 1.672.5 cm high-resolution

K 1824 GHz 1.111.67 cm

from Germankurz, meaning 'short';limited use due to absorption by water

vapour, so Ku and Ka were used insteadfor surveillance. K-band is used fordetecting clouds by meteorologists, and

by police for detecting speedingmotorists. K-band radar guns operate at24.150 0.100 GHz.

Ka 2440 GHz 0.751.11 cm

mapping, short range, airportsurveillance; frequency just above K

band (hence 'a') Photo radar, used totrigger cameras which take pictures oflicense plates of cars running red lights,operates at 34.300 0.100 GHz.

Mm 40300 GHz7.5 mm

1 mm

milli metre band, subdivided as below.The frequency ranges depend onwaveguide size. Multiple letters areassigned to these bands by differentgroups. These are from Baytron, a nowdefunct company that made testequipment.

Q 4060 GHz7.5 mm

5 mmUsed for Military communication.

V 5075 GHz 6.04 mm Very strongly absorbed by atmosphericoxygen, which resonates at 60 GHz.

W 75110 GHz 2.7 4.0 mm

Used as a visual sensor for experimental

autonomous vehicles, high-resolution

meteorological observation, and

imaging.

UWB 1.610.5 GHz18.75 cm

2.8 cm

Used for through-the-wall radar and

imaging systems.

2.4 Radar equation

The powerPrreturning to the receiving antenna is given by the radar equation:

Pr = (PtGtArF4)/ (4)2 * Rt2 *Rr2

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..2.1

Where

Pt = transmitter power

Gt = gain of the transmitting antenna

Ar= effective aperture (area) of the receiving antenna

= radar cross section, or scattering coefficient, of the target

F= pattern propagation factor

Rt = distance from the transmitter to the target

Rr= distance from the target to the receiver.

In the common case where the transmitter and the receiver are at the same

location,Rt =Rr and the termRtRr can be replaced byR4, whereR is the range. This

yield:

This shows that the received power declines as the fourth power of the range,

which means that the reflected power from distant targets is very, very small.

The equation above with F = 1 is a simplification for vacuum without

interference. The propagation factor accounts for the effects of multi path and

shadowing and depends on the details of the environment. In a real-world situation,

path loss effects should also be considered.

2.5 Range-Resolution

In apulse compression system, the range-resolution of the radar is given by the

length of the pulse at the output-jack of the pulse compressing stage. The ability to

compress the pulse depends on the bandwidth of the transmitted pulse (BW tx) (not by

its pulse width). As a matter of course the receiver needs at least the same bandwidth

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to process the whole spectrum of the echoes and c0 is speed of light at which all

electromagnetic waves propagate, Sr is range sensitivity.

.2.2

2.5.1 Polarization_9_7_ _

In the transmitted radar signal, the electric field is perpendicular to the

direction of propagation, and this direction of the electric field is the polarization of

the wave. Radars use horizontal, vertical, linear and circular polarization to detect

different types of reflections. For example, circular polarization is used to minimize

the interference caused by rain. Linear polarization returns usually indicate metal

surfaces.Random polarization returns usually indicate a fractal surface, such as rocks

or soil, and are used by navigation radars.

2.5.2 Interference

Radar systems must overcome unwanted signals in order to focus only on

the actual targets of interest. These unwanted signals may originate from internal and

external sources, both passive and active. The ability of the radar system to overcome

these unwanted signals defines its signal-to-noise ratio (SNR). SNR is defined as the

ratio of signal power to the noise power within the desired signal. In less technical

terms, SNR compares the level of a desired signal (such as targets) to the level ofbackground noise. The higher a system's SNR, the better it is in isolating actual targets

from the surrounding noise signals.

2.5.3 Noise

Signal noise is an internal source of random variations in the signal, which

is generated by all electronic components. Noise typically appears as random

variations superimposed on the desired echo signal received in the radar receiver. The

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Sr c0

--------

2 BWtx
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lower the power of the desired signal, the more difficult it is to discern it from the

noise (similar to trying to hear a whisper while standing near a busy road). Noise

figureis a measure of the noise produced by a receiver compared to an ideal receiver,

and this needs to be minimized.Noise is also generated by external sources, most importantly the natural

thermal radiation of the background scene surrounding the target of interest. In

modern radar systems, due to the high performance of their receivers, the internal

noises is typically about equal to or lower than the external scene noise. An exception

is if the radar is aimed upwards at clear sky, where the scene is so "cold" that it

generates very little thermal noise.

2.5.4 Clutter

Clutter refers to radio frequency (RF) echoes returned from targets which

are uninteresting to the radar operators. Such targets include natural objects such as

ground, sea,precipitation (such as rain, snow or hail), sand storms, animals (especially

birds), atmospheric turbulence, and other atmospheric effects, such as ionosphere

reflections, meteor trails, and three body scatter spike. Clutter may also be returned

from man-made objects such as buildings and, intentionally, by radar countermeasuressuch aschaff.

There are several methods of detecting and neutralizing clutter. Many of

these methods rely on the fact that clutter tends to appear static between radar scans.

Therefore, when comparing subsequent scans echoes, desirable targets will appear to

move and all stationary echoes can be eliminated. Sea clutter can be reduced by using

horizontal polarization, while rain is reduced with circular polarization (note that

meteorological radars wish for the opposite effect, therefore using linear polarization

the better to detect precipitation). Other methods attempt to increase the signal-to-

clutter ratio.

Constant False Alarm Rate (CFAR, a form ofAutomatic Gain Control, or

AGC) is a method relying on the fact that clutter returns far outnumber echoes from

targets of interest. The receiver's gain is automatically adjusted to maintain a constant

level of overall visible clutter. While this does not help detect targets masked by

stronger surrounding clutter, it does help to distinguish strong target sources. In the

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past, radar AGC was electronically controlled and affected the gain of the entire radar

receiver. As radars evolved, AGC became computer-software controlled, and affected

the gain with greater granularity, in specific detection cells.

