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CHAPTER -1
OVERVIEW
1
1.1 Introduction
The basic purpose of radar is to detect the presence of an object of interest
and provide information concerning the object’s location, motion, size and other
parameters. The first task is called the problem of target detection. And the second
one is referred to as parameter estimation. For good detection a radar needs a large
peak signal power to average noise power ratio, So/No, at the time of the target's
return signal. Thus for good detection many radars 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, a radar needs
short pulses. 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 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.
Both the analog and digital pulse compression technologies have been
successfully exploited for achieving best results. Range resolution with low
probability of intercept can be achieved by employing multiple digital phase codes of
greater length having good auto correlation and cross correlation properties, synthesis
of which is a nonlinear multivariable optimization problem.
2
1.2 Aim of the project
There are two objectives of this project:
1. To design an optimum signal which improves range resolution with
minimum autocorrelation side lobes
2. The other main emphasis of this project has been to write a
MATLAB program which plots the output figures of input signal,its
autocorrelation function and ambiguity function depending upon the
type of overlaying selected.
1.3 Methodology
Our project mainly consists of three sections. The function of each section is as given
below.
Section One: This section deals with the introduction to basic pulse compression
technique, its advantages and the resulting high side lobes which decreases the
range resolution by masking the nearby target
Section Two: This section introduces to the various input signals and orthogonal
codes used in the code for overlaying to reduce the side lobes resulting from un-
modulated signal without any overlay
Section Three: In this section, we present output obtained from the simulation of
code for all combinations of overlaying the input signal with different orthogonal
codes presented in section two. Figures are followed with conclusion
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1.4 Organization of the work
This project report is mainly organized into 4 chapters.
Chapter 1 deals with the introduction of the project presenting the aim and
methodology of the project. The significance and applications are also cited in the
same chapter.
Chapter 2 introduces the concepts related to radar and existing pulse
compression techniques. It also discusses the limitations of these existing techniques.
Chapter 3 deals with the explanation of various orthogonal codes used in the
simulation. It also presents the flowchart of the MATLAB code written.
Chapter 4 deals with the results obtained through simulations of the code
presented with all the combinations possible and output figures of each combination is
presented in this chapter along with conclusion
4
CHAPTER -2
INTRODUCTION
TO
PULSE COMPRESSION
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2.1 Introduction
Radar (RAdio Detection And Ranging) is an electromagnetic system for the
detection and location of reflecting objects such as aircraft, ships, spacecraft, vehicles,
people, and the natural environment. It operates by radiating energy into space and
detecting the echo signal reflected from an object, or target. The reflected energy that
is returned to the radar not only indicates the presence of a target, but by comparing
the received echo signal with the signal that was transmitted, its location can be
determined along with other target-related information Radar can perform its function
at long or short distances and under conditions impervious to optical and infrared
sensors. It can operate in darkness, haze, fog, rain, and snow. Its ability to measure
distance with high accuracy and in all weather is one of its most important attributes.
The basic principle of radar is illustrated in Figure 2.1
Figure 2.1
A transmitter generates an electromagnetic signal that is radiated into space by
an antenna. A portion of the transmitted energy is intercepted by the target and
reradiated in many directions. The reradiating directed back towards the radar is
collected by the radar antenna, which delivers it to a receiver [1]. There it is
processed to detect the presence of the target and determine its parameters.
Thus there are two tasks performed by radar [2]
1. Target detection: It is accomplished by transmitting an electromagnetic
signal and detecting, in the unavoidable system noise, the wave reflected by
the target.
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2. 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.
Target detection and parameter estimation are difficult practical problems,
particularly if the target is small and at a great distance. In principle, however, both
tasks are quite simple when one is dealing with a single target. In order to ensure its
detection, the designer merely has to make certain that the received signal contains
sufficient energy to be visible in the noise. If the parameters of the transmitted signal
are chosen properly, adequate signal energy also guarantees measurement precision.
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.
Because detection, parameter estimation, and resolution are parts of the same
measurement process, one must be careful to avoid confusing these terms. Target
detection refers to the task of recognizing a return signal in the system noise.
Similarly, measurement precision concerns the effect of noise in that it describes the
uncertainty that the interfering noise causes in the value of the parameter. In contrast,
resolution depends on the interference from other targets. Although a high signal to
noise ration ensures a good detection performance and high measurement precision, it
is merely a prerequisite for target resolution. This means that the problems of
resolution can be treated separately from those of detection
2.2 Resolution
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.
2.2.1 Types of Resolution:
i). Range Resolution:
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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= ………. .(2.1)
Where ΔR=range resolution (in meters),
c=velocity of propagation (in meters/sec) and
=processed target’s pulse width (in sec).
Figure 2.2 Range resolution and processed pulse width
Without pulse compression, the processed pulse width is approximately equal to that
of the transmitted pulse. With pulse compression the processed pulse is narrower than
that of the transmitted pulse. The effective bandwidth of any pulse is approximately
the reciprocal of the pulse processed width. Thus range resolution can also be
described in bandwidth terms.
The transmitted matched Bandwidth (B) =1/τ-----------------------------------------------
(2.2)
Hence ΔR=c/(2B) (meters)---------------------------------
(2.3)
The degree of range resolution depends upon certain parameters
a) Width of transmitted pulse.
b) Types and sizes of targets.
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c) Efficiency of receiver and indicator.
For small range resolution we require large bandwidth or shorter pulse width.
2. Angular and cross range resolution:
Cross-range is the linear dimension perpendicular to the axis of the antenna, specified
as azimuth (horizontal) or elevation (vertical cross range). Cross range resolution
gives the radar the ability to separate multiple targets at the same range.
Resolution in cross range is determined by the antenna’s effective beam width, with
narrow antenna beams resolving more closely spaced targets. The criterion for cross-
range resolution is that the targets at the same range separated by more than the
antenna beam width are resolved. Those separated by less than the beam width are
not.
Figure 2.3 shows the principle of cross range resolving with antenna beams, in this
case with a scanning search radar. The upper portion of the figure shows the
relationship between the antenna beams and the targets. The lower portion shows the
relative amplitudes of numerous consecutive echoes from the targets as the antenna
scans by them.
Figure 2.3 Cross range resolution with antenna beams.
Cross range resolution ΔX R (meters)---------------------------------------------------
(2.4)
Where ΔX = the cross range resolution (meters),
R=target range (in meters),
= the antenna beam width (in radians).
The beam width of the antenna is related to its length and the wave length of the
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electromagnetic wave by the following relationship.
----------------------------------------------------
(2.5)
Where = signals wave length,
De = the effective length of the antenna in the
direction
in which beam width is being calculated
(azimuth/elevation)
The effective size of the antenna is typically about 0.7 times its actual size.
3) Enhanced cross range resolution:
As with the range resolution techniques exist for enhancing cross range resolution
beyond that available with reasonably sized real antennas. These processes depend
upon sequentially viewing targets from several different locations with the small real
antenna.
In the figure 2.4 , the antenna is moved to position 1, transmitter pulsed, and signal
stored. It is then moved to the position 2, 3 and 4 and the process is repeated. After
observing the target’s space from the four positions, the four signals are processed in
such a way as to make the effective antenna length the distance it moved. Thus the
cross-range resolution is improved. This concept is applied in Synthetic Aperture
Radar (SAR) and Inverse Synthetic Aperture Radars (ISAR). SAR techniques are
particularly well adapted to situation where the radar moves rapidly and the target is
stationary, such as air-borne radar viewing the target on the ground. ISAR technique
is applied where the radar is stationary and the targets move rapidly fast. ISAR is
sometimes used to analyze formations of aircraft from the ground-based or ship-board
radars to determine how many aircraft are in the formation.
