Final Work

CHAPTER 1

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CHAPTER 1

INTRODUCTION

1.1 RADAR FUNDAMENTALS

Radar is an object detection system that uses electromagnetic waves to

identify the range, altitude, direction, or speed of both moving and fixed objects such

as aircraft, ships, motor vehicles, weather formations, and terrain. The term RADAR

was coined in 1940 by the U.S. Navy as an acronym for RAdio Detection And

Ranging. The term has since entered the English language as a standard word, radar,

losing the capitalization. Radar was originally called RDF (Range and Direction

Finding) in the United Kingdom, using the same acronym as Radio Direction Finding

to preserve the secrecy of its ranging capability.

A radar system has a transmitter that emits radio waves. When they come into

contact with an object they are scattered in all directions. The signal is thus partly

reflected back and it has a slight change of wavelength (and thus frequency) if the

target is moving. The receiver is usually, but not always, in the same location as the

transmitter. Although the signal returned is usually very weak, the signal can be

amplified through use of electronic techniques in the receiver and in the antenna

configuration. This enables radar to detect objects at ranges where other emissions,

such as sound or visible light, would be too weak to detect. Radar uses include

meteorological detection of precipitation, measuring ocean surface waves, air traffic

control, police detection of speeding traffic, military applications, or to simply

determine the speed of a baseball.

Radar is a system used to detect and locate reflecting objects. It transmits

electromagnetic energy into space and processes the received echo to detect, not only

the presence of a target, but also other target related information such as range, size

etc., One of the most commonly used radar signal is a series of pulses, usually

rectangular in shape, modulating a sine wave. The reverberation signal carries three

types of information.

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The round trip time from send to receive reveals the range of the object.

The frequency shift of the signal reveals the relative velocity (Doppler) of the

object.

The strength and phase shift of the reverberation reveals the nature of the

object.

To attain these values, there are two tasks performed by radar

Target detection: It is accomplished by transmitting an electromagnetic signal via

an antenna and detecting, in the unavoidable system noise, the wave reflected by

the target.

Parameter estimation: If the returned signal of adequate strength is received, it is

further analyzed to determine target distance, velocity, shape of target, number of

targets and so forth. This analysis is referred to as parameter estimation.

The real test of radar capability comes when detection and parameter estimation must

be performed for several targets simultaneously. This is the problem of target

resolution.

1.2 AIM OF THE PROJECT

The aim of this project is to design and develop high resolution waveforms

using Non Linear Frequency Modulation(NLFM) Technique.

1.3 THE BASIC RADAR RANGE EQUATION

The radar range equation says,

(4)

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Where Smin is given by

(5)

As we can see, for good detection many types of radar seek to transmit long-

duration pulses to achieve high energy, since transmitters are typically operated near

their peak power limitation. On the other hand, for good range resolution, radar needs

short pulses.

The shorter pulses (of shorter pulse-widths) have smaller pulse width have

wide spectral bandwidth and good range resolution as well. This gives it the

advantages of good bandwidth and good range resolution.

But on the other hand, shorter pulses need more peak power. The shorter the

pulse gets, the more should its peak power be increased so that enough energy is

packed into the pulse. High Peak Power makes the design of transmitters and

receivers difficult since the components used to construct these must be able to

withstand the peak power. One way to overcome the problem of high peak power is to

convert the short pulse into a long one and using some form of modulation to increase

the bandwidth of the long pulse so that the range resolution is not compromised.

The longer pulses, on the other hand, have better doppler resolution (since,

lower the bandwidth, the more precise will the estimation of the Doppler falling band)

and because of the higher pulse width, not much higher peak power is needed for high

energy transmissions. Hence, the longer pulses have the advantages of higher Doppler

resolution and lesser peak power transmission in high energy environments.

These divergent needs of long pulses for detection and short pulses for range

accuracy in measurements prevented early radars from simultaneously performing

both functions well.

As radar development progressed, and emphasis changed from merely getting

things to work, to getting things work in an optimum or near optimum manner, new

concepts came into being that laid the foundations of waveform design as an integral

part of the radar system development. During World War II, Woodward indicated that

the transmitted pulse could be designed to be as wide as necessary to meet the energy

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requirements of the system and that after the detectability requirements had been

satisfied, the range resolution conditions could be met by coding the transmitted

signal with wide band modulation information. This technique is referred to as pulse

compression and is used widely in Airborne Radar applications where high peak

power is undesirable

Pulse compression is a generic term that is used to describe a wave shaping

process that is produced as a propagating waveform is modified by the electrical

network properties of the transmission line. The pulse is frequency modulated, which

provides a method to further resolve targets which may have overlapping returns.

Pulse compression originated with the desire to amplify the transmitted impulse

(peak) power by temporal compression. It is a method which combines the high

energy of a long pulse width with the high resolution of short pulse width.

However, to meet the maximized target-detectability requirements of radar

receiver, the output peak signal-to-mean noise ratio has to be maximized. For this, we

use a linear network called ‘matched filter’.

1.4 MATCHED FILTER

For good detection, radar needs a large peak signal power to average noise

power ratio (So/No) at the time of the target's return signal. Hence, at the initial stages

of the receiver, a matched filter is used to

Increase the received the signal power to average noise power ratio (So/No).

Obtain the information carried by the matched filtering of reverberation signal.

The output of the matched filter is the cross-correlation between

The received signal plus noise and

A replica of the transmitted signal.

The frequency response of the linear time-invariant filter that maximizes the output

peak SNR is

H(f) = GaS*(f) exp(-j2πftm)

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And the output peak SNR from this matched filter is

(SNR) max ≤ 2E/N0

which means that the matched filter output SNR output depends only on the total

energy of the received signal and the noise power per unit spectral bandwidth. The

matched filter has the interesting property that no matter what the shape, duration, or

bandwidth of the input signal waveform, the maximum ratio of the output peak SNR

is simply twice the energy E contained in the received signal divided by the noise

power per unit bandwidth N0.

When the SNR received is large (as it must be for detection), the output of the

matched filter can usually be approximated by the auto-correlation function of the

transmitted signal; that is, the noise is ignored. This assumes that there is no Doppler

shift so that the received echo signal has the same frequency as the transmitted signal,

which might not be the case in many radar systems.

Hence, the output of the matched filter is re-considered as the cross-correlation

between the

Doppler-shifted received signal and

The transmitted signal, with the noise being ignored.