Clutter may also originate from multipath echoes from valid targets due toground reflection, atmospheric ducting or ionosphere reflection/refraction. This clutter

type is especially bothersome, since it appears to move and behave like other normal

(point) targets of interest, thereby creating a ghost. In a typical scenario, an aircraft

echo is multipath-reflected from the ground below, appearing to the receiver as an

identical target below the correct one. The radar may try to unify the targets, reporting

the target at an incorrect height, or - worse - eliminating it on the basis ofjitteror a

physical impossibility. These problems can be overcome by incorporating a groundmap of the radar's surroundings and eliminating all echoes which appear to originate

below ground or above a certain height. In newer Air Traffic Control (ATC) radar

equipment, algorithms are used to identify the false targets by comparing the current

pulse returns, to those adjacent, as well as calculating return improbabilities due to

calculated height, distance, and radar timing.

2.5.5 Jamming

Radar jamming refers to radio frequency signals originating from sources

outside the radar, transmitting in the radar's frequency and thereby masking targets of

interest. Jamming may be intentional, as with an electronic warfare (EW) tactic, or

unintentional, as with friendly forces operating equipment that transmits using the

same frequency range. Jamming is considered an active interference source, since it is

initiated by elements outside the radar and in general unrelated to the radar signals.

Jamming is problematic to radar since the jamming signal only needs to

travel one-way (from the jammer to the radar receiver) whereas the radar echoes travel

two-ways (radar-target-radar) and are therefore significantly reduced in power by the

time they return to the radar receiver. Jammers therefore can be much less powerful

than their jammed radars and still effectively mask targets along the line of sight from

the jammer to the radar (Main lobe Jamming). Jammers have an added effect of

affecting radars along other lines of sight, due to the radar receiver's side lobes ( Side

lobe Jamming).

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2.6 TYPES OF RADARS

Although there is no single way to characterize a radar, here we do so by

means of what might be the major feature that distinguishes one type of radar from

another.

Pulse radar: This is radar that radiates a repetitive series of almost-rectangular pulses.

It might be called the canonical form of a radar, the one usually thought of as a radar

when nothing else is said to define a radar.

High-resolution radar. High resolution can be obtained in the range, angle, or

Doppler velocity coordinates, but high resolution usually implies that the radar has

high range resolution. Some high-resolution radars have range resolutions in terms offractions of a meter, but it can be as small as a few centimetres.

Pulse compression radar. This is a radar that uses a long pulse with internal modu-

lation (usually frequency or phase modulation) to obtain the energy of a long pulse

with the resolution of a short pulse.

Continuous wave (CW) radar. This radar employs a continuous sine wave. It almost

always uses the Doppler frequency shift for detecting moving targets or for measuringthe relative velocity of a target.

FM-CW radar. This CW radar uses frequency modulation of the waveform to allow

a range measurement.

Surveillance radar. Although a dictionary might not define surveillance this way,

surveillance radar is one that detects the presence of a target (such as an aircraft or a

ship) and determines its location in range and angle. It can also observe the target over

a period of time so as to obtain its track.

Moving target indication (MTI). This is pulse radar that detects moving targets in

clutter by using a low pulse repetition frequency (PRF) that usually has no range

ambiguities. It does have ambiguities in the Doppler domain that result in so-called

blind speeds.

Pulse Doppler radar. There are two types of pulse Doppler radars that employ either

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high or medium PRF pulse radar. They both use the Doppler frequency shift to extract

moving targets in clutter. A high PRF pulse doppler radar has no ambiguities (blind

speeds) in Doppler, but it does have range ambiguities. A medium PRF pulse doppler

radar has ambiguities in both range and Doppler.

Tracking radar. This is a radar that provides the track, or trajectory, of a target.

Tracking radars can be further delineated as STT, ADT, TWS, and phased array

trackers as described below:

Single Target Tracker (STT). Tracks a single target at a data rate high enough to

provide accurate tracking of a man overing target. A revisit time of 0.1 s (data rate of

10 measurements per second) might be typical. It might employ the mono pulsetracking method for accurate tracking information in the angle coordinate.

Automatic detection and tracking (ADT). This is tracking performed by sur-

veillance radar. It can have a very large number of targets in track by using the

measurements of target locations obtained over multiple scans of the antenna. Its data

rate is not as high as the STT. Revisit times might range from one to 12 seconds,

depending on the application.

Track-while-scan (TWS). Usually a radar that provides surveillance over a narrow

region of angle in one or two dimensions, so as to provide at a rapid update rate

location information on all targets within a limited angular region of observation. It

has been used in the past for ground-based radars that guide aircraft to a landing, in

some types of weapon-control radars, and in some military airborne radar.

Phased array tracker. An electronically scanned phased array can (almost) con-tinuously track more than one target at a high data rate. It can also simultaneously

provide the lower data rate tracking of multiple targets similar to that performed by

ADT.

Imaging radar. This radar produces a two-dimensional image of a target or a scene,

such as a portion of the surface of the earth and what is on it. These radars usually are

on moving platforms.

Side looking airborne radar (SLAR). This airborne side looking imaging radar pro-

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vides high resolution in range and obtains suitable resolution in angle by using a

narrow beam width antenna.

Synthetic aperture radar (SAR). SAR is a coherent* imaging radar on a moving

vehicle that uses the phase information of the echo signal to obtain an image of a

scene with high resolution in both range and cross-range. High range resolution is

often obtained using pulse compression.

Inverse synthetic aperture radar (ISAR). ISAR is coherent imaging radar that uses

high resolution in range and the relative motion of the target to obtain high resolution

in the doppler domain that allows resolution in the cross-range dimension to be

obtained. It can be on a moving vehicle or it can be stationary.

Weapon control radar. This name is usually applied to a single-target tracker used

for defending against air attack.

Guidance radar. This is usually a radar on a missile that allows the missile to home

in, or guide itself, to a target.

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2.7 INFORMATION AVAILABLE FROM ARADAR

Detection of targets is of little value unless some information about the

target is obtained as well. Likewise, target information without target detection is

meaningless.