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Figure2.4 Enhanced Cross Range resolution
4) 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. 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.
2.2.2 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.
1) Range resolution: Usually easier to separate (resolve) multiple targets in range
than in angle.
2) Range accuracy: Radar capable of good range resolution is also capable of good
range accuracy.
3) Clutter reduction: Increased target-to-clutter ratio is obtained by reducing the
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amount of distributed clutter with which the target echo signal must compete.
4) Interclutter visibility: With some types of "patchy" land and sea clutter, a high
resolution radar can detect moving targets in the clear areas between the clutter
patches.
5) Glint reduction: Angle and range tracking errors introduced by a complex target
with multiple scatterers are reduced when high range-resolution is employed to
isolate (resolve) the individual scatterers that make up the target.
6) 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.
7) 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.
8) 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.
9) 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.
10) 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.
11) 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.
2.2.3 Following are the limitations of the short pulse radar:
1. Since the spectral bandwidth of a pulse is inversely proportional to its
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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
likelihood of interference to and from other users of the electromagnetic spectrum.
2. In some high-resolution radars 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 the detection decision is made
by an operator.
3. Wide bandwidth can also mean less dynamic range in the receiver because
receiver noise power is proportional to bandwidth.
4. 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.
5. 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 waveguide dimensions are small.
If the peak power is limited by break down, the pulse might not have sufficient
energy.
2.3 Pulse compression
Pulse compression can be defined as a technique that allows radar to utilize a
long pulse to achieve large radiated energy but simultaneously obtaining range
resolution of short pulse. Theoretically, in pulse compression, the code is modulated
on to the pulsed waveform during the transmission. At the receiver, the code is used
to combine the signal to achieve a high range resolution.
Pulse compression can be clearly understood by this explanation. It is the method of
breaking the unwanted constraint between range and resolution. Pulse compression
radar transmits a modulated pulse, which is both long (having good range
characteristics) and wideband (having good range resolution criterion). The received
echo signals it possessed in a matched filter that compresses long pulse to a width of
1/B where `B' is spectral bandwidth. The pulse compression becomes attracting when
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peak power required of short pulse radar can’t be achieved with practical transmitter.
Since the spectral band width of a pulse is inversely proportional to its width. The
bandwidth of short pulse is large .The generation of long pulse from some short pulse
by expender having same bandwidth. These we achieve a long pulse with large
bandwidth. This process is called pulse compression.
2.3.2 Necessity
For border or field area surveillance a portable radar system with reasonable
range is required. These characteristics are archived only by employing pulse
compression tech why because in pulse compression tech we need low power for
transmission, which reduces size of equipment. Such as BFSR used by Indian Army
for the purpose.
2.3.3 Advantage
a) More efficient use of average power available at the radar transmitter and,
in some cases, avoidance of peak power problems in the high power sections of the
radar transmitter.
b) Transmission of long pulse using average power capability of system
c) The average power may be increased without increasing PRF hence it
decreases range ambiguity.
d) Reduction of vulnerability to certain types of interfering signals that don’t
have same properties as that of the coded waveform.
e) Increased system resolving capability, both in range and velocity. In the
case of range resolution, the generation of extremely fast rise time and high peak
power signals is bypassed using pulse compression techniques.
f) Good range accuracy.
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2.3.4 Limitation:
The pulse compression have the range side lobes which might be taken mistakenly as
true signal and also can missed the weak echo signal from target.
2.4 Types of pulse compression techniques:
Chart 2.1 shows the classification of pulse compression techniques.
2.5 FM Techniques:
If a long-duration pulse is frequency modulated, its spectrum can have a wider
bandwidth than if no FM were present. Since increasing bandwidth corresponds to
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waveforms with decreasing effective duration, the potential exists for a long-duration,
large-bandwidth, and pulse to be converted to a short-duration, "effective" pulse. In
effect we seek to "squeeze" the long pulse into a short pulse. If energy can be
conserved, we can even expect the shorter "compressed" pulse to increase in
peak amplitude compared to its amplitude as a long pulse. These effects can all
be achieved by a signal processing technique called pulse compression. The actual
signal processor consists of a matched filter that is often followed by a second filter
that minimizes certain undesired responses from the matched filter.
To visualize the process of pulse compression, imagine that a long pulse (duration T)
has a linearly varying instantaneous angular frequency i(t) with time, as shown in
Fig.2.5a. At the start of the pulse, the carrier cycles at a rate o – (Δ ω/2). In the pulse
center the frequency is o, the nominal carrier’s angular frequency. At the pulse’s end
frequency increases to o + (Δ ω /2). The total frequency deviation over time T is Δ
(rad/s). This pulse is applied to a pulse compression filter that has a constant-
modulus transfer function but a phase that corresponds to a linearly
decreasing envelope delay as shown in figure 2.5b .
We may visualize the low frequencies that enter the filter first as being delayed more
than those that enter later. If the slope (T seconds of delay change over an angular
frequency span of Δ ) is a match to the input signal's FM, all the frequencies can
be thought of as emerging at the same time and "piling up" in the output. The
response can be larger in amplitude, as shown in figure 2.5c. However, because the
input’s bandwidth is large, these frequencies can "pile up" for only a short time and
the output quickly decreases from the peak in relation to the reciprocal of bandwidth.
The duration of the main response is smaller than T by the factor 1/ΔfT , where Δf
=Δ /(2 ) and ΔfT is called the time-bandwidth product or the pulse compression
ratio of s(t). Similarly peak power is larger by a factor ΔfT. Outside the region of
main response, undesired response occurs for a time duration T on each
side of the main response.
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Figure 2.5a) A long duration pulse with linear FM.b) The compression filter and its Linear delay characteristicsc) The output pulse.
Sidelobes are unwanted by-products of the pulse compression process. Their form
and amplitudes depend greatly on the type of FM used and whether the
modulated pulses s(t) contains any shaping (tapering of amplitude) across the pulse.
2.5.1 Two principle methods exist for reduction of sidelobes:
1) In this method the output of the pulse compression filter is passed through
compensation, or weighting filter, specifically designed to reduce the sidelobes at the
expense of some loss in signal-to-noise ratio. These filters are sometimes called
mismatch filters because they don’t preserve the performance of the matched filter.
The fundamental idea in side lobe reduction is to choose a mismatch filter transfer
function such that the final output signal spectrum has a taper that leads to a
waveform with low side lobes.
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The table 2.1 gives examples of weighting, peak side lobes that result, and other
properties of output wave form. The mismatched filter loss can generally be kept to
about 1 dB when the peak side lobe level is reduced to 30 dB below the peak
response. It is the loss that is tolerated in order to achieve lower time side lobe levels.
Table no: 2.1.
S.no. Weighting functionPeak
sidelobe(dB)
Loss
(dB)
Main
beam
width
1 Uniform -13.2 0 1.0
2* 0.33+0.66cos2( ) -25.7 0.55 1.23
3*cos2( ) -31.7 1.76 1.65
4*
0.16+0.84 cos2( )-34.0 1.0 1.4
5*
0.08+0.92 cos2( )
(Hamming)
-42.8 1.34 1.5
*2, 3, 4, 5 are cosine power weightings whose general weighting function is given
below.