The shape of the projected signal acts the filter's ability to recognize the

return. For example, if the signal resembles a time shifted version of itself, the filter

might incorrectly determine the range of the object. If multiple signals are at the

surrounding at the same time, the filter must reject all non-matched signals.

Hence, we can say that the filter's ability to reject a non-matched waveform

depends on how little the non-matched waveform resembles time and frequency shifts

of the filter's matched signal.

1.5 PULSE COMPRESSION

Now that, the detectability requirements can be achieved by the use of the

matched filter, we now proceed to the pulse compression coding to achieve range

resolution conditions.

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High-resolution can be obtained by the short pulse, but there can be limitations

to the use of short pulse as discussed above.

A short pulse has a wide spectral bandwidth. A long pulse can have the same

spectral bandwidth as a short if the long pulse is modulated in frequency or phase.

The modulated long pulse with its increased bandwidth B is compressed by the

matched filter of the receiver to a width equal to 1/B. This process is called ‘pulse

compression’.

The pulse compression can be described as the use of a long pulse of width T

to obtain the resolution of a short pulse by modulating the long pulse to achieve a

bandwidth B (>> 1/T), and processing the modulated long pulse in a matched filter to

obtain a pulse width τ approximately equal to 1/B.

Dividing a longer pulse into ‘N’ number of sub-pulses and then doing the

phase or frequency modulation of these sub-pulses achieve the pulse compression. It

helps us maintain the pulse width unchanged besides increasing the bandwidth. This

increase in the bandwidth as a result of pulse compression is defined as ‘pulse

compression ratio’.

The pulse compression ratio is defined as the ratio of the long pulse width T to

the compressed pulse width τ, or T/ τ. It can also be written as BT (which cannot be

used when amplitude weighting is used on the received signal).

The receiver gain depends proportionally on this time-bandwidth product of

the received waveform. In the case of the uncompressed pulse transmission, the

receiver gain is unity because of the fact that there is no pulse compression and

consequently time, bandwidth is inversely proportional to each other. But when the

pulse compression is done, it results in an improved receiver gain more than unity is

obtained because the pulse compression factor of order N2 (where N is the number of

sub-pulses) can be achieved. In practical radar systems, pulse compression ratio might

be as small as 10 or even greater than 105. However, typically pulse compression ratio

values can be in the range of 100 to 300.

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1.5.1 TARGET RESOLUTION OF RADAR

Resolution is the ability to separately detect multiple targets or multiple

features on the same target, as opposed to reporting multiple targets as a single

detection. Targets are resolved in four dimensions although not necessarily by all

radars: Range, Azimuth, Elevation and Doppler shift.

Of these the most important dimensions that the target resolution is mainly dealt with

are range and doppler shift of the received signal.

a) Range Resolution

It is the ability to separate multiple targets at the same angular position, but at

different ranges. Range resolution is the function of radar’s radio frequency signal

bandwidth, with wide bandwidths allowing targets closely spaced in range being

resolved. To be resolved in range, the basic criteria are that targets must be separated

by at least the range equivalent of the width of the processed echo pulse.

ΔR= (1)

ΔR = range resolution (in meters),

c = velocity of propagation (in meters/sec)

= processed target’s pulse width (in sec).

The range resolution is often called as delay resolution because the range is based on

the delay in the reception of the transmitted pulse.

b) Doppler Resolution

Doppler resolution is the ability to separate targets at the same range, azimuth

and elevation but moving at different radial velocities. It is particularly useful in

identifying the targets by separating the net target motion from the spectral

components caused by rotating pieces of the targets.

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The criterion for Doppler resolution is that the Doppler frequencies must differ

by at least one cycle over the time of observation. Doppler resolution thus is a

function of the time over, which signal, is gathered for processing. Long data

gathering times (referred to as the look or dwell time) results in smaller Doppler

frequencies being resolved.

The Range to a target in a radar is determined by the two-way travel time for

the radar pulse to travel the distance to the target and back. Denoting the speed of

light as c and the range

as R, the two-way travel time is given by:

(2)

Once a radar pulse, has been transmitted, sufficient time must elapse for the

reflected pulse to arrive, so that the next pulse can be transmitted. If the reflected echo

arrived after the next pulse is transmitted, range ambiguity results. Thus, the quantity

Ru defined by:

(3)

denotes the Maximum unambiguous Range of the radar, where Tp denotes the

pulse repetition frequency. The term duty cycle of a radar denotes the ratio of the

time the radar is transmitting to the total time that it could have transmitted. For a

radar with Tp = 1ms and pulse duration of 1 s, the duty cycle is 0.001. A shorter radar

pulse is better than a longer one, since the radar is not transmitting while the echo

arrives. A pulse of duration tp = 1 s extends for a distance of ctp = 300m in space.

Two equal targets separated in range with a distance of more than half is value

are said to be resolvable in range. In this case, two targets separated by 150m are

resolvable in range (the factor of two appears because the radar signal has to travel

back and forth after the reflection).

Hence, it is obvious that good range resolution can be achieved with a

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shorter pulse.

But on the other hand, shorter pulses need more peak power. The shorter the

pulse gets, the more should its peak power be increased so that enough energy is

packed into the pulse. High Peak Power makes the design of transmitters and

receivers difficult since the components used to construct these must be able to

withstand the peak power. One way to overcome the problem of high peak power is to

convert the short pulse into a long one and using some form of modulation to increase

the bandwidth of the long pulse so that the range resolution is not compromised. This

technique is called Pulse Compression and is used widely in Airborne Radar

applications where high peak power is undesirable.

1.5.2 RANGE RESOLUTION WITH SHORT PULSE RADAR

High range resolution (ΔR small), as might be obtained with the short pulse, is

important for many radar applications which are listed below.

a) Range resolution: Usually easier to separate (resolve) multiple targets in range

than in angle.

b) Range accuracy: Radar capable of good range resolution is also capable of good

range accuracy.

c) Clutter reduction: Increased target-to-clutter ratio is obtained by reducing the

amount of distributed clutter with which the target echo signal must compete.

d) Inter-clutter visibility: With some types of "patchy" land and sea clutter, high

resolution radar can detect moving targets in the clear areas between the clutter

patches.

e) Glint reduction: Angle and range tracking errors introduced by a complex target

with multiple scatterers are reduced when high range-resolution is employed to

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isolate (resolve) the individual scatterers that make up the target.

f) Multipath resolution: Range resolution permits the separation of the desired

target echo from the echoes that arrive at the radar via scattering from longer

propagation paths, or multipath.