2.7.1 Range.

Probably the most distinguishing feature of a conventional radar is its

ability to determine the range to a target by measuring the time it takes for the

radar signal to propagate at the speed of light out to the target and back to the

radar. No other sensor can measure the distance to a remote target at long range

with the accuracy of radar (basically limited at long ranges by the accuracy of the

knowledge of the velocity of propagation). At modest ranges, the precision can be

a few centimetres. To measure range, some sort of timing mark must be

introduced on the transmitted waveform. A timing mark can be a short pulse (an

amplitude modulation of the signal), but it can also be a distinctive modulation of

the frequency or phase. The accuracy of a range measurement depends on the

radar signal bandwidth: the wider the bandwidth, the greater the accuracy. Thusbandwidth is the basic measure of range accuracy.

2.7.2 Radial Velocity.

The radial velocity of a target is obtained from the rate of change of range

over a period of time. It can also be obtained from the measurement of the Doppler

frequency shift. The speed of a moving target and its direction of travel can be

obtained from its track, which can be found from the radar measurements of the

target location over a period of time.2.7.3Angular Direction.

One method for determining the direction to a target is by determining the

angle where the magnitude of the echo signal from a scanning antenna is

maximum. This usually requires an antenna with a narrow beam width (a high-

gain antenna). Air-surveillance radar with a rotating antenna beam determines

angle in this manner. The angle to a target in one angular dimension can also be

determined by using two antenna beams, slightly displaced in angle, and

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comparing the echo amplitude received in each beam. Four beams are needed to

obtain the angle measurement in both azimuth and elevation.

2.7.4 Size and Shape.

If the radar has sufficient resolution capability in range or angle, it can

provide a measurement of the target extent in the dimension of high resolution.

Range is usually the coordinate where resolution is obtained. Resolution in cross

range (given by the range multiplied by the antenna beam width) can be obtained

with very narrow beam width antennas. However, the angular width of an antenna

beam is limited, so the cross-range resolution obtained by this method is not as

good as the range resolution. Very good resolution in the cross-range dimension

can be obtained by employing the doppler frequency domain, based on SAR

(synthetic aperture radar) or ISAR (inverse synthetic aperture radar systems),

There needs to be relative motion between the target and the radar in order to

obtain the cross-range resolution by SAR or ISAR. With sufficient resolution in

both range and cross-range, not only can the size be obtained in two orthogonal

coordinates, but the target shape can sometimes be discerned..

2.7.5 Signal-to-Noise Ratio.

The accuracy of all radar measurements, as well as the reliable detection

of targets depends on the ratio E/No, where E is the total energy of the received

signal that is processed by the radar and No is the noise power per unit bandwidth

of the receiver. Thus E/No is an important measure of the capability of a radar.

2.7.6 The Doppler Shift in Radar.

The importance of the doppler frequency shift began to be appreciated for

pulse radar shortly after World War II and became an increasingly importantfactor in many radar applications. Modern radar would be much less interesting or

useful if the doppler effect didnt exist. The doppler frequency shift fd can be

written as

fd=2Vr/=(2vcos)/

2.3

where Vr = v cos q is the relative velocity of the target(relative to the radar) in m/s, v is the absolute velocity of the

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target in m/s, is the radar wavelength in m, and is the angle

between the targets direction and the radar beam.

The doppler frequency shift is widely used to separate moving targets

from stationary clutter. Such radars are known as MTI (moving target indication),AMTI (airborne MTI), and pulse doppler. All modern air-traffic control radars, all

important military ground-based and airborne air-surveillance radars, and all

military airborne fighter radars take advantage of the Doppler Effect. Yet in

WWII, none of these pulse radar applications used doppler. The CW (continuous

wave) radar also employs the Doppler effect for detecting moving targets, but CW

radar for this purpose is not as popular as it once was. The HF OTH radar could

not do its job of detecting moving targets in the presence of large clutter echoes

from the earths surface without the use of doppler.

2.8 Applications of Radar

2.8.1 Military Applications.

Radar was invented in the 1930s because of the need for defence against

heavy military bomber aircraft. The military need for radar has probably been its

most important application and the source of most of its major developments,including those for civilian purposes.

The chief use of military radar has been for air defense operating from

land, sea, or air. It has not been practical to perform successful air defense without

radar. In air defense, radar is used for long-range air surveillance, short-range

detection of low-altitude pop-up targets, weapon control, missile guidance, non

cooperative target recognition, and battle damage assessment. The proximity fuze

in many weapons is also an example of radar. An excellent measure of the success

of radar for military air defense is the large amounts of money that have been

spent on methods to counter its effectiveness.

These include electronic counter measures and other aspects of electronic

warfare, anti radiation missiles to home on radar signals, and low cross-section

aircraft and ships. Radar is also used by the military for reconnaissance, targeting

over land or sea, as well as surveillance over the sea.

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On the battlefield, radar is asked to perform the functions of air

surveillance (including surveillance of aircraft, helicopters, missiles, and

unmanned airborne vehicles), control of weapons to an air intercept, hostile

weapons location (mortars, artillery, and rockets), detection of intrudingpersonnel, and control of air traffic.

The use of radar for ballistic missile defence has been of interest ever

since the threat of ballistic missiles arose in the late 1950s. The longer ranges, high

supersonic speeds, and the smaller target size of ballistic missiles make the

problem challenging. There is no natural clutter problem in space as there is for

defence against aircraft, but ballistic missiles can appear in the presence of a large

number of extraneous confusion targets and other countermeasures that an attacker

can launch to accompany the re entry vehicle carrying a warhead. The basic

ballistic missile defence problem becomes more of a target recognition problem

rather than detection and tracking. The need for warning of the approach of

ballistic missiles has resulted in a number of different types of radars for

performing such a function. Similarly, radars have been deployed that are capable

of detecting and tracking satellites.

A related task for radar that is not military is the detection and

interception of drug traffic. There are several types of radars that can contribute to

this need, including the long-range HF over-the-horizon radar.