Weighting function W ( ) =k+ (1-k) cosn ( )
Where n is an integer this function includes the cosine power weighting
functions .The time response function for this class of spectrum response functions is
given by
g (t) =
Data has been calculated for non integer values of n in the range from n=1.8 to n=2.2,
for various for values of k, to evaluate the effect of errors in the hamming response
function. This data also provides approximate results on the effects of parameter
18
errors for the -40 dB Taylor weighting function.
2) In second method, non linear frequency modulation is used which offers the
advantage over linear FM of producing low time side lobes using a constant
amplitude waveform and a theoretically lossless matched filter. It doesn’t experience
the loss in the signal to noise ratio associated with matched filter used to reduce side
lobes in a linear FM pulse compression system. Constant amplitude envelope allows
efficient generation of high power. Non linear rate of change of frequency performs
the same role as amplitude weighing of the spectrum. If less time is spent over some
part of the spectrum, it is equivalent to reduced amplitude of the spectrum. In
addition, there is no significance widening of the corresponding pulse. When a
symmetrical non linear FM is used the ambiguity diagram is that of a thumb tack; i.e.,
it has a single peak rather than a ridge (a symmetrical waveform is one where the
frequency increases during the first half of the pulse and decreases in a similar manner
during the second half of the pulse or vice-versa.) hence, symmetrical non linear FM
is more sensitive to large doppler shifts and is not doppler tolerant. A non
symmetrical waveform utilizes only one half of the symmetrical waveform and has
some of the Range-Doppler coupling characteristic of the linear FM. Non linear FM
waveforms result in more system complexity than linear FM.
2.6 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
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binary phase coding.
2.6.1 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.
2.6.2 Following are some of the types of binary phase codes:
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 2.6 (a)
Figure 2.6(a)
The (+) indicates 0 phase and (-) indicates radian phase. Its autocorrelation
20
function, which is the output of the matched filter, is shown in Figure 2.6(b).
Figure 2.6(b)
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.)
Figure 2.6(c)
In Figure 2.6(c) 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 2.2 below.
Table 2.2
SNO Code length Code ElementsSide lobe level
dB
1 2 +-,++ -6.0
2 3 ++- -9.5
3 4 ++-+,+++- -12.0
4 5 +++-+ -14.0
21
5 7 +++--+- -16.9
6 11 +++---+--+- -20.8
7 13 +++++--++-+-+ -22.3
There are none greater than length 13; hence, the greatest pulse-compression 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 2.7 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 2.7
(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
22
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 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
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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 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: [Natthenson]
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:
24
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
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.
2.7 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
25
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.
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
6) Poly phase codes:
26
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.
2.8 Conclusion
The pulse compression technology was developed in late 1960s and since then
there has been considerable advancement in this field. With advent of pulse
compression technology the radar system and their application in military and non-
military field have undergone a sea change. The trade off between signal transmitted
and range resolution was solved and that too without encountering excessively high
27
peak powers; which could cause electrical breakdowns. Both the analog and digital
technologies have been successfully exploited for achieving best results. Range
resolution with low probability of intercept can be achieved by employing digital
phase codes of greater length having good auto correlation and cross correlation
properties, synthesis of which is a nonlinear multivariable optimization problem.
Various optimization techniques which could be used for this purpose are being
developed which are discussed in the following chapters.
28
CHAPTER-3
EXPLANATION OF VARIOUS INPUT
SIGNALS
AND
OVERLAYING CODES USED IN PROJECT
29
3.1.INTRODUCTION TO INPUT SIGNALS USED:
There are five signals which are used as input signals in our project .they are:
1 .Un-modulated (constant frequency) train of pulses.
2 .Barker phase coded pulse train.
3. LFM (linearly frequency modulated) pulse train
4. Costas array embedded in pulse train
5. Modified Costas array embedded pulse train.
Let us have a brief discussion on each one of them.
3.1.1.UN-MODULATED (CONSTANT FREQUENCY) TRAIN OF PULSES: .
Before we start considering the train of un-modulated pulses, let us just notice
a single pulse first..
The complex envelope of a constant-frequency (or un-modulated) pulse appears in
Fig. 4.1 and is given by:
FIGURE 3.1: Complex envelope of a constant-frequency pulse.
The ambiguity function is obtained by using (4.1) in
30
Solving the integrals and taking absolute value yields
FIGURE 4.2 Partial ambiguity function of a constant-frequency pulse of length T.
FIGURE 4.2 Partial ambiguity function of a constant-frequency pulse of length T.
F-:ONST
FIGURE 3.2: Partial ambiguity function and Zero-delay cut of the AF of a pulse.
The first two quadrants of the ambiguity function are plotted in Fig. 3.2. Figure 3.2
clearly shows the triangular zero-Doppler cut of the ambiguity function. The delay
response reaches zero at the pulse width T. The zero-delay cut is less obvious from
Fig. 3.2
and is plotted separately in Fig. 4.4. The first Doppler null is at the inverse of the
31
pulse duration: namely, Ix(0, l/T)] = 0. We can therefore approximately state that the
delay resolution is T and the Doppler resolution is l/T.
A single uncompressed pulse cannot usually provide sufficient range
resolution or velocity (Doppler) resolution. Satisfactory range resolution will be
reached using pulse compression and acceptable velocity resolution by using a
coherent pulse train. One more function of interest is the spectrum of the complex
envelope of the signal. The voltage spectral density is the Fourier transform of u(t).
Because in the constant-frequency pulse u(t) is a real constant, the Fourier transforms
of u(t) and mod(u(t)) exhibit the same result, and Fig. 4.4 also describes the
magnitude of the voltage spectral density of our signal. Here again we see poor
performance. The signal occupies its bandwidth inefficiently, with relatively high
spectral side lobes. We saw poor performances of a constant-frequency rectangular
pulse in many aspects: poor range resolution, poor Doppler resolution and high
Doppler side lobes, and inefficient spectral use.
Now let us observe a train of pulses..
The complex envelope of a train of N identical pulses is described by…
The time plot of pulse train is as:
"'I(;URE 4.16 Envelope of a I;oherenl pulse Iraill
Figure3.3: Envelope of a coherent pulse train
Figure 3.4: Partial ambiguity function of a coherent train of N= 6 pulses.
32
Figure 3.5: Zero-delay cut of the ambiguity function of six pulses.
The coherent pulse train provides independent control of the delay and Doppler
resolutions that was not possible in the single pulse case. The delay resolution is
controlled by the pulse duration T, while the Doppler resolution is controlled by the
total signal length N Tr. On the other hand, the Doppler and delay ambiguities are
tied; bt) the are functions of the pulse repetition interval T,. Note that their product,
which is the area of the rectangle connecting four recurrent lobes, is given by T, .1/ T,
= 1. This trade-off between Doppler (velocity) and delay (range) ambiguities is an
inherent difficulty in radar. It is the cause for the design parameter referred to as tow,
medium. or high pulse repetition frequency (PRF).
3.1.2.BARKER PHASE CODED PULSE TRAIN:
One of the early methods for pulse compression is by phase coding. We start
from a pulse of duration T . The pulse is divided into M bits of identical duration tb =
T/M, and each bit is assigned (or coded) with a different phase value. The complex
envelope of the phase-coded pulse is given by:
where um = exp(jφm) and the set of M phases {φ1, φ2, . . . , φM} is the phase code
associated with u(t).
The binary phase coded sequence of 0,π values that result in equal low side
lobes after passage through the matched filter is called BARKER CODE.