g) Multipath height-finding: When multipath due to scattering of radar energy from

the earth’s surface can be separated from the direct-path signal by high range-

resolution, target height can be determined without a direct measurement of elevation

angle.

h) Target classification: The range, or radial, profile of a target in some cases can

provide a measure of target size in the radial dimension. From the range profile one

might be able to sort one type of target from another based on size or distinctive

profile, especially if the cross-range profile is also available.

i) Doppler tolerance: With a short-pulse waveform, the doppler-frequency shift from

a moving target will be small compared to the receiver bandwidth; hence, only a

single matched filter is needed for detection, rather than a bank of matched filter

search tuned for a different Doppler shift.

j) ECCM: Short-pulse radar can negate the effects of certain electronic

countermeasures such as range-gate stealers, repeater jammers, and decoys. The

wide bandwidth of the short-pulse radar can, in principle, provide some reduction in

the effects of broadband noise jamming and reduce the effectiveness of some

electronic warfare receivers and their associated signal processing.

k) Minimum range: A short pulse allows the radar to operate with a short minimum

range. It also allows reduction of blind zones (eclipsing) in high-prf radars.

There can be limitations; however, to the use of a short pulse. They are,

a) Since the spectral bandwidth of a pulse is inversely proportional to its width, the

bandwidth of a short pulse is large. Large bandwidth can increase system

complexity, make greater demands on the signal processing, and increase the

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likelihood of interference to and from other users of the electromagnetic spectrum.

b) In some high-resolution radar the limited number of resolution cells available with

conventional displays might result in overlap of nearby echoes when displayed, which

results in a collapsing loss if an operator makes the detection decision.

c) Wide bandwidth can also mean less dynamic range in the receiver because

receiver noise power is proportional to bandwidth.

d) A short-pulse waveform provides less accurate radial velocity measurement than

if obtained from the Doppler-frequency shift. It spite of such limitations, the short

pulse waveform is used because of the important capabilities it provides.

e) A serious limitation to achieving long ranges with short-duration pulses is that a

high peak power is required for large pulse energy. The transmission line of high

peak power radar can be subject to voltage breakdown (arc discharge), especially at

the higher frequencies where wave guide dimensions are small.

If the peak power is limited by break down, the pulse might not have sufficient

energy.

1.5.3 PULSE COMPRESSION TECHNIQUES

As it was already mentioned, the pulse compression is done either by phase or

frequency modulation of the longer pulse which is divided into ‘N’ sub-pulses. There

are two general classes of the pulse-compression techniques:

Phase-Coding techniques

Frequency-Coding techniques

1.6 METHODOLOGY

Our project mainly consists of three sections. The function of each section is as given

below.

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Section I:

This section deals with linear frequency modulation technique .Here amplitude

weighting is done to reduce the side lobe level.

Section II:

This section introduces to non linear frequency modulation using symmetrical

waveform. Here each and every windowing function is considered, their weights are

applied as frequency weights and auto correlation and ambiguity figures are plotted

and studied.

Section III:

In this section, we deal with the non linear frequency modulation using asymmetrical

waveform. Here different asymmetrical weights are considered and autocorrelation

plots are plotted to compare the amount of side lobe reduced near main lobe as the

aim is to design high resolution waveforms. To do this, the help of Matlab-7 is taken.

1.7 ORGANIZATION OF THE WORK

This project is mainly organized into 4 chapters.

Chapter 1 deals with the introduction of the project presenting the aim and

methodology of the project.

Chapter 2 introduces the concepts related to phase coding techniques. It gives an

insight of binary phase coding like Barker codes, shift register codes etc.. as well as

poly phase coding techniques.

Chapter 3 deals with the linear frequency modulation technique.

Chapter 4 presents a good comparison of non linear frequency modulation using

symmetrical as well as asymmetrical waveforms .

Chapter 5 deals with results discussion.

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CHAPTER 2

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CHAPTER 2

2.Phase coded pulse compression:

Changes in phase can also be used to increase the signal bandwidth of long

pulse for purposes of pulse compression. A long pulse of duration T is divided into N

sub pulses each of width . An increase in bandwidth is achieved by changing the

phase of each sub pulse since the rate of change of phase with time is frequency. In

multiple target environments it may be significant that the distribution of the time side

lobes of phase coded words is different from that of linear FM pulse compression.

The time side lobes of linear FM are maximum immediately adjacent to the main lobe

and decrease with distance from the main peak unless some unusual form of tapering

is used. This is not generally true for phase codes. Depending on the class of phase

code considered, the side lobes may be fairly uniform, or they may actually exhibit a

tendency to be relatively low near the main lobe. A common form of phase change is

binary phase coding.

2.1 Binary phase coding:

In binary phase coding the phase of each sub pulse is selected to be either 0 or

radians according to some specified criterion. If the selections of the 0, phases

are made at random, the waveform approximates a noise modulated signal and has a

thumb tack like ambiguity function. The output of matched filter will be a compressed

pulse of width τ and will have peak N times greater than that of long pulse. The pulse

compression ratio equals the number of sub pulses N=T/τ BT, where the bandwidth

B 1/τ. The matched filter output extends for a time T on either side of peak response.

The unwanted, but unavoidable, portions of the output waveform other than the

compressed pulse are known as time side lobes.

2.2 Following are some of the types of binary phase codes:

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1) Barker Codes: Barker code is a special type of code which belongs to a class of

sophisticated signal. They are widely used in radar system because of its following

properties

a) Low power and relatively high energy.

b) Sharp auto correlation function and relatively low side lobes

Barker codes are originally developed for radar application. This code is a

subset of pseudorandom code with a length up to 13. The property that makes them

popular for application in radars is known as the one shot correlation. Barker codes

are the perfect codes compared to the other codes this is because all the side lobes in

Barker codes are either zero or 1. Hence we can conclude here that all side lobes

are very low. Therefore a high discrimination can be obtained.

The Barker code of length N=13 is shown in Figure 2.1

Figure 2.1 Barker code of length N=13

The (+) indicates 0 phase and (-) indicates radian phase. Its autocorrelation

function, which is the output of the matched filter, is shown in Figure 2.2.

Figure 2.2 Output of matched filter

There are six equal time-sidelobes to either side of the peak, each at a level

22.3 dB below the peak. (The sidelobe level of the Barker codes is l/N2 that of the

peak signal.)