2.8.2 Remote Sensing of the Environment.

The major application in this category has been weather observation radar

such as the Nexrad system whose output is often seen on the television weather

report. There also exist vertical-looking wind-profiler radars that determine wind

speed and direction as a function of altitude, by detecting the very weak radar echofrom the clear air. Located around airports are the Terminal Doppler Weather

Radar (TDWR) systems that warn of dangerous wind shear produced by the

weather effect known as the downburst, which can accompany severe storms.

There is usually a specially designed weather avoidance radar in the nose of small

as well as large aircraft to warn of dangerous or uncomfortable weather in flight.

Another successful remote-sensing radar was the downward-looking

space borne altimeter radar that measured worldwide the geoid (the mean sealevel, which is not the same all over the world), with exceptionally high accuracy.

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There have been attempts in the past to use radar for determining soil moisture and

for assessing the status of agriculture crops, but these have not provided sufficient

accuracy. Imaging radars in satellites or aircraft have been used to help ships

efficiently navigate northern seas coated with ice because radar can tell whichtypes of ice are easier for a ship to penetrate.

2.8.3 Air-Traffic Control.

The high degree of safety in modern air travel is due in part to the

successful applications of radar for the effective, efficient, and safe control of air

traffic. Major airports employ an Airport Surveillance Radar (ASR) for observing

the air traffic in the vicinity of the airport. Such radars also provide information

about nearby weather so aircraft can be routed around uncomfortable weather.

Major airports also have a radar called Airport Surface Detection

Equipment (ASDE) for observing and safely controlling aircraft and airport

vehicle traffic on the ground. For control of air traffic en route from one terminal

to another, long-range Air Route Surveillance Radars (ARSR) are found

worldwide.

The Air Traffic Control Radar Beacon System (ATCRBS) is not a radar

but is a cooperative system used to identify aircraft in flight. It uses radar-like

technology and was originally based on the military IFF (Identification Friend or

Foe) system.

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2.8.4 Other Applications.

A highly significant application of radar that provided information not

available by any other method, was the exploration of the surface of the planet

Venus by an imaging radar that could see under the ever-present clouds that maskthe planet.

One of the widest used and least expensive of radars has been the civil marine

radar found throughout the world for the safe navigation of boats and ships. Some

readers have undoubtedly been confronted by the highway police using the CW

doppler radar to measure the speed of a vehicle. Ground penetrating radar has been

used to find buried utility lines, as well as by the police for locating buried objects

and bodies. Archaeologists have used it to determine where to begin to look for

buried artefacts. Radar has been helpful to both the ornithologist and entomologist

for better understanding the movements of birds and insects. It has also been dem-

onstrated that radar can detect the gas that is often found over underground oil and

gas deposits.

A long-range radarantenna, known as ALTAIR, used to detect and track

space objects in conjunction with ABM testing at the Ronald Reagan Test Site on

the Kwajaleinatoll.

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3.MATCHED FILTER

In telecommunications, a matched filter (originally known as a Northfilter) is obtained bycorrelating a known signal, ortemplate, with an unknown signal

to detect the presence of the template in the unknown signal. This is equivalent to

convolving the unknown signal with a conjugated time-reversed version of the

template (cross-correlation). The matched filter is the optimal linear filter for

maximizing the signal to noise ratio (SNR) in the presence of additive stochastic

noise. Matched filters are commonly used in radar, in which a known signal is sent

out, and the reflected signal is examined for common elements of the out-going signal.

Pulse compression is an example of matched filtering. Two-dimensional matched

filters are commonly used in image processing, e.g., to improve SNR for X-ray

pictures.

3.1 Derivation of the matched filter

The following section derives the matched filter for a discrete-time

system. The derivation for a continuous-time system is similar, with summations

replaced with integrals.

The matched filter is the linear filter with impulse response h(n),

maximizes the output signal-to-noise ratio.

..3.1

Though we most often express filters as the impulse response of

convolution systems, as above (see LTI system theory), it is easiest to think of the

matched filter in the context of the inner product, which we will see shortly.

We can derive the linear filter that maximizes output signal-to-noise ratio

by invoking a geometric argument. The intuition behind the matched filter relies on

correlating the received signal (a vector) with a filter (another vector) that is parallel

with the signal, maximizing the inner product. This enhances the signal. When we

consider the additive stochastic noise, we have the additional challenge of minimizingthe output due to noise by choosing a filter that is orthogonal to the noise.

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Let us formally define the problem. We seek a filter, h, such that we

maximize the output signal-to-noise ratio, where the output is the inner product of the

filter and the observed signalx.

Our observed signal consists of the desirable signals and additive noise v:

.3.2

Let us define the covariance matrix of the noise, reminding ourselves that

this matrix has Hermitian symmetry, a property that will become useful in the

derivation:

.3.3

Where .H denotes Hermitian (conjugate) transpose and E denotes

expectation. Let us call our output,y, the inner product of our filter and the observed

signal

such that

3.4

We now define the signal-to-noise ratio, which is our objective function,

to be the ratio of the power of the output due to the desired signal to the power of the

output due to the noise:

.3.5

We rewrite the above:

_3_1_7493.6

We wish to maximize this quantity by choosing h. expanding the denominator of our objectivefunction, we have

.3.7

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Now, ourSNR becomes

.3.8

We will rewrite this expression with some matrix manipulation. The

reason for this seemingly counterproductive measure will become evident shortly.

Exploiting the Hermitian symmetry of the covariance matrixRv, we can write

...3.9

We would like to find an upper bound on this expression. To do so, we

first recognize a form of the Cauchy-Schwarz inequality:

.3.10

Which is to say that the square of the inner product of two vectors can only be as large as the product ofthe individual inner products of the vectors. This concept returns to the intuition behind the matched

filter: this upper bound is achieved when the two vectors a and b are parallel. We resume our

derivation by expressing the upper bound on ourSNR in light of the geometric inequality above:Our valiant matrix manipulation has now paid off. We see that the expression for our upper bound canbe greatly simplified:

3.11

We can achieve this upper bound if we choose,

3.12

Where is an arbitrary real number. To verify this, we plug into our expression for the output SNR:

Thus, our optimal matched filter is

..3.13

We often choose to normalize the expected value of the power of the filter

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output due to the noise to unity. That is, we constrain

.3.14

This constraint implies a value of, for which we can solve:

.3.15

Yielding

3.16

giving us our normalized filter,

3.17

If we care to write the impulse response of the filter for the convolution

system, it is simply the complex conjugate time reversal ofh.