Probably the most famous family of phase codes is named Barker, after its
inventor (Barker, 1953). Originally, the Barker codes were designed as the sets of M
33
binary phases yielding a peak-to-peak side lobe ratio of M. For example, the
autocorrelation function of the M = 13-element Barker code is shown in Fig.
All known binary sequences yielding a peak-to-peak sidelobe ratio of M were
reported by Barker (1953) and Turyn (1963) and are given in Table 6.2. Although it
was only proved that no binary Barker codes exist for 13 <M < 1, 898, 884 and that
no binary Barker codes exist for all odd M >13 (Turyn, 1963; Eliahou and Kervaire,
1992), it is common belief that no Barker codes exist for all M > 13. In Barker and
other phase-coded signals, the instantaneous phase switching causes extended spectral
sidelobes. The effect of a phase-switching slope on bandwidth and autocorrelation
function is discussed in Section 6.8. A Barker signal is used as an example.
FIGURE 3.6: Autocorrelation function of phase-coded pulse using 13-element barker code
3.1.3LINEAR FREQUENCY MODULATED (LFM) PULSE:
Linear frequency modulation (LFM) is the first and probably still is the most
popular pulse compression method. It was conceived during World War II,
independently on both sides of the Atlantic, as can be deduced from German, British,
and U.S. patents (Cook and Bemfeld, 1967; Cook and Seibert, 1988). The basic idea
is to sweep the frequency band B linearly during the pulse duration T
34
The complex envelope of a linear-FM pulse is given by
Figure 3.7: Linear-FM signal.
The instantaneous frequency f(t) is obtained by differentiating the argument of
the exponential,
The instantaneous frequency is indeed a linear function of time. The frequency slope
k has the dimension S-2. The ambiguity function (AF) is obtained by applying
property 4 to the AF of an unmodulated pulse.
The phase and frequency of the complex envelope are shown in Fig The effective
time-bandwidth product of the signal is kT2 = BT = 10, where B is the total frequency
deviation. Note that the total deviation of the normalized frequency plot is BT, and the
total phase deviation is BT*π/4
35
FIGURE 3.8: Partial ambiguity function of linear-FM pulse
FIGURE 3.9: Phase and frequency characteristic of the LFM pulse used in above figure
36
FIGURE 3.10: ZERO-DOPPLER CUT OF AF OF LFM WITH TIME BANDWIDTH PROUCT=10.
Adding line,tr frequency modulation has increased the bandwidth and thus
improved the range resolution of the signal by a factor equal to the time-bandwidth
product. However. relatively strong sidelobes remain in the autocorrelation function
(ACF), as seen, for example, in Fig The ACF is related to the power spectral density
of the signal through the Fourier transform. ACF sidelobes can be reduced by shaping
the spectrum.
3.1.4.COSTAS ARRAY EMBEDDED PULSE TRAIN:
Costas coding, which results in a rather randomlike frequency evolutionJohn
P. Costas (1984) suggested a discrete frequency coding that is practically the opposite
of the linear law used in LFM. The difference is demonstrated by the binary matrices
in Fig. 5.1. The columns represent M contiguous time slices (each of duration tb), and
the rows represent M distinct frequencies, equally spaced (by _f ). In both signals we
find one dot in each column and in each row. This means, respectively, that at any one
of the M time slices, only one frequency is transmitted, and each frequency is used
only once.
37
FIGURE 3.11: Binary matrix representation of quantized linear FM (left) and Costas coding
(right).
The hopping orders described in Fig. 5.1 are only two out of M! possible
orders that meet the restriction of one dot per column and per row. Th hopping order
strongly affects the ambiguity function (AF) of the signal. The AF can be predicted
roughly by overlaying a copy of the binary matrix on itself, and then shifting one
relative to the other according to the desired delay (horizontal shifts) and Doppler
(vertical shifts). When a given delay–Doppler shift results in a coincidence of N
points, the ambiguity function is expected to yield a peak of approximately N/M at the
corresponding delay–Doppler coordinate.
In the LFM case it is easily observed that only delay and Doppler shifts of
equal number of units [τ = mtb, ν = m_f , m = 0,±1, . . . ,±(M − 1)] will cause an
overlap of dots, and the number of coinciding dots will be N = M − |m|. This hints at a
diagonal ridge in the ambiguity function, along the line ν = _f τ/tb. What is unique for
a Costas signal is that the number of coinciding dots cannot be larger than one for all
but the zero-shift case, where all dots coincide (N = M). This property implies a
narrow peak of the AF at the origin and low sidelobes elsewhere. If _f = 1/tb, the
exact AF values at the grid points will be either 1 or 0, according to the corresponding
number of coinciding dots. By grid points we refer to delay and Doppler shifts which
are integer multiples of tb and _f , respectively. An example of an overlap in a Costas
signal of length 7 is shown in Fig.
38
(A).Costas Signal Definition and Ambiguity Function:
Construction algorithms for Costas signals were discussed by Golomb and
Taylor (1984). However, a simple approach is to perform an exhaustive search of all
possible signals of a given dimension M. Checking if a signal is Costas can be
performed easily with the help of the difference matrix. An example is shown in Fig.
5.3. The coding sequence, the order of frequencies used, is a concise way to describe
the coding matrix. With regard to the difference matrix, note that the top row and the
leftmost column are headings and not part of the matrix. The element of the
difference matrix in row i and column j is…
where ai is the ith element of the coding sequence. The remaining locations (where i
+j >M) are left blank. Equation (5.1) says that the first row is formed by taking
differences between adjacent elements in the coding sequence, the second row by
taking differences between next-adjacent elements, and so on. How the sidelobe
matrix (Fig. 5.5) is derived from the difference matrix will be demonstrated by an
example. Consider the first element in the first row of the difference matrix (D1,1 =
a2 − a1 = 7 − 4 = 3). It says that at a positive normalized delay of 1 there is a
coincidence if the normalized Doppler shift is 3. This result will prompt adding 1 to
the value accumulated in the {delay = +1, Doppler = +3} location of the sidelobe
matrix. For the signal to be Costas, there should not be accumulated values larger than
1. Another way to say the same thing: If all the elements in a row of the difference
matrix are different from each other, the signal is Costas. Since the serial number of a
row in the difference matrix represents the normalized delay (a column) in the
sidelobe matrix, how do we fill the columns representing negative delays? To
complete the left-hand side of the sidelobe matrix, we simply apply the AF rule of
symmetry with respect to the origin. The number of 1’s in the sidelobe matrix is M(M
− 1). The number of 0’s in the sidelobe matrix is 3M(M − 1). To complete the
resemblance between the sidelobe matrix and the grid point values of the ambiguity
function, we add a value of M (= 7) at the origin. This is the only nonzero entry in the
zero-delay column and the zero-Doppler row. This brings the total number of
39
elements to (2M − 1)2. Finally, to normalize the peak value to 1, we divide all the
entries by M.
finally,it can be said that when pulse train is frequency modulated with large no.of
costas array elements,we can get ACF with very much reduced sidelobes.
TABLE 3.1: Examples of Welch-Constructed Costas Coding Sequences for M = 2
TO M=18
FIGURE 3.12: Partial ambiguity function of a Costas signal with code sequence {4 7 1 6 5 2 3}.
40
FIGURE 3.13: Autocorrelation function of a Costas signal with code sequence {4 7 1 6 5 2 3}.