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Figure 2.3 Tapped delay line to generate Barker codes

In Figure 2.3 is shown schematically a tapped delay line that generates the

Barker code of length 13 when the input is from the left. The same tapped delay-line

filter can be used as the receiver matched filter if the received signal is applied from

the right. The Barker codes are listed in Table 2.1 below.

There will be sidelobes on either side of the peaks of the auto-correlation

function. These are called Range Sidelobes, since they are present in the Range

Profile. There are many methods for reducing the range sidelobes and the differ

based on which encoding method is used in the radar. In the case of chirp radar,

windowing is used to reduce the sidelobes and in the case of binary-phase coded

pulse compression, proper selection of the codes will reduce the sidelobes, the

length of the code being a constant.

SNO Code length Code ElementsSide lobe level

dB

1 2 +-,++ -6.0

2 3 ++- -9.5

3 4 ++-+,+++- -12.0

4 5 +++-+ -14.0

5 7 +++--+- -16.9

6 11 +++---+--+- -20.8

7 13 +++++--++-+-+ -22.3

Table 2.1 Barker codes for different lengths

There are none greater than length 13; hence, the greatest pulse-compression

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ratio for a Barker code is 13. This is a relatively low value for pulse compression

applications.

2) Linear Recursive Sequences (or) Shift-Register Codes:

The limitation of a maximum length of 13 segments is a serious one, since it

doesn’t allow complete decoupling of average power from resolution, a principle aim

of pulse compression systems. When a pulse compression ratio larger than 13 is

required, some other criterion for selecting the 0, phases is needed. One method for

obtaining a set of random-like phase codes is to employ a shift register with

feedback and modulo-2 addition that generates a pseudorandom sequence of zeros

and ones of length 2n-1, where n is the number of stages in the shift register. An n-

stage shift register consists of n consecutive two-state memory units controlled by a

single clock. The two states considered here are 0 and 1. At each clock pulse, the

state of each stage is shifted to the next stage in line. Figure 2.4 shows a seven-

stage shift register used to generate a pseudorandom sequence of zeros and ones of

length 127. In this particular case, feedback is obtained by combining the output of

the 6th and 7th stages in a modulo-2 adder.

Figure 2.4 pseudorandom sequence generator of length 127

(In a modulo-2 gate or two input Ex-or gate, the output is zero when

both inputs are same [(0, 0) or (1, 1)] and the output is one when they are not

the same. It is equivalent to base-two addition with only the least-significant bit

carried forward.) An n-stage binary device has a total of 2n different possible states.

The shift register cannot, however, employ the state in which all stages are zero

since it would produce all zeros thereafter. Thus an n-stage shift register can

generate a binary sequence of length no greater than 2n-1 before repeating. The

actual sequence obtained depends on both the feedback connection and the initial

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loading of the shift register. When the output sequence of an n-stage shift register

is of period 2n-1, it is called a maximal length sequence, or m-sequence. This type

of waveform is also known as a linear recursive sequence (LRS), pseudorandom

sequence, pseudo noise (PN) sequence, or binary shift-register sequence. They are

linear since they obey the superposition theorem. When applied to phase-coded pulse

compression, the zeros correspond to zero phase of the sub pulse and the ones

correspond to radians phase. There can be more than one maximal length

sequence, depending on the feedback connection. For example, 18 different maximal

length sequences, each of length 127, can be obtained with a seven-stage shift register

by using different feedback connections. With the proper code, the highest (power)

sidelobe can be about 1/2N that of the maximum compressed-pulse power. A 24-dB

sidelobe can be available with a sequence of length 127. Not all maximal length

codes, however, have this Iowa value of peak sidelobe. For example, 45 with N =

127, the highest sidelobe of the various maximal length sequences can vary from 18

to 24 dB below the peak. It is generally said that the more usual maximum sidelobe

of a "typical" maximal-length shift register sequence is approximately l/N that of the

peak response. In the above example with N =127, this is 21 dB. As mentioned

above, a completely random selection of the phases usually results in a side lobe

approximately 2/N below the peak; the typical maximal-length shift-register

sequence might have a side lobe of 1/N, and the best of the maximal-length

sequences might approach 1/2N. By comparison, the Barker codes have a peak side

lobe 1/N2 below the peak. Sometimes the term code is used and at other times the

term sequence is used to describe the phases of the individual sub pulses of a phase-

coded 'waveform. Both terms are found in the literature and are often interchangeable

when discussing in pulse compression, as is the practice in this section. The shift-

register codes fit several of the tests for randomness. They are called pseudo random

since they may appear to be random, but they are actually deterministic once the

shift-register length and feedback connections are known. The fact that a pulse

compression sequence is random or pseudorandom does not mean it will produce the

lowest time- sidelobes at the output of the matched filter. For instance, the Barker

code of length 13 in Table for Barker codes as seen above is a good sequence (for its

length) in that it produces a -22.3dB peak side lobe, but it is not what is usually

thought of as "random." It does not satisfy the balance property of a random

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sequence (the number of ones differs from the number of zeros by at most one); nor

does it satisfy the run property (among the number of runs of ones and zeros in each

sequence, one half are of length two, one quarter of length three, and so forth);

nor does it satisfy the correlation property (when the sequence is compared term by

term with any cyclic shift of itself, the number of agreements differs from the number

of disagreements by at most one). Thus the Barker codes are not random in the

above sense, but they produce the lowest sidelobes for their length. It should be no

surprise, therefore, to find that there are better binary sequences than the shift-

register sequences.

3)Code concatenation:

Another somewhat different approach to achieve longer codes with higher

compression ratios is the process of code concatenation or combination. In this

approach, one utilizes whatever codes are available and codes the transmit pulse at

two or more levels so that each segment of code again coded with another phase code.

This has been called Barker-squad code or combined Barker coding when utilized

with Barker codes. The properties of such codes were calculated by Hollis. He

combined a Barker code of length 4 with a code of length 13 in two ways when each

bit of 13 bit word was coded into 4 bits, the zero Doppler auto correlation function of

the waveform yielded 4 side peaks of amplitude 13 located at range offsets of 1, 3

segments and 12 peaks of amplitude 4 . When each bit of the 4 bit code was coded

into 13 bits, the same number of side peaks of amplitude greater than unity appeared,

but the location of the side peaks of amplitude 13 occurred at offsets of 13 and 39

sequences. The main peak of the auto correlation function in both cases was 52. The

first combination maybe useful if the expected interference is considerable separated

in range from target. General properties of concatenated codes have been compiled by

Cohen.