Though we have derived the matched filter in discrete time, we can extend

the concept to continuous-time systems if we replace Rv with the continuous-time

autocorrelation function of the noise, assuming a continuous signal s(t), continuous

noise v(t), and a continuous filterh(t).

3.2 Alternative derivation of the matched filter

Alternatively, we may solve for the matched filter by solving our

maximization problem with a Lagrangian. Again, the matched filter endeavors to

maximize the output signal-to-noise ratio (SNR) of a filtered deterministic signal in

stochastic additive noise. The observed sequence, again, is

.3.18

With the noise covariance matrix,35

.3.19

The signal-to-noise ratio is

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.3.20

Evaluating the expression in the numerator, we have

..3.21

and in the denominator,

..3.22

The signal-to-noise ratio becomes

..3.23

If we now constrain the denominator to be 1, the problem of maximizing

SNR is reduced to maximizing the numerator. We can then formulate the problem

using a Lagrange multiplier:

..3.24

..3.25

..3.26

..3.27

which we recognize as an eigenvalue problem

..3.28

SincessH

is of unit rank, it has only one nonzero eigenvalue. It can beshown that this eigenvalue equals

..3.29

yielding the following optimal matched filter

..3.30

This is the same result found in the previous section.

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3.3 Frequency-domain interpretation

When viewed in the frequency domain, it is evident that the matched filter

applies the greatest weighting to spectral components that have the greatest signal-to-

noise ratio. Although in general this requires a non-flat frequency response, theassociated distortion is not significant in situations such as radar and digital

communications, where the original waveform is known and the objective is to detect

the presence of this signal against the background noise.

3.4 Example of matched filter in radar and sonar

Matched filters are often used in signal detection .As an example, suppose

that we wish to judge the distance of an object by reflecting a signal off it. We maychoose to transmit a pure-tone sinusoid at 1 Hz. We assume that our received signal is

an attenuated and phase-shifted form of the transmitted signal with added noise.

To judge the distance of the object, we correlate the received signal with a

matched filter, which, in the case ofwhite (uncorrelated) noise, is another pure-tone 1-

Hz sinusoid. When the output of the matched filter system exceeds a certain threshold,

we conclude with high probability that the received signal has been reflected off the

object. Using the speed of propagation and the time that we first observe the reflectedsignal, we can estimate the distance of the object. If we change the shape of the pulse

in a specially-designed way, the signal-to-noise ratio and the distance resolution can

be even improved after matched filtering: this is a technique known as pulse

compression.

Additionally, matched filters can be used in parameter estimation

problems .To return to our previous example; we may desire to estimate the speed of

the object, in addition to its position. To exploit the Doppler effect, we would like to

estimate the frequency of the received signal. To do so, we may correlate the received

signal with several matched filters of sinusoids at varying frequencies. The matched

filter with the highest output will reveal, with high probability, the frequency of the

reflected signal and help us determine the speed of the object. This method is, in fact,

a simple version of the discrete Fourier transform (DFT). The DFT takes anN-valued

complex input and correlates it with Nmatched filters, corresponding to complex

exponentials at N different frequencies, to yield N complex-valued numbers

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corresponding to the relative amplitudes and phases of the sinusoidal components.

3.5 Example of matched filter in digital communications

The matched filter is also used in communications. In the context of a

communication system that sends binary messages from the transmitter to the receiver

across a noisy channel, a matched filter can be used to detect the transmitted pulses in

the noisy received signal.

Imagine we want to send the sequence "0101100100" coded in non polarNon-return-to-zero (NRZ)through a certain channel.Mathematically, a sequence in NRZ code can be described as a sequence of unit pulses or shiftedrectangular functions, each pulse being weighted by +1 if the bit is "1" and by 0 if the bit is "0".

Formally, the scaling factor for the kth bit is,

3.31

We can represent our message, M(t), as the sum of shifted unit pulses:

3.32

Where Tis the time length of one bit.

Thus, the signal to be sent by the transmitter is

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If we model our noisy channel as an AWGN channel, white Gaussian noise is added to the signal. Atthe receiver end, for a Signal-to-noise ratio of 3dB, this may look like:

A first glance will not reveal the original transmitted sequence. There is a high power of noise relativeto the power of the desired signal (i.e., there is a low signal-to-noise ratio). If the receiver were tosample this signal at the correct moments, the resulting binary message would possibly belie theoriginal transmitted one.To increase our signal-to-noise ratio, we pass the received signal through a matched filter. In this case,the filter should be matched to an NRZ pulse (equivalent to a "1" coded in NRZ code). Precisely, theimpulse response of the ideal matched filter, assuming white (uncorrelated) noise should be a time-reversed complex-conjugated scaled version of the signal that we are seeking. We choose

3.33

In this case, due to symmetry, the time-reversed complex conjugate of

h(t) is in fact h(t), allowing us to call h(t) the impulse response of our matched filterconvolution system.

After convolving with the correct matched filter, the resulting signal,

Mfiltered(t) is,

Where * denotes convolution.

This can now be safely sampled by the receiver at the correct sampling

instants, and compared to an appropriate threshold, resulting in a correct interpretation

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of the binary message.

4.PULSE COMPRESSION

Pulse compression is a signal processing technique mainly used in radar,sonar andecho grapy to augment the range resolution as well as the signal to noise

ratio. This is achieved by modulating the transmitted pulse and then correlating the

received signal with the transmitted pulse.