3.1.5.MODIFIED COSTAS PULSES:
A modified Costas pulse combines Costas frequency coding with LFM within
each Costas
element (bit). Adding LFM allows increasing the size of the frequency step beyond
the nominal ¢fCostas = §1=tb. The increased frequency step results in wider
bandwidth, hence higher pulse compression. One of several discrete relationships
must exist between the LFM bandwidth B, the bit duration tb, and the increased
frequency step ¢fMod. Costas, in order to
nullify the ACF grating lobes that would show up without the LFM. The polarity of
the LFM slope need not be fixed.
3.2: INTRODUCTION TO OVERLAYING CODES USED:
Achieving Doppler resolution in radar usually requires a coherent train of P
pulses. In most cases the train is constructed by repeating the same compressed pulse.
It was recently reported that overlaying the P identical pulses with an orthogonal set
of P sequences, each one constructed of M elements, will remove completely the
sidelobes of the autocorrelation function (ACF) over the delay range ts =< (TAU) =<l
T, where ts is the duration of one element of the sequence, which is referred to as a
slice, and T = Mts is the pulse duration. The penalty for adding phase modulation is
spectrum broadening, typical of conventional phase coding. One method to reduce the
spectral width of phase coded radar signals, while maintaining constant envelope, is
41
the “derivative phase (DP) modulation” . We apply this method, instead of the phase-
coded orthogonal overlay, and check the resulting ACF and Subsections(A) and (B)
briefly describe the orthogonal overlay concept and the derivative phase (DP) method.
3.2.1.ORTHOGONAL PHASE OVERLAY MATRIX:
The orthogonal set was implemented using phase modulation. An example of
a P-by-M binary orthogonal set, where P =M =8, can be described st C:\Program
Files\Java\jre1.6.0_06\lib C:\Program Files\Java\jre1.6.0_06\libarting with the phase
matrix
The actual orthogonal set is given by the matrix
Clearly, the elements of A get only two values: +1 and -1.
Recall that the matrix A is said to be orthogonal when the dot product between any
two
columns of A is zero, implying that the matrix ATA is diagonal. Note also that
orthogonal P-by-M matrices A exist only for M · P. The overlay is implemented by
phase modulating the pth pulse by the pth row of A. One problem caused by adding a
binary phase-coded overlay, is the broadening of the spectrum.
3.2.2.DERIVATIVE PHASE OVERLAY MATRIX:
DP modulation differs from conventional binary phase modulation by
replacing phase jumps with phase slopes (frequency steps). The frequency steps are so
42
designed that at the end of the slice duration ts the accumulated phase change is the
desired 0 or ¼. Zero phase accumulation is obtained by splitting the slice into two bits
ts =2tb; during the first bit the frequency step is ¢f =1=4tb yielding accumulated
phase of 2πδftb = π/2; during the second bit the frequency step is -δf yielding
accumulated phase of – 2; hence, zero total phase accumulation during a slice.
Phase accumulation of π(or -π) is achieved by maintaining
the frequency step of -δf during the entire slice. There are several variations to DP. In
the one to
be used here the split slice, in which the frequency modulation (FM) is[δf,-δf ]is used
in the first slice of a sequence, and whenever the current slice is identical to the
previous slice.[-δf,-δf] is used when the current slice is different from the previous
slice. Matrix below presents the FM (DP) matrix corresponding to the orthogonal
phase-coded matrix.(given above)
In the phase-coded overlay , it was straightforward to show that the overlay (namely
A) was orthogonal. In the frequency-coded overlay described above, the meaning of
orthogonality is not so simple. The test will have to be the removal of the
autocorrelation sidelobes. Because the suggested new overlay involves FM, it is of
special interest to test it with signals that are already frequency modulated.
Thus, in our project we have written a MATLAB program which allows us to select
any of the five input signals…and also provides the choice to select no overlaying,
orthogonal phase overlaying or derivative phase overlaying on the selected input
signal.
43
It plots the ambiguity function, autocorrelation function and the modulated input
signal after overlaying (i.e., amplitude. phase and frequency are plotted).
NOTE: The program written to get plots (i.e., ambiguity function, ACF, modulated
input signal after overlaying) of all the possible inputs is kept in the appendix of
above documentation.
44
CHAPTER-4
RESULTS
AND
CONCLUSIONS
45
RESULTS:
1.FOR UNMODULATED PULSE :
(a)with no overlay:
As we can observe that the ambiguity function obtained from this
input has many sidelodes (and it is often called as BED of NAILS) .The nails refer to
the mesh of recurrent lobes, at intervals of Tr in delay and 1/Tr in doppler.
The zero doppler cut, is a set of triangles, all with a base of
2T,but with linearly decreasing height.
REASON: Around t=0,all 13 pulses Rxed overlaps the 13 references pulses, around
T=Tr only 12 pluses overlap and so on.
Because the pulses are identical, the correlation between individual
pulses maintains the triangular shape. Only the heights of the triangle change to
reflect the decreasing no. of pulses invloved in cross correlation between the Rxed
and references signals.
The coherent pulse train provides independent control of delay and
doppler resolutions that is not possible in single pulse case. The delay resolution is
controlled by pulse duration ‘T’, while the doppler resolution is controlled by the
total signal length [N(Tr)].On the other hand the doppler and delay ambiguities are
tied. both are functions of pulse repetition interval( Tr ) note that their product,
which is the area of the rectangle connecting four recurrent lobes is given by
Tr.1/Tr=1.This trade off between Doppler (velocity) and delay (range) ambiguities is
an inherent difficulty in radar.
And we can observe that there are many side lobes in the AF of un-
modulated pulse train. These side lobes can be reduced by overlaying methods.
46
(b)with orthogonal phase overlay:
An example of a P-by-M binary orthogonal set(used in our program), where P
=M =8, can be described starting with the phase matrix
The actual orthogonal set is given by the matrix
One problem caused by adding a binary phase-coded overlay, is the
broadening of the spectrum. An example is given using a train of 8 constant frequency
(i.e., un-modulated) pulses, with and without binary overlay. The relationships
between the slice width ts, the pulse width T, the pulse repetition interval Tr and the
total signal duration PTr are:
The large duty cycle (T=Tr =0:4) was selected in order to simplify the drawings. Fig.
presents the well-known ACF and magnitude spectrum of a coherent train of 13 un-
modulated pulses. The spectrum’s first local null is at fPTr = 1, with a major null at f
=1/T, namely at fPTr = 20.
Adding binary phase modulation with a slice width equal to ts = T=8 should
broaden the spectrum by a factor of 8. Because of the instant phase transition at slice
boundary, the spectrum exhibits an extended skirt, decaying at a rate of 6 dB/octave.
Thus it can be said orthogonal phase overlaying undoubtedly decreases sidelobes but
it broadens the spectrum.
47
(c)with derivative phase overlay:
The FM (derivative phase –DP) matrix, corresponding to the orthogonal
phase-coded matrix used in above section is:
Note that there are 92 “-1” in above matrix and only 36 “+1”. This implies that the
spectrum of the complex envelope of the signal containing this type of overlay will be
shifted downward in frequency. The recurrent lobes of the ACFs are also affected by
the specific frequency coding overlay In the phase-coded overlay defined in (a), it was
straightforward to show that the overlay
(namely A) was orthogonal. In the frequency-coded overlay described by (c), the
meaning of orthogonality is not so simple. The test will have to be the removal of the
autocorrelation sidelobes.