4) Other Binary Sequences:

Computer search has shown that the longest code with side- lobe level of 2 is

of length 28; 46, 47. The longest code with sidelobe level 3 is 51; 48and the longest

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codes for levels 4 and 5 are 69 and 88 respectively. It should be quoted that the

above sidelobe levels are almost 25 dB for code lengths varying from 51 to 88.

These sidelobe levels are better than the 1/2N values of the best maximal-length

sequences.

2.3 Poly phase codes:

The phases of the sub pulses in phase-coded pulse compression need not be

restricted to the two levels of 0 and . When other than the binary phases of 0 or ,

the coded pulses are called polyphase codes. They produce lower sidelobe levels

than the binary phase codes and are tolerant to doppler frequency shifts if the

doppler frequencies are not too large. An example is the Frank polyphase code

defined by an M by M matrix as shown below.

The numbers in the matrix are each multiplied by a phase equal to 2 /M

radians (or 360/M deg). The polyphase code starts at the upper left-hand corner of

the matrix, and a sequence of length M2 is obtained. The pulse compression ratio is

M2 = N, the total number of sub pulses. Frank conjectured that for large N, the

highest sidelobe of a polyphase code relative to the peak of the compressed pulse is 2 N 10*(pulse compression ratio). In the above example with N = 25, the peak

side lobe is 23.9 dB. Since the rate of change of phase is a frequency, examination of

the matrix indicates that the frequencies of the Frank code change linearly with time

in a discrete fashion. The Frank codes can be thought of as approximating a stepped

linear-FM waveform. The ambiguity diagram for a polyphase code is similar to that

of a linear-FM waveform, but there can be a 3 to 4dB loss.

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2.4 Other pulse compression codes:

1) Ternary codes: Ternary code is another type of code that can be used to represent

information and data. However, ternary code uses 3 digits for representation of data.

Therefore ternary code may also be called as three alphabet code. This code consists

of 1 0 and -1.

2) QuinQuenary code: This code can also be applied to represent data. It uses five

arguments to represent information. Therefore this code can also be known as five

alphabet code. Quin quenary code usually uses alphabets 2, 1and 0.

3) Multi level code: This code is unrestricted code which actually means that the

alphabet will not only be restricted to integers but also to non integer values.

Example is 1, 1.2, 0, 7.8,-4, 1,-9.8.

4) Huffman Codes

So far every pulse compression wave form discussed is of constant amplitude

across the uncompressed pulse. The signal bandwidth is increased by phase or

frequency modulation rather than by amplitude modulation. The Huffman codes on

the other hand consist of elements that vary in amplitude as well as in phase. When

the Doppler shift is zero they produce autocorrelation function with no side lobes on

the time axis except for a single unavoidable side lobe at both ends of the compressed

wave form. The level of these two end side lobes is a design trade off. In one

example, a Huffman code of length 64 with no Doppler shift has a side lobe at each

end that is 56 db below the peak. As with other methods for obtaining zero or low side

lobes the volume under the ambiguity diagram must remain constant which means

that higher side lobes will appear else where in the Doppler domain. The side lobes

also degrade if the tolerance in the amplitude and phase are not maintained

sufficiently high.

23

5) Complementary Codes

It is possible to find pairs of equal length phase coded pulse in which the side

lobes of the auto correlation function of one all the negative of the other. If the

autocorrelation function from the out put of the matched filter are added, the algebraic

sum of the side lobes will be zero and main response will be 2 N where N is the no, of

elements in each of the two codes these are called complementary codes a galaxy

code after the person who first reported their existence of described how to construct

them. Theoretically, there are no side lobes on the time axis when complimentary

codes are employed. Complementary codes can be obtained with either binary or poly

phase sequences.

These are two problems however those limit the use of complimentary codes.

1) The first is that the two codes have to be transmitted on two separated pulse,

detected separately then subtracted, any delay that occurs during this time between

two pulses can result in incomplete correlation of the side lobes transmitting the two

codes simultaneously at two different frequencies does not solve the problem since

the target response can very with frequency.

2) The second problem is that the side lobes all not zero after cancellation when

there is a Doppler frequency shift so that the ambiguity diagram will con trains other

regions with high side lobes has series practical difficulties is not as differentiae as if

might seen at first glance

24

CHAPTER 3

25

CHAPTER 3

3.1 Linear Frequency Modulation

From 1980s,with the development of modern digital technology and VLSI

technology rapidly, the modern digital technology has been universalized. Combined

the digital with the traditional frequency synthesis technology, direct digital frequency

synthesis is generated. It is an important revolution in the area of frequency synthesis.

DDS consists of phase accumulator, sine ROM, DAC and LPF. There are some

advantages of DDS, such as frequency conversion fast, resolution high, keeping phase

continuous during frequency conversion, implementing many modulated function and

microprocessor controlling easily. Recently, with the rapid development, DDS is

applied to the area of radar, communication, anti-electronics and instruments tested,

and so on. One of an important application is radar signal. Based on DDS technology,

LFM signal has good performance of pulse compression, anti-jamming capability and

have a thick skin to Doppler shift. Now LFM signal is one of the most widely used

pulse compression signal in radar.This paper sets about from the generation of the

LFM signal, studies around the LFM signal technology to do some work on the theory

and practice. Linear Frequency Modulation (LFM) matched

filtering results in range sidelobes. These sidelobes are often objectionable because

they may mask small targets or may be mistaken for targets themselves. Various

methods of sidelobe control are investigated and their performance is measured. The

26

methods of sidelobe reduction include Dual Apodization, Spatially Variant

Apodization, and Leakage Energy Minimization.

Linear Frequency Modulation (LFM), also known as chirp, is the most

common continuous phase pulse compression technique due to its ease of generation

and insensitivity to Doppler shifts. The matched filter output from LFM has large

time, or range, sidelobes to either side of the peak response. The large sidelobes are

often objectionable since a large target might mask nearby, smaller targets. In

addition, large sidelobes might be mistaken for separate targets. These objectionable

sidelobes can be suppressed by linear and nonlinear sidelobe reducing techniques,

known by the general term apodization.

At this pulse compression method the transmitting pulse has a linear FM

waveform. This has the advantage that the wiring still can relatively be kept simple.

However, the linear frequency modulation has the disadvantage that jamming signals

can be produced relatively easily by so-called „Sweeper”.

The block diagram on the picture illustrates, in more detail, the principles of a pulse

compression filter.