In addition to these advantages, narrow pulse widths also assist radar when

operating in a cluttered environment. Radar also has an ability to perform limited

target classification if operating with sufficiently narrow pulse widths, or sufficiently

fine range resolution.

Extremely narrow pulse widths result in wide receiver bandwidths and the associated

problems with noise. Large receiver bandwidths effectively de-sensitise the radar

receiver and either force the transmitter to transmit higher levels of peak power to

compensate, or accept the consequential reduction in range. There are always limits on

the amount of peak power available from the transmitter, and invariably a reduction in

pulse width leads to a reduction in the maximum range of the radar.

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In short, narrow pulse widths are desirable, but they are not always feasible. Pulse-

compression radars make use of specific signal processing techniques to provide most

of the advantages of extremely narrow pulses widths whilst remaining within the peak

power limitations of the transmitter.There are numerous waveforms suitable for use with pulse compression including

binary or phase coding and linear frequency modulation.

4.1 CONCEPT OF PULSE COMPRESSION

The block diagram for a pulse-compression radar is very similar to that of the

standard pulse radar, and a simplified block diagram is shown in Figure 4.1.

Figure 4.1

The block diagram shows the frequency modulator responsible for generating thefrequency-modulated (chirp) pulse. In addition to generating the transmitted pulse, thefrequency modulator also plays a role in the design of the pulse-compression filter.The pulse-compression filter is an example of a matched filter because the filter isspecially designed to recognise the characteristics of the transmitted pulse as they arereturned to the receiver in the form of reflected pulses.To that end, the filter has been matched to the transmitted waveform. Received pulseswith similar characteristics to the transmitted pulse are recognised by the matchedfilter where other received signals pass relatively unnoticed by the receiver.4.2 Linear Frequency-Modulated Waveforms

The power of the pulse compression concept comes from the waveforms used. Weconcentrate on a popular pulse compression waveform called the linear frequency-

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modulated (or chirp) pulse. The actual pulse train from a pulse-compression radar is

the same as for any pulse radar. To the casual observer, the pulse train looks like an

amplitude- modulated sinusoidal signal. To an extent, this is true, however the

sinusoidal signal has now been frequency-modulated as well as amplitude-modulated.The modulation within each pulse (in this case, frequency-modulation) is the critical

element of the pulse compression waveform. The modulation provides the basis and

power of the compression concept. As stated earlier, the same modulation provides the

basis for the design of the pulse-compression filter.

The below figure shows two ways to represent the pulses in a pulse train from

linear FM pulse-compression radar.

Figure 4.2(a) shows the modulated sinusoidal signal that is transmitted by the

pulse-compression radar. The pulse is characterised by its pulse width, which in the

case of a pulse-compression radar is called the uncompressed pulse width, T. This

pulse width is one of the critical characteristics of the pulse-compression radar. Figure

4.2 (b) shows the frequency change within the pulse as a function of time.

The characteristic of interest in Figure 4.2(b) is the bandwidth of the modulation

within the pulse, B. The bandwidth is simply the difference between the highest and

lowest frequencies within the uncompressed pulse.

To recognise the presence of the uncompressed pulse, the pulse-compression filter

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performs a correlation between the received pulse and the transmitted pulse. In this

context, correlation is a signal processing term but it is directly analogous to the

common English use of the term. The pulse-compression filter is simply looking for a

strong correlation between what was transmitted and what was received. When awaveform similar to the waveform shown in Figure 4.2(a) is passed through the

matching pulse-compression filter, an interesting pulse called a sinc pulse results as

the output of the filter. We have already encountered the sinc pulse and, therefore

know that a sinc has a shape described by (sinx/x). An example of a sinc pulse as it

applies to the output of the pulse-compression filter is shown in Figure below and is

characterised by a very narrow and tall central pulse surrounded by gradually

decaying signals. The height and the width of the central pulse of the sinc pulse from

the pulse-compression filter are dependent upon the bandwidth and pulse width of the

uncompressed pulse.

From Figure 4.3, the width of the sinc pulse is inversely proportional to the

bandwidth of the uncompressed pulse and the height is proportional to the product of

the bandwidth and uncompressed pulse width.

The output of the pulse-compression filter forms the input into the detector section

of the pulse-compression radar. It is therefore desirable to have a very narrow and tall

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pulse (just as it is in a standard pulse radar system). The main points to note from

Figure are that the input to the filter is a relatively broad and low-power pulse. The

output pulse, however, is very narrow and strong; two very desirable characteristics

from a pulse radar.The output of the pulse-compression filter shown in Figure 4.3 represents the

amplitude of a signal rather than its power. To be consistent with the radar range

equation, the output of the pulse-compression filter is converted into power that is

taken as the square of the amplitude.

When the signal in Figure 4.3 is converted into power, we see that the peak value of

the pulse becomes the product of the modulation bandwidth, B and the uncompressed

pulse width, T. This is known as the pulse-compression ratio of the pulse-compression

radar.

Where B is the bandwidth of the modulation within each pulse

in hertz and T is the uncompressed pulse width in seconds. As with radar antennas, we

normally consider the usable portion of the sinc pulse to be half the null-to-null points

in Figure above.

To that end, the width of the compressed pulse is simply the inverse of the modulation

bandwidth, B.

..4.2

where B is the bandwidth of the modulation within each pulse in hertz.

Pulse compression ratios in the hundreds are common in modern pulse-

compression radar.

Elementary pulse-compression radar has been simulated using the MATLAB to

demonstrate the real power of the signal processing technique behind pulse

compression.

Figure 4.4 shows an uncompressed chirp pulse representing the transmitted pulse,

and the output of the corresponding pulse-compression filter when that pulse is

processed. Note that the two graphs in Figure 4.4 are drawn on the same time scale

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(horizontal axis) to show how the process has compressed Figure 4.4 (b) into a very

narrow pulse. Unfortunately, it is not possible to draw the amplitude axes (vertical

axis) on the same scale due to the magnitude difference between Figure 4.4 (a) and

(b). The amplitude of the pulse in Figure 4.4 (b) is approximately 500 times largerthan Figure 4.4 (a).