Because the suggested new overlay involves FM, it is of special interest to
test it with signals that are already frequency modulated. In coming sections, it is
applied to Costas, linear FM (LFM), and modified Costas.
NOTE: The same above discussion regarding orthogonal overlay and derivative phase
overlay will not be again done in further sections.
2.BARKER PHASE CODED PULSE:
On raw constant frequency pulse train.we are first implementing phase coding
using the 13-element predefined barker code.
48
It can be clearly observed from the AF plot that barker phase modulated pulse
train gave sidelobes which have low strength and they are of equal strength in its
ACF.
By overlaying the orthogonal phase matrix on barker phase modulated input
the sidelobes can be reduced to a great extent and the derivative phase overlaying is
more efficient in removal of sidelobes.
3.COSTAS ARRAY EMBEDDED PULSE TRAIN:
Since costas array is a algorithmically predefined set of discrete frequencies
which is designed to reduce sidelobes.
The ambiguity plot of costas input gives very improved output i.e., sidelobes
reduction is quiet evident in this case of input.
Overlaying orthogonal phase matrix has given better result, but overlaying
derivative phase matrix of orthogonal phase on costas frequency modulated input
gave us the best result so far.. Fig. shows how much the wide bandwidth Costas signal
narrows the ACF mainlobe. The pulse compression of a 16 element Costas signal is
162 = 256. Indeed the mainlobe width is
approximately tb/16 = T/256.
For comparison the ambiguity function of the same Costas pulse train without
an overlay is shown in Fig. 12. It seems incredible how the small frequency dither that
the DP overlay adds to the Costas frequencies makes such a profound change in the
ambiguity function sidelobe pattern near zero-Doppler.
4.LINEAR FREQUENCY MODULATED PULSE TRAIN:
In this case we give an input signal in which the ON period of pulse in pulse
train is having constantly (i.e., linearly) increasing frequency. And when the
ambiguity plot of it is observed it is clear that linear frequency modulation has
eliminated the sidelobes effectively. But the performance of this is less than that of
costas array.
49
Adding linear frequency modulation has increased the bandwidth and thus
improved the range resolution of the signal by a factor equal to the time-bandwidth
product. However, relatively strong sidelobes remain in the autocorrelation function
(ACF), as seen, for example, in Fig. The ACF is related to the power spectral density
of the signal through the Fourier transform. ACF sidelobes can be reduced by shaping
the spectrum.
When orthogonal phase is overlayed on it,(since OP overlaying provides linear
independence),thus the resulting ambiguity function is an improvement over the one
with no overlay.
And when derivative phase code(i.e.,frequency code) is overlayed on LFM
pulses.it reduced the width of main lobe and Comparing the ACF of a train of
identical LFM pulses without overlay reveals that indeed, with overlay, the sidelobes
beyond 1tau1> 3tb were removed, and the recurrent lobe was reduced. However, most
of the remaining sidelobes, over 1tau1 < 2tb, are higher than without overlay.
Finally, the ambiguity function of the LFM train with derivative phase overlay
shows that the slice with sidelobes 1tau1 < 3tb is actually a strip that extends to higher
Doppler frequencies.
5.MODIFIED COSTAS MODULATED PULSE TRAIN:
A modified Costas pulse combines Costas frequency coding with LFM within
each Costas element (bit). Adding LFM allows increasing the size of the frequency
step beyond the nominal
The increased frequency step results in wider bandwidth, hence higher pulse
compression. One of several discrete relationships must exist between the LFM
bandwidth B, the bit duration tb, and the increased frequency step ¢fMod. Costas, in
order to nullify the ACF grating lobes that would show up without the LFM. The
polarity of the LFM slope need
not be fixed. In our example we use the relationships
50
The LFM slope polarity alternates between bits whose frequency slots are
adjacent, in order to reduce their relatively large spectral overlap
Since the variation of frequency in modified costas is random and it’s a
combination of discrete and linear change , therefore the output plot i.e., ambiguity
plot has less sidelobes (but comparatively more sidelobes when compared to costas
modulated pulse train)
When orthogonal phase is overlayed on it result gives an improvement in side
lobe reduction. and when derivative phase is overlayed i.e., frequency of modified
costas and the deviation in frequency (i.e.,δf,-δf depending on change in orthogonal
phase components) are combined,the resulted ambiguity function has less sidelobes
and compressed main lobe.
Note The output will vary when different costas arrays of same order are used in
combination with linearly increasing frequency. And therefore there can be a few
combinations of costas input which can give even more better results
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4.2: OUTPUT WAVEFORMS
1. FOR UNMODULATED PULSE TRAIN
FIG 4.1 : OUTPUT WAVEFORMS FOR UNMODULATED PULSE TRAIN WITHOUT ANY OVERLAY
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FIG 4.2:OUTPUT WAVEFORMS FOR UNMODULATED PULSE TRAIN ORTHOGONAL PHASE OVERLAY
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FIG 4.3 : OUTPUT WAVEFORMS FOR UNMODULATED PULSE TRAIN WITH DERIVATIVE PHASE OVERLAY
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2.OUTPUT WAVEFORMS FOR 13 BIT BARKER CODE:
FIG 4.4 : OUTPUT WAVEFORMS FOR 13 BIT BARKER CODE WITHOUT OVERLAY
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FIG 4.5 : OUTPUT WAVEFORMS FOR 13 ELEMENT BARKER CODE WITH ORTHOGONAL PHASE OVERLAY
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FIG 4.6 : OUTPUT WAVEFORMS FOR 13 ELEMENT BARKER CODE WITH DERIVATIVE PHASE OVERLAY
3.:OUTPUT WAVEFORMS FOR 8 ELEMENT COSTAS ARRAY:
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FIG 4.7 : OUTPUT WAVEFORMS FOR 8 ELEMENT COSTAS ARRAY WITHOUT ANY OVERLAY
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FIG 4. 8 : OUTPUT WAVEFORMS FOR 8 ELEMENT COSTAS ARRAY WITH ORTHOGONAL PHASE OVERLAY
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FIG 4. 9 : OUTPUT WAVEFORMS FOR 16 ELEMENT COSTAS ARRAY WITH DERIVATIVE PHASE OVERLAY
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4. OUTPUT WAVEFORMS FOR LFM PULSES:
FIG 4.10 :OUTPUT WAVEFORMS FOR LFM PULSES WITHOUT ANY OVERLAY
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FIG 4.11 : OUTPUT WAVEFORMS FOR LFM PULSES WITH ORTHOGONAL PHASE OVERLAY
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FIG 4.12 : OUTPUT WAVEFORMS FOR LFM PULSES WITH DERIVATIVE PHASE OVERLAY
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5. OUTPUT WAVEFORMS FOR MODIFIED COSTAS ARRAY OF PULSES:
FIG 4. 13 : OUTPUT WAVEFORMS FOR MODIFIED COSTAS ARRAY OF PULSE WITHOUT ANY OVERLAY
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FIG 4.14 : OUTPUT WAVEFORMS FOR MODIFIED COSTAS ARRAY OF PULSES WITH ORTHOGONAL PHASE OVERLAY
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FIG 4. 15 : OUTPUT WAVEFORMS FOR MODIFIED COSTAS ARRAY OF PULSES WITH DERIVATIVE PHASE OVERLAY
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CONCLUSION:
Thus it can now be concluded that we have observed the output plots of all
(i.e., 15 ) possible input combinations and The basic fact which can be said is
practically the same ACF sidelobes removal can be achieved when implementing
orthogonal overlay using DP, which is frequency coding, rather than the original
binary phase-coded overlay.