27

Figure.3.1 Pulse compression process

The compression filter are simply dispersive delay lines with a delay, which is

a linear function of the frequency. The compression filter allows the end of the pulse

to „catch up” to the beginning, and produces a narrower output pulse with a higher

amplitude.

As an example of an application of the pulse compression with linear FM

waveform the RRP-117 can be mentioned.

Filters for linear FM pulse compression radars are now based on two main types.

Digital processing (following of the A/D- conversion).

Surface acoustic wave devices .

Figure 3.2: View of the Time-Side-Lobes

3.2Time-Side-Lobes

The output of the compression filter consists of the compressed pulse

accompanied by responses at other times (i.e., at other ranges), called time or range

sidelobes. The figure shows a view of the compressed pulse of a chirp radar at an

oscilloscope and at a ppi-scope sector.

Amplitude weighting of the output signals may be used to reduce the time

sidelobes to an acceptable level. Weighting on reception only results a filter

„mismatch” and some loss of signal to noise ratio.The linear frequency modulation, or

chirp, waveform has a rectangular amplitude modulation with pulsewidth T and a

linear frequency modulation with a swept bandwidth B applied over the pulse. The 28

time-bandwidth product of the LFM waveform is equal to TB, where TB is the

product of pulsewidth and swept bandwidth. The 3-dB width of the compressed pulse

at the output of the matched filter is τ3= 0.886/B, for large values of time-bandwidth

product. The peak time sidelobe level of the compressed pulse is –13.2 dB.

The sidelobe levels are an important parameter when specifying a pulse

compression radar. The application of weighting functions can reduce time sidelobes

to the order of 30 db's.

LFM is a form of phase modulation that yields a transmitted signal time-

bandwidth product that is greater than one. A signal modulated using LFM has a

phase function

(1)

where γ is the chirp frequency rate and f0 is the center frequency. A signal

modulated using LFM is given by

(2)

where τ is the uncompressed pulse width. The frequency domain spectrum of

(2) takes the form

(3)

Where

(4)

29

And

(5)

The functions given in (5), C(X) and S(X), are the Fresnel Integrals . In the

frequency domain, the matched-filter transfer function, H(ω), is the complex

conjugate of the spectrum of the signal to be processed. In general terms it is given by

(6)

where S(ω) is the spectrum of the input signal s(t), Td is a delay constant

required to make the filter physically realizable, and k is a normalization constant.

The corresponding time domain relationship between the input signal and the matched

filter is given by

(7)

The LFM reference signal matched filter output, formed by correlating the

input signal and the reference signal, takes a form as given in (8)

(8)

LFM Waveform Examples. Figure below shows the magnitude of the

autocorrelation function as a function of relative time delay τ for doppler shifts‡ of –

0.5 MHz, 0 and 0.5 MHz, pulse width T = 10 μs, swept bandwidth B = 1 MHz, and

LFM slope a = B/T = 0.1 MHz/μs. A doppler shift of fd = B/2 = 0.5 MHz causes the

peak of the correlation function to move to t = fdT/B = 5 μs. Figure 3.3 shows the

result when the pulse width is increased to 100 μs to yield a waveform with an LFM

slope equal

3.1 LFM Waveform Time Delay and Range Resolution Widths

30

3.3. LFM waveform autocorrelation function (T = 10 μs, B = 1 MHz, TB = 10)

50 μs, an increase of a factor of ten compared to the result for a 10-μs pulse width. to 0.01 MHz/μs. In this case, a Doppler shift of 0.5 MHz shifts the peak of autocor-relation function to τ = 50 μs, an increase of a factor of ten compared to the result for a 10-μs pulse width.

31

3.4.LFM waveform autocorrelation function (T = 100 μs, B = 1 MHz, TB = 100)

3.3 Frequency Domain Weighting for LFM Time Side lobe Reduction:

A frequency domain weighting filter is used following the matched filter for

time side lobe reduction. Taylor weighting provides a realizable approximation to the

ideal Dolph-Chebyshev weighting, which achieves the minimum mainlobe width for a

given value of peak time side lobe level. The frequency response of the equivalent

low pass filter for the Taylor weighing filter is

(9)

where Fm is the Taylor coefficient and n is the number of terms in the weighting

function. The compressed pulse response at the output of the weighting filter is given

by

(10)

As discussed below, the compressed pulse response is based on the assump-

tion that the time-bandwidth product of the LFM waveform is much greater than unity

. The filter matching loss for Taylor weighting is given by Klauder et al.1 as

(11)

Figure 3.5 shows a comparison of the compressed pulse response for three fre-

quency domain weighting types: Curve A is for uniform weighting where W( f ) = 1

32

FIGURE 3.5 Comparison of compressed-pulse shapes for three frequency-domain

weighting functions

Table below shows the peak time sidelobe level, 3-dB and 6-dB compressed-

pulse widths, and filter matching loss for the three weighting function types.

Table 3.2

The phase modulation introduced in the transmitted pulse allows compression

of the received pulse by matched filtering. The long received signal of length τ is

compressed into a narrow signal of width 1/β, where β is the modulatedphase spectral

bandwidth of the transmitted pulse. For LFM, the chirp frequency rate γ is equal to

β/τ. The pulse compression ratio is a measure of the degree to which the pulse is

compressed, and is given by βτ. The pulse compression ratio is also known as the

time-bandwidth product .

The LFM matched filter output is a good approximation to a sinc function for

large pulse compression ratios. For low compression ratios, the matched-filter output,

has irregularly spaced sidelobes, and does not approximate the sinc function well .

Fig. 3.6 shows the LFM matched filter output for a compression ratio of 42. At this

33

compression ratio, the sinc-like behavior is evident, but the sidelobes are irregularly

spaced.

Figure3.6:The LFM matched filter output

34

LFM increases the bandwidth and subsequently improved the range resolution

of the signal by a factor equal to the time bandwidth product. However, relatively

high sidelobes remain in the autocorrelation function. Such autocorrelation function is

unacceptable in some radar applications, where more than one target is present, giving

rise to echo of di®erent amplitudes. Three major techniques have been implemented

to obtain lower sidelobes level, i.e., weighting in time domain, frequency domain

weighting andNLFM . Time domain weighting is equals to the amplitude modulation

of the transmitted signal. However such implementation will lead to reduction of

transmitted power, and therefore a signal to noise ratio loss will occur. Frequency

domain weighting spectrum shaping using well knows weighting windows such as

Hann and Hamming. However the implementation of weighting windows may lead to

the penalty of main lobe broadening. NLFM waveform is designed such that its

matched filter response satisfies the side lobe requirements. Since the receiver is

matched with to the signal shape, no mismatch losses as in time domain weighting

and frequency domain weighting.