It is fair to say that the results in Figure 4.4 are not realistic because the pulse passed

through the matched filter has not suffered from noise or attenuation. Noise and

attenuation are a real problem when operating radar systems, so the exercise has been

repeated incorporating both random noise and signal attenuation, and is shown in

Figure 4.5.

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The output of the matched filter in Figure 4.5 (b) is now much noisier than thematched filter output in Figure 4.4 (b), which reflects the effects of the noise andattenuation of the input pulse. With that said, the output is still impressive from a peak

Amplitude and pulse-width perspective and the target is clearly visible.

4.3Required energy to transmit signal

The instantaneous power of the transmitted pulse is P(t) = | s | 2(t). The energy put into that signalis

E= p(t) dt = A2.T .4.3

Similarly, the energy in the received pulse is Er = K2A2T. If is the

standard deviation of the noise, the signal-to-noise ratio (SNR) at

the receiver is:

.4.

4

The SNR augments with the pulse duration, if other parameters are frozen.

This goes against the resolution requirements, since generally one wants a large

resolution.

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5. PULSE COMPRESSION TECHNIQUES

5.1 Pulse compression by linear frequency modulation (chirping)

5.1.1 Basic principles

How can one have a large enough pulse (to still have a nice SNR at the

receiver) without having a lousy resolution? This is where pulse compression

enters the picture. The basic principle is the following:

A signal is transmitted, with a long enough length so that the energy

budget is correct. This signal is designed so that after matched filtering, the widthof the inter correlated signals is smaller than the width obtained by the standard

sinusoidal pulse, as explained above (hence the name of the technique: pulse

compression).

In radar or sonar applications, linear chirps are the most typically used

signals to achieve pulse compression. The pulse being of finite length, the

amplitude is a rectangle function. If the transmitted signal has a duration T, beginsat t = 0 and linearly sweeps the frequency band fcentered on carrierf0, it can

be written:

.5.1

Remark: the chirp is written that way so the phase of the chirped signal

(that is, the argument of the complex exponential), is:

.5.2

Thus the instantaneous frequency is (by definition):

.5.3

Which is the intended linear ramp going from f0 f / 2 at t = 0 to f0 + f /

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2 at t = T.

5.1.2 Cross-correlation between the transmitted and the received signal

As for the "simple" pulse, let us compute the cross-correlation betweenthe transmitted and the received signal. To simplify things, we shall consider that

the chirp is not written as it is given above, but in this alternate form (the final

result will be the same):

.5.4

Since this cross-correlation is equal (save for the Kattenuation factor), tothe autocorrelation function ofsc', this is what we consider:

.5.5

It can be shown that the autocorrelation function ofsc' is:

.5.6

Function is the triangle function, its value is 0 on

, it augments linearly on [ 1 / 2,0] where it

reaches its maximum 1, and it decreases linearly on [0,1 / 2] until it reaches 0

again

The maximum of the autocorrelation function of sc' is reached at 0.

Around 0, this function behaves as the sin c term. The -3 dB temporal width of

that cardinal sine is more or less equal toT' = 1 / f. Everything happens as

if, after matched filtering, we had the resolution that would have been reached

with a simple pulse of duration T'. For the common values off, T' is smaller

than T, hence the "pulse compression" name.

Since the cardinal sine can have annoying side lobes, a common practiceis to filter the result by a window (Hamming, Hann, etc). In practice, this can be

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done at the same time as the adapted filtering by multiplying the reference chirp

with the filter. The result will be a signal with slightly lower maximum amplitude,

but the side lobes will be filtered out, which is more important.

The distance resolution reachable with a linear frequency modulation of a pulse on a

bandwidth fis: where c is the speed of the wave.

Ratio is the pulse compression ratio. It is generally greater than 1

(usually, its value is 20 to 30).

Example (chirped pulse): transmitted signal in red (carrier 10 hertz, modulation

on 16 hertz, amplitude 1, and duration 1 second) and two echoes (in blue).

Before matched filteringAfter matched filtering: the echoes

are shorter in time.

5.1.3 SNR augmentation through pulse compression

The energy of the signal does not vary during pulse compression.

However, it is now located in the main lobe of the cardinal sine, whose width is

approximately . If P is the power of the signal before compression

and P' the power of the signal after compression, we have:

.5.7

This yield:

.5.8

Besides, the power of the noise does not change through inter correlation since it

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is not correlated to the transmitted pulse (it is totally random). As a consequence:

After pulse compression, the power of the received signal can be

considered as being amplified byT.f. This additional gain can be injected in the

radar equation.

Example: same signals as above, plus an additive Gaussian white noise ( = 0.5)

Before matched filtering: the signal is hiddenin noise

After matched filtering: echoes becomevisible.

5.2. Pulse compression by phase coding

There are other means to modulate the signal. Phase modulation is a

commonly used technique; in this case, the pulse is divided in N time slots of duration

T/N for which the phase at the origin is chosen according to a pre-established

convention. For instance, it is possible not to change the phase for some time slots

(which comes down to just leave the signal as it is, in those slots) and de-phase the

signal in the other slots by (which is equivalent of changing the sign of the signal).

The precise way of choosing the sequence of{0,} phases is done according to a

technique known as Barker codes. It is possible to code the sequence on more than

two phases (poly phase coding). As with a linear chirp, pulse compression is achieved

through intercorrelation.

5.2.1 Advantage

The advantages of the Barker codes are their simplicity (as indicated

above, a de-phasing is a simple sign change), but the pulse compression ratio islower than in the chirp case and the compression is very sensitive to frequency

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changes due to theDoppler effect

5.2.2 Barker code

A Barker code is a sequence of N values of +1 and 1,

aj

for

such that

.5.9

for all [1].

Here is a table of all known Barker codes, where negations and reversals of the codes

have been omitted. A Barker code has a maximum autocorrelation of 1 (when codes

are not aligned). Longer Barker-like codes exist; there is a 28 baud sequence which

has sidelobes no larger than 2, and which thus has better RMS performance than the

codes below. The table below shows all known Barker codes; it is conjectured that no

other perfect binary phase codes exist.