The added frequency coding is especially attractive for signals in which the
pulse compression was obtained originally through FM. The DP overlay was
demonstrated on four types of frequency modulated pulse train signals: un-modulated
pulses, Costas pulses, LFM pulses, and modified Costas pulses.
Orthogonal overlay using DP produced complete removal of almost the
same portion of the near sidelobes, as did phase-coded overlay. It also drastically
attenuated (below ¡20 dB) the recurrent lobes. Finally, its spectrum decayed
faster than when phase-coded overlay was used.
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APPENDIX
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MATLABCODE:clc;clear all;z1=[zeros(1,28)];%array of 28 zeros used for f_basicz=zeros(1,20);%array of 20 zeros used as off periodp=[ones(1,8),z];%basic pulsept=[p,p,p,p,p,p,p,p,p,p,p,p,p];%train of 13 pulsesb=[p,p,p,p,p,-p,-p,p,p,-p,p,-p,p];%barker code of 13 elementsfcb=[1 2 5 7 6 4 8 3,z];%costas array of order 8 & zerosfc=[fcb,fcb,fcb,fcb,fcb,fcb,fcb,fcb,fcb,fcb,fcb,fcb,fcb];%costas for 13 pulsesflfmb=[1 2 3 4 5 6 7 8,z];%linearly increasing frequency for lfm%flfmb=[.062 .0651 .0682 .0713 .0744 .0775 .0806 .0837,z];flfm=[flfmb,flfmb,flfmb,flfmb,flfmb,flfmb,flfmb,flfmb,flfmb,flfmb,flfmb,flfmb,flfmb];%lfm for 13 pulses%fmcb=[1.062 2.0651 5.0682 7.0713 6.0744 4.0775 8.0806 3.0837,z];fmcb=[2 4 8 11 11 10 15 11 ,z];%modified lfm for pulsefmc=[fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,fmcb,]; %modified lfm for pulse trainop=[ 1 1 1 1 1 1 1 1,z 1 1 -1 -1 -1 -1 1 1,z 1 -1 -1 1 1 -1 -1 1,z 1 1 1 1 -1 -1 -1 -1,z 1 -1 1 -1 1 -1 1 -1,z %orthogonal phase overlay matrix for 13 pulses 1 1 -1 -1 1 1 -1 -1,z 1 -1 1 -1 -1 1 -1 1,z 1 -1 -1 1 -1 1 1 -1,z 1 1 1 1 1 1 1 1,z 1 1 -1 -1 -1 -1 1 1,z 1 -1 -1 1 1 -1 -1 1,z 1 1 1 1 -1 -1 -1 -1,z 1 -1 1 -1 1 -1 1 -1,z ];opr=[op(1,:),op(2,:),op(3,:),op(4,:),op(5,:),op(6,:),op(7,:),op(8,:),op(9,:),op(10,:),op(11,:),op(12,:),op(13,:)]; oppt=opr.*pt;%orthogonal phase overlayed on pulse train dp= 0.25.*[1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1,z 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1,z 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1,z 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1,z 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1,z 1 -1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 1 -1,z 1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1,z%DERIVATIVE PHASE OVERLAY 1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1,z MATRIX FOR 13 PULSES 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1,z 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1,z 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1,z 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1,z 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1,z ];
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dpr=[dp(1,:),dp(2,:),dp(3,:),dp(4,:),dp(5,:),dp(6,:),dp(7,:),dp(8,:),dp(9,:),dp(10,:),dp(11,:),dp(12,:),dp(13,:)]; p1=[ones(1,16),z]; pt1=[p1,p1,p1,p1,p1,p1,p1,p1,p1,p1,p1,p1,p1]; b1=[p1,p1,p1,p1,p1,-p1,-p1,p1,p1,-p1,p1,-p1,p1]; fcb1=[1 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6,z]; fc1=[fcb1,fcb1,fcb1,fcb1,fcb1,fcb1,fcb1,fcb1,fcb1,fcb1,fcb1,fcb1,fcb1];% flfmb1=[.062 .062 .0651 .0651 .0682 .0682 .0713 .0713 .0744 .0744 .0775 .0775 .0806 .0806 .0837 .0837,z];flfmb1=[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16,z]; flfm1=[flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1,flfmb1];% fmcb1=[1.062 1.062 2.0651 2.0651 5.0682 5.0682 7.0713 7.0713 6.0744 6.0744 4.0775 4.0775 8.0806 8.0806 3.0837 3.0837,z];fmcb1=[2 5 12 14 18 11 22 19 25 24 19 19 17 26 17 22 ,z]; fmc1=[fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1,fmcb1]; fcode=1; gcode=input('want to overlay....yes => press 1 & no => press 0: ')%FOR ENTERING CHOICE TO OVERLAY OR NOT if gcode==0 %for entering choice for input signal g=input('enter the choice for input signal..i.e.,press:\n 1. for unmodulated pulse train..\n 2. for 13 element barker coded pulse..\n 3. for order 8 costas array..\n 4. for lfm pulses..\n 5. for modified costas array of pulses.. ')if g==1 u_basic=pt; f_basic=[z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1];elseif g==2 u_basic=b; f_basic=[z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1];elseif g==3 u_basic=pt; f_basic=fc;elseif g==4 u_basic=pt; f_basic=flfm;else u_basic=pt; f_basic=fmc;end else g1=input('enter the choice for input signal..i.e.,press:\n 1. for unmodulated pulse train..\n 2. for 13 element barker coded pulse..\n 3. for order 8 costas array..\n 4. for lfm pulses..\n 5. for modified costas array of pulses.. ') %to have a choice for overlaying signal.. g2=input('for orthogonal phase overlay press 1: \n for derivative phase overlay press 2:');
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if g2==1 if g1==1 u_basic=oppt; f_basic=[z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1];
elseif g1==2 u_basic=b.*opr; f_basic=[z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1,z1];elseif g1==3 u_basic=oppt; f_basic=fc;elseif g1==4 u_basic=oppt; f_basic=flfm;else u_basic=oppt; f_basic=fmc; end else if g1==1 u_basic=pt1; f_basic=dpr;elseif g1==2 u_basic=b1; f_basic=dpr;elseif g1==3 u_basic=pt1; f_basic=fc1+dpr;elseif g1==4 u_basic=pt1; f_basic=flfm1+dpr;else u_basic=pt1; f_basic=fmc1+dpr; end end end m_basic=length(u_basic); F=5%input(' Maximal Doppler shift for ambiguity plot [in units of 1/Mtb] (e.g., 1)= ? ');K=50%input(' Number of Doppler grid points for calculation (e.g., 100) = ? ');df=F/K/m_basic; T=.5%input(' Maximal Delay for ambiguity plot [in units of Mtb] (e.g., 1)= ? ');N=50%input(' Number of delay grid points on each side (e.g. 100) = ? ');sr=50%input(' Over sampling ratio (>=1) (e.g. 10)= ? ');r=ceil(sr*(N+1)/T/m_basic); if r==1 dt=1;
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m=m_basic; uamp=abs(u_basic); phas=uamp*0; phas=angle(u_basic); if fcode==1 phas=phas+2*pi*cumsum(f_basic); end uexp=exp(j*phas); u=uamp.*uexp;