Linear amplitude weighting functions applied to the time-domain LFM

matched filter output can be used to suppress time sidelobes. One class of weighting

functions that is used in general sidelobe suppression schemes is the family of cosine-

on-pedestal functions, which include the rectangular, Hamming, and Hann windows.

These windows may be implemented as convolutions with Dirichlet kernels,

operating on the principle of sidelobe structure cancellation. Partial cancellation of

sidelobes is achieved if the transform of the unweighted aperture has regularly spaced

zeros and sidelobes in which the carrier frequency is successively shifted by π, like

the sinc function . The matched filter output of a LFM signal is sinc-like only for

large compression ratios. Hence, small pulse compression ratio LFM matched filter

output signals do not achieve significant sidelobe cancellation through cosine-on-

pedestal weighting.

Weighting functions can be constructed using other sidelobe canceling

techniques and optimality criteria. The prolate -spheroidal wave functions of zero-

35

order maximize the energy in a band of frequencies. This family of functions is

parameterized over the time-bandwidth product, and is approximated by the Kaiser

window . Since the Kaiser window makes no assumptions about the periodicity of

sidelobes, it results in sidelobe reduction for non-sinc-like signals.

In general, a linearly weighted matched filter response is given by

where F{•} is the Fourier Transform, F-1{•} is the inverse Fourier Transform, w(t) are

the weights, sLFM( t) is the time domain LFM return echo signal, and rLFM(t) is the

reference signal used for matched filtering.

3.4 RANGE SIDELOBES AND LINEAR APODIZATION

Rectangular window:

w = rectwin(21);

36

Hamming window:

w = hamming(21);

Hanning window:

w = hann(21)

Kaiser window:37

w = kaiser(200,2.5);wvtool(w)

38

CHAPTER 4

39

CHAPTER 4

NON LINEAR FREQUENCY MODULATION

4.1 Introduction:

The non-linear FM waveform has several distinct advantages. The non-linear

FM waveform requires no amplitude weighting for time-sidelobe suppression since

the FM modulation of the waveform is designed to provide the desired amplitude

spectrum, i.e., low sidelobe levels of the compressed pulse can be achieved without

using amplitude weighting.

4.2 Types Of Waveforms:

Matched-filter reception and low sidelobes become compatible in this design.

Thus the loss in signal-to-noise ratio associated with weighting by the usual

mismatching techniques is eliminated.

A symmetrical waveform has a frequency that increases (or decreases) with time during the first half of the pulse and decreases (or increases) during the last half of the pulse. A non symmetrical waveform is obtained by using one half of a symmetrical waveform

Figure4.1 Symmetrical waveform Figure4.2 Non-symmetrical waveform

40

The nonlinear-FM waveform has several distinct advantages over LFM. It

requires no frequency domain weighting for time sidelobe reduction because the FM

modulation of the waveform is designed to provide the desired spectrum shape that

yields the required time sidelobe level. This shaping is accomplished by increasing

the rate of change of frequency modulation near the ends of the pulse and decreasing

it near the center. This serves to taper the waveform spectrum so that the matched

filter response has reduced time sidelobes. Thus, the loss in signal-to-noise ratio

associated with frequency domain weighting (as for the LFM waveform) is

eliminated.

If a symmetrical FM modulation is used with time-domain amplitude

weighting to reduce the frequency sidelobes, the nonlinear-FM waveform will have a

thumbtack-like ambiguity function. A symmetrical waveform typically has a

frequency that increases (or decreases) with time during the first half of the pulse and

decreases (or increases) during the last half of the pulse. A nonsymmetrical waveform

is obtained by using one-half of a symmetrical waveform. However, the

nonsymmetrical waveform retains some of the range-doppler coupling of the linear-

FM waveform.

The NLFM waveform is generated using a continuous phase modulation

technique.. Increasing the frequency modulation rate near the ends of the transmitted

pulse and decreasing it near the center tapers the transmitted spectrum so that the

matched filter response has reduced sidelobes. This characteristic of the NLFM

waveform sometimes necessitates processing using multiple matched filters offset in

doppler shift to achieve the required time sidelobe level. Because of the doppler

sensitivity of the ambiguity function, the nonlinear frequency modulation waveform is

useful in a tracking system where range and doppler frequency are approximately

known, and the target doppler shift can be compensated in the matched filter. The

nonsymmetrical NLFM waveform is used in the MMR system, for example, which

detects and tracks ordnance such as mortars, artillery, and rockets.

41

4.3 Taylor Response:

To achieve a –40-dB Taylor compressed pulse response, for example, the fre-

quency-versus-time (frequency modulation) function of a nonsymmetrical NLFM

waveform of bandwidth B is

(1)

Figure4.3 Symmetric LFM and NLFM Ambiguity Function

Other NLFM waveforms that have been utilized in radar include the nonsymmetrical

sine-based and tangent-based waveforms. For the sine-based waveform, the

relationship between time and frequency modulation is given as

42

(2)

where T is the pulsewidth, B is the swept bandwidth, and k is a time sidelobe

level control factor. Typical k values are 0.64 and 0.70, which yield time sidelobe

levels of –30 dB and –33 dB, respectively. Figure 4.4 is a plot of peak time sidelobe

level as a function of the time sidelobe control factor k, for various TB products, for

this NLFM waveform.

Figure 4.4:Peak time sidelobe level for a sine-based NLFM waveform as a function of

k-factor

The frequency modulation-versus-time function for a tangent-based waveform is

given as

(3)

where T is the pulsewidth, B is the swept bandwidth, and β is defined as

(4)

43

where α is a time sidelobe level control factor.

When α is zero, the tangent-based NLFM waveform reduces to an LFM wave-

form. However, a cannot be made arbitrarily large because the compressed pulse tends

to distort. The ambiguity function of a NLFM sine-based waveform is shown in

Figure 4.5. It can be noted that this ambiguity function is more thumbtack-like in

nature than for an LFM waveform, indicating that this waveform is more doppler

sensitive than the LFM waveform.

Table given below provides a comparison of NLFM waveforms with

weighted and unweighted LFM for different values of the target radial velocity in

terms of peak and average time sidelobe levels and SNR loss. The NLFM waveform

shows better performance in terms of SNR loss and peak time sidelobe level (TSL)

than the LFM waveform. The TSL level does not degrade appreciably for the LFM

waveform for higher radial velocities, demonstrating the higher doppler tolerance of

LFM.