Figure 5.2

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Figure 5.3

Known Barker Codes

Length PSL in db Codes

2 -6.0 +1 1 +1 +1

3 -9.5 +1 +1 1

4 -12.0 +1 +1 1 +1 +1 +1 +1 1

5 -14.0 +1 +1 +1 1 +1

7 -16.9 +1 +1 +1 1 1 +1 1

11 -20.8 +1 +1 +1 1 1 1 +1 1 1 +1 1

13 -22.3 +1 +1 +1 +1 +1 1 1 +1 +1 1 +1 1 +1

Barker codes of length 11 and 13 are used in direct-sequence spread

spectrum and pulse compression radarsystems because of their low autocorrelation

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properties. The +ve and -ve amplitudes of the pulses forming the Barker codes imply

the use of biphase modulation; that is, the change of phase in the carrier wave is 180

degrees.

A Barker code resembles a discrete version of a continuous chirp, anotherlow-autocorrelation signal used in other pulse compression radars. Pseudorandom can

be thought of as cyclic Barker Codes, having perfect (and uniform) cyclic

autocorrelation side lobes. Very long pseudorandom number sequences can be

constructed.

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6. CHAOTIC MAPPING METHODS

6.1 Chaotic signal:

A chaotic frequency modulated (FM) sine wave is an example of a chaotic signal that

can yield higher transmitted mean power when peak power limited transmitters are

used .

In general, a frequency modulated (FM) signal provides high resolution, high

transmitting power and low design cost.

If the FM signal is random it can lead to a low probability of intercept and interference

frequency range and simplicity of generation.

6.2 Chaotic waveforms:

They are deterministic (defined by an iterative map or differential equations), and can

therefore be practically implemented. They are non periodic, which suggests there

are potential advantages in security and can be used as (infinitely) large sequences.

They are sensitive to initial conditions (SIC) so that the behaviour of two systems with

small difference in the initial system state (or) a parameter diverge exponentially in

time. A solution to the power efficiency challenge can be provided by chaoticsystems.

6.3 Mapping methods:

Chaotic map where chosen for analysis, the parameter for each map are chosen so that

the map falls in its chaotic regime

Different maps: logistic map, quadratic map, exponent map, Bernoulli map, hopping

map, chebyshev map, congruent map etc.

map g! (x)logistic r(1-2x)

quadratic 8x

exponent (1-Bx)exp(B(A-x))

tent r x0

Bernoulli r x0

chebyshev Cos(Aacrcos(x(n)) A>2

congruent B xA

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6.4 Chebyshev chaotic equation

The one dimension chaotic mapping is defined as followsXn+1 = f (xn) .6.1

We use chebyshev chaotic mapping

Xn+1 = cos (Aacrcos (xn)) A>2 .6.2

To simulate and analyze our method, the density of its orbit point is

(x) = 1/1-x^2 -1


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Fig 6.1 (b) x0 = 0.1 1

The threshold for the binary codes is done as below

X (n) >0 xx (n) = 1 . 6.4

X (n) = 0.3 xx (n) =1 ..6.6

X (n)


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R(k) = x (n) x(n+k) limits from n= zero to N-1-k ..6.8

From the autocorrelation pattern, the discriminator (D) can be formed as,

D = R (0)/max(R (k)) where k 0 ..6.9

For all the lengths,

The performance parameters o of chebyshev mapping binary and ternary codes which

is discrimination factor has been estimated with and with out windowing functions

and the results are compared. At every lengths the best sequence having the highest

discrimination factor are found.

.

7. SIDELOBE REDUCTION USING WINDOW FUNCTIONS

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The main disadvantage of pulse compression is the appearance of side lobes in the

autocorrelation function which will mask the weak reflections from other targets, this

can be over come by reducing the side lobes, there are various techniques for this

purpose, and one such is the windowing techniques.

The following window functions have been applied.

7.1. Hamming -Hamming window

Syntax

w = hamming (L)

w = hamming (L,'sflag')

Description

w = hamming (L) returns an L-point symmetric Hamming window in the column

vector w. L should be a positive integer. The coefficients of a Hamming window

are computed from the following equation.

..7.1

The window length is .

w = hamming (L,'sflag') returns an L-point Hamming window using the window

sampling specified by 'sflag', which can be either 'periodic' or 'symmetric' (thedefault). The 'periodic' flag is useful for DFT/FFT purposes, such as in spectral

analysis. The DFT/FFT contains an implicit periodic extension and the periodic

flag enables a signal windowed with a periodic window to have perfect periodic

extension. When 'periodic' is specified, hamming computes a length L+1 window

and returns the first L points. When using windows for filter design, the

'symmetric' flag should be used.

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Note If you specify a one-point window (L=1), the value 1 is returned.

Examples

Create a 64-point Hamming window and display the result in WVTool:

L=64;

Wvtool (hamming (L))

7.2. hann -Hann (Hanning) window

Syntax

w = hann(L)

w = hann(L,'sflag')

Description

w = hann(L) returns an L-point symmetric Hann window in the column vector w.L must be a positive integer. The coefficients of a Hann window are computed

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from the following equation.

..7.2

The window length is .

w = hann(L,'sflag') returns an L-point Hann window using the window sampling

specified by 'sflag', which can be either 'periodic' or 'symmetric' (the default). The

'periodic' flag is useful for DFT/FFT purposes, such as in spectral analysis. The

DFT/FFT contains an implicit periodic extension and the periodic flag enables asignal windowed with a periodic window to have perfect periodic extension. When

'periodic' is specified, hann computes a length L+1 window and returns the first L

points. When using windows for filter design, the 'symmetric' flag should be used.

Examples

Create a 64-point Hann window and display the result in WVTool:

L=64;

wvtool(hann(L))

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Binary and Ternary High Resolution Codes Generation Using CHEBYSHEV CHAOTIC METHOD in MATLAB

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