else % i.e., several samples within a bit dt=1/r; % interval between samples ud=diag(u_basic); ao=ones(r,m_basic); m=m_basic*r; u_basic=reshape(ao*ud,1,m); % u_basic with each element repeated r times uamp=abs(u_basic); phas=angle(u_basic); u=u_basic; if fcode==1 ff=diag(f_basic); phas=2*pi*dt*cumsum(reshape(ao*ff,1,m))+phas; uexp=exp(j*phas); u=uamp.*uexp; endend t=[0:r*m_basic-1]/r;tscale1=[0 0:r*m_basic-1 r*m_basic-1]/r;dphas=[NaN diff(phas)]*r/2/pi;% this block is for ACFacfun=20*log10(abs(xcorr(u)));acfun=acfun-max(acfun);acfun=max(acfun,-90);acfun=acfun(1:(length(acfun)+1)/2);acfun=fliplr(acfun);scalet=[0:length(acfun)-1]/(length(acfun)-1)*t(length(t)); figure(1), clf, hold off %set(AX,'XMinorGrid','on') %subplot(111);plot(scalet,acfun);xlabel('{\it\tau}/\itt_b');ylabel('Autocorelation [dB]');%title(titlest);axis([0 max(scalet)/5 -90 0]) grid on% denominator of scalet determine the x axes scale
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% plot the signal parametersfigure(2), clf, hold off subplot(3,1,1)plot(tscale1,[0 abs(uamp) 0],'linewidth',1.5)ylabel(' Amplitude ')axis([-inf inf 0 1.2*max(abs(uamp))]) subplot(3,1,2)plot(t, phas,'linewidth',1.5)axis([-inf inf -inf inf])ylabel(' Phase [rad] ') subplot(3,1,3) % original %subplot(1,1,1)plot(t,dphas*ceil(max(t)),'linewidth',1.5)axis([-inf inf -inf inf])xlabel(' \itt / t_b ')ylabel(' \itf * Mt_b ') % calculate a delay vector with N+1 points that spans from zero delay to ceil(T*t(m))% notice that the delay vector does not have to be equally spaced but must have all% entries as integer multiples of dt dtau=ceil(T*m)*dt/N;tau=round([0:1:N]*dtau/dt)*dt; % calculate K+1 equally spaced grid points of Doppler axis with df spacingf=[0:1:K]*df; % duplicate Doppler axis to show also negative Dopplers (0 Doppler is calculated twice)f=[-fliplr(f) f]; % calculate ambiguity function using sparse matrix manipulations (no loops) % define a sparse matrix based on the signal samples u1 u2 u3 ... um% with size m+ceil(T*m) by m (notice that u' is the conjugate transpose of u)% where the top part is diagonal (u*) on the diagonal and the bottom part is a zero matrix%% [u1* 0 0 0 ... 0 ] % [ 0 u2* 0 0 ... 0 ]% [ 0 0 u3* 0 ... 0 ] m rows% [ . . . ]% [ . . . ]% [ . 0 0 . ... um*]% [ 0 0 ] % [ . . ] N rows% [ 0 0 0 0 ... 0 ]%mat1=spdiags(u',0,m+ceil(T*m),m);
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% define a convolution sparse matrix based on the signal samples u1 u2 u3 ... um% where each row is a time(index) shifted versions of u.% each row is shifted tau/dt places from the first row % the minimal shift (first row) is zero% the maximal shift (last row) is ceil(T*m) places% the total number of rows is N+1% number of columns is m+ceil(T*m) % for example, when tau/dt=[0 2 3 5 6] and N=4%% [u1 u2 u3 u4 ... ... um 0 0 0 0 0 0]% [ 0 0 u1 u2 u3 u4 ... ... um 0 0 0 0]% [ 0 0 0 u1 u2 u3 u4 ... ... um 0 0 0]% [ 0 0 0 0 0 u1 u2 u3 u4 ... ... um 0]% [ 0 0 0 0 0 0 u1 u2 u3 u4 ... ... um] % define a row vector with ceil(T*m)+m+ceil(T*m) places by padding u with zeros on both sidesu_padded=[zeros(1,ceil(T*m)),u,zeros(1,ceil(T*m))]; % define column indexing and row indexing vectorscidx=[1:m+ceil(T*m)];ridx=round(tau/dt)'; % define indexing matrix with Nused+1 rows and m+ceil(T*m) columns % where each element is the index of the correct place in the padded version of uindex = cidx(ones(N+1,1),:) + ridx(:,ones(1,m+ceil(T*m))); % calculate matrixmat2 = sparse(u_padded(index)); % calculate the ambiguity matrix for positive delays given by %% [u1 u2 u3 u4 ... ... um 0 0 0 0 0 0] [u1* 0 0 0 ... 0 ]% [ 0 0 u1 u2 u3 u4 ... ... um 0 0 0 0] [ 0 u2* 0 0 ... 0 ]% [ 0 0 0 u1 u2 u3 u4 ... ... um 0 0 0]*[ 0 0 u3* 0 ... 0 ]% [ 0 0 0 0 0 u1 u2 u3 u4 ... ... um 0] [ . . . ]% [ 0 0 0 0 0 0 u1 u2 u3 u4 ... ... um] [ . . . ]% [ . 0 0 . ... um*]% [ 0 0 ] % [ . . ] % [ 0 0 0 0 ... 0 ]%% where there are m columns and N+1 rows and each element gives an element % of multiplication between u and a time shifted version of u*. each row gives% a different time shift of u* and each column gives a different entry in u.%uu_pos=mat2*mat1;
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clear mat2 mat1; % calculate exponent matrix for full calculation of ambiguity function. the exponent% matrix is 2*(K+1) rows by m columns where each row represents a possible Doppler and% each column stands for a different place in u.e=exp(-j*2*pi*f'*t); % calculate ambiguity function for positive delays by calculating the integral for each% possible delay and Doppler over all entries in u.% a_pos has 2*(K+1) rows (Doppler) and N+1 columns (Delay)a_pos=abs(e*uu_pos');y=max(max(a_pos))%h=max(a_pos);% normalize ambiguity function to have a maximal value of 1a_pos=a_pos/max(max(a_pos)); %a_pos=a_pos/m; % can be normalised to length of sequence % use the symmetry properties of the ambiguity function to transform the negative Doppler% positive delay part to negative delay, positive Dopplera=[flipud(conj(a_pos(1:K+1,:))) fliplr(a_pos(K+2:2*K+2,:))]; % define new delay and Doppler vectors delay=[-fliplr(tau) tau];freq=f(K+2:2*K+2)*ceil(max(t)); % exclude the zero Delay that was taken twicedelay=[delay(1:N) delay((N+2):2*(N+1))];a=a(:,[1:N (N+2):2*(N+1)]); % plot the ambiguity function and autocorrelation cut[amf amt]=size(a); % create an all blue color mapcm=zeros(64,3); cm(:,3)=ones(64,1); %cm=zeros(64,3); %cm(:,1)=ones(64,1);%cm=rand(64,3)figure(3), clf, hold offmesh(delay, [0 freq], [zeros(1,amt);a]) hold onsurface(delay, [0 0], [zeros(1,amt);a(1,:)]) colormap(cm)
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view(-40,50)axis([-inf inf -inf inf 0 1])xlabel(' {\it\tau} /{\itt_b}','Fontsize',12);ylabel(' {\it\nu} *{\itNt_b}','Fontsize',12);zlabel(' |{\it\chi}({\it\tau},{\it\nu})| ','Fontsize',12);hold off 4.16 Envelope of al
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