FIGURE 4.5 Ambiguity function of a sine-based symmetrical NLFM waveform

44

TABLE 4.1 Comparison of Linear FM and Nonlinear FM Waveform Performance

Fig. 2 shows the 40 dB Taylor NLFM matched filter output for a compression ratio of

42. The actual sidelobe levels achieved are higher than the –40 dB level desired, due

to the small pulse compression ratio. The phase modulation function for NLFM is

given by

(5)

Figure 4.6:40db Taylor NLFM matched filter output.

45

One of the primary disadvantages of the nonlinear-FM waveform is that it is

less doppler tolerant than the LFM. In the presence of doppler shift, the time sidelobes

of the pulse-compressed NLFM tend to increase compared to those of the LFM.

Figure 4.7 Flow diagram of Windowing process

46

CHAPTER 5

RESULTS AND CONCLUSION

Symmetrical Waveforms:

5.1 Bartlett window

5.1.1 Signal plot of Bartlett window for 21 samples

47

5.1.2 Ambiguity plot of Bartlett window

48

5.1.3 Auto correlation plot of Bartlett window

5.2 Blackman Harris window

49

5.2.1 Signal plot of Blackman Harris window for 21 samples

5.2.2 Ambiguity plot of Blackman Harris window

50

5.2.3 Autocorrelation plot of Blackman Harris window

5.3 Chebyshev Window:

51

5.3.1 Signal plot of Chebyshev for 21 samples

5.3.2 Ambiguity plot for Chebyshev window

52

5.3.3 Autocorrelation plot of Chebyshev window

5.4GaussianWindow:

53

5.4.1 signal plot of Gaussian window

5.4.2 Ambiguity plot of Gaussian window

54

5.4.3 Autocorrelation plot of Gaussian window

55

5.5NuttelWindow:

5.5.1 signal plot of Nuttel window for 21 samples

5.5.2 Ambiguity Plot of Nuttel window

56

5.5.3 Autocorrelation plot of Nuttel window

57

5.6TaylorWindow:

5.6.1 Signal plot of Taylor window for 21 samples

5.6.2 Ambiguity plot of Taylor window

58

5.6.3 Autocorrelation plot of Taylor window

59

5.7 Modified Exponential Weights:

5.7.1

Signal plot

Ambiguity plot60

Autocorrelation plot

61

5.7.2

Signal plot

Ambiguity plot

62

Autocorrelation plot

63

5.7.3

Signal plot

Ambiguity plot

64

Autocorrelation plot

65

Comparison of Autocorrelation plots of modified Exponential weights:

(a)autocorrelation plot for modified exponential weights shown in fig5.7.1

(b)autocorrelation plot for modified exponential weights shown in fig5.7.2

(c)autocorrelation plot for modified exponential weights shown in fig5.7.3

66

Comparison of Ambiguity plots of modified Exponential weights

(a)Ambiguity plot of exponential weights shown in fig 5.7.1

(b)Ambiguity plot of exponential weights shown in fig 5.7.2

(c)Ambiguity plot of exponential weights shown in fig 5.7.3

67

Modified Exponential Weights :

Weights for figure 5.7.1:

-3.2038 -3.0189 -2.8416 -2.6713 -2.5077 -2.3504 -2.1990 -2.0530 -1.9122 -1.7762 -1.6446 -1.5172 -1.3935 -1.2733 -1.1563 -1.0422 -0.9307 -0.8215 -0.7144 -0.6090 -0.5052 -0.4027 -0.3011 -0.2003 -0.1000 0 0.1000 0.2003 0.3011 0.4027 0.5052 0.6090 0.7144 0.8215 0.9307 1.0422 1.1563 1.2733 1.3935 1.5172 1.6446 1.7762 1.9122 2.0530 2.1990 2.3504 2.5077 2.6713 2.8416 3.0189 3.2038

Weights for figure 5.7.2:

-5.5808 -5.1792 -4.8029 -4.4502 -4.1193 -3.8086 -3.5166 -3.2418 -2.9829 -2.7386 -2.5077 -2.2891 -2.0818 -1.8847 -1.6968 -1.5172 -1.3450 -1.1795 -1.0197 -0.8649 -0.7144 -0.5673 -0.4231 -0.2809 -0.1401 0 0.1401 0.2809 0.4231 0.5673 0.7144 0.8649 1.0197 1.1795 1.3450 1.5172 1.6968 1.8847 2.0818 2.2891 2.5077 2.7386 2.9829 3.2418 3.5166 3.8086 4.1193 4.4502 4.8029 5.1792 5.5808

Weights for figure 5.7.3:

-148.4064 -121.5022 -99.4743 -81.4386 -66.6713 -54.5798 -44.6788 -36.570 -29.9307 -24.4918 -20.0357 -16.3838 -13.3895 -10.9325 -8.9142 -7.2537 -5.8843 -4.7511 -3.8086 -3.0189 -2.3504 -1.7762 -1.2733 -0.8215 -0.4027 0 0.4027 0.8215 1.2733 1.7762 2.3504 3.0189 3.8086 4.7511 5.8843 7.2537 8.9142 10.9325 13.3895 16.3838 20.0357 24.4918 29.9307 36.5709 44.6788 54.5798 66.6713 81.4386 99.4743 121.5022 148.4064

68

5.8 Conclusion:

This paper presents the detailed description of NLFM signal in terms of pulse

compression. The main characteristics of NLFM waveform include various weighting

functions obtained from different windowing techniques and exponential functions,

and its autocorrelation function can be approximated by a modified taylor windowing

weights was found to be -36 dB sidelobes. The NLFM functions are attractive since

they are capable of reducing sidelobes level with simple implementation scheme. The

simulation results show an improvement of 3 dB to 12 dB depends on the

configuration applied. It has been shown that the Modified NLFM signal capable of

achieving better sidelobe reduction as compared to LFM signal.

A highest sidelobe suppression of 38.2 dB can be achieved. The simulation

results show the autocorrelation function exhibits attenuated sidelobes to less than -35

dB in one of the NLFM waveform.In summary, the NLFM has been demonstrated to

be an effective technique for sidelobes suppression. More importantly, its

implementation scheme is to achieve without any SNR-robbing. The methods

presented above are general enough to be used to assess the performance of proposed

non-linear FM waveform radars.

69